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THESE EN COTUTELLEpresentee a

l’UNIVERSITE PAUL VERLAINE – METZ

par

Abedalhasan BREIDIpour l’obtention des grades de:

DOCTEUR DE L’UNIVERSITE PAUL VERLAINE – METZ

DOCTEUR DE L’UNIVERSITE LIBANAISE

Discipline: Physique

Specialite: Physique de la matiere condensee

Temperature – pressure phase diagrams,structural and electronic properties

of binary and pseudobinary semiconductors:an ab initio study

Diagrammes de phase temperature – pression,proprietes structurales et electroniques

des semiconducteurs binaires et pseudo-binaires:un etude ab initio

Soutenue le 16 juin 2011

devant le Jury compose de:

M Niels CHRISTENSEN Professeur a l’Universite de Aarhus (Danemark) RapporteurM Jozef DENISZCZYK Dr. habil. a l’Universite de Silesie (Pologne) RapporteurM Mahmoud KOREK Professeur a l’Universite Arabe de Beyrouth (Liban) RapporteurM Andrei POSTNIKOV Professeur a l’Universite Paul Verlaine – Metz Directeur de theseM Fouad EL HAJ HASSAN Professeur a l’Universite Libanaise Directeur de theseM Igor ABRIKOSOV Professeur a l’Universite de Linkoping (Suede) ExaminateurM Jean-Louis BRETONNET Professeur a l’Universite Paul Verlaine – Metz ExaminateurM Abbas HIJAZI Professeur a l’Universite Libanaise ExaminateurM Alain POLIAN Professeur a l’Universite Paris 6 Examinateur

I would like to dedicate this thesis to my loving parents ...

c© 2011Abedalhasan BREIDIAll rights reserved

Acknowledgements

First and foremost, I would like to thank my advisors for their guidance andsupport throughout my thesis. I count myself as very fortunate for having themas research mentors and collaborators during these years.

I wish to acknowledge our collaborator at our Laboratoire de Physique desMilieux Denses (LPMD): Olivier Pages. He put me on the road of this thesisand enriched it through advice and unrelenting support and was always willingto share his deep knowledge and understanding of semiconductor alloys. I amalso indebted to all the members of the LPMD community who have helped mein my research endeavours over the years.

I wish to thank Jozef Deniszczyk for hosting me at his laboratory at the Sile-sian University in Katowice and who responded to my computational inquirieswith interest and guided me through my first steps to use the Alloy TheoreticAutomated Toolkit (ATAT), and his PhD student Andrzej Wozniakowski for in-teresting discussions about aloy thermodynamics. I wish to acknowledge thesupport of our collaborators of the Laboratoire de Physique Moleculaire et desCollisions (LPMC), particularly Boghos Joulakian and Claude Dal Cappello forbeing receptive to my interrogations and letting me use their computational fa-cilities.

I extend special thanks to my long-time officemates, Jihane Souhabi, NarjesMortazavi and my colleagues in other laboratories joining the same institute:Nichola Barthen, Gregory Hamm, Karim Khalouk and Mohamad Mouas ... :All of you have been wonderful; you made my work so much more enjoyable,especially in the weekly Futsal soccer sessions !!

Thanks to our Ecole Doctoral SESAMES and Universite Paul Verlaine for fi-nancing my several travels and attendances of workshops and tutorials in Barcelona,Oxford and Zaragoza, Berlin, Katowice and Strasbourg. I am thankful for Regionde Lorraine for partly financing this thesis. I also acknowledge the support ofour Ecole Doctoral des Sciences et Technologies (EDST) and Laboratoire desPhysiques des Materiaux (LPM) at the Universite Libanaise.

All my deepest thoughts and gratitudes go to my family for their unconditionallove and constant support across the distance and throughout the years. Yourdetermination and optimism have been a constant source of inspiration to me.You made this possible.

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Temperature–pressure phase diagrams,

structural and electronic properties

of binary and pseudobinary semiconductors:

an ab initio study

Abstract

In semiconductor research, an isovalent substitution on a sublattice, eithercationic or anionic one, is a convenient tool to tune lattice parameters, electronicor elastic properties. External hydrostatic pressure is another such tool, whichmay either enhance or inhibit the effects of alloying, and in any case offers an ad-ditional “coordinate” for scanning and probing the properties of semiconductors.Moreover, external pressure may induce a sequence of structural phase transitions,which in their turn may result in unusual band structures and lattice-dynamicalproperties. As energies of different phases (needed to construct phase diagrams)and force constants (responsible for vibrational spectra in crystals) are ground-state properties, they can be reliably obtained from first-principles calculationsdone within the density functional theory (DFT). The present work outlines suchstudies realized with different and, in part, complementary calculation methods.Comparative analysis of possible phases of pure ZnS and ZnSe compounds underhydrostatic pressure, done using a highly precision APW+lo method (realized inthe WIEN2k code), helped to refine the sequence of pressure-induced phase tran-sitions and to resolve some earlier controversies in theory works, related to theuse of different exchange-correlation schemes. The calculation of phonon disper-sions in the same compounds under pressure, done in linear-response formalism(abinit code), helped to identify dynamical instabilities associated with soften-ing of acoustic modes in some parts of the Brillouin zone, indicating precursorsof phase transition. The vibrational spectra under pressure were then subject tostudy under alloying, in the (Zn,Be)Se system; calculations done using the frozen-phonon method as employed in Siesta code. The thermodynamic stability of theknown competing phases (zincblende and wurtzite), in the alloy Cd(S,Se), is in-vestigated using pseudopotential method (VASP), where the special quasirandomstructures (SQS) formalism is employed to efficiently represent the disorder ofthe considered phases. On this basis, it became possible to give full structuralcharacterization, and to discuss the temperature-concentration phase diagram.

Keywords:DFT, lattice vibrations, phase transitions, mixed semiconductors

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Diagrammes de phase temperature–pression,

proprietes structurales et electroniques

des semiconducteurs binaires et pseudo-binaires:

un etude ab initio

Resume

Concernant la recherche sur les semiconducteurs, la substitution isovalente surun sous-reseau, qu’il soit cationique ou anionique, permet d’ajuster les parametresde structure de maille et les proprietes electroniques ou elastiques. La pressionhydrostatique externe peut augmenter ou inhiber les effets produits par cettesubstitution, et, en tout cas, offre un degre de liberte supplementaire pour varier,ou sonder, les proprietes des semiconducteurs. Par ailleurs, la pression exterieurepeut induire une sequence de transitions structurales de phase, qui a leur tourpeuvent induire une modification de la structure de bande et des proprietes dedynamique du reseau inhabituelles. Puisque les energies des phases en presenceainsi que les constantes de force sont des proprietes que relevent de l’etat fonda-mental, elles peuvent etre obtenues de maniere fiable a partir de calculs “premiersprincipes” effectues au sein de la theorie de la fonctionnelle de la densite (DFT).Le present travail concerne de telles etudes d’ordre theorique, realisees avec di-verses methodes de calcul, qui sont, en partie, complementaires.

L’analyse comparative des phases en competition dans les composes purs ZnSet ZnSe sous pression hydrostatique, realisee en utilisant la methode de calculde haute precision APW+lo (mise en oeuvre dans le logiciel WIEN2k), a permisde preciser la sequence des transitions de phase induites sous pression exterieureet de resoudre certaines controverses theoriques, liees a l’utilisation de schemasdifferents d’echange-correlation. Le calcul des dispersions des phonons dans l’undesdits composes (ZnSe) sous pression, effectue dans le cadre du formalisme dela reponse lineaire via le code abinit , a permis d’identifier les instabilites dy-namiques associees au “ramollissement” des modes acoustiques dans certainesparties de la zone de Brillouin, correspondant au pressentiment d’une transitionde phase. Les spectres de vibration sous pression ont ensuite ete simules dans unsysteme mixte, en l’occurrence (Zn,Be)Se, en utilisant la methode des “phononsgeles” et le code Siesta. Enfin, la stabilite thermodynamique des phases encompetition, zincblende et wurtzite, du semiconducteur mixte Cd(S,Se) a eteetudiee en utilisant le logiciel VASP selon la methode du pseudopotentiel, encombinaison avec le formalisme des structures quasialeatoires speciales (SQS)permettant de representer efficacement le caractere desordonne des phases con-siderees. Il a ete possible, sur cette base, de realiser la caracterisation structuralecomplete, et de discuter le diagramme de phase temperature–concentration.

Mots cles:DFT, vibrations de reseau, transitions de phase, semiconducteurs mixtes.

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Le travail englobe, apres une Introduction, les sept chapitres numerotes, lesConclusions generales et les quatre annexes. L’Introduction explique brievementle besoin de systematiser les resultats des etudes des proprietes electroniques etvibrationels des semi-conducteurs II-VI (purs et mixtes) sous pression, et ex-plique dans quelle logique cette information devient interessante et abordable parmoyens des calculs premiers principes, comme dans le travail donne. Tels calculspeuvent etre regardes comme une sorte d’experience numerique; ils completent,ou permettent d’interpreter, dans certaines situations les “vraies” experiences.Le niveau de precision pratique dans la solution des equation fondamentales dela mecanique quantique est celle de la theorie de la fonctionnelle de la densite(Density Functional Theory – DFT). Cette theorie est capable de decrire avectres bonne precision les proprietes liees a l’etat fondamental d’un materiau. Cesproprietes sont, par exemple, les changements de l’energie totale en fonction dela modification de la structure cristalline, ce que en principe ouvre une voie ala modelisation des diagrammes de phase sous pression. L’organisation du cal-cul premiers principes permet de definir facilement les conditions microscopiques(composition chimique strictement donnee, la pression exterieure) qui soient diffi-cile d’obtenir, ou de controler, dans l’experience. Les calculs peuvent etre utilisespour la recherche des nouveaux phases de haute pression, ainsi que pour identifieret caracteriser les barrieres d’activation (au passage d’une phase a l’autre).

Parmi les materiaux abordes dans le travail, il y a deux composes binairespurs, – ZnS et ZnSe, – et des alliages semiconducteurs pseudobinaires Cd(S,Se),(Zn,Be)Se et In(As,P). Par consequent, les deux axes de recherche principalesdans le these sont (a) les transitions de phases induites par pression dans lescomposes binaires et (b) l’effet de la temperature et la composition sur la stabilitedes phases dans les alliages pseudobinaires. Le premier but est caracterise parla construction du diagramme des phases sur la base de comparaison de leursenthalpies. En plus, le changement de la dispersion de phonons sous pressiondans un de ces systemes est etudie sur la base de calcul de la dynamique dereseau dans l’approche de reponse lineaire.

Le seconde but (b) qui concerne des alliages pose une question de modelisationdes phases mixtes; une approche bien etablie pour le choix de phases representa-tives est celle de “structures speciales quasi-aleatoires” (Special QuasirandomStructures – SQS). Un certain nombre de tels structures ont ete generes et utilisespour d’en tirer les resultats de calcul premiers principes vitales pour la construc-tion du diagramme des phases.

Le reste de l’Introduction annonce le continu de travail, chapitre par chapitre.Le premier chapitre, Entree dans le travail, s’ouvre (Sec. 1.1., Structures

cristallines) par un expose des phases pertinentes pour ce travail, notamment lazinc blende (ZB) et wurtzite (WZ), visualisees dans la Fig. 1.1. Les formules utilespour les distances interatomiques et angles entre les liaisons en chaque phase sont

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donnees dans le Tab. 1.1. Ensuite, je donne un contexte historique des etudessur les semiconducteurs mixtes. La difference est soulignee entre, d’un cote, desalliages semiconducteurs binaires du type AxB1−x, ou A et B sont les elements dela 4eme colonne du tableau periodique, tels comme Si–C ou Si–Ge, et, d’un autrecote, des alliages pseudobinaires du type AxB1−xC, C etant un sous-reseau parfait(soit cationique soit anionique), par exemple, GaxIn1−xAs ou ZnSxSe1−x. Si lesstructures cristallines de deux composes parents, AC et BC, sont identiques, lesstructures intermediaires ont typiquement la meme structure. La variation duparametre du reseau moyen en fonction de la concentration x a typiquement uncaractere lineaire, en obeissant la loi de Vegard, v. Eq. (1.1). Ceci a ete confirmedans l’experience (par diffraction des rayons X, dont un exemple est donne dansFig. 1.2) pour un grand nombre de systemes. En ce que concerne l’ajustement deslongueurs des liaisons A–C, B–C dans un systeme mixte, la diffraction des rayonsX ne permet pas etudier ces proprietes locales, mais heureusement une methodespectroscopique, celle de Extended X-ray Absorption Fine Structure (EXAFS), enest plus puissante. Depuis des travaux de Mikkelsen Jr. et Boyce en 1983, il estconnu que les distances premiers voisins (liaisons anion–cation) ne changent pasbeaucoup dans un systeme mixte par rapport de leurs valeurs respectives dans lescristaux parents – v. Fig. 1.3, extraite du travail originale de Mikkelsen Jr. andBoyce. La definition des parametres de relaxation pour les longueurs de liaison,qui caracterisent le decalage de ces derniers d’un comportement prevu par la loide Vegard, est donnee par Eq. (1.3). Fig. 1.4 montre l’evolution de distancesentre des deuxiemes voisins, anion–anion et cation–cation, dans le systeme mixteInxGa1−xAs, d’apres les resultats de Mikkelsen Jr. et Boyce. Il en resulte que lesdistances As–As restent “bimodales”, alors clairement separes en deux groupes,le plus court As–Ga–As et le plus longue As–In–As, tandis que toutes les troistypes de distances cation–cation, In–In, Ga–Ga et In–Ga, suivent grossierementla loi de Vegard.

En ce que concerne la description theorique des semiconducteurs mixtes, apresavoir mentionne l’approximation du cristal virtuel (virtual crystal approxima-tion – VCA), la plus simplisitique et moins interessante a cette fin, l’approchenommee “l’isodeplacement modifie des elements aleatoires” (modified random-element isodisplacement – MREI) du a Chang et Mitra, est mentionnee. Cetteapproche, dans l’esprit de l’observation experimentale de Mikkelsen Jr. andBoyce, postule l’existence d’une longueur de liaison unique pour chaque especede la liaison (cation–anion) donne. Ceci a des consequences immediates pourl’interpretation des vibrations (phonons) dans un cristal mixte. Cette interpre-tation, connue comme une paradigme “1 liaison – 1 mode”, a ete contesteerecemment dans les travaux de Pages et al., qui a analyse certaines limitationsmarquantes du ledit modele de Chang et Mitra. Pages et al. ont soulignel’importance de l’environnement locale (une enrichement local du materiau en

v

l’un ou en l’autre compose parent, du aux variations locales de la concentration)et l’observation, dans plusieurs cas, de (au moins) deux modes de vibration bienresolus, derives du meme espece de la liaison chimique. Le travail actuel rend unecontribution a l’analyse de ce type du comportement vibrationel pour un systememixte, BexZn1−xSe sous pression, aborde dans le Chapitre 6.

Suite a cette breve introduction dans la thematique des alliages, le chapitre1 se tourne (Sec. 1.2., Transitions de phase induites pas pression) versle role de l’etude sous pression dans la science des materiaux. Ils rendent possi-ble une creation des nouvelles phases, mais aussi ouvrent des riches perspectivesdans la science fondamentale, grace a la manifestations de certains phenomenesinteressants: (de)localisation electronique, effets du transport, couplage des pro-prietes electroniques et vibrationelles, proprietes de symetrie caches et releves,decrites par la theorie de groupes. Meme lorsque on se limite a des systemessemiconducteurs, le champ de recherche correspondent est tres vaste. Apres avoirexpose les techniques d’experience utilisees dans les etudes sous pression (dia-mond anvil cells pour generer la pression et end stations des sources synchrotronequipees par diffractometres des rayons X afin de dechiffrer les structures), leChapitre 1 entre dans la discussion de thermodynamique des alliages, mettantle cadre sur l’analyse ulterieure (sur le systeme CdSxSe1−x dans le Chapitre 7).Le point de depart est l’equation de l’energie libre de Gibbs G(p, T ), donnee parEq. (1.4). Leur parties essentielles accessibles dans des calcul premiers principessont l’energie interne (l’energie totale, calculee a la temperature zero pour les con-ditions structurelles, telles comme pression ou volume, donnees an avance), et lacontribution vibrationelle a l’energie de Gibbs (accessible, dans l’approximationquasi-harmonique, a partir de la densite de modes de vibration).

Une difference est expliquee entre des transitions de phase du premier et dudeuxieme ordre, qui sont classees selon l’ordre de la derivee de l’energie libre deGibbs G(p, T ) en la pression p, soit ∂G/∂p = V (volume) soit ∂2G/∂2p = B(compressibilite), qui subit une discontinuite lors la transition. On note queles changements de la symetrie lors de la transition peuvent etre identifies parmethodes de la theorie de groupes en faisant usage de la theorie de Landau detransitions de phases.

Dans la discussion qui suit sur les transitions de phase, une distinction est faiteentre une transition displacive et reconstructive. Dans la premiere, les groupesd’espace de deux phases concernes se trouvent dans une relation “groupe – sous-groupe”; un parametre d’ordre peut etre identifie qui lie, dans l’esprit de la theoriedes transitions de phase de Landau, les deux situations avant et apres la tran-sition, et qui quantifie alors une distortion de la phase plus symetrique vers laphase moins symetrique. Par contre, dans le cas oppose d’une transition re-constructive, les deux phases de depart et finale ne sont pas connectes commegroupe et sous-groupe. Plutot, il existe un trajet entre les deux structures termi-

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nales qui passe a travers d’une phase intermediaire dont le groupe d’espace estun sous-groupe de deux phases terminales. L’identification d’un tel trajet n’estpas forcement unique; notamment l’Annexe A donne un exemple detaille d’untelle analyse. L’existence d’une barriere a surmonter afin d’arriver d’une phase al’autre explique pourquoi certains transitions sont evitees dans le sens cinetique.

Les transitions de phase de premiere ordre peuvent etre soit displacive soitreconstructive, tandis que les transitions de seconde ordre sont typiquement dis-placive, et associes a un emoulement d’un mode de vibration acoustique transver-sale dans un certain point de la zone de Brillouin. Telles transitions ne changentpas le nombre de coordination d’un anion ou un cation, et sont reversibles. Cepen-dant les transitions de premiere ordre sont parfois irreversibles, ou autrementexposent une hysteresis.

Une attention et attiree au fait que la pression de transition avisee par uneexperience peut dependre fortement sur la methode utilisee, du aux differencesdans l’echelle de localisation dont la technique en question est sensible. Typ-iquement la photoluminescence et la diffusion de Raman donnent les valeurs tropbasses tandis que la diffraction des rayons X surestime la pression de transition,et l’absorption des rayons X donne les estimation intermediaires.

Un effet bien connu de l’application de la pression est de transformer lesliaisons molles, du type van der Waals, dans les liaisons dures. Une autre ob-servation interessante est que les sequences de phases qui ont ete trouvees pourles elements legers sous pression ont tendance a se reproduire aussi avec leurshomologues plus lourds, mais aux pressions moins elevees.

La partie finale du Section 1.2 expose les phases typiques pour semiconduc-teurs binaires II-VI et III-V sous pression. L’etat fondamental est typiquementrepresente soit par le structure ZB soit par WZ. Les deux phases sont souventtres proches en energie et sont alors concurrentes dans la diagramme des phases;une faible preference vers soit l’une soit l’autre depend des details de la liaisonchimique, degre de covalence etc. – les exemples sont discutes. En principe,la structure de ZB est preferee dans les semiconducteurs avec les liaisons pluscovalents. Sous pression, la structure de sel de cuisine (rocksalt, RS) s’emergeeventuellement dans presque toutes les semiconducteurs (les exceptions sont dis-cutes), precedee par les phase intermediaires, cinnabar ou Cmcm. Les structuresRS et Cmcm sont montrees dans Fig. 1.5. Un revue des travaux precedents surles structures conclue la section 1.2.

La section finale du premier chapitre, 1.3. Equilibre des phases dans lesalliages pseudobinaires – une breve introduction, s’adresse a la generali-sation de l’approche thermodynamique, introduite deja dans la Section 1.2, versles systemes mixtes. La difficulte principale dans ce contexte est un choix efficacedes configurations representatives, qui sont a priori tres nombreuses, dans le for-mat pratique des calculs abordables. Le concept des “structures speciales quasi-

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aleatoires” (special quasirandom structures – SQS), suggere par Wei et Zungeren 1990, s’a rendu tres utile pour diverses calculs microscopiques pour les al-liages. En plus que faciliter le choix de structure dans une simulation ad hocpour une concentration donnee, dont les exemples se trouvent dans le chapitre 7,les SQS font parti des algorithmes, generalement compris comme “methodologieCALPHAD”, qui permettent un calcul systematique des diagrammes de phase,dans une approche combinee. D’abord, les calculs premiers principes permet-tent de construire et enrichir la “base des donnees” des energies en fonction deconfiguration microscopique (dans la “technique d’expansion en clusters” – clus-ter expansion); ensuite la simulation dy type Monte Carlo avec un hamiltonienmodele de l’alliage ainsi parametrise permet de faire une tres bonne statistique.(Cette derniere partie n’a ete achevee dans le travail actuel).

Le deuxieme chapitre, Fondaments de la theorie et les approximationsutilisees, s’adresse aux aspects pratiques des calculs realises. La section 2.1:Calculs ab initio de la structure electronique couvre les details du calculde la structure electronique pour un systeme quelconque. La base generale descalculs est celle de la Theorie de la fonctionnelle de la densite (Density FunctionalTheory – DFT), qui a commence par les travaux de Hohenberg et Kohn de 1964et Kohn et Sham de 1965. Puisque ces fondaments sont vraiment bien connus,leur expose dans la these epargne les details et se limite a un guide a travers leminimum necessaire, fourni des citations des articles de review utiles.

La solution pratique des equations de Kohn–Sham dans Eq. (2.3) demandede faire un choix d’une methode numerique; le caractere de nos systemes d’etudecomme objects massifs justifie l’usage des conditions aux limites periodiques (im-poses, dans le cas d’un alliage, sur une supermaille eventuellement grande) et,par consequent, la presence du vecteur k dans l’espace reciproque comme unnombre quantique valide. La relation entre les valeurs du k d’une supermailleet ceux du reseau prototype demande cependant un soin dans l’inerpretation etdans la discussion. L’usage des divers approximations pour l’echange-correlation(XC), notamment soit l’approximation de la densite locale (Local Density Approx-imation, LDA) soit l’approximation du gradient conjugue (Generalized GradientApproximation, GGA) est discute.

La suite est une discussion entre les codes de calcul et des logiciels a utiliser, cesderniers, notamment le code WIEN2k, VASP, Siesta et abinit, sont expliqueset compares. Cette gamme de methodes comprends un code toutes electrons(WIEN2k), qui est tres precis dans les calculs de l’energie totale a travers unelarge variation des parametres de reseau, donc bien adapte pour la simulationdes diagrammes de phase sous pression (Chapitre 4). Le code abinit permetle calcul des dispersion des phonons dans n’importe quel point de la zone deBrillouin, ce qu’est souhaitable pour les etudes de l’instabilite dynamique souspression (Chapitre 5). Siesta, capable de traiter d’assez grandes supermailles

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sans etre limite par le manque de symetrie, est efficace dans le calcul la structurerelaxee et les phonons geles. Pour le code VASP, une interface existe avec uncoquille (ATAT) qui permet faire des calculs thermodynamiques.

Quelques sections du Chapitre 2 sont dedies au traitement des effets de la dy-namique de reseau. La section 2.2: Modele du reseau statique et ses limita-tions explique le context dans lequel ces effets peuvent etre negliges. La section2.3: L’approximation adiabatique justifie la couplage standard, d’apres Bornet Oppenheimer, des degres de liberte electroniques et ioniques. Touts calculsdynamiques dans ce travail ont ete faits dans l’approximation de Born et Op-penheimer. La section 2.4: Un mot d’introduction sur un phonon rappellel’essentiel sur le definition, et interpretation, des phonons du point de vue descalculs pratiques ab initio, explique comment les constantes de forces necessairespour les calculs dynamiques peuvent-ils etre accumulees. Les tailles des matricesresultantes (a diagonaliser) sont expliquees, et une petite discussion est ajoutee,en vue des calculs pour supermailles elargies dans les chapitres qui suivent, sur larelation entre le vecteur d’onde soit correspondant au “vrai” centre de la zone deBrillouin du cristal, soit “plie” vers q = (0, 0, 0) de la supercellule a partir d’unautre point de la zone de Brillouin.

La section 2.5: Formalisme de l’approximation harmonique comprendla deduction de l’equation dynamique (2.24) pour les modes de vibration, avecleurs pulsations ωα(q) dependants du nombre d’onde q et avec polarisationscorrespondantes α, a partir des equations Newtoniennes et en utilisant l’approxi-mation harmonique pour decoupler les modes differents. Le probleme se reduitensuite a la construction de la matrice dynamique de l’Eq. (2.25). Ce dernierpeut etre obtenu par le methode directe (ou, “deplacements finies”, ou “phononsgeles”), de facon a quelle les elements de la matrice dynamique soient evalues pardifferences finies, comme [changement de force sur un atome i]/[deplacement d’unatome j]. Une telle option est implementee dans le code Siesta. Une autre possi-bilite est un calcul “direct” des elements de la matrice dynamique pour n’importequelle valeur du vecteur d’onde, en utilisant la theorie de perturbation en 2eme

ordre. Cette approche est implementee dans le code abinit.La Section 2.6: Diagramme de phase d’un alliage pseudo-binaire re-

vient au traitement thermodynamique de l’energie libre necessaire pour la con-struction d’une diagramme de phases, deja aborde brievement dans la Sec. 1.2,maintenant dans le contexte plus pratique, et specifiquement confine aux al-liages pseudo-binaires AxB1−xC. Je discute le parametrisation de l’energie librede Helmholtz, F , en fonction de la concentration x, et donne l’expression explicitepour l’entropie configurationale. Ensuite la forme generale de la fonction F (x)est discutee pour les cas de l’enthalpie de melange positive et negative. Dansle premier cas, illustre par Fig. 2.5, le passage des courbes spinodale et binodaleest discute, et la presence de lacune de miscibilite expliquee. Pour l’enthalpie

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de melange negative (Fig. 2.6), une lacune de miscibilite ne se manifeste pas;cependant, autour de la concentration x = 0.5, on note une region de l’existencede la phase ordonne ABC2.

La section 2.7: Modeles analytiques des diagrammes de phase couvreles deux cas speciaux mais importants du point de vue de la theorie: un modelede la solution ideale, i.e., le cas de l’energie de melange nulle, ΔU = 0, et lecas nomme “modele strictement regulier”, avec un comportement parabolique del’energie en fonction de la concentration, ΔU ∼ x(1 − x).

La section 2.8: Calculs premiers principes des proprietes thermo-dynamiques porte sur les details de l’evaluation de l’entropie electronique etvibrationelle, necessaires pour l’evaluation des termes respectives dans l’energielibre donnee par Eq. (2.37). L’approximation quasi-harmonique est expliquee, etles exemples de son utilisation (expansion thermique des metaux, d’apres le re-view de van de Walle et al.) sont donnes: Fig. 2.7 montre la variation de l’energielibre du compose Al2Cu en fonction de temperature, et Fig. 2.8 donne les courbesde l’expansion thermique calculee, aussi en fonction de la temperature, pour deuxmetaux purs, sodium et aluminium.

Le troisieme chapitre, Etudes premiers principes de gap bowing dansles systemes mixtes InAs1−xPx, ouvre la presentation des resultats originals dutravail. Le but principal de cette partie de travail est de decrire la variation de labande interdite en fonction de composition. La deviation, parfois assez grande, dece comportement de la loi de Vegard est caracterise par le parametre nomme gapbowing. Dans le systeme InAs1−xPx, aborde dans ce chapitre, la bande interditevarie entre 0,36 eV pour InAs et 1,35 eV pour InP, se que tombe dans le domainespectrale d’infrarouge. Par consequent les semiconducteurs mixtes In(As,P), avecleur largeur de la bande interdite ajustable, sont devenu interessantes pour con-struction des detecteurs de l’infrarouge et, en telle qualite, ont servi objets deplusieurs etudes recents. Dans mon travail, le calcul de la structure electroniquea ete effectue en utilisant le methode full-potential linearized augmented planewaves (FP-LAPW). Deux types du potentiel d’echange-correlation ont ete utilise:la GGA dans la realisation de Perdew – Burke – Ernzerhof, une option “standard”dans les calculs DFT qui demandent une haute precision de l’energie totale, etun scheme de Engel et Vosko, qui et connu de pouvoir enlargir les predictions decalcul pour la bande interdite, vers les valeurs plus proches a l’experience, au prixde deterioration de l’energie totale. Les deux schemas sont alors souvent utilisesen parallele et en comparaison.

L’aspect alliage etait pris en compte dans le calcul par l’usage se supermaillesordonnes, comptants soit 8 soit 16 atomes. La distribution des anions (As, P)sur leurs sites du sous-reseau cubique a faces centrees pour les concenrations de25 et 75% correspondant a la structure ordonnee du type Cu3Au. La supermailleutilise pour la concentration de 50% a ete similaire a la structure de chalcopyrite.

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Pour chaque concentration, le volume de supermaille a ete optimise (par varia-tion homogene du parametre de reseau, sans ajuster les coordonnes internes). Lecalcul a demontre une decroissance graduelle du parametre de reseau a partir de6,189 A pour InAs (experience: 6,036 A) vers 5,964 A pour InP (exp.: 5,861),accompagnee par une croissance du module de rigidite a partir de 49 GPa pourInAs (exp.: 59 GPa) vers 60 GPa pour InP (exp.: 74). Les resultats pour con-centrations intermediaires sont inclus dans le Tableau 3.1. La Fig. 3.1 montreque, tandis que la variation du parametre de reseau est bien lineaire, la variationdu module de rigidite en fonction de la concentration demontre une deviationimportante de la linearite.

La bande interdite calculee etait nulle en InAs d’apres les deux schemas del’echange-correlation utilises (Fig. 3.2; Tab. 3.2); pour les concentrations eleveesde phosphor, la schema de Engel-Vosko donne des valeurs positives, augmentantjusqu’a 1,003 eV pour InP (toujours inferieurs a l’experience). Dans le schemade Perdew–Burke–Ernzerhof, la bande interdite ne s’ouvre pas que a 75% dephosphore. Malgre la difference dans les valeurs absolues de la bande interditeavec l’experience, il est connu que les tendances de la variation, en fonction de laconcentration, sont souvent bien reproduites, c’est qui est justifie par les calculs.Les parameres decrivant la variation de la bande interdite sont ensuite analyses etdecomposes en leurs composants dus aux divers mecanismes physiques de gap bow-ing, suite au contexte mis precedement dans les travaux par Bernard et Zunger.La parametrisation correspondante est systematisee dans le Tab. 3.3.

Le quatrieme chapitre, la diagramme de phases des composes semi-conducteurs du type II-VI, ZnSe et ZnS, en fonction de la pression,decrit une analyse detaillee de la construction du diagramme de phases de deuxsemiconducteurs binaires. La variable principale ici est la pression, et une ques-tion qui se pose concerne la serie de phases cristallins qui se changent l’une al’autre des que la pression s’augmente. Considerant la phase ZB aux condi-tions ambiantes, les deux systemes ZnSe et ZnS arrivent a la structure RS souspression d’ordre de quelques GPa. Les travaux existent, surtout dans le reg-istre calculs premiers principes, qui modelisent l’effet de pressions encore plushautes, portants le systeme au-dela de la phase RS. Mes priorites, par contre,se concernent la simulation des phase intermediaires entre la ZB et la RS. Lesdeux phases “suspects”, deja mentionnees dans un grand nombre de travauxexperimentaux et theoriques, sont cinnabar et SC16. Ce que reste a verifier etaitune precise attribution de ces deux phases sur le diagramme de phases (pres-sions de transition etc.) et leur stabilite relative, vue un assez grand nombre descalculs precedents, dont les resultats sont parfois en contradiction. Les calculsfaits parfois dans le schema “tous electrons” ou parfois dans l’approche pseu-dopotentiel, en utilisant parametrisations differentes pour l’energie d’echange-correlation, typiquement sans optimiser les coordonnees internes des phases com-

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plexes (cinnabar et SC16), ont signale l’emergence de soit une soit l’autre enleur qualite des phases intermediaires entre ZB et RS. La situation experimentalen’est pas claire non plus; les transitions vers une phase hexagonale (apparem-ment cinnabar) ont ete indiquees lors de l’enlevement de la pression (downstroke),mais pas dans la transition inverse. Dans cette perspective, l’etude actuelle a etefaite en utilisant le methode FP-LAPW tous-electrons de haute precision, deuxparametrisations pour l’echange-correlation standards (LDA et GGA), et prenanten compte tous les degres de liberte a optimiser lors de calcul (parametres dereseau et les coordonnes internes). Cette motivation primaire et une revue destravaux precedents est compris dans la Section 4.1: Introduction. La sectionsuivante 4.2: Methode et details de calcul precise les cutoffs relevants enondes plans et points k, soumises aux tests meticuleux. La Section 4.3: ZnSe;reflexions de debut et resultats des calculs de l’energie totale s’ouvrepar les discussions sur les structures cristallins concernes (dont plus de detailssont elabores dans l’Annexe A). L’observation importante est que la structurecinnabar, avec les valeurs des coordonnes internes u = v = 1

3et le rapport des

parametres cristallins c/a =√

6, devient identique avec le RS. Lors d’une optimi-sation de la structure sans contraintes, pour les valeurs differentes de la pressionhydrostatique, les courbes energie(pression) de deux phases RS et cinnabar sejoignent, a partir d’une valeur critique de pression, suite d’une transition de phasede 2eme ordre. Dans les travaux precedents ou les coordones internes n’etaient paslibres mais fixes a leurs caracteristiques pour la phase cinnabar aux conditionsambiantes, la transition de phase n’apparait pas, et les courbes energie (volume)de deux phases se croisent sans se joindre.

Fig. 4.1 montre les courbes pour les quatre phases concernees, d’apres lescalculs effectues avec deux types d’echange-correlation, et Fig. 4.2 – les courbescorrespondantes de l’enthalpie statique (relatif a la phase ZB). Tandis que laforme des courbes d’energie sur Fig. 4.1 (ou, autrement, les pentes des lignesde l’enthalpie dans Fig. 4.2) ne different pas beaucoup entre les cas LDA etGGA, leur placement exacte n’est pas le meme. Comme un resultat, on trouveune region etroite des pressions pour la stabilite de la phase cinnabar entre lesdeux phases ZB et RS d’apres le calcul en GGA, mais pas avec LDA. En plus,les intersections des lignes de l’enthalpie pour diverses phases sont legerementdecalees en GGA vers les plus hautes pressions, par rapport au cas LDA. Cecipermet de comprendre des ambiguıtes dans les resultats des calculs anterieurs etd’estimer une certaine “marge de tolerance” vis-a-vis de nos resultats actuels. Ala difference de notre prediction concernante la stabilite de la phase cinnabar, laregion de stabilite presumee de la phase SC16 se trouve dans les deux diagrammesde phase, calcules selon la GGA bien que selon la LDA, meme si les largeurs deces regions different: 2,5 GPa, a partir de la pression estimee de transition a

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10,8 Gpa d’apres LDA, contre l’intervalle de 4,3 GPa, a partir de transition a11,8 GPa, dans le cas de GGA. Pour comparaison, la region de stabilite de lastructure cinnabar est plus etroite – entre 13,35 et 13,9 GPa, et uniquement enconsiderant les calculs en GGA seulement.

L’observation experimentale que la phase cinnabar existe lors de l’enlevementde pression est interprete, selon nous, parce que la transition ZB ↔ RS n’est passymetrique: le passage a travers la phase cinnabar est plus simple dans un sensque dans le sens oppose, donc la transition de phase sous pression dans le cyclerepete se realise, sur l’echelle microscopique, le long d’un trajet “triangulaire”.Cette idee est explique avec plus de details dans l’Annexe A. Une autre observa-tion interessante est que la phase SC16 echappe jusqu’a maintenant sa detectionexperimentale, malgre son placement energetique, d’apres le calcul, au-dessousde la phase cinnabar. L’interpretation plausible de ce fait-la, deja evoquee dansla litterature, est que la phase SC16 est “cinetiquement evitee” (kinetically hin-dered), par force de la barriere energetique qu’il faut traverser pour arriver acette phase-la. Notons que l’analyse statique manifestee par les figures 4.1 – 4.2n’adresse pas que les energies de chaque phase separement, sans rien dire sur lestrajets microscopiques qui transforment une phase dans l’autre. Les exemplesexistent deja de preparation la phase SC16 a haute pression et temperature et deleur preservation suivante comme une phase metastable aux conditions ambiantesdans GaAs.

Le reste de la Section 4.3 explique les relations microscopiques entre les phaseset expose les details des changements structurales (longueurs de liaison; angles)sous pression, lorsque on modifie graduellement le volume de la maille primitive.Notamment le pli dans les valeurs de parametres marquant la “condensation”de la phase cinnabar vers la RS fait objet de la discussion. La Fig. 4.3 donnele schema de l’environnement locale (tetraedrique distorte) autour d’un anion etd’un cation dans la structure cinnabar, pour mieux comprendre la transformationde cette derniere dans la structure RS (Fig. 4.4). La Fig. 4.5 montre la variationdes parametres cristallins a, c de la phase cinnabar, en fonction de pression,rendant visible une singularite qui se manifeste lors de la transition cinnabar→ RS. La largeur de la bande interdite augmente sous pression dans les phasesZB et cinnabar (etant plus petite dans cette derniere, v. Fig. 4.6); la phase RSreste metallique. La structure de bandes electronique dans la phase cinnabar,calcule avec GGA et LDA, est presentee dans la Fig. 4.7. La Fig. 4.8 centraliseles longueurs des liaisons et angles des liaisons dans la phase cinnabar en fonctionde la pression et rend visible, elle aussi, le collapsus vers la RS a la pressioncritique. Ensuite, la sructure SC16 est expliquee en plus de details. Fig. 4.9montre un dessin de sa maille primitive; Fig. 4.10 donne une variation de cescoordonnes internes sous pression, Fig. 4.11 – changement des longueurs et desangles de liaison. Les details de la structure cristalline et relations explicites

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entre les coordonnes internes at longueurs/angles de liaison sont donnees dansl’Annexe B.

La Section 4.4: sulfure de zinc decrit la situation autour du second systemecouvert par le Chapitre 4; malgre la similarite de ses proprietes chimiques aveccelles de ZnSe, le diagramme de phase de deux systemes n’est pas exactement lememe. Notamment la structure de l’etat fondamental est WZ, qui transforme versZB sous pression. Les courbes volume – energie de ces deux phases (Fig. 4.12) sontpresque indiscernable sur l’echelle qui permet de resoudre la transition suivanteinduite par pression, vers la phase RS, atteinte a 15 GPa a peu pres (d’apresle calcul avec LDA, v. Fig. 4.13). Il y a un certaine ambiguıte a propos del’existence de la phase SC16 intermediaire entre ZB et RS; elle etait preditedans certains calculs mais pas trouvee dans l’experience. L’existence de cinnabarcomme une autre phase intermediaire, differemment de la situation dans ZnSe,n’etait pas confirme ni dans les calculs ni dans l’experience. Le but du calculactuel est d’essayer resoudre ces controverses et construire le diagramme de phasesen fonction de la pression en utilisant le meme methode de calcul ainsi preciseque dans l’analyse sur ZnSe.

Les courbes de l’energie totale en fonction de volume sont montres dans laFig. 4.12; les enthalpies de phases, par rapport de ZB, – dans Fig. 4.13. On noteque la phase SC16 peut etre thermodynamiquement stable d’apres le calcul enLDA seulement, tandis que le calcul GGA indique la transition directe a partirde ZB vers le RS. La phase cinnabar reste instable partout selon les deux ap-proximations pour l’echange-correlation, la LDA comme la GGA. Les resultatsactuels de mon travail sont conformes avec les meilleurs experiences pas seule-ment qualitativement, mais donnent aussi une bonne estimation de la pressionde transition ZB vers RS: a peu pres 16.3 GPa, a comparer avec 14.5 – 15.5 GPacomme valeurs experimantales.

Le cinquieme chapitre, Instabilite dynamique dans la phase zinc-blendede ZnSe, est dedie a l’etude des mecanismes intrinseques de la transformationstructurelle a partir de la phase ZB du ZnSe, et a la discussion sur le trajet mi-croscopique possible d’une telle transformation. Une approche statique abordeedans le chapitre precedent ne permet pas d’analyser que le bilan energetique dediverses phases separement, dans leurs situations metastables, en laissant horsde discussion les possibilites d’un passage d’une phase a l’autre. Le chapitre enquestion s’occupe d’une situation quand une phase metastable devient instable,susceptible d’entrer dans une re-structuration. Il faut preciser que je ne discutepas ici qu’une transition de phase displacive. Une tendance vers une telle transi-tion se manifeste par une forte abaissement (jusqu’au zero) de la frequence d’unmode acoustique de vibration pour une valeur quelconque du vecteur d’onde qhors du centre de la zone de Brillouin. Dans cette situation, un de trois criteres dela stabilite d’un reseau, formules par Peierls, n’est plus satisfait. La premiere regle

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de Peierls dit que toutes les forces sur tous les atomes doivent etre egale a zero; ladeuxieme demande est que le materiel soit stable par rapport aux deplacementsmacroscopiques telles que la compression/expansion, ou poussee; la troisiemeregle est que le materiau soit stable vis-a-vis chaque petite deplacement atom-ique a partir de sa structure d’equilibre. Cette derniere demande dit autrementque toutes les frequences des phonons doivent etre reelles (et positives), avec uneseule exclusion de trois modes acoustiques de frequence nulle a q = 0. Le casoppose de frequence(s) imaginaire(s) signifie que la hyper-surface d’energie totaleen fonction de toutes deplacements atomique possibles, a partir de la structured’equilibre, ait la courbure negative le long d’(au moins) un sens dans l’espacedes deplacements, et alors le systeme trouverait un moyen d’abaisser son energietotale par rapport a l’equilibre presume. La pression est un parametre externequi peut influencer la condition d’equilibre et induire une instabilite structurelle,manifestee par un mode mou dans le spectre des vibrations. La valeur de pressionqui induit la frequence du mode concerne d’approcher zero donne une indicationd’une condition critique pour la transition de phases; le vecteur d’onde concerne,pour quelle une telle instabilite a lieu, permet de juger la forme de la mailleprimitive du materiau au-dela de la transition.

La Section 5.2: Methode, parametres de calcul; tests de precisionexplique que le calcul a ete effectue en utilisant le methode de la reponse lineaire,

afin de pouvoir tracer la dispersion des phonons dans le 1ere zone de Brillouin.Une telle approche est realisee dans le logiciel abinit , un code qui utilise despseudopotentiels (et donc ne sont pas du type “tous-electrons”) et des ondesplanes pour base des fonctions. D’ailleurs j’explique brievement dans cette Sec-tion comment estimer les vecteurs de translation de la superstructure resultantede la transition de phase a partir du vecteur d’onde dans la zone de Brillouin quicaracterise le mode de vibration devenant mou. L’optimisation de la precision ducalcul concerne un choix de deux parametres essentiels, le cutoff d’ondes planes,estime suffisant avec la valeur de 32 Ha, et la finesse de la grille des pointsk, trouvee suffisante avec 8×8×8 divisions. L’analyse de la precision de cal-cul comprend une comparaison des courbes pression–volume (donc d’equationd’etat) obtenues avec les deux methodes, WIEN2k et abinit , qui est en effettres bonne (Fig. 5.1). Pour l’echange-correlation, la GGA a ete utilisee dans laparametrisation de Perdew–Burke–Ernzerhof. En plus que la similarite de deuxcourbes, les valeurs du volume d’equilibre obtenus par les deux methodes sont tresproches (162,3 Bohr3 dans abinit , a comparer avec 160,8 Bohr3 par WIEN2k),ainsi que les valeurs de bulk modulus (55 GPa dans les deux cas).

La Section 5.3: Resultats et discussion donne des exemples des courbesde dispersion des phonons le long les intervalles “habituelles” entre les pointsde symmetrie dans la zone de Brillouin de la structure ZB, calcules pour deux

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valeurs de la pression (12 et 19 GPa), respectivement avant et au-dela l’instabilitequi se manifeste autour du point X (Fig. 5.2). Dans le deuxieme cas, l’instabiliteest ainsi prononcee avec des frequences de vibration imaginaires (indiquees parles valeurs negatives dans la figure) dans un certain voisinage du point X. Unabaissement de la frequence sous pression se manifeste aussi autour du pointL, mais cette instabilite n’est encore pas completement acheve a 19 GPa. LaFig. 5.3 montre une serie des valeurs de frequence calculees au point X pour diversvaleurs de la pression; l’interpolation du zero de cette fonction permet d’estimerfacilement la valeur critique de pression responsable pour l’instabilite, que rend18 GPa a peu pres. La discussion suivante concerne le type de superstructureissue de la transition. Apparemment l’instabilite dans X implique une structuretetragonale, avec des cotes a√

2× a√

2× a, ou a est le parametre cubique du reseau

ZB. Le volume de la maille primitive est ainsi double.On trouve une certaine similarite entre ce type de doublage de la maille

cristalline avec la structure Cmcm, expliquee dans la Fig. 1.5. L’analyse mi-croscopique des deplacements atomiques lors de la transition displasive peut etrefait en regardant le vecteur propre du mode de vibration qui devient mou.

Le sixieme chapitre, Effet de la pression sur la dynamique de reseaudans (Zn,Be)Se, decrit une recherche combinee (theorie – experience, avec par-ticipation des equipes francaise et indienne) sur le systeme mixte avec une substi-tution sur le sous-reseau cationique, et notamment ma contribution dans le misen oeuvre des calculs et leur interpretation. La motivation pour cette rechercheest due du fait que la dynamique de reseau des semiconducteurs pseudobinairesconstitue un point fort de la recherche au sein de mon laboratoire d’accueil enFrance lors du travail de these; une approche dite modele de percolation et avanceepar Olivier Pages a deja permis d’expliquer les details des spectres de vibrationdes alliages binaires en fonction de la concentration. Le but principal de cetteapproche est, lors de la modelisation des alliages, de depasser les limitations del’approximation de cristal virtuel, qui neglige des fluctuations de concentrationa l’echelle mesoscopique. Lorsque on prend en compte qu’une espece de la liai-son chimique donnee peut se trouver dans l’environnement variable d’un point al’autre du cristal, localement riche ou pauvre en tel ou l’autre composant chim-ique present dans un alliage, ces differences dans l’entourage cristallin ont pourconsequence la diversification des longueurs de liaison en equilibre et, dans leprochain pas de l’argumentation, a une certaine diversification du mode de vi-bration caracteristiques pour une liaison en question. En somme on parle d’unconcept “une liaison – deux modes”, car les deux modes de vibration sont souventdiscernables en les spectres de vibration ou, d’apres l’approximation du cristalvirtuel, on ne peut attendre qu’une.

Les alliages (Zn,Be)Se sont deja objet d’etudes experimentales (par la spectro-scopie Raman) et des calculs premiers principes. Une structure bi-modale du

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spectre Raman, surtout dans le domaine spectrale des vibrations du Berillium,a ete detectee, expliquee dans le cadre du modele de percolation, et quantita-tivement confirmee dans les calculs premiers principes. Maintenant la pressionexterieure est un parametre important susceptible d’affecter les tendances ob-serves et d’en tirer certaines consequences a propos des mecanismes elastiques.

Une observation interessante est que les deux composes parents de l’alliage(Zn,Be)Se, ZnSe et BeSe, existent dans la phase ZB sous conditions ambiantes,mais manifestent une comportement differente sous pression: tandis que ZnSe setransforme dans la phase RS, le BeSe prend la structure hexagonale du type NiAs.Les pressions critiques de transition sont tres differentes: ∼56 GPa pour BeAs (ZB→ NiAs) et ∼13 GPa pour ZnSe (ZB → RS). Une question interessante concernele comportement des liaisons Be–Se lorsque leur cristal hote rich en ZnSe subitune transition vers la phase RS. La part experimentale du travail (faite par GopalPradhan et Chandrabhas Narayana du Jawaharlal Neru Centre for Advanced Sci-entific Research a Jakkur, Inde) comporte les etudes par spectroscopie Raman dujeu des modes caracteristiques de vibration dans le systeme mixte ; la separation(jusqu’a Δ ∼40 cm−1, entre les peaks principales dans la region spectrale de BeSe)ou, au contraire, la jonction des peaks sous pression. La Fig. 6.1. donne un ex-emple de l’evolution des spectres pour un alliage representative, Zn0,76Be0,24Se.Trois singularites sont indiquees, dont une (“C”), correspondante a une jonctiondes modes initialement separes dans le domaine spectrale de Be–Se a partir dela pression de 14 GPa a peu pres, fait l’objet de l’analyse theorique. Les calculspremiers principes ont ete faits sur une supercellule cubique de 64 atomes, dontsoit une soit deux impuretes de Be (dans le dernier cas, le deux etant voisinsdu meme anion Se) dans le cristal de ZnSe; le calcul a ete fait en utilisant lemethode Siesta dans l’approche LDA. Lorsque la pression a ete variee comme unparametre exterieur, la structure cristalline a ete chaque fois relaxee, et les modesde vibration calcules par le methode de phonons geles. Le but de ces simulationsconsiste en estimation quantitative des parametres essentiels du modele de perco-lation – la frequence du mode d’impurete isolee et la separation entre les modesdifferents de vibration (“le long de la chaıne” ou “perpendiculaire a la chaıne”) dedeux impuretes, en fonction de la pression externe. J’ai demontre que, effective-ment, le calcul reproduit une observation experimentale de la jonction des modesde vibration Be–Se sous pression. Grace a une possibilite d’analyser le caracterede tous modes de vibration d’apres le calcul, on peut noter que la jonction dedeux modes arrive suite au “refroidissement” progressive des liaisons Be–Se dansune petite region ou ils sont obligees de s’adapter a la structure “non naturelle”de l’environnement cristallin ZnSe.

Comme une illustration de decalage des frequences de vibration en fonction delongueurs de liaison, les resultats des calculs realises auparavant par une thesardede O. Pages, Jihane Souhabi, pour le systeme Ga(As,P), en utilisant la meme

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taille des supercellules que les susmentionnees, sont montres en Fig. 6.2. Meme sila substitution dans ce cas est effectuee sur le sous-reseau anionique, l’effet prin-cipal reste identique: on detecte un decalage entre des frequences de vibrationassociee a une impurete isolee de P dans GaAs et une paire de tels impuretes, avecun “doublet de percolation” d’ordre Δ ∼12 cm−1. Un tel doublet se manifestelorsque la liaison concernee (Ga–P, alors la plus courte dans l’alliage en question)vibre dans deux environnements differents, un riche en GaAs et un autre richeen GaP. La situation dans (Zn,Be)Se est largement similaire. La Fig. 6.2 montreschematiquement les modes de vibration associes aux impuretes soit isoles soitinteragissants mutuellement, et le placement des lignes de vibration correspon-dantes a ces deux situations. Cette analogie avec l’alliage Ga(As,P) permet decomprendre l’origine de l’anomalie “C” dans le spectre de (Zn,Be)Se sous pres-sion, prenant en compte que le motif de vibration classe comme “bond bending”par rapport aux impuretes est au meme temps une vibration “bond stretching”de point de vue du cristal principal. Alors “C” est une manifestation d’une“congelation” graduelle des liaisons Be–Se qui se trouvent dans les domainesminoritaires “riches en BeSe” lorsque ces derniers sont obliges d’accepter unestructure RS imposee par les regions ZnSe, qui est “non naturelle” pour BeSe.

Le septieme chapitre, Modelisation computationnelle des proprietesstructurelles, electroniques et thermodynamiques de l’alliage Cd(S,Se),explique la construction du diagramme de phases concentration–temperature etdiscute de la variation des parametres structuraux pour les compositions in-termediaires. La motivation pour le choix des systemes est la suivante. CdSet CdSe sont les semiconducteurs de grande bande interdite, qui trouvent leur us-age dans l’opto-electronique, surtout l’optique non-lineaire, et dans la fabricationdes diodes electroluminescentes et des lasers. Un besoin d’avoir une largeur de labande interdite precisement controlee peut etre accede par variation de la compo-sition. Les systemes mixtes CdSxSe1−x peuvent etre fabriques dans toute gammede concentrations x = 0−1, ce que resulte en une bande interdite d’entre ≈2,44 eVpour CdS et ≈1,72 eV pour CdSe. Les longueurs d’onde correspondentes sontde 509 jusqu’a 720 nm, que couvre presque tout l’intervalle visible de spectre.Un materiel ideal par rapport a son efficacite opto-electronique serait de suivreune demi-voie entre le CdS (susceptibilite elevee; temps de reponse long) et CdSe(temps de reponse plus court mais susceptibilite moins haute). Un grand nom-bre d’etudes concentres sur l’usage possible de Cd(S,Se) pour photovoltaıqueet photoelectrochimie existent deja. De plus, Cd(S,Se) semble d’etre une bonnesubstance active pour la construction des lasers semiconducteurs dans le domainevisible.

Les proprietes structurelles et electroniques des systemes purs CdS, CdSeet des superstructures de compositions intermediaires, generees dans le cadrede l’approche SQS pour supercellules des 32 atomes, ont ete etudiees par le

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methode PW-PP realisee dans le logiciel VASP. Suite a l’optimisation illimiteedes parametres de reseau et des coordonnes internes, la variation des longueursde liaison est intermediaire entre la resultat “naıf” de l’approximation du cristalvirtuel (c’est-a-dire, la variation lineaire en fonction de la concentration) et lalimite de Pauling (ou, autrement, de la relaxation complete), etant plus proche ace dernier (Fig. 7.1). La Fig. 7.2 montre la variation des parametres de relaxationdes liaisons, donnes par Eq. (1.3). En ce que concerne le placement relative lesdeuxiemes voisins, presentee dans la Fig. 7.3, on note que les distances anion-anion suivent la tendance lineaire (loi de Vegard) en fonction de la concentration,tandis que les distances Cd–Cd se separent nettement en deux groupes, vis-a-visde l’anion, soit S soit Se, qui les connecte. Ceci confirme la tendance, aussi con-nue pour les autres semiconducteurs mixtes, que le reseau anionique, lorsque laconcentration varie, comprime ou dilate de la facon presque rigide.

La section 7.4. Transition de phase induite par la composition et lebowing de la bande interdite concerne la competition de deux structures,ZB et WZ, de sorte que l’energie totale de WZ est bien superieure a celle deZB dans le CdSe, mais inferieure dans le CdSe. Une transition presque continue(sauf une certaine ambiguıte liee au resultat pour a la concentration de 50%) estevidente dans la figure 7.4. De toute facon, les calculs impliquent une transitionZB → WZ pour x entre 0,25 et 0,5 (de S, dans CdSxSe1−x) tandis que les sourcesexperimentales donne des valeurs x=0,4 ou 0,6. En outre, la structure de la bandeinterdite (Fig. 7.5) est caracterisee dans cette section par le parametre de bandgap bowing b, explique auparavant dans le Chapitre 3. Une comparaison est faiteavec les calculs precedents. Suite au fait que le desaccord entre des parametres dereseau de CsS et CdSe compte 4% seulement, le band gap bowing est assez faibledans leurs alliages – 0,25 dans le WZ et 0,32 dans le ZB, d’apres le calcul actuel.

La section 7.5: Proprietes thermodynamiques est centrale pour la dis-cussion sur la preference energetique relative aux deux phases, ZB et WZ, etla construction theorique du diagramme de phases composition / temperature.L’approche utilisee est assez standarde, suite des travaux de van de Walle etCeder, Wei et Zunger, et des autres auteurs. Le point de depart est le cal-cul de l’enthalpie de formation des phases concurrentes, de la facon similaire decelle decrite dans le Chapitre 4 pour les systemes ZnSe, ZnS purs, mais main-tenant prenant en compte la concentration variable. Les calculs pour concentra-tions intermediaires x=0,25; 0,5; 0,75 ont ete faits en utilisant les superstructuresspeciales quasi-aleatoires (SQS), generes apres la suggestion de Wei – Zunger –Ferreira – Bernard, contenants dans notre cas 32 atomes. La Fig. 7.6 montre lesenthalipes de formation (exprimees par rapport des valeurs pour deux systemesparents purs); une transition, a une concentration intermediaire, d’un regimede preference du phase ZB (CdSe pur) vers la dominance du phase WZ (CdSpur) est evidente. Le fittage polynomial de l’enthalpie calculee en fonction de la

xix

concentration est ensuite utilise pour la construction du diagramme de phases.Les vibrations de reseau, eventuellement capables de donner une contribu-

tion importante a l’entropie de formation d’un alliage, ont ete calcules, toujourspour les memes supercellules SQS, par la methode de phonons geles. L’evolutiondes spectres de densite des modes de vibration est montre dans la Fig. 7.7. Lesdetails de calcul technique de la contribution correspondente a l’entropie sontdecrites dans l’Annexe C, mais les calculs memes n’ont pas ete effectues dansle cadre de ce travail. Sans contribution vibrationnelle a l’entropie, mais enprenant bien en compte l’entropie configurationelle de l’alliage, les courbes bin-odale et spinodale sont construites, qui representent des courbes dans le planconcentration–temperature et qui separent les domaines de stabilite, instabilite,ou meta-stabilite sur le diagramme de phases (Fig. 7.8 pour la structure ZB;Fig. 7.9 pour WZ).

Les Conclusions generales recapitulent la motivation dans le choix de su-jet du travail, decrivent le deroulement de ce dernier, et detaillent ces resultatsessentiels chapitre par chapitre.

L’Annexe A se porte sur les relations structurelles entre les phases ZB, RS etcinnabar, et notamment sur les trajets de transformations structurelles a partirde chacune de ces phases vers les deux autres. Cette analyse est largement baseesur les travaux de Hudrun Sowa, qui a suggere deux types de transformationZB ↔ RS plausibles et “conservatrice”, ne modifiant en route que graduellementet legerement les longueurs des liaisons et les coordinations des atomes voisins.Une telle transformation connecte une phase ZB, decrite, dans l’arrangementhexagonale, par le groupe d’espace R3m (Nr 160) ou encore, dans la symetriereduite, par le groupe d’espace R3 (Nr 146), a travers la structure de cinnabardecrite par le groupe d’espace soit P31 soit P32, vers le RS. Il faut noter quela structure RS emerge comme un cas special de cinnabar, avec les valeurs descoordonnes internes u, v = 1/3 et le rapport des parametres de reseau c/a =

√6.

Un autre trajet, egalement suggere par H. Sowa, porte la structure ZB directementvers la RS, sans quitter le groupe d’espace soit P31 soit P32 mais ne passantpas par la structure de cinnabar. Une consequence eventuellement importantedans le contexte du travail precedent est une possibilite d’organiser un trajet“triangulaire” ZB ↔ cinnabar ↔ RS ↔ ZB, probablement responsable pour lecomportement irreversible (donc differente dans “upstroke” et “downstroke”) lorsdes etudes des transitions de phase sous pression dans ZnSe (voir Chapitre 4).

L’Annexe B exprime les longueurs de liaison cation–anion et angles entre lesliaisons (cation–anion–cation ou anion–cation–anion) dans les deux structurescristallines, de types cinabar et SC16 , en fonction des parametres cristallo-graphiques (parametres de reseau, coordonnes internes) de ces phases-la. Cecipermet de “traquer” les modifications de la structure, comme obtenues en coursde l’optimisation par le methode WIEN2k, de la facon palpable, permettant ainsi

xx

une comparaison plus facile avec l’experience. Une discussion bien detaillee sur lastructure est donnee pour deux groupes d’espace enantiomorphes, P3221 (N◦ 154)et P3121 (N◦ 152), dont chacun permet une description alternative de la phasedu type cinnabar.

L’Annexe C presente une deduction detaillee de la formule pour l’energie li-bre de Helmholtz dans le cas d’un systeme des oscillateurs quantiques (phonons),subis a la statistique de Bose–Einstein, Eq. (C.1), y compris leur energies de zeropoint. L’expression resultante est celle qui figure dans les formules de l’equilibrethermodynamique dans le Chapitre 7. La deduction est bien generale, de sorteque toute la specificite d’un materiel soit prise en compte par son spectre de vibra-tions (densite des modes de vibration). On admet que ce spectre est calcule dansl’approximation harmonique et alors pour la temperature nulle; tout l’effet dela temperature dans la description thermodynamique (diagramme de phases) estpris en compte par la fonction de Bose et alors dans le contexte quasi-harmonique.Apres avoir obtenue l’expression pour l’energie interne (Eq. C.2; l’energie totaledes oscillateurs), les expressions sont obtenues pour la chaleur specifique (C.3),fonction de partition (C.4) et ensuite l’energie libre. La differentiation de cettederniere en temperature permet d’exprimer l’entropie (C.10). Le seul parametrenon triviale caracterisant le systeme des oscillateurs est leur densite de modes,disponible d’apres le calcul des “phonons geles” ou par reponse lineaire. Les sug-gestions pratiques sont faites comment realiser le calcul ayant un spectre discretde vibrations, resultant d’un calcul de phonons pour le vecteur d’onde q = (0, 0, 0)de la supercellule (C.17).

L’Annexe D concerne les modalites techniques du tracement des courbes(x, T ) en cours de la construction du diagramme de phase pour un alliage; unsujet couvert par la Section 2.6 du texte principal. Il s’agit notamment dedeux sortes de courbes. La courbe spinodale est un lieu commun de points dis-tincts [x1(T ), x2(T )] , dans lesquels l’energie libre de Helmholtz F (x, t), construited’apres l’Eq. (D.1) pour une temperature quelconque T , est touchee par une tan-gente commune. La courbe binodale dans le plan (x, T ) transverse les points ou ladeuxieme derive ∂2F (x, T )/∂x2 soit null. Chacun de ces deux especes de courbe(x, T ) prend la forme de deux branches, schematiquement indiques en Fig. 2.5.L’energie libre (D.1) consiste en l’energie totale, issue du calcul premiers principeset parametrise en fonction de x dans la forme polynomiale, et de l’autre part pro-portionnelle a l’entropie, qui a, en son tour, une contribution combinatorique (caril s’agit d’un alliage de substitution), et une contribution vibrationnelle (v. An-nexe C). Le tracement pratique des branches binodales consiste en la resolution,pour chaque T , d’un systeme de deux equations non-lineaires (D.4). Ceci est or-ganise par le methode iterative de Newton, concretisee en Eq. (D.5) – (D.7). Letracement de la courbe spinodale procede, pour chaque valeur de x, en la solutionpour T qui assure ∂2F (x, T )/∂x2 = 0.

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Publication des resultats

[1] A. Breidi, B. Amrani and F. E. H. Hassan. First-principles calculations onthe origins of the gap bowing in InAs1−xPx alloys. Physica B: Condensed Mat-ter 404 (20), pp. 3435–3439 (2009). DOI: 10.1016/j.physb.2009.05.029.

[2] F. El Haj Hassan, A. Breidi, S. Ghemid, B. Amrani, H. Meradji andO. Pages. First-principles study of the ternary semiconductor alloys(Ga,Al)(As,Sb). Journal of Alloys and Compounds 499 (1), pp. 80 – 89(2010). DOI: 10.1016/j.jallcom.2010.02.121.

[3] A. Breidi, A. V. Postnikov and F. El Haj Hassan. Cinnabar and SC16 high-pressure phases of ZnSe: An ab initio study. Phys. Rev. B 81 (20), p. 205213(2010). DOI: 10.1103/PhysRevB.81.205213.

[4] G. K. Pradhan, C. Narayana, O. Pages, A. Breidi, J. Souhabi, A. V. Post-nikov, S. K. Deb, F. Firszt, W. Paszkowicz, A. Shukla and F. El Haj Has-san. Pressure-induced phonon freezing in the Zn1−xBexSe alloy: A studyvia the percolation model. Phys. Rev. B 81 (11), p. 115207 (2010).DOI: 10.1103/PhysRevB.81.115207.

[5] G. K. Pradhan, C. Narayana, M. N. Rao, M. D’Astuto, S. L. Chaplot,O. Pages, A. Breidi, J. Souhabi, A. Postnikov, S. K. Deb, T. Ganguli,A. Polian, G. Bhalerao, A. Shukla, F. Firszt and W. Paszkowicz. The phononpercolation scheme for alloys: Extension to the entire lattice dynamics andpressure dependence. Japanese Journal of Applied Physics 50 (5), p. 05FE02(2011). DOI: 10.1143/JJAP.50.05FE02.

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Contents

List of abbreviations 4

List of figures 5

Introduction 7

1 Preliminaries of work 111.1 Crystal structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.1.1 Zincblende, Diamond and Wurtzite Structures . . . . . . . 111.1.2 Bulk alloys: general observations . . . . . . . . . . . . . . 131.1.3 Alloy structure determined by EXAFS . . . . . . . . . . . 15

1.2 Pressure-induced phase transitions . . . . . . . . . . . . . . . . . 181.2.1 Role of pressure in materials science: a brief introduction . 181.2.2 Thermodynamical and structural issues . . . . . . . . . . . 201.2.3 Pressure-induced phase transitions

in binary semiconductors . . . . . . . . . . . . . . . . . . . 231.3 Phase equilibria in pseudobinary alloys: a brief introduction . . . 26

2 Basics of theory and approximations used 292.1 Ab initio electronic structure calculations . . . . . . . . . . . . . . 29

2.1.1 Density functional theory . . . . . . . . . . . . . . . . . . 292.1.2 Solving techniques . . . . . . . . . . . . . . . . . . . . . . 342.1.3 Bloch Theorem . . . . . . . . . . . . . . . . . . . . . . . . 342.1.4 Brillouin zone . . . . . . . . . . . . . . . . . . . . . . . . . 352.1.5 Electronic-structure methods . . . . . . . . . . . . . . . . 362.1.6 Accuracy of DFT calculations . . . . . . . . . . . . . . . . 432.1.7 Fixed symmetry studies of phase stability . . . . . . . . . 432.1.8 Finite temperature effects on phase stability . . . . . . . . 46

2.2 Static lattice model and its limitations . . . . . . . . . . . . . . . 472.3 Adiabatic approximation . . . . . . . . . . . . . . . . . . . . . . . 482.4 An introductory word about phonon . . . . . . . . . . . . . . . . 50

1

Contents

2.5 Harmonic approximation formalism . . . . . . . . . . . . . . . . . 522.5.1 Nonperturbative methods . . . . . . . . . . . . . . . . . . 542.5.2 Perturbative approach . . . . . . . . . . . . . . . . . . . . 56

2.6 Pseudobinary alloy phase diagram . . . . . . . . . . . . . . . . . . 572.7 Phase diagram analytical models . . . . . . . . . . . . . . . . . . 61

2.7.1 Ideal-solution model . . . . . . . . . . . . . . . . . . . . . 612.7.2 Zeroth approximation . . . . . . . . . . . . . . . . . . . . . 62

2.8 First-principles calculations of thermodynamic properties . . . . . 62

3 First-principles study of gap bowing in InAs1−xPx alloys 673.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.2 Method and results . . . . . . . . . . . . . . . . . . . . . . . . . . 683.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . 68

3.3.1 Structural properties . . . . . . . . . . . . . . . . . . . . . 683.3.2 Gap bowing and its origins . . . . . . . . . . . . . . . . . . 70

3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4 Pressure-induced phase diagram of the II-VI semiconductorcompounds: ZnSe and ZnS 754.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.2 Methods and technical details . . . . . . . . . . . . . . . . . . . . 774.3 ZnSe: preliminary considerations and total-energy results . . . . . 78

4.3.1 Cinnabar structure . . . . . . . . . . . . . . . . . . . . . . 824.3.2 SC16 structure . . . . . . . . . . . . . . . . . . . . . . . . 884.3.3 Open questions . . . . . . . . . . . . . . . . . . . . . . . . 914.3.4 Conclusion on ZnSe . . . . . . . . . . . . . . . . . . . . . . 92

4.4 Zinc Sulfide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 924.4.1 Experimental and theoretical evidence;

relation to ZnSe and other materials . . . . . . . . . . . . 934.4.2 Calculation results . . . . . . . . . . . . . . . . . . . . . . 94

5 Dynamical instability of zincblende phase in ZnSe 975.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975.2 Method, parameters, accuracy tests . . . . . . . . . . . . . . . . . 985.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . 100

6 Pressure effect on lattice dynamics in (ZnBe)Se 1036.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1036.2 Experimental details and ab initio methods . . . . . . . . . . . . 1056.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . 1066.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

2

Contents

7 Computational modelling of structural, electronic andthermodynamic properties of Cd(S,Se) alloy 1117.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1117.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1127.3 Structural characterizations . . . . . . . . . . . . . . . . . . . . . 1127.4 Composition-induced phase transition and bandgap bowing . . . . 1157.5 Thermodynamic properties . . . . . . . . . . . . . . . . . . . . . . 117

7.5.1 Enthalpy of formation . . . . . . . . . . . . . . . . . . . . 1187.5.2 Lattice vibration calculations . . . . . . . . . . . . . . . . 1207.5.3 Alloy phase diagram . . . . . . . . . . . . . . . . . . . . . 121

General conclusions 125

A Relation between zincblende, rocksalt and cinnabar structures 127A.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

A.1.1 ZB structure . . . . . . . . . . . . . . . . . . . . . . . . . . 127A.1.2 RS structure . . . . . . . . . . . . . . . . . . . . . . . . . . 128A.1.3 Relation between ZB and RS . . . . . . . . . . . . . . . . 129A.1.4 B9 structure . . . . . . . . . . . . . . . . . . . . . . . . . . 129

A.2 Path from ZB via cinnabar to RS . . . . . . . . . . . . . . . . . . 130A.3 Path from ZB directly to RS . . . . . . . . . . . . . . . . . . . . . 133A.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

B Bonds and angles in cinnabar and SC16 phases 137B.1 Cinnabar phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137B.2 SC16 structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

C Phonon energy expression 143

D Calculation of binodal and spinodal linesin the alloy phase diagram 147

References 149

Index 177

3

List of abbreviations

APW augmented plane wave (method)CPA coherent potential approximationDAC diamond anvil cellDFT density-functional theoryDOS density of statesEXAFS extended x-ray absorption fine structureFP-LAPW full-potential linearized augmented plane wave (method)GGA generalized gradient approximation (in DFT)HF Hartree–Fock (approximation; method)IR infraredLCAO linear combinations of atomic orbitals (method)LDA local density approximation (in DFT)LO longitudinal optical (phonon)MREI modified random-element isodisplacementMT muffin-tin (geometry; sphere; approximation)NN nearest neighboursNNN next-nearest neighboursPAW projector-augmented-wave (method)PP pseudopotentialPW plane wave(s)RS rocksalt (structure)R.T. room temperatureSiesta “Spanish Initiative for Electronic Structure calculations

with Thousands of Atoms” (method)SQS special quasirandom structure(s)TO transversal optical (phonon)USPP ultrasoft pseudopotentialVASP Vienna ab-initio simulation packageVCA virtual crystal approximation (for alloys)VFF Valence Force Field (method)WZ wurtzite (structure)XC exchange-correlation (energy; potential)XRD X-ray diffractionZB zinc blende (structure)ZC zone-center (phonon)

4

List of figures

1.1 Zincblende and wurtzite crystal structures . . . . . . . . . . . . . 12

1.2 X-ray diffraction peaks of GaAs, InAs and (Ga,In)As . . . . . . . 14

1.3 Nearest-neighbours distances in (In,Ga)As from EXAFS . . . . . 16

1.4 Next-nearest neighbours distances in (In,Ga)As from EXAFS . . . 17

1.5 Rocksalt and Cmcm structures . . . . . . . . . . . . . . . . . . . 25

2.1 Division of space into muffin-tin spheres and interstitial region . . 41

2.2 Calculated and experimental volumes in binary compounds . . . . 44

2.3 Typical output data from a series of total-energy calculations . . . 46

2.4 Schematic view of a supercell representing a frozen phonon . . . . 55

2.5 Schematic mixing free energies, miscibility gap, and spinodal curvesfor positive mixing enthalpy . . . . . . . . . . . . . . . . . . . . . 59

2.6 Schematic picture of mixing free energies as a function of alloycomposition, for negative mixing enthalpy . . . . . . . . . . . . . 61

2.7 Temperature dependence of the free energy of θ and θ′ phases inAl2Cu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

2.8 Thermal expansion of selected metals calculated within the quasi-harmonic approximation . . . . . . . . . . . . . . . . . . . . . . . 66

3.1 Calculated lattice constant and bulk modulus in InAs1−xPx . . . . 70

3.2 Band gap in In(As,P) calculated with GGA and EV-GGA . . . . 71

4.1 Total energy vs. volume for ZnSe in different phases . . . . . . . . 79

4.2 Static enthalpy per atom in different phases of ZnSe . . . . . . . . 79

4.3 Scheme of distorted tetrahedric coordinations around anion andcation sites in the cinnabar structure . . . . . . . . . . . . . . . . 82

4.4 Internal coordinates in the cinnabar phase of ZnSe . . . . . . . . . 83

4.5 Variation of lattice parameters in the cinnabar phase of ZnSe . . . 84

4.6 Band gap evolution with pressure in four ZnSe phases . . . . . . . 85

4.7 Band structure and total density of states as calculatedin the cinnabar phase of ZnSe within GGA and LDA . . . . . . . 86

5

List of figures

4.8 Bond lengths and bond angles under different pressuresin the cinnabar structure of ZnSe . . . . . . . . . . . . . . . . . . 87

4.9 A side view of the cubic primitive cell of the SC16 structure . . . 894.10 Internal coordinates and lattice parameter in the SC16 phase

of ZnSe as optimized at different applied pressures . . . . . . . . . 904.11 Bond lengths and bond angles in the SC16 phase of ZnSe . . . . . 904.12 Total energy vs. volume for ZnS in different phases . . . . . . . . 954.13 Static enthalpy per atom in different phases of ZnS . . . . . . . . 95

5.1 Volume-pressure curves for ZB ZnSe with abinit and WIEN2k . . 995.2 Phonon dispersion curves of ZnSe crystal in cubic ZB structure,

at two pressures, before and after the ultimate softening at X . . 1005.3 TA phonon frequency at X point of the BZ for ZnSe,

depending on volume and on pressure . . . . . . . . . . . . . . . . 101

6.1 Pressure-dependent Raman spectra of Zn0.76Be0.24Se . . . . . . . . 1076.2 Ab initio ZC TO-DOS of 2-imp. Be in ZnSe and P motif in GaAs;

vibration patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

7.1 Variation of the bond lengths in CdSxSe1−x alloy with composition,from calcultions in ZB supercells . . . . . . . . . . . . . . . . . . 114

7.2 Variation of the relaxation parameters in ZB CdSxSe1−x alloywith composition . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

7.3 Variation of anionic and cationic NNN distances in CdSxSe1−x . . 1157.4 Total energies of CdSxSe1−x WZ supercells relative to those

in corresponding ZB supercells . . . . . . . . . . . . . . . . . . . . 1167.5 Variation of optical bandgap of CdSxSe1−x as function of x. . . . 1177.6 Formation enthalpy of CdSxSe1−x in WZ and ZB structures as

function of composition x . . . . . . . . . . . . . . . . . . . . . . 1197.7 Phonon DOS of ZB and WZ CdSxSe1−x, for x=0, 1/4, 1/2, 3/4

and 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1217.8 Free energy variation with alloy composition in ZB structure of

CdSxSe1−x-ZB at 180 K; x − T phase diagram . . . . . . . . . . . 1237.9 Same as Fig 7.8, for WZ structure. . . . . . . . . . . . . . . . . . 124

6

Introduction

Out of different subjects I came across during the work on my thesis, this manu-script converged gradually onto first-principles studies of electronic properties,equilibrium structure and lattice dynamics in II-VI semiconductors at ambientconditions and under pressure. A bunch of controversial results exists in whatregards their corresponding behaviour and the issues of their thermodynamicphase stability. I discuss my own and other people’s results in the broad contextof information available on a number of related compounds, from both experi-ment and theory, in the hope to offer general conclusions about the nature ofthe enthalpy – pressure and temperature – composition phase diagrams of thesesemiconductor materials.

The first-principles calculations done here are, in a sense, complementary toexperimental studies. Approximative and limited as they are, they yield never-theless informations not available from direct measurements. For example, onecan computationally simulate a study for an unstable phase which helps in iden-tifying its competitivity with the other ones. Moreover, the calculations give aninprecedented insight into how the atoms shuffle under pressure (via an inspectionof their phonon modes eigenvectors). Due to an absolute control, in a calculation,over physical conditions and chemical compositions of the system, a certainty ex-ists at least about the structures, therefore the calculations serve as a powerfultool to verify and probe experimental results. It is moreover feasible to enforcecalculations to “operate” at extreme conditions (high pressure or temperature),complicated if at all possible in experiment. The calculation can, with a bit ofluck, be used to predict new high-pressure phases. Experimental informationabout the actual path by which a transition takes place is very hard to obtain,but first-principles calculations may allow to guess and explore transition pathsand activation barriers.1

The leitmotiv of this thesis is phase stability of semiconductor compoundsand alloys, probed by performing first-principles calculations within the contextof the Density Functional Theory (DFT). The concerned materials belong to twocategories: pure compounds, ZnS and ZnSe, and pseudobinary alloys Cd(S,Se),

1In this thesis I do not probe possible activation barriers between different structures.

7

Introduction

(Zn,Be)Se and In(As,P). The main axis of the thesis splits into two major lines,investigating (a) pressure-induced phase transitions in the binary compoundsand (b) temperature-concentration effect on pseudobinary alloy phase stability.The first task is illustrated by constructing the pressure-induced phase diagramusing enthalpy comparison method (often called “standard method”) and es-tablishing phonon dispersion curves under pressure using the DFT-based latticedynamics (linear response method). The second task is illustrated by forming thetemperature-concentration (x−T ) and the composition-induced phase transitiondiagrams of the considered alloys’ phases. In parallel, diverse properties are alsoaddressed with adequate importance to those of the major lines: structural, elec-tronic, optical and thermodynamic. The interest in such properties is mostly dueto their connection to, or indispensability to understand/explain, the main taskof the thesis mentioned above. Full structural characterizations are done for allphases that the relevant materials would assume, e.g., lattice parametes, atomicinternal coordinates, bond lengths and bond angles versus pressure (for binarysemiconductors) or alloy composition (for pseudobinary alloy), that was achievedby relaxing all structural degrees of freedom with respect to the total energy.For each phase, I calculate the electronic band structure, electronic density ofstates, and optical band gap evolution under pressure, or under alloying compo-sition. The formation enthalpies and, consequently, the Helmholtz free energyof mixing of pseudobinary alloys, which are essential thermodynamic propertieshelpful for establishing the solubility and instability limits in the temperature-concentration diagram, are determined for different potential phases. I carefullytreat alloy randomness by employing the special quasirandom structures (SQS)method generating representative supercells corresponding to different structuralconfigurations at some sensitive anionic concentrations. When calculating theHelmholtz free energy, the ionic thermal vibration energy contribution is also cal-culated using the “frozen phonon” method within the harmonic approximation.The pressure-induced lattice dynamics changes are probed uniquely for one alloy.

This dissertation is organized as follows:The first (following) chapter, “Preliminaries of Work”, is a bibliographic re-

view of the addressed issues in this thesis.In the second chapter, “Experiment and theory methods”, experimental tech-

niques and their existing challenges are discussed in the first place, then first-principles electronic structure calculations are intensively addressed. As I fo-cused on the physical and technical sides of how DFT is applied to unravel andelucidate phase stability in binary compounds and pseudobinary alloys, I inten-tionally escaped DFT mathematical formulations and details which are not thegoal of this thesis. In doing so I tried to avoid banalities in outlining the DFTand redundancies. Theoretical methods needed to investigate pressure-inducedphase transitions and pseudobinary alloy phase diagrams are extensively pre-

8

Introduction

sented. The failure of the static model and the importance of the adiabatic oneare discussed, and harmonic / quasiharmonic approximation, direct and linearresponse methods to solve lattice dynamics problems are explained in a broadcontext, emphasizing their crucial role in solving problematic issues addressed bythe following chapters.

The third chapter, “First-principles study of the gap bowing in InAs1−xPx

alloys”, exclusively deals with the results of structural and electronic-propertiessimulations for the In-As-P pseudobinary alloy, namely variation of its structuralparameters (lattice constant, bulk modulus) and optical band gap with alloyingcomposition, providing a detailed procedure to understand the deviation of theband gaps from linearity.

In the fourth chapter, “Pressure-Induced phase diagrams of the II-VI semi-conductor compounds: ZnSe and ZnS”, the enthalpy comparison method is em-ployed to study thermodynamic phase stability of a set of selected structures,while applying external pressure on pure ZnSe and ZnS semiconductors. Fullstructural characterizations are given for the low symmetry intermediate-pressurestructures, cinnabar and SC16. Electronic band structures, electronic density ofstates and the optical band gap are calculated for each phase at different appliedpressures.

The fifth chapter, “Dynamical instability of zincblende phase in ZnSe”, ex-plores the effectiveness of the linear response method in revealing the structuralinstabilities of ZnSe-zincblende phase under pressure. It features the importanceof this increasingly appreciated method in studying structural phase transitions.

The sixth chapter “Pressure-driven phase transition effect on ZnBeSe lat-tice dynamics”, which explains my immediate calculation results in the contextof joint multipartner experimental / theoretical work, elucidates the pressure-induced change in phonon mode behavior for the highly contrasted (in whatregards vibration properties) Zn-Be-Se alloy when approaching the zincblende-rocksalt transition, a known one for the Zn-Se matrix but unnatural “from thepoint of view of” the Be-Se bonds. Calculations were done using the “frozenphonon” method at several applied hydrostatic pressures and for different atomicsubstitutions on the cationic (Zn/Be) sublattice.

The concluding seventh chapter “Computational modelling of structural, elec-tronic and thermodynamic properties of Cd(S,Se) alloy” outlines the results ofDFT-based thermodynamic modelling of concentration – temperature phase dia-gram in random-type Cd-S-Se ternary system, taking into account its both possi-ble zincblende and wurtzite structures. Disordering effects were simulated by em-ploying the SQS methodology. Structural and electronic (optical band gap) prop-erties are modelled for both structures (ZB and WZ), and composition-inducedphase transition is obtained. Then, I turn to a calculation of the (x, T )-phasediagram of CdSSe. Care is taken to determine the ionic vibration frequencies,

9

Introduction

since a recent study done on semiconductor has shown that including such con-tribution can modify the solubility limits curve and shift the critical temperatureof the order-disorder transition on the substitutional sublattice. This was doneby using the “frozen phonon” method, within the harmonic approximation.

10

Chapter 1

Preliminaries of work

1.1 Crystal structures

As we’ll deal in the following with binary semiconductors and their alloys, areview of corresponding structures would be in place. I start with those of puresemiconductors, really well known (at ambient pressure), and then proceed towhat is revealed, concerning the structures of nearly random bulk alloys, byextended X-ray absorption fine-structure spectroscopy (EXAFS).

1.1.1 Zincblende, Diamond and Wurtzite Structures

These are structures with tetrahedral coordination of each atom, favoured by thesp3 hybridization and hence common in semiconductors, but rare otherwise.

The diamond structure (A4 according to Strukturbericht, space group Fd3mNr. 227) consists of two interpenetrating face-centered-cubic (fcc) sublatticesdisplaced from each other by (1

414

14)a, where a is the cube edge length, referred

to as lattice constant. Besides namely diamond (tetrahedral carbon, attributedto wide-gap semiconductors), the both known “conventional” elementary semi-conductors, Si and Ge, and moreover the semimetal gray tin (α-Sn) assume thisstructure.

The zincblende structure (ZB in the following, named after the ZnS min-eral, called also sphalerite) has its two sublattices occupied by different chemicalspecies, e.g., an anion and a cation in binary semiconductors. This structure(Fig. 1.1, left panel) appears in Strukturbericht as B3; its space group is F 43m(Nr 216).

Wurtzite (WZ in the following, B4 of Strukturbericht, space group P63mc Nr.186) is a hexagonal “analogue” of ZB, different from the latter by rotating thecoordination of each second atom (say, each anion) around a selected anion-cationbond (running along the z axis) by 60◦ (Fig. 1.1, right panel). As a result, when

11

Chapter 1. Preliminaries of work

Figure 1.1: Zincblende (left) and wurtzite (right) crystal structures.

looking along [001] the bonds are seen in the “eclipsed” configuration, whereasin ZB, looking along the cubic [111] axis, they would be in the “staggered” one.The hexagonal space group of WZ allows an internal coordinate u, which has themeaning of the selected (that along [001]) bond length in terms of the c latticeconstant. The primitive cell has two double (cation+anion) layers, stacked asABABAB... (Zincblende, seen as hexagonal structure with c‖[111], would have6 atoms per unit cell and ABCABC... stacking of double layers). The lengthrelations between ZB and WZ are shown in Table 1.1. The “ideal” WZ (allbond lengths and angles as in ZB) corresponds to aw = azb/

√2; cw = 2azb/3;

cw/aw =√

8/3; u = 3/8.Since ZB and WZ may have almost identical bond lengths, i.e. nearest

neighbours (NN) distances, and bond angles, differing only in coordination ofnext-nearest neighbours (NNN), their total energies (per formula unit) are usuallyvery close, so that these phases are strongly competitive. In order to lower itstotal energy, the WZ phase makes use (to larger or smaller extent, depending onthe compound in question) of an option to slightly deviate its c/a and u valuesfrom their “ideal” values. In fact those semiconductors known to be definitelymore stable in the WZ phase exhibit noticeable deviations from ideal c/a and uvalues – see Lawaetz [1972].

A natural complication within ZB- or WZ-type semiconductors occurs wheneither the cation, or the anion, or both sublattices are occupied by different chem-ical species. If such substituents belong to the same column of the periodic tableand are thus characterized by identical nominal valence, we have what is referredto as (mixed) III-V or II-VI semiconductor. Components of the same valenceare typically freely miscible on a given sublattice, and populate its sites in moreor less random fashion. Heterovalent substitutions are also possible, but they

12

1.1. Crystal structures

Table 1.1: Relations between ZB and WZ structures; NN: nearest neighbours(cations to anions); NNN: next-nearest neighbours (cations to cations or anionsto anions).

Structure;lattice parameters bond angles neighbours

zincblende (azb) 6× arccos(−13) ≈ 109.5◦ 4NN at azb

√3/4

12NNN at azb/√

2

wurztite (aw, cw; u) 3× arccos(−√

xx+1

)1NN at ucw

3× arccos(

2x−12x+2

)3NN at

√a2

w/3 + c2w(1

2−u)2

6NNN at aw

x = 3(12−u)2(cw/aw)2 6NNN at

√a2

w/3 + c2w/4

demand to respect certain rules, notably the “octet rule”: each cation or anionmust have a complete shells of 8 electrons, shared with its four neighbours. Thisis, in general case, only possible at certain concentrations and enforces a certainordering in the placement of constituent atoms over sublattices. For example, aAIIBVI ZB semiconductor may allow a diversification AII→CI+DIII at the cationlattice, resulting in a chalcopyrite-type CDB2 lattice (space group I 42d). Furtherternary and multinary compounds, structurally being derivatives of ZB or WZlattice, may result. Their common feature is a tetrahedral coordination, more orless distorted due to diversification of cation-anion bond lengths.

1.1.2 Bulk alloys: general observations

We confine us henceforth to the situaton of homovalent alloying, when, withina given structure type, a given atomic position may host two different chemicalspecies, distributed more or less randomly according to concentration x. We shalldistinguish between binary semiconductor alloys AxB1−x, where A and B bothbelong to the IV column of the periodic table (e.g. Si–C or Si–Ge), and pseudobi-nary alloys AxB1−xC, where the C sublattice remains complete and ordered (withmany III-V or II-VI examples: GaxIn1−xAs, ZnSxSe1−x, etc.) If the alloying isproceeding from two parent compounds having similar structure, then the under-lying crystal structure remains the same throughout broad concentration ranges,

13

Chapter 1. Preliminaries of work

Figure 1.2: X-ray (422) diffraction peaks of GaAs, InAs, and five alloy compo-sitions, using Cu Kα1 − Kα2 radiation, indicating that the alloys are chemicallyhomogeneous (Fig. 1 from Mikkelsen Jr. and Boyce [1983]).

as can be proven by X-ray diffraction analysis, but the average lattice constantand bond lengths would typically undergo some changes. An example of thisbehaviour is seen in Fig. 1.2, taken from Mikkelsen Jr. and Boyce [1983], whichshows an example of diffraction spectrum, concerning the (422) line of (Ga,In)Asalloy: roughly the same structure of diffraction lines gradually drifts with con-centration, revealing the gradual change of corresponding interplanar distance.Otherwise, i.e., if the parent compounds’ structure are different, then the formedalloy, inevitably, suffers a composition-induced phase transition at some alloyingcomposition value, which is the case of our CdSxSe1−x alloy (see chapter 7).

The lattice parameters a of isostructural alloys as the GaxIn1−xAs in theabove figure, determined from X-ray diffraction (XRD), are found to be wellapproximated by the concentration weighted average of those of the constituents,which is usually referred to as the Vegard’s law [Vegard, 1921]:

aAxB1−xC = x aAC + (1 − x) aBC . (1.1)

A more profound analysis would demand to describe somehow the microscopics ofthe alloy and to make e.g. some conclusions about cation-anion bond lengths. Thesimplest approximation for an alloy (not necessarily a semiconductor one) at themicroscopic scale is the virtual crystal approximation (VCA). According to it, the

14

1.1. Crystal structures

concentrationally weighted species at the “mixed” sublattice are represented byidentical fictitious, “somehow averaged” atoms, thus yielding an ordered virtualcrystal. Obviously, in this approximation only one bond length value emerges ata given concentration, instead of some scattering of bond lengths, as is detected inreality (see below). Moreover the chemical identity of constituents is smeared out,and the corresponding electronic structure is either not discussed at all, or hasa rather dubious meaning. Correspondingly, this is a quite poor approximation,which is nevertheless sometimes used, mostly for treating the chemically veryclose substitutions, even in modern literature.

The next step in sophistication would be to preserve the identity of A andB atoms in the AxB1−xC alloy, but treat each one as an impurity in otherwisehomogeneous effective host crystal. From the solution of two impurity problems,the description of averaged effective host can be recovered, and updated till self-consistency. This level of modeling is sometimes still referred to as virtual crystalapproximation, although, in the context of numerous theoretical studies of al-loys, it would be more appropriate to call it coherent potential approximation(CPA). It is immediately obvious that this approximation level, dealing with twoseparate impurity problems in the effective crystal, would allow for different av-erage A–C and B–C bond lengths. Indeed, such distinction is readily confirmedby experiment, starting from the works by Mikkelsen Jr. and Boyce [1982] andMikkelsen Jr. and Boyce [1983], revealing that the (different) bond lengths fromparent binary compounds have a strong tendency to survive in a mixed crystal.

It can be noted moreover that, in specific relation to phonon problematics inmixed semiconductor crystals, the first usual discussed approximation appears asthe modified random-element isodisplacement (MREI) one of Chang and Mitra[1971]. However, a discussion on the shortcomings of MREI in the treatment oflattice dynamics can be found in [Pages et al., 2008].

1.1.3 Alloy structure determined by EXAFS

XRD is an efficient method of structure characterization in periodic systems; insemiconductor alloys the variation of bond lengths may vary strongly dependingon local (microscopic) configuration, and, in general, does not exhibit an orderedpattern. Therefore the XRD may reveal, at most, the average lattice parame-ters (as obvious from Fig. 1.2) but not bond lengths. An alternative structure-sensitive technique is Extended X-ray Absorption Fine Structure (EXAFS), thatis, an analysis of broad-range (up to 500 – 1000 eV beyond the absorption thresh-old) spectrum of X-ray absorption. The “fine structure” of such spectra bearsinformation about backscattering of electrons, excited from atomic core statesinto the conduction band, from neighbouring atoms, and hence helps to decipher(in a complicated mathematical procedure) the local atomic structure. Since ab-

15

Chapter 1. Preliminaries of work

Figure 1.3: Nearest-neighbours distances in InxGa1−xAs alloys as a functionof alloy concentration x measured by Mikkelsen Jr. and Boyce [1983] (Fig. 6thereof). The diagonal straight line is the virtual crystal value given by Eq. (1.2).

sorption thresholds of different atoms are typically well separated in energy, themethod is element-specific.

Mikkelsen Jr. and Boyce [1983] studied InxGa1−xS alloy by EXAFS and afterdetailed analysis of all the K-edges of Ga, In, and As in their compounds GaAsand InAs, and in their alloys, they arrived at the following conclusions:

(1) The individual nearest-neighbor (NN) Ga-As and In-As bond lengths d inthe alloy are close to the pure crystal values than to the concentration weightedaverage value,

d(x) = (1 − x)d0AC + xd0

BC , (1.2)

where d0AC and d0

BC are, respectively, the bond lengths of the constituent AC andBC compounds. However, the average lattice constant agrees with Vegard’s law.Fig. 1.3 summarizes the results for the NN bond lengths. If one defines relaxationparameters ΓAC(x) and ΓBC(x) as

ΓAC(x) =dAC(x) − d0

BC(0)

d0AC(0) − d0

BC(0)(1.3)

and similarly for ΓBC(x), then Γ is about 75% more or less throughout all xvalues. In Eq. (1.3), dAC(x) and dBC(x) are the NN bond lengths in the alloy.

16

1.1. Crystal structures

Figure 1.4: Second-neighbour distances in As-As second neighbor distance forInxGa1−xAs as a function of composition, after Mikkelsen Jr. and Boyce [1983](Figs. 8 and 9 thereof). The middle curve is VCA-based interpolation. Left panel:As-As distances; the shorter one corresponding to As-Ga-As bonds and the longerone correspponding to As-In-As bonds. Right panel: Ga-Ga, In-In, and In-Gadistances. The cation-cation distances are seen to approach the VCA values, thesolid line.

(2) The As-As next-neighbor distances are also found to be “bimodal”: thealloy As-As distance in the As-In-As configuration is longer than that in the As-Ga-As one, as shown in the left panel of Fig. 1.4. This figure also shows that theAs-As distances are nearly fully relaxed; that is, they preserve their pure crystalvalues.

(3) The In-In, Ga-Ga, and In-Ga second neighbor distances are closer to theVCA values than the pure crystal values, as shown in the right panel of Fig. 1.4.

EXAFS experiments have been performed for many pseudobinary semiconduc-tor alloys. Their results give us a rough picture of the way a bulk semiconductoralloy fits together to fill space; the substituted atoms retain their fcc arrangementwith VCA lattice constant, while the atoms on the other (complete) sublatticearrange themselves in their local enviroment to minimize the strain energy causedby bond distortions.

While the gross structure of bulk semiconductor alloys, as measured by theX-ray diffraction patterns, resembles virtual (average) crystals, the detailed bondlengths, as revealed by EXAFS experiments, have significant deviations from thevirtual crystal values. A rough view of an A1−xBxC bulk alloy is one where thesubstituted atoms A and B occupy (nearly at random) perfect fcc sublattice sites,while the C atoms assume positions that minimize the local strain energy. Whilebulk semiconductor alloys are disordered, many epitaxially grown alloys exhibit a

17

Chapter 1. Preliminaries of work

partial long-range ordering into CuAu, chalcopyrite, CuPt, or more complicatedstructures. The bulk alloy bond lengths, when compared with those calculatedfor ordered alloys, are closer to the chalcopyrite than the CuAu or the CuPtstructure. However, the real structure of a semiconductor alloy is neither totallydisordered nor perfectly ordered and, to a great extent, is influenced by growthand annealing processes.

1.2 Pressure-induced phase transitions

1.2.1 Role of pressure in materials science: a brief intro-duction

As we care in solid state physics or in materials science about probing proper-ties of already existing materials, or designing new ones, we notice that pressure,along with temperature, is an extremely important, powerful, and flexible leverfor such purposes. It allows to squeeze atoms (or molecules) closer, overcomeenergy barriers which might be too high at ambient conditions, and force thematerials into new crystallographic arrangements. Employed together with tem-perature, pressure allows to explore phase diagrams in a systematic way, huntingfor peculiar structures and unusual atom coordinations. When it comes to thestudy of alloys, the concentration is yet additional variable spanning the space ofpossible phases. In fact, chemical doping and external pressure have in some casessimilar, and in other complementary effect, as they both come down to producinginternal stress, which might be relieved through a structural transformation. Thefascinating about effect of pressure is that it does not affect individual atomsdirectly, in a universally predictable way. Hence the variety of phase transfor-mations in materials which might seem only minutely different in their electronicstructures.

The study of phase transitions under pressure is not only of interest for prac-tical materials research, but it offers a rich playground in physics: group theory,basic electronic structure theory, lattice dynamics, and transport. To pursue re-lations between different phases is a fine exercise in the group theory; electronicproperties exhibit effects of drastic (de)localization, and certain lattice vibrationsmay indicate a (kinetic) way into a new phase. This playground is very vast;the present work deals only with some semiconductor structures, addressing onlysome aspects of their electronic structure.

As temperature, pressure also plays a key role in determining the thermody-namics of the materials. In a different way than temperature, pressure affectsoverall interatomic and intermolecular distances and hence the density. The vari-ation of interatomic distances allows to scan atomic and molecular interactions

18

1.2. Pressure-induced phase transitions

and reveal exotic structural, electronic, optical, elastic and thermal properties.Actually, nowadays, the field of study of matters under high pressure is no morerestricted to ‘hard’ physical science; it has extended to affect our immediateneeds or daily life interest, ranging from commercial treatment of food stuffs toinvestigating the origins of life. It is a rapidly developing field that is receivingincreasing attention especially due to the use of the diamond anvil cell (DAC)[Eremets, 1996], the advent of modern (third-generation) synchrotron facilities[Hemley et al., 2005], well suited to the use of the DAC, and the theoretical ad-vances in quantum mechanical computations – see, e.g., Yin and Cohen [1980]and Payne et al. [1992]. Additionally, experimental advances in laser-heatingtechnology [Errandonea et al., 2003] have paved the way to the study of mat-ter at high pressures and high temperatures like those present in the interior ofplanets.

On the other hand, high-pressure experiments are used as a tool to tunethe electrical and magnetical properties of low- and high-temperature supercon-ductors who possess high-technological interest. At the same time, and from afundamental point of view, huge efforts are concentrated on understanding thevery basic principles governing structures crystallization, and how to infer themfor given conditions of pressure, temperature and composition. In the search forthose fundamental principles, the knowledge of the structure of matter at ex-treme conditions (pressures and temperatures) is of invaluable help to test recentadvances in quantum mechanical computations – see, e.g., Trimarchi and Zunger[2007], Ogitsu [2007], Pickard and Needs [2007], Oganov and Glass [2008], Behleret al. [2008a], Behler et al. [2008b].

Among diverse interests in high-pressure science, there is a quest for achiev-ing new or exotic properties that don’t exist at ambient pressure. Interestinglyenough, a high-pressure phase that may be enforced under high pressure existssometimes as a metastable phase at the ambient pressure. The most remarkableexample of this is probably made by diamond, which is a metastable phase ofcarbon at ambient conditions, the ground-state phase being graphite. Diamondwith its tetrahedral bonding can be obtained from hexagonal graphite at highpressures and temperatures, that is known since considerable time, has been sim-ulated from first principles by Scandolo et al. [1995], and has been synthesizedin nanocrystalline and microcrystalline forms near ambient pressure conditions– see Gogotsi et al. [2001] and references therein. It is worth noting that highpressure science developing era started by the interest in synthesizing new strongmaterials at high pressures [McMillan, 2002], notably growing artificial diamond.Nowadays, new ultrahard materials are being synthesized, like noble metal ni-tride PtN [Gregoryanz et al., 2004] which seems to be PtN2 indeed, as discussedby Horvath-Bordon et al. [2006], and many other materials of M3N4 (M = C, Si,Ge, Sn, Ti, Zr, Hf) composition [Horvath-Bordon et al., 2006]. Remarkably, a

19

Chapter 1. Preliminaries of work

recent theoretical study predicted a larger bulk modulus than diamond displayedby a high-pressure phase of sp3-bonded carbon nitride (C3N4) [He et al., 2006]; aprediction not yet experimentally supported. Nice reviews on the high-pressuresynthesis of ultrahard materials have been published by Horvath-Bordon et al.[2006] and Brazhkin [2007].

Pressure is capable to drastically redistribute electron density, thus changingthe nature of chemical bonds and converting insulators into metals (includinghydrogen) and soft bonds into stiff ones. Moreover, pressure available now atlaboratories can induce changes in the free energy of materials that exceed thoseof the strongest chemical bonds present at ambient pressure (> 10 eV). Onecan say that this adds a new dimension to the Periodic Table, allowing to varyaffinities, electronegativities, and reactivities of elements.

1.2.2 Thermodynamical and structural issues

The outcome of competition between two phases, say I and II, depends on theirGibbs free energies, or (as sometimes equivalently called) free enthalpies,

G(p, T ) = U(V ) + pV − TS + Gvib(p, T ) + Ge(p, T ) . (1.4)

Apart from explicit effect of pressure p, the equilibrium volume V and total (in-ternal) energy U at equilibrium may differ for different phases, and entropy S,with its combinatorial and vibrational parts (see details below), may differ as well.For structurally perfect compounds (not alloys), combinatorial entropy does notenter, and differences in vibrational parts can often be neglected in comparisonwith differences related to U and V . The terms Gvib, the thermal contributionfrom ionic movement, including zero-point motion, and Ge, the thermal contribu-tion from excited electrons, are explicitly added in Eq. (1.4) for completeness. Atzero temperature, the globally thermodynamically stable phase is the one withthe lowest enthalpy. Such comparison (neglecting temperature effects) is oftenused, with variable success, on the basis of ground-state ab initio calculations,leaving aside the temperature. An example of such analysis in the present workis that outlined in Chapter 4, for pressure-induced phase transitions in ZnS andZnSe. A more sophisticated phase analysis, including temperature effects, shouldbe based on Gibbs free energies, necessarily providing some modeling for thephonon-driven term Gvib.

In pressure-induced phase transitions, as in any others, a distinction can bedone between first-order and second-order ones, depending on whether the dis-continuity occurs in the first or second-order derivative of G with respect topressure p. Since ∂G/∂p = V , a first-order phase transition is accompanied by avolume collapse in the unit cell between the two phases. Correspondingly, since

20

1.2. Pressure-induced phase transitions

∂2G/∂p2 = B, a second-order phase transitions do not show a change in the unitcell volume between the two phases, but a discontinuity in the compressibilityB. Experimentally, it is sometimes very difficult to distinguish a weak first-orderphase transition from a second-order one. However, the symmetry changes insecond-order phase transitions can be traced by group-theoretical methods basedin the Landau theory of phase transitions, conveniently digested e.g. by Erran-donea [2007]; Errandonea and Manjon [2009]; Evarestov and Smirnov [1993].

In what regards structure or atomic arrangements upon pressure, the solid-solid phase transitions can be classified into displacive or reconstructive. In adisplacive (also called diffusionless) transition, the atoms depart from their ini-tial positions somehow, typically conserving their coordination number withoutbreaking their atomic bonds, nor undergoing a long-range atomic diffusuion. Suchtransitions do often involve a small strain. Probably the most important featureof a displacive transition is that the space groups of its initial and final phases arerelated by a group-subgroup relationship. Quite in the spirit of the Landau the-ory of phase transitions, an order parameter can be identified which controls howeither the structure under pressure “escapes” from a higher-symmetry phase (saycubic) into a lower-symmetry one (say tetragonal), or, on the contrary, is forcedinto a more symmetric phase. The other species, a reconstructive transition, con-nects the initial and final phases not related by a group-subgroup relationship bya path throughout which the crystal symmetry is lower, so that the “underway”space group is a subgroup to both end phases. This is like passing a watershedbetween two energy valleys, rather than riding down from one valley into a deeperone as in displacive scenario, and would obviously demand some kinetic energyto overcome the barrier. The search for common subgroup between two distinctspace groups is an isolated mathematical / crystallographic problem [Capillaset al., 2007]; however the identification of a really plausible path among manyformally possible ones, that would mean minimizing the energy barrier and hencenot too drastic local distortions and bond breakings, is a very difficult task.

First-order phase transitions happen to be either of reconstructive or dis-placive type, while second-order phase transitions usually are of the displacivetype, and can be associated with a softening of transverse accoustical mode(s)at some point in the Brillouin zone. Also, second-order phase transitions in-volve no change in the cationic (or anionic) coordination number and are usuallyreversible (initial phase is recovered after pressure release). On the contrary,first-order phase transitions may experience a change in the cationic (anionic) co-ordination number, and are moreover sometimes irreversible, or otherwise show arelatively large hysteresis (difference in the phase transition pressure at increasingand decreasing pressure).

Transition pressure in a first-order reconstructive type may be not unique,i.e., indicate two different values in the upstroke and downstroke. The corre-

21

Chapter 1. Preliminaries of work

sponding hysteresis is typically well observed in large first-order reconstructivephase transitions of covalent materials, characterized by large kinetic barriersthat impede the transition between two phases at the equilibrium pressure. A re-markable case is the diamond: whereas the theoretical study reports a transitionfrom the graphite to the diamond structure at 1.7 Pa at room temperature (RT),the experimental transition is observed in the interval 5-9 GPa only, at 1200-2800K [Mujica et al., 2003], indicating the presence of large kinetic barrier hinderingits transition at the predicted pressure; this barrier is overcome at high pressuresand temperatures. In such cases, the experimental estimate of the transitionpressure would often be set in the middle of the hysteresis interval. Then thisexperimental pressure can be compared to the predicted one, assuming that thetheoretical calculations do not account for the kinetic barriers.

It is quite important to draw the attention here to the fact the the experimen-tal equilibrium thermodynamic transition pressure may strongly depend on themethods used for its determination. For instance, usually photoluminescence andRaman scattering measurements give systematically lower pressure values, X-rayabsorption measurements give intermediate ones, whereas XRD measurementstend to overestimate the transition pressure. The reason for these differences isrelated to whether the methods in question probe the transition on a more localor less local scale. The merits of the different methods for locating the thresholdsof reconstructive phase transitions have been already discussed in the literature[Besson et al., 1991; Manjon et al., 2006].

A well known effect of applying pressure is changing soft bonds (those dueto van der Waals forces) into stiff ones at high pressure; it has been observedempirically that long bonds, typically being softer, are more compressible thanshort ones. The effect of pressure is not restricted to inducing changes in the bondlength, but goes beyond this to compress the ions; anions (usually having largeionic radii) are more compressible than cations – see Bastide [1987]; Fukunagaand Yamaoka [1979] and references therein. Hence we came up with twofold ruleconcerning the effect of pressure: (1) upon increasing it, we induce a reduction ininteratomic distances and atomic sizes yielding an increase in the cation-cationrepulsive forces [Otto et al., 1991; Sleight, 1972]; (2) a comparatively strongerdecrease of the anion sizes compared to the cationic ones leads to an enhancementof the packing efficiency of anions in the cationic sublattice. As a result we cansay that upon pressure, the material’s density is augmented, cation coordinationincreases, and the materials compressibility is largely reduced due to enhancedrepulsion between cations.

All pressure-induced phase transitions have some common features in view ofthe observation that light elements at elevated pressures tend to pass through thephases typical for heavy elements at smaller pressures. A straightforward exampleis given by group-IV elements (C, Si, Ge, and Sn). While carbon crystallizes in

22

1.2. Pressure-induced phase transitions

the hexagonal graphite structure at ambient pressure and undergoes a pressure-induced phase transition to the diamond structure, Si and Ge, having diamondstructure as their ground state at ambient conditions, adopt, in their turn, theβ-Sn as their high-pressure structure, which is, further on, the native one of Snat ambient. This similarity puts us a step ahead in understanding the systematicbehaviour of pressure-induced phase transitions.

A common apparent paradox related to applying pressure is that, whereashigh-pressure phase have usually smaller volume than low-pressure phase, it isoften characterized by larger mean interatomic distances. The explanation is thathigh pressure often enforces higher coordination numbers. For example, the C-Cdistance in (triple coordinated) graphite is 1.42 A (the interlayer distance beingmuch larger, ∼3.4 A) while in tetragonally coordinated diamond it amounts to1.54 A.

High pressure is applied to small laboratory samples in a controlled mannerusing devices such as DAC (see Chapter 2). The static pressure applied in a DACis a continuously variable parameter which can be used for systematic studies ofthe properties of solids, as interatomic distances vary along. A phase transitioncan be most easily detected if the change in whatever parameters, either dis-continuous or continuous, is accompanied by a change in crystal symmetry. Theproperties of the high-pressure phases may be very different from those under nor-mal conditions. A broad consensus was reached by the end of the 1980s that underincreasing pressure the materials adopt high-symmetry structures with increasedcoordination number, as discussed before. However, recent experiments haveshown that previously unexpected lower-symmetry phases are formed at interme-diate pressures, like cinnabar and SC16 – see Nelmes and McMahon [1998]. Thesediscoveries have been made possible by advances in the resolution of high-pressurepowder XRD and an advent of more sophisticated data analysis techniques. Theanalysis of crystallographic results normally yields the simplest structure consis-tent with the data. Modern high-resolution data are now subject to much morestringent analysis techniques, allowing the identification of previously undetectedor overlooked symmetry-breaking distortions. Presumably, at very high pres-sures simple high-symmetry structures are ultimately adopted, but the pressuresat which they occur are in many cases much higher than previously thought.

1.2.3 Pressure-induced phase transitionsin binary semiconductors

We turn in the following to more specific discussion of binary semiconductorsof III-V and II-VI families, and which structures they are known to acquire atpressure-induced phase transitions. Two important reviews in this sense are those

23

Chapter 1. Preliminaries of work

by Nelmes and McMahon [1998] and Mujica et al. [2003].The ground-state phase in semiconductors is largely represented by previously

discussed ZB and WZ. The WZ phase, whenever detected/discussed, turns out tobe almost indistinguishable from ZB, on the energy scale of transitions in furtherphases. The ZB and WZ structures are so close in many aspects that, wheneverone of them emerges as the ground state for whichever compound, the other isusually also accessible, as a metastable one. The overall trend, concerning thecompetition of ZB and WZ as ground-state phases, is that the former is favouredin more covalent semiconductors (AlSb, AlAs, AlP, GaSb, GaAs, GaP, InSb,InAs, InP, ZnS, ZnSe, ZnTe, CdTe, CdSe, HgSe, HgTe), and the latter – inmore ionic ones (AlN, GaN, InN, BN, ZnO, CdS, CdSe, AgI, BeO, α-SiC). Aninteresting observation is that those binary semiconductors whose ground stateis WZ typically have c/a ratios slightly below the “ideal”

√8/3≈ 1.633 value

while those where the WZ structure is only metastable typically have c/a ratiosslightly exceeding the “ideal” [Chelikowsky and Phillips, 1978; Yeh et al., 1992].In both cases, the bonding in the WZ structures is distorted slightly from idealtetrahedral. Further on, the most ionic binary wide-gap semiconductors (MgO,CdO, HgO) and most of the binary alkali halides (I-VII family) crystallize inthe rocksalt stucture (RS in the following, B1 of Strukturbericht, space groupFm3m).

The experimental sequence of pressure-induced phase transition of ZB-typecompounds does not fully follow early suggestions by Chelikowsky [1986]: ZB→ RS → double β-Sn → CsCl. A rather common observation is however that,from about 20 GPa on, the RS structure emerges as the stable one. Whereasa good number of works has been dedicated to further high-pressure transitionsfrom the RS phase [Kirin and Lukacevic, 2007; Mujica et al., 2003], the ZB – RSintermediary regime is rich in ambiguities, due to difficulties of both experimentalcharacterization and ab initio simulations. One reason for such difficulties is thatintermediary phases possess relatively low symmetry and large unit cell size.

A ZB semiconductor may undergo a phase transition either to RS structureproperly speaking, or to the Cmcm phase (an orthorhombic distortion of RS –see Fig. 1.5), whereas the double β-Sn structure was in fact so far not observed inneither III-V nor II-VI compounds [Mujica et al., 2003]. Realistic transition fromZB to RS, as discussed by Smith and Martin [1965] and Karzel et al. [1996], is notstraightforward, that was elucidated in special discussions – see Sowa [2003, 2005]and references therein. Common candidates for intermediate phases between ZBand RS are hexagonal cinnabar (B9 in the Strukturbericht) and cubic SC16.

The cinnabar structure, named so after the mercury sulphide mineral andbelonging to either the P3121 (Nr 152) or P3221 (Nr 154) space group, appearsin rare examples also as the ground-state phase; however, internal coordinatesin these examples (notably HgS) are quite different from those that appear in

24

1.2. Pressure-induced phase transitions

Figure 1.5: (Fig. 3 from Radescu et al. [2011]) The RS structure (left panel)and orthorhombic Cmcm structure (right panel). The Cmcm structure can beobtained from the RS structure by shearing alternate (001) planes in the [010]direction, which in turn results in puckering of the [100] atomic rows and anorthorhombic deformation of the cell.

high-pressure phases of “conventional” semiconductors, as we found for ZnSe-cinnabar 1. The structural relation of cinnabar to ZB and RS phases, of which theformer can be considered as an intermediate structure, is explained in AppendixA. The cinnabar structure can be otherwise considered as a distortion of the Sisubarray in quartz2 and related to the structure of the CrSi2 alloy [Santamarıa-Perez and Vegas, 2003; Santamarıa-Perez et al., 2005].

The absence of the high-pressure RS phase in some compounds, like GaSb,InSb, AlAs, GaAs, and GaP, and the direct transition to the Cmcm phase isapparently related to the ionicity of the substances in question: the most ionicexhibit a larger tendency towards the RS phase [Ozolins and Zunger, 1999]. Kimet al. [1999] and Zunger et al. [2001] have also discussed the systematic absenceof the CsCl-high pressure phase in several pure semiconductors. Lopez-Solanoet al. [2007] have theoretically predicted the existence of a new InAs-high pres-sure phase, super-Cmcm, above the Cmcm structure. A fine review on II-VIsemiconductor phases is that of Shchennikov and Ovsyannikov [2007]. The mech-anisms of the above transitions are still not clear and depend on the material.The mechanisms of the first-order ZB-to-RS and WZ-to-RS phase transitionshave been discussed in a number of compounds [see, e.g. Cai and Chen, 1999;Catti, 2001, and references therein].

The advance of quantum mechanics in the domain of (electronic) structurestudies started as early as since 1930s, to continuously develop methods aptto reliably evaluate, and compare, the energies of solids. Yin and Cohen [1980]

1Full characterization of this phase is given in chapter 4.2Here is a fine web source: http://www.quartzpage.de/gen struct.html.

25

Chapter 1. Preliminaries of work

were probably the first who tried first-principles electronic structure computationsto investigate high pressure phases of Silicon(Si); the results reported were ingood agreement with the then available experimental data. This early successof the theory promoted both theoretical and experimental work in the field. Abig number of excellent calculations have been carried on II-VI, III-V and IVsemiconductors.

Some overview of methods is given in chapter 2, and the assessment of actualcalculation state of art for different systems – in corresponding chapters outliningresults of the present work.

1.3 Phase equilibria in pseudobinary alloys: a

brief introduction

A big dream in computational materials science, crowned so far by not so manysuccess stories but invariably attractive, is a design of a new material, departingfrom its given target properties, towards the optimal composition and processingsteps. To achieve this goal, it is necessary to have a thorough understanding ofhow atomic features influence macroscopic behaviour. In particular, one mustunderstand how alloy properties may change with composition, temperature andpressure. Computational thermodynamics approaches known as the CALPHAD1

methodology [Lukas et al., 2007] have been highly successful in studying sucheffects on alloy properties and in calculating the phase equilibria in complex,multi-component, industrial alloys. These methods rely on databases of freeenergies, obtained from an optimization process involving experimental thermo-dynamic data combined with observed phase diagrams. Once the database built,the computational thermodynamics programs perform minimization of the multi-component free energy functional of interest to predict phase equilibria. Howeverthese methods can not predict the existence of phases that are not already inthe database. First-principles theory methods are generally quite good in re-producing, or predicting, ground-state properties, such as relative total (inter-nal) energies of different phases, and their elastic parameters. Such calculationsdo, however, typically relate to zero temperatures and perfectly ordered (albeit

1Computer Coupling of Phase Diagrams and Thermochemistry (CALPHAD) aims to pro-mote computational thermodynamics through development of models to represent thermody-namic properties for various phases which permit prediction of properties of multicomponentsystems from those of binary and ternary subsystems, critical assessment of data and their incor-poration into self-consistent databases, development of software to optimize and derive thermo-dynamic parameters and the development and use of databanks for calculations to improve un-derstanding of various industrial and technological processes. This work is disseminated throughthe CALPHAD journal and its annual conference. Website: http://www.calphad.org/.

26

1.3. Phase equilibria in pseudobinary alloys: a brief introduction

probably quite complicated) phases. The difficulties, on the way to bring suchcalculations closer to reality, are that i) only the structures already known can beprobed, and ii) any inclusion of temperature, atomic vibrations, or other typesof disorder needs an employment of methods of statistical physics, is not a priorisimple for such complex, microscopically large, and only numerically tractablesystems. A practical way out of this dilemma is in constructing databases of freeenergies, obtained from an optimization process involving experimental thermo-dynamic data combined with observed phase diagrams, and/or mapping the datafrom ab initio calculations onto model Hamiltonians for further statistical (e.g.,Monte Carlo) simulations.

During the last decades, important theoretical and algorithmic advances intheory of condensed matter and in ab initio calculations have boosted the devel-opment of such approaches. With the continuing rapid increases in computationalpower, theses advances enable us to tackle problems of practical importance. Asa result, ab initio modelling is starting to pay off in a dramatic way, bridging thegap between fundamental research and materials engineering. Such an approachto modelling alloy phase equilibria however raises some specific questions. Ab ini-tio calculations are (ideally) supposed, and not rarely advertized, to be performedby solving the Schrodinger equation, hence deducing interatomic interactions andthe resulting equilibrium structures that follow using only the atomic number ofthe constituents as input. The reality is typically less glorious, and importantapproximations have to be adopted for the sake of practical feasibility. One dif-ficulty arises at the level of solving many-electron problem for any multiatomicsystems, to which end the density functional theory (DFT, see chapter 2) offers alargely accepted way of advancement. Another issue concerns specifically alloys– the number of constituents, the distribution of different atoms over sites, andhow to cope with prohibitively high number of sampling configurations. We covereverywhere in the present work only substitutional alloys. Practical calculationson them are typically done using supercells – reasonably large repetitive (latticeperiodic) units over sites of which the constituent atoms, in quantities given byalloy concentration, are distributed, in a number of possible configuration givenby combinatorics. Beyond the simplest and smallest supercells, the number ofpossible configurations rapidly exhaustes any facilities of first-principles calcula-tion. The present work, therefore, in its parts where alloying is an issue, makesuse of special quasirandom structures (SQS, see Wei et al. [1990]; Zunger et al.[1990]), or a gradually increasing hierarchy of constructued small supercells incluster expansion technique – see, e.g., Sanchez [2010].

An additional complication is related to physical contributions to the freeenergy of a phase, with different entropy terms: purely combinatorial and vibra-tional ones. Depending on a system, e.g., contributions due to magnetic disordermay further play on crucial role in the stabilization of a phase. At this time, a

27

Chapter 1. Preliminaries of work

full accounting for all physical contributions and their couplings from ab initiocalculations is not a straight forward matter.

Predicting phase stability with absolute certainty is currently an intractableproblem in materials science. When calculating phase equilibria, we often needto know phase boundaries to within a few degrees. Even for simple cases, abinitio calculations do not have this exceedingly high accuracy necessary to suchdeterminations.

The complexity of materials of interest combined with the need to achievehigh accuracy defines the twofold role of ab initio calculations to designing ther-modynamic properties:

(a) ab initio calculations combined with statistical mechanics must be usedto unravel the main physical contributions to the free energy of a phase andto predict what will happen when composition, temperature and pressure arevarying.

(b) ab initio calculations can be combined with the CALPHAD methodologyto form a hybrid ab initio/computational-thermodynamics method. In recentyears, the lattice stability data for pure elements and the formation energiesof compounds from ab initio methods have been used to supplement or modifyexisting thermodynamic databases. This approach is particularly useful whenthere is little or no information on the phases that take part in equilibria in thesystem under study.

28

Chapter 2

Basics of theory andapproximations used

2.1 Ab initio electronic structure calculations

Experimental data are not always conclusive about the details of phase compo-sition or of crystal structure, hence the motivation to check the hints given byexperiments in first-principles calculations. The role of first-principles calcula-tions is not just to reproduce, or predict, the structure, but to understand whythat or other crystal structure takes over under certain conditions. In the studiesof solids, first-principle calculations practically imply the use of the density func-tional theory (DFT). Without going into fine details, I remind in the followingsome basic issues of DFT essential for understanding the organization of calcu-lations done in the present work. A single outstanding reference for immersinginto the subject (among many others available) in the book by Martin [2004a].Speaking specifically about simulating semiconductors under pressure, the pre-viously cited work by Mujica et al. [2003] can be pointed out as a good recentreview.

2.1.1 Density functional theory

DFT is one of the most widespread and successful quantum mechanical ap-proaches used in both “applied” materials science and for getting adequate nu-merical answers to various “more fundamental” problems of condensed matterphysics and chemistry. The traditional applicability domain of DFT is the de-scription of ground-state properties in periodic solids. As ever more demandingcalculations became technically possible, an “expansion” towards defective andotherwise non-periodic systems became possible. The development of accuratealgorithms eventually permitted to achieve “chemical accuracy” in calculations,

29

Chapter 2. Basics of theory and approximations used

i.e., the precision in evaluating energy differences at the scale of ∼μeV/at, aboutsufficient to meaningfully predict the barriers in chemical reactions. (The studyof barriers also plays role in analyzing pressure-driven phase transitions). Anexpansion of DFT happens nowadays in domains as distant from the essentialquantum mechanics as microbiology, on one side, and mineralogy, on the other.

DFT offers a way to address a many-body problem (which resists an exactsolution already starting from 3 bodies) for a special case of a (multi-electron)system being in its ground state. While a priori a grave limitation, this caseapplies however to a big number of practically important problems; moreover theways to treat excited states by DFT are also permanently under development.It was recognized in a pioneering work of Hohenberg and Kohn [1964] that thetotal energy, along with being obviously related to (i.e., being functional of) themany-body (N -body) wave function of system, Etot = Etot[Ψ(r1, r2, . . . , rN)],can be uniquely characterized in much more restricted manner, that is, being afunctional of the one-particle density ρ(r) of N -particle system:

Etot = E [ρ(r)] , ρ(r) = N

∫|Ψ(r, r2, . . . , rN)|2 dr2. . .drN . (2.1)

The rest of DFT, in a nutshell, are “speculations” about how such universalfunctional may in principle look like, and how it can be best approximated fordifferent practical purposes. As a huge literature exists on this subject, and thepresent work does not deal with foundations of DFT nor with development ofmethods, I’ll strip the discussion of DFT to a necessary minimum.

It would make sense to point out that the DFT did not appear out of the blue,but inherited something from previous theoretical development over decades, no-tably, the Thomas–Fermi theory (which casts total energy in the form depend-ing on particle density), and the Hartree–Fock (HF) theory, which introducedthe concept of exchange energy (and, correspondingly, exchange potential) in amulti-electron system.

An essential push towards the large-scale of DFT came with the work by Kohnand Sham [1965] who suggested to represent the many-body density by a sumover contributions from non-interacting quasiparticles (obeying the Fermi-Diracstatistics with distribution function fFD and chemical potential μ1), namely – atzero temperature – the N quasiparticles whose states φi (Kohn–Sham functions,in the following) have the lowest one-electron energies εi:

ρ(r) =N∑i

|φi(r)|2 =∞∑i

fFD(εi − μ) |φi(r)|2 (2.2)

1For more details see section 2.8.

30

2.1. Ab initio electronic structure calculations

The one-particle wavefunctions and energies needed to realize the above represen-tation of many-particle density follow then from the set of Kohn–Sham equations{

−�2Δ

2m+ vext(r) + vH[ρ(r)] + vxc[ρ(r)]

}φi(r) = εi φi(r) , (2.3)

written here in the form explicitly separating electrostatic external potential (e.g.,from point nuclei or external fields) vext, the electrostatic potential of electron sys-tem interacting with itself (including self-interaction), that is often referred to asHartree potential, hence vH, and the exchange-correlation potential vxc, whichcorrects for self-interaction (and has its analogy in the HF approach as exchangepotential), but allows, in principle, to go further than the Hartree–Fock and in-corporate correlation effects. As can be worth reminding, the HF theory is byconstruction a mean-field one, that deals with averaged electron densities and ne-glects the electrons being point objects, which (somehow simplifying, in classicalsense) pose themselves at different time moments in different mutual standings.Taking such correlation effects into account brings us, at least potentially, beyondthe limitations of the Hartree–Fock approach.

Another important notion concerning the DFT vs. HF and exchange vs. cor-relation issues is that of Slater determinant. By construction, the HF approachaims at yielding the best description of many-electron system possible with asingle Slater determinant. Such single-determinant treatment is also in principlecapable to take exchange fully into account, but none of correlation. A systematicimprovement of this paradigm demands to allow mixing of many Slater determi-nants, i.e. configuration interaction. The DFT way is not a priori constrainedby such systematics; all the subtleties and practical details of its use are pushedinto a definition of exchange-correlation (XC) potential, vex of Eq. (2.3). Someproperties (norm, asymptotics) of the exact (unknown) XC potential are in factknown, whereas practical recipes for vxc mostly come nowadays from convenientanalytical fit to numerical results obtained by sufficiently accurate methods (e.g.,quantum Monte-Carlo) on “ideal” reference system, e.g., electron liquid of differ-ent density. It is important to emphasize that the DFT philosophy, in differenceto semiempirical methods, do not encourage the use of parameters tuned ad hocto one or another system; the parametrization of vxc is supposed to work for broadclass of systems and tasks. However, given an experience over decades of usingDFT, it is known that some approximations for vxc “work” better than the otherswith respect to those or other materials and problems. Moreover there is a tradeof accuracy needed against sophistication of algorithms and hence computationalload. Therefore some discussion on common XC potentials “on the market” willbe in place here, in view of the subsequent discussion of results.

The most broadly used approximations for vxc, and the only ones used inthe present work are, local density approximation (LDA) and generalized gradient

31

Chapter 2. Basics of theory and approximations used

approximation (GGA). Each of both exists in a number of flavours.In LDA the contribution to the exchange-correlation energy from a point

in space is taken as if it would be from a uniform electron gas at the densityappropriate for that point. This is, probably, the simplest realistic approximation.The domain of applicability of the LDA has been found to extend far beyond theuniform electron gas, and accurate results could have been obtained even for quiteinhomogeneous systems. The reasons why LDA performs so well are quite wellunderstood [Jones and Gunnarsson, 1989].

However, in some situations the LDA is inadequate (predicting too smallequilibrium volume, wrong ground-state magnetic or other structure, etc.) Theattempts to improve LDA involved taking into account spatial variations of den-sity around the r point in question, in the form of gradients. However, theformal gradient expansion was not particularly useful, and more sophisticatedschemes, which intentionally build in desired normalization, asymptotics, etc.,have ultimately established themselves for practical calculations. Such schemesare generally referred to as generalized gradient approximation (GGA).

For both LDA and GGA, many practical interpolation schemes are known, andit is a good habit to specify the one specifically used in a calculation. Each of theseboth notations stands for a certain distinct level of accuracy; variations withineach class are less important. GGA helped to fix many shortcomings of LDApredictions, notably those related to magnetism. However, even as GGA is moresophisticated by construction, it is not automatically superior in practical cal-culations: equilibrium lattice parameters, while being somehow underestimatedin LDA, often come out slightly overestimated in GGA; GGA calculations aremore demanding in what regards numerical accuracy, cutoffs etc. and less stable(due to numerical differentiation). In fact, apparently due to effective cancella-tion of errors, LDA remains interesting and alive for first-principles calculations,at least in relation to condensed matter. The problems of LDA in describing thechemistry of molecules are more grave, see Jones and Gunnarsson [1989], and itis desirable to improve upon it. Ideas on how to improve on the LDA include,besides the GGA – [see, e.g., Perdew et al., 1996], orbital-dependent functionals,e.g., exact exchange potentials (Gross et al. [1996]; Gross et al. [1994]). One ofthe motivations for carrying out theoretical studies of materials under pressureis to investigate how well approximate functionals such as the LDA and GGAperform in describing the wide range of chemical bonding that occurs in high-pressure phases.

The exchange-correlation term in Eq. (2.3) incorporates all the difficulties re-lated to the original many-body problem and the task of finding a functional thatembodies the required information seems just as hopeless as that of calculatingthe exact many-body wave function for hundreds of electrons. What saves thisapproach is that very oversimplified approximate functionals like the local-density

32

2.1. Ab initio electronic structure calculations

approximation (LDA) work correctly at least for metallic systems. However ithas some shortcomings mostly due to the tendency of overbinding, which causetoo small lattice constants and too high cohesive energies. GGA approximationimproves predicted binding and dissociation energies. However both approxima-tions fail to describe correlated materials, like oxide materials and/or systemscontaining f -electrons, e.g., plutonium. A lot of work has been invested over lastyears to design new functionals able to describe these systems. Among them,the LDA+U scheme [Anisimov et al., 1997] which combines a DFT descriptionof extended states with a Hartree-Fock like treatment of the Coulomb repulsionU between states in narrow d- or f -bands. Such an approach leads to a signif-icant improvement in the ground state properties of oxides based on transitionmetals (e.g., NiO) and actinides (UO2, PuO2 etc.). The dynamical mean-fieldtheory [see, e.g., Lichtenstein et al., 2001] may be considered as a many-bodyextension of DFT that attempts to capture the true many-body physics of theelectron-electron interaction. It provides a meaningful description of structuraland spectral properties of many different strongly and moderately correlated sys-tems – see, e.g., Held [2007] for a review. At this time, no unified “post-DFT”theory exists, and users will need to decide with care which improvement strategyis best suited for a property that can not be accurately computed at the DFTlevel. A variety of mature DFT codes [Martin, 2004b] are nowadays available toprovide an accurate framework for calculating the relative energetics of compet-ing structures in solidsfor a wide range of materials. In practical terms, standardDFT calculations are actually “limited” (by common sense and parameters oftypical hardware rather than by foundations of theory) to structures containingsay around 500 atoms, and sequential runs of ab initio Molecular Dynamics – totime scales less than 1 ns. It is clear that for most cases in materials science, di-rect applications of DFT to the calculation of thermodynamic properties remainintractable. Effective strategies have nonetheless been developed to extrapolatefrom the size and time-scale limitations imposed by DFT.

Among a great number of textbooks, monographs and reviews dedicated toDFT, I’d single out only some which helped me to get a consistent idea of thisdifficult subject: a good general review of DFT and its applications by Hafner[2000]; a book that covers DFT and many of the features that pop up in ac-tual computational work (k-space, integration methods, computational aspects,examples of computational studies on structural properties, phonons, thermody-namical quantities, etc.): Pisani [1996]; two books that go really to the bottomof subject: Dreizler and Gross [1990] and Parr and Yang [1989].

33

Chapter 2. Basics of theory and approximations used

2.1.2 Solving techniques

The Kohn-Sham equations (2.3), once the exchange-correlation potential is spec-ified, make a system of integro-differential equations. Their analytical solutionmight only be possible in some special model cases, so that the numerical ap-proach is the only practical alternative. The latter can be achieved either by“discretizing” Kohn-Sham functions, charge density and potential on some con-venient spatial mesh, or by expanding relevant functions over a convenient systemof continuous basis functions. Both approaches are, in fact, not that different:spatial discretization is equivalent to decomposition over a basis of sharp localized“spikes” or cubic polynomials [Beck, 2000; Chelikowsky et al., 1994; Pask et al.,1999; Tsuchida and Tsukada, 1995]; moreover a combination of, say, discretizedpotential with smooth basis-expanded Kohn-Sham functions is quite common.Whatever the realization, the practical solution is very often (but not always; theKorringa–Kohn–Rostoker method is a well known counter example) reduced toa generalized diagonalization problem. The size of matrices to diagonalize is animportant measure of the numerical complexity of tasks, and the diagonalizationalgorithms greatly contribute to the progress of methods.

In the hierarchy of differences which define the specificity of methods, themain division lines go along the following questions:(i) Is the system finite (molecule, cluster) or periodic and hence infinite? (Com-binations of finiteness in some dimensions and periodicity in others are possible).This would decide whether the Bloch theorem will be used (see below).(ii) Are all electrons in all atoms, from 1s shells upwards, included in the solu-tion of Kohn–Sham equations (that makes an all-electron calculation), or onlythe electrons is some upper shells which contribute to chemical bonding? In thelatter case, the inner core states are treated in an approximative way: they are, assuch, non-existent in the calculation, but they give rise to pseudopotential (PP)which modifies the Kohn–Sham equations for upper (valence) electrons.

As concerns (i), only bulk crystalline systems (hence periodic, albeit probablypossessing large supercells) are considered in the present work. As concerns (ii),both all-electrons and PP calculations were done in different cases; this issue isaddressed in some more detail in subsection 2.1.5.

2.1.3 Bloch Theorem

The Bloch theorem (which, in different textbooks, is casted in different state-ments) essentially follows from spatial periodicity of perfect infinite crystal, onwhich the Born – von Karman periodic boundary conditions are imposed, i.e.,from the fact that the effective crystal potential allows a 3-dimensional discrete

34

2.1. Ab initio electronic structure calculations

Fourier transform:

Veff(r) =∑m

Veff(Gm) exp(iGm·r) , (2.4)

where Gm are the crystal reciprocal vectors. This has the consequence that theKohn–Sham functions appear in the Bloch form, a complex exponent times alattice periodic function:

φi,k(r) = exp(ik·r)ui,k(r) , (2.5)

ui,k(r + R) = ui,k(r) , (2.6)

R = n1a1 + n2a2 + n3a3 (2.7)

(ai: translation vectors; ni: integer numbers). The equations (2.3) on the Kohn–Sham functions φi(r) are then re-formulated as those on the corresponding Blochamplitudes uk(r). The size of the system Eq. (2.3) to solve hereby reduces tremen-dously: from [number of states for all electrons in crystal] to [number of statesfor all electrons in one unit cell]. However, each solution correspond to a givenk value which “lives” in the first Brillouin zone. In order to collect the chargedensity by summing over occupied states as in Eq. (2.2) and go for the nextKohn–Sham iteration, one needs, strictly speaking, to know the solutions for alldifferent [number of unit cells in crystal] k values. Fortunately the variationsof Kohn–Sham functions throughout the Brillouin zone usually are convenientlysmooth, so that an interpolation over a quite small number of k points is accurateenough. This “k-mesh convergence” must be tested in advance, in view of givensystem, properties calculated and the accuracy needed.

2.1.4 Brillouin zone

The first Brillouin zone is (symmetrically re-arranged) unit cell in the reciprocalspace, where different allowed k values are densely situated. As mentioned above,practical calculation schemes replace the “exact” summation over all k valuesby an approximate integration over their selected coarse grid. The issues ofimportance in this relation are:(i) the possibly optimal choice of such coarse grid;(ii) the exact technique of integration / summation / sampling;(iii) the issues of variation of the Brillouin zone size in calculations like ours,where typically large supercells are constructed (and this can be done in differentways).

Concerning (i), the density of k-mesh is a technical parameter of calculation;it rests to the user’s choice, and the convergence of results, as the number (andhence density) of k-points increases, must be carefully checked. It might be not

35

Chapter 2. Basics of theory and approximations used

immediately clear how to guarantee a “good” homogeneous placement of pointover Brillouin zones of substances with fancy (tilted, elongated, low-symmetric)unit cells. A work by Moreno and Soler [1992] is very instructive in this sense,covering also the issue of “shifted” vs. “unshifted” k-meshes.

Concerning (ii), the general experience says that the integration over theBrillouin zone, assuming linear interpolation of data (eigenvalues, eigenvectors)between the sampling points is usually quite efficient in terms of reaching thedesired convergence of results, as compared to conventional sampling. i.e., asum over values at different k-points, taken with corresponding weights. Theprocedure of integration of linearly-interpolated functions over the Brillouin zone,known as the tetrahedron method, has been suggested by Jepson and Anderson[1971] and refined by Blochl et al. [1994]; a generalization over more sophisticatedintegration, with interpolated functions in both nominator and denominator, canbe found in Molenaar et al. [1982]. However, the accurate integration is especiallyimportant for metals, where some bands cross the Fermi energy, demanding toaccurately define the occupied portion of band. In semiconductors and insulators,where bands are either fully occupied or fully empty, the simple sampling worksacceptably well. For it, some standard choices of points and weights is coveredby Baldereschi [1973], Chadi and Cohen [1973] and Monkhorst and Pack [1976],which last paper became a de facto standard.

As for the issue (iii), it is a priori clear that choosing a large supercell (that isoften needed for simulating defects, or different configurations in alloys), results incorrespondingly diminishing the Brillouin zone, with correspondingly more bandsbeing folded into it. (This argument holds for both electron bands and phononbranches). Because of the increased number of bands, the same information willbe preserved by maintaining the density of k-points, in fact reducing their numberwhen coming to larger supercells. In practical terms, the necessary density of k-mesh can be controlled by a single parameter, expressed e.g. as a radius of spherein the real space, as suggested by [Moreno and Soler, 1992]. Once this parametertested and optimized for a given chemical substance, there is no need to repeatthe tests for each different supercell.

2.1.5 Electronic-structure methods

The overall approach of DFT with Kohn–Sham equations and a specific form ofXC potential can be realized and brought to numerical solution on a number ofways. The resulting calculation methods differ in many aspects, of which themost pronounced are the following:(i) Whether all electrons of a system, from the deepest core states upwards, areincluded in the construction of Kohn–Sham orbitals and the iterative solution ofKohn–Sham equations (2.2; 2.3), or only the highest valence states are taken into

36

2.1. Ab initio electronic structure calculations

account, whereas the inner atom states are considered unaffected. This makesthe distinction between all-electron and PP methods;(ii) what is the numerical basis to describe the Kohn–Sham functions in theprocess of resolving the integro-differential equations (2.3): is this a system of“universal” functions, like e.g. planewaves (PW) or wavelets, or some kind of 3-dimensional numerical representation (e.g., superposition of atom-centered radialfunctions), or a decomposition into conveniently chosen atom-centered bases (e.g.,Gaussian-type functions).

In the following, the methods used in the calculations within the present workare briefly explained, in their standing among the other methods to solve Kohn–Sham equations available “on the market”.

In Kohn-Sham DFT the interacting problem is mapped onto a set of single-particle equations familiar from band-structure theory. The well-developed ma-chinery of band-structure theory can therefore be used to solve them, and the fieldof “band-structure calculations” has evolved into the modern field of “total-energycalculations.” The computational method chosen to solve the Kohn-Sham equa-tions is of great importance. One would like to obtain equal accuracy for high-and low-symmetry phases, and it is enormously advantageous to be able to calcu-late the derivatives of the total energy with respect to the positions of the atomsand the shape of the unit cell, i.e., the forces on the atoms and the stress tensor.Preeminent in these regards is the PW method, in which the periodic parts of theKohn-Sham wave functions are represented by Fourier series. Since “true” (i.e.,all-electron) Kohn-Sham functions are strongly oscillating inside atoms, wherepotentials of ion cores are particularly deep, their Fourier expansion would go toprohibitively high number of PWs. PWs are therefore used almost exclusivelywith PP, which replace the core electrons and make smooth pseudovalence wavefunctions. Care must be taken when dealing with systems in which polarizablesemi-core electrons (the outermost occupied d electrons in the core for the IIBand IIIA families) play a role in the chemical bonding, as is the case, for example,in GaAs. A proper account of such semi-core effects requires the explicit treat-ment of these orbitals as hybridized states in the valence band [Fiorentini et al.,1993; Serrano et al., 2000], although for many purposes it is perfectly acceptableto consider these orbitals to be frozen. The calculations done in the present workconcerning phase transitions in perfect crystalline compounds are performed usingfull-potential augmented plane waves method with local orbitals (FP-LAPW+lo)method [see Singh, 1994], which is known to yield ultimate and controllable accu-racy in all-electron calculations within the conventional DFT. To mention othermethods, one should state that the vast majority of the calculations, in the basicliterature, were performed using PW-PP methods. Another widely used schemeis the full-potential linearized muffin-tin orbital method (FP-LMTO), historicallysuggested by O.K. Andersen simultaneously with LAPW [Andersen, 1975]; see

37

Chapter 2. Basics of theory and approximations used

also Andersen et al. [1987, 2000].

Other DFT implementations of the all-electron and PP methods use basis setsconsisting of linear combinations of Gaussian orbitals or linear combinations ofatomic orbitals (LCAO), in particular for Hartree-Fock (HF)-type calculations. Ifthe calculations are performed carefully enough, the PWPP, FP-LMTO, and FP-LAPW methods normally give results in very good agreement As an example,consider Si-VI, the elusive high-pressure phase of Si intermediate between thesimple hexagonal (sh) Si-V phase and the hexagonal-close-packed (hcp) Si-VIIphase. Si-VI was first observed in 1984 [Olijnyk et al., 1984], but its structure(Cmca) has only recently been resolved [Hanfland et al., 1999]. Both LMTOcalculations [Ahuja et al., 1999; Christensen et al., 1999] and PWPP calcula-tions [Mujica et al., 2001a,b] confirm the existence of an interval of stability forthe Cmca phase with sh/Cmca and Cmca/hcp coexistence pressures that agreevery well both among the different calculations: [33(2), 41(5)] GPa [Christensenet al., 1999]; (34.0, 41.5) GPa [Ahuja et al., 1999]; (36.0, 43.5) GPa [Mujicaet al., 2001a,b] and with the experimental values for the onset of the sh → Cmcatransition (∼38 GPa) and Cmca→ hcp transition [∼42 GPa; Hanfland et al.,1999]. This agreement is all the more outstanding if one realizes that an errorof ∼20 meV per atom in the calculation of the difference in the energies of thephases would result in the total disappearance of the calculated interval of sta-bility of the Cmca phase (see Fig. 2 of Christensen et al. [1999])! A similar levelof agreement exists for the structural parameters of the Cmca phase.

Pseudoptential

Large energy cutoffs must be used to include planewaves that oscillate on shortlength scales in real space. This is problematic because the tightly bound coreelectrons in atoms are associated with wavefunctions with exactly this kind ofoscillation. From a physical point of view, however, core electrons are not es-pecially important in defining chemical bonding and other chemical propertiesof materials; these properties are dominated by the less tightly bound valenceelectrons. From the earliest development of planewave methods, it was clearthat there could be great advantages in calculations that approximated the coreelectrons in a way that could reduce the number of planewaves necessary in calcu-lation. The most important approach to reducing the computatoinal burden dueto core electrons is to use PP. Conceptually, the PP approach takes into accountthe screening of bare nucleus charge by inner (core) electrons and redefines thusthe potential felt by outer (valence) electrons. The core electrons are thus com-pletely removed from the band structure calculation; they are “hidden” in theconstruction of the PP. The screening of core states introduced in PP formalismis not a straightforward (local) screening of a point charge by the cloud of core

38

2.1. Ab initio electronic structure calculations

electrons, that would be a too crude approximation. Instead, the screening is de-fined in a more sophisticated way, to satisfy certain physical conditions imposedon valence states, one quite important thereof being the norm conservation fora given wave function. This results in PP being non-local, i.e., dependent on theorbital quantum number, and different for different wave functions. The mostimportant consequence for wave functions of valence states from using the PPformalism is that they do not have to be orthogonal to (now missing) core statesand thus have no nodes in their radial part; in fact such pseudofunctions are a bitphysically “wrong”. However, this was exactly an intention in the constructionof PP: to suppress strong oscillations of wave functions in the inner core regionof each atom, whereas the requirement of norm conservation assures that thecharge carried by a given pseudofunction is the same as that of the true wavefunction it replaces. Therefore, the care must be only taken to guarantee that,apart from the “fault” of missing nodes, the rest of the description – the shapeof wave functions at the periphery of atoms, the overlap of wave function, theresulting band structure, – makes sense. Such “taking care” is incorporated intothe work of construction and tuning of a PP. Typically, a PP is developed foran isolated atom of a given element, but the result can then be used reliablyfor calculations that place this atom in any chemical enviroment without furtheradjustement of the PP. This desirable property is referred to as the transferabilityof the PP. Current DFT codes do often contain a library of PP that includes anentry for almost every element in the periodic table.

The details of a particular PP define a minimum energy cutoff that should beused in calculations including atoms associated with that PP, in view of oscilla-tions that the resulting (pseudo) wave function would exhibit. These oscillationsdepend on the depth and shape (smoothness) of the PP. Pseudopotentials requir-ing high cutoff energies are said to be hard, while more computationally “easy”PPs with low cutoff energies are soft. There is a certain trade between softnessand transferability of a PP, somehow contradictory properties in the process ofPP construction. A certain breakthrough, allowing to take the best of both, ispossible at the expense of lifting the condition of norm conservation and some-how complicating the whole formalism; this is the idea of ultrasoft pseudopotential(USPP) invented by Vanderbilt [1990] and described by Laasonen et al. [1991,1993].

An approach to cope with core states which is somehow based on the PPheritage but goes further is the projector-augmented-wave (PAW) method origi-nally introduced by Blochl [1994] and later adapted by planewave calculations byKresse and Joubert [1999]. This method keeps the core states in the “frozen” form(i.e., not changing from one iteration to another) and maintains the simplicityof PP approach in the decsription of valence states which have to be orthogonalto the core ones. Kresse and Joubert [1999] performed an extensive comparison

39

Chapter 2. Basics of theory and approximations used

of USPP, PAW, and all electron calculations for small molecules and extendedsolids. Their work shows that the well constructed USPPs and the PAW methodsgive results that are essentially identical in many cases and, just as importantly,these results are in good agreement with all-electron calculations. In materi-als with strong magnetic moments or with atoms that have large differences inelectronegativity, the PAW approach gives more accurate results than USPPs.

As a good introduction into the PP formalism and implementations, the bookby Singh [1994] and a review by Pickett [1989] can be recommended, among abig number of other sources.

All-electron schemes

Historically and didactically very important, and moreover very broadly used infirst-principles studies on semiconductors, the PP formalism has nevertheless ob-vious limitations: (i) as core states are excluded from band structure calculationand put into the free-atom part (generation of PP), no event involving both coreand valence-band states, e.g., X-ray emission or absorption can not be addressed;(ii) the error introduced by presuming the core states excluded from chemicalbonding may be a priori not negligible; (iii) the conventional formulation of PPmethods allows only one pseudofunction per l-channel, i.e., the states with dif-ferent principal quantum number but the same orbital quantum number can notbe attributed to the valence-band states simultaneously. Yet in some cases theexplicit inclusion of, say, outer-shell 4s and semicore 3s states in the solution ofbandstructure problem is essential for obtaining correct hierarchy of total ener-gies, or accurate forces. To overcome such difficulties, all-electron schemes haveto be used, in place of PP ones.

The all-electron method used in the present work, the full-potential linearizedaugmented plane wave method with local orbitals (FP-LAPW+lo), a descendentof old augmented plane waves (APW) method by Slater [1937], realized in theWIEN2k code [Blaha et al., 2001], uses the muffin-tin geometry : the space is di-vided into two types of regions (see Fig. 2.1) where different bases are used: atom-centered numerical radial functions, adjustable from one iteration to the other,multiplied by spherical harmonics to account for angular coordinates, within non-overlapping muffin tin spheres (MT) Sα, centered at the atomic positions, andPWs, generated with vectors of reciprocal lattice up to maximal chosen cut-off magnitude, in the interstitial region. At sphere boundaries, both kinds ofexpansions for the Kohn-Sham functions have to match along with their radialderivatives, that imposes the conditions on the expansion coefficients, to be solvedby matrix diagonalization.

In a nutshell, this is an example of dual basis method, whose justificationis as follows: within the interstitial region, the potential is nearly constant and

40

2.1. Ab initio electronic structure calculations

Figure 2.1: Division of a unit cell into muffin-tin spheres and the interstitialregion, for a case with two atoms. The black dot is the origin of the axis system(which may but need not to coincide with the nucleus of an atom). Figure takenfrom Cottenier [2002] with the permission of the author.

the electrons behave as almost free ones, thereby allowing the use of PWs for abetter description of corresponding wave functions, whereas near the nucleus, theother limiting cases is taken, that of electrons bound in a central potential. Suchpotential need not to be spherical symmetric, but can be expanded in sphericalharmonics around the given center, with numerical radial functions as expansioncoefficients. Similarly, in the interstitial region the potential does not have to beflat (as in the early formulation of the APW method by Slater [1937]), but canbe warped and described by 3-dim. Fourier expansion. The details of formalism,about matching of all relevant coefficients, are straightforward but cumbersome.The “old” APW method is well described by Loucks [1967], Mattheiss et al. [1968]and Dimmock [1971]. Further development of method, including linearizationissues [Andersen, 1975; Koeling and Arbman, 1975], is well covered by the bookof Singh and Nordstrom [2010]. Still further development, notably a very usefulinclusion of additional “local orbitals” to the basis, is described by Madsen et al.[2001]; Sjostedt et al. [2000], and Schwarz et al. [2002].

41

Chapter 2. Basics of theory and approximations used

Codes presently used

In the calculations within the present work, four different methods have beenapplied: WIEN2k1 Blaha et al. [2001], VASP,2 Siesta,3 and abinit.4 They allproduce, as a primary result, electronic (band) structure and forces on atoms,that allows to perform optimization of crystal structures. Their results can betherefore compared; however, they all have different strong (and weak) pointsand are best chosen to serve (slightly) different purposes. According to theabove classification, we note that WIEN2k, as earlier stated, is an example ofall-electron method; whereas the other three are not, and use PP. Concerning thechoice of basis, we note that VASP [Kresse and Furthmuller, 1996a,b] and abinit[Gonze et al., 2002] use PWs, Siesta [Ordejon et al., 1996; Soler et al., 2002] usesatom-centered numerical functions of strictly finite extension, whereas WIEN2k,the most sophisticated and time-consuming among the four methods, as can berepeated, uses a mixed representation for Kohn–Sham functions: numerical solu-tions around each atom, within its muffin-tin sphere, matched with plane-waveexpansion in the interstitial region between the spheres. The details of methodsare readily given by the documentation provided at each code’s internet site. Itlooks from the said that VASP and abinit fall into the same category of methods;fine details, apart from different implementations, are that VASP allows the useof USPP and PAW, the latter technique permitting to handle the deep core statesbetter than in a conventional PP scheme. In its turn, abinit allows to calculateforce constants in linear-response formalism (for arbitrary q value), and henceconstruct phonon dispersions.

Our use of different codes comes as a need to address specific properties ofbinary compounds and pseudobinary alloys. For instance, I use the WIEN2k codeto simulate numerically sensitive phase diagram of pure systems, while apply thelinear response method, as iplemented in abinit code, to establish phonon banddispersion curves5 (also for the same pure systems) needed to examine the dynam-ical stability of the predicted stable phases by the enthalpy comparison method6

(studied with WIEN2k) and to identify the new stable phases. Regarding theissue of simulating thermodynamic stability of larger systems, than merely smallordered unit cells, like different alloy supercells, I employ VASP code as it is fa-mous for yielding very good numerical accuracy together with pretty reasonable

1WIEN2k homepage: http://www.wien2k.at2VASP homepage: http://www.vasp.at3Siesta homepage: http://www.icmab.es/siesta4The abinit code is a common project of the Universite Catholique de Louvain, Corning

Incorporated, and other contributors. abinit homepage: http://www.abinit.org5see chapter 56see section 2.1.7

42

2.1. Ab initio electronic structure calculations

time cost.1 Finally, I use “frozen phonon” approach provided in Siesta code (seesection 2.5.1) to calculate zone-center (ZC) phonon density of states (DOS) indifferent alloy-imitating supercells.

2.1.6 Accuracy of DFT calculations

The energy differences between many of the competing phases are very small, anda resolution of a few meV per atom is sometimes required to distinguish theirrelative stabilities. The accuracy of the LDA or GGA is orders of magnitudeworse than this, so how is it possible for such calculations to give reasonableresults? The answer is that one relies on a cancellation of errors between theenergies of the competing phases, which appears to work extremely well as longas the nature of the bonding is similar. One also requires cancellation of theinevitable numerical approximations in the calculated energies of the competingphases, which are mostly due to the finite basis sets used to describe the wavefunctions and the approximations used in the Brillouin-zone integrations.

To illustrate the accuracy of DFT calculations, I show in Fig. 2.2 the exper-imental equilibrium volumes at zero pressure for different materials and thosecalculated using the LDA. It is generally known that LDA “tends to overbind”and predicts equilibrium volumes at zero temperature by up to ∼5% smaller thanin reality. Fig. 2.2 shows that the agreement is in fact much better than that, forbig number of semiconductor systems.

2.1.7 Fixed symmetry studies of phase stability

A major problem encountered in many fields of science is the difficulty to obtain aglobal minimum. It might be relatively easy to pinpoint a bunch of local minimafor which the structure is stable. Investigating crystal thermodynamic stabilitythrough optimizing the total internal energy with respect to volume, or enthalpywith respect to pressure, restricts the results to set of a priori selected phases,while in reality Nature generally explores far more possible structures than ourselected set of phases, and often prefers to assign a global minimum to yet anotherphase. Certainly, this standard widely used approach bounds the effectivness ofthe computations, though many successful predictions of new phases have beenachieved by analogy, or by reasonable extrapolation.

There are two widely adopted techniques, called enthalpy comparison method,to investigate pressure-induced phase transition and identify stable phases. Thefirst one is to optimize the total internal energy of some selected structures withrespect to structural parametrs at constant volume, and then construct their

1Simulating such supercells in WIEN2k is much more time consuming.

43

Chapter 2. Basics of theory and approximations used

Figure 2.2: Comparison between calculated (using the LDA) and experimen-tal equilibrium volumes per atom for the low-pressure phase of several ANB8−N

compounds. Figure taken from [Mujica et al., 2003] with the permission of theauthor.

enthalpies (H = U + PV )1. In the second one, the enthalpy of the chosenstructures is minimized at constant pressure. The lower enthalpy correspondsfor almost zero values of the forces and stress tensors, and, consequently, thestructure with lowest enthalpy, at a given pressure, is the most stable amongother phases [Pfrommer et al., 1997].

It is unlikely that one can predict many phase transitions by analyzing localstability, because the vast majority of transitions are first order, i.e., the tran-sition is from one locally stable phase to another one. However, starting froman equilibrium structure, one can carry out Density Functional Perturbations

1I apply this method to investigate pressure-induced phase diagram of ZnS and ZnSe semi-conductors; details in chapter 4.

44

2.1. Ab initio electronic structure calculations

Theory calculations1 [Baroni et al., 2001; Gonze, 1997; Gonze and Lee, 1997]to obtain the phonon dispersion curves of the initial phase at different appliedpressures. These curves are rich in information regarding the structural stabilityof the considered phase, i.e., they provide a decisive clue for possible displacivephase transitions. This is based on the fact that a crystal is said to be locallystable if all of its phonon frequencies are real, and its elastic constants satisfycertain inequalities. The softening of transversal acoustic branches is a precur-sor of structure instability. Then, the task is to find the crystal structure ofthe new stable phase by analyzing the phonon eigenvectors. These eigenvectorscarry information about the direction of displacement of the atoms. Dependingon such displacements, one constructs possible paths along which, by distortingan unstable crystal structure, the crystal can find a way into a new stable phase.Establishing the energy curves versus the distortion parameter of different pathsenables to choose between several possibly stable phases. Such calculations havebeen performed for several group-IVA, IIIA-VA, and IIB-VIA semiconductors byKim et al. [1999]; Ozolins and Zunger [1999] and moreover by Kirin and Lukacevic[2007]; Lukacevic [2011]; Lukacevic et al. [2010]. These developments have notso far been used very widely in theoretical studies of phase stability, but it isclear that they offer considerable advantages and will prove very important inthe future. I refer again to this method in Sec. 5.

Fig. 2.3 illustrates a typical energy-volume E(V ) and enthalpy-pressure H(p)diagrams for four phases of ZnS material2. The left panel imitates typical outputdata from a series of total-energy calculations with fixed symmetry and helps tounderstand how the transition pressures can be derived thereof.

Phases A and B have equal enthalpies at the two points EA(VA) and EB(VB),respectively, where the common tangent touches the energy-volume curves shownin the left panel of Fig. 2.3. The negative of the slope of the common tangent givesthe equilibrium pressure, to be equal in both phases on transition, peq(A/B) =−(EB−EA)/(VB−VA). In the right panel of Fig. 2.3, the corresponding enthalpy-pressure relations are illustrated. The two H(p) curves for phases A and B crossat the pressure peq(A/B). The phases c and d are not stable in any pressure range.However, the enthalpy of phase c is rather close to those of phases A and B nearpeq(A/B), and it is conceivable that the effects of temperature or an improvementin the calculations could result in the emergence of a narrow field of stability forphase c. Even if phase c is unstable, it is possible that the A → c transition couldbe observed at a slightly larger pressure if the A → B transition were suppressedby a large activation barrier. Conversely, a c ← B transition could be favorableon decrease of pressure from phase B if the A ← B transition were impeded.

1This method is applied in chapter 5 to investigate dynamic stability of relevant structures.2Results not yet published; details in chapter 4.

45

Chapter 2. Basics of theory and approximations used

80 100 120 140

Volume per atom (a.u.3)

-0.38

-0.36

-0.34

-0.32

Tot

al e

nerg

y pe

r at

om +

2195

(R

y) ABcd

12 14 16 18

Pressure (GPa)

-2

0

2

4

Ent

halp

y pe

r at

om r

elat

ive

to B

3 (m

Ry)

ABcd

Figure 2.3: Typical output data from a series of total-energy calculations. Leftpanel: energy-volume curves for four phases of a ZnS (see chapter 4). Thecurves correspond to fitting of the calculated points [in the present case, us-ing a Murnaghan-type expression [Murnaghan, 1944]. Right panel: enthalpy vs.pressure curves for the same four phases. Labels A, B, c and d, correspond to ZB,RS, simple cubic (SC16) and cinnabar phases, respectively.

In both cases, phase c would exist only as a metastable phase. Of course, aproper theoretical study of these possibilities would require the investigation ofthe actual mechanisms of the transitions.

2.1.8 Finite temperature effects on phase stability

The above analysis based on zero-temperature calculation results is useful forgetting a general idea of phase diagram and a fairly good estimate of transitionpressures. At elevated temperatures the results may need to be refined, takinginto account, in place of enthalpy (H = U + PV ), the Gibbs free energy (1.4),that has to be adjusted by the (temperature-dependent) vibrational contributionGν :

G = U + pV − TS + Gν . (2.8)

In what concerns the electronic system, the temperature effect on its internalenergy (via smearing out of the Fermi step function) can be usually neglectedin semiconductors for temperatures below typical melting points. The entropyof the electronic system, under this assumption, is non-zero only in disorderedalloys, but not in perfect periodic structures.

The effect of vibrations on the free energy is, in principle, complicated. The in-clusion of vibrations adds new states to the (previously purely electronic) system,

46

2.2. Static lattice model and its limitations

which can be populated in different ways, with an impact on internal energy andentropy. A number of complications (anharmonicity, electron-phonon coupling,vibronic effects) may arise.

In order to keep the problem feasible, and as a reasonable workaround, a quasi-harmonic approximation is generally accepted for practical calculations. Thisapproximation assumes that individual harmonic modes ων(q) of branches ν andwave vectors q are induced according to Bose–Einstein statistics, with mean num-ber of phonons

nω =1

e�ω

kBT − 1, (2.9)

which are additive and do not interact. For a practical calculation, merely theknowledge of density of vibration modes g(ω) is needed – a function which canbe accurately enough calculated, or realistically estimated. A straightforwardderivation1 leads in the following expressions for free energy

Gν = kBT

ωmax∫0

ln

(2 sinh

�ω

2kBT

)g(ω)dω (2.10)

and entropy:

S = kB

ωmax∫0

[(nω + 1) ln(nω + 1) − nω ln nω

]g(ω)dω . (2.11)

2.2 Static lattice model and its limitations

After this very brief excursion into the domain of vibrations and elevated tem-peratures, we need to recollect corresponding effects within a more systematiccontext, that is the subject of several following sections.

Many phenomena in solid-state physics can be understood on the basis of astatic model. In this model, the atoms of the solid are taken to constitute a fixed,rigid, immobile periodic array. Within this framework it is, for example, possibleto account for a wealth of equilibrium properties of metals dominated by thebehaviour of conduction electrons. To some extent it is also possible to accountfor the equilibrium properties of ionic and molecular insulators. The static modelis, of course, an approximation to the actual ionic configuration, because theatoms or ions are not fixed to their equilibrium positions, but, instead, oscillateabout them with an energy which is governed by the temperature of the solid.

1The details of derivation are given in Appendix C

47

Chapter 2. Basics of theory and approximations used

In classical theory the static lattice models can therefore be valid only atzero temperature. In quantum theory, even at zero temperature, the “genuinely”static lattice model is not valid, because according to the uncertainty principleΔx ΔP≥h, localized ions possess some non-vanishing mean square momentum.

The dynamic of atoms in solids is responsible for many phenomena whichcan not be explained within the static lattice model. The basic shortcomingsof the lattice model can be grouped into three broad categories: (i) failures toexplain equilibrium properties; (ii) failures to explain transport properties; (iii)failures to explain the interaction of various types of radiation with the solid. Inthe context of the present work, primarily the shortcomings in the descriptionof lowest excitations from equilibrium, able to affect relations between phases,should be of immediate concern.

2.3 Adiabatic approximation

In reality, we are dealing with nuclei and electrons, so atoms are not regardedas merely point masses with no structure. The question arises to which extentthe motion of the nuclei are independent of the motion of electrons. The “core”electrons can be assumed to move rigidly with the nuclei in the course of lat-tice vibrations, but the wave functions of the valence electrons do not remainunaffected.

Traditionally by now is to consider the ion cores and the valence electrons asindependent constituents of the solid, what is known as the adiabatic approxima-tion, or, the Born–Oppenheimer one; Born and Oppenheimer [1927].

The essential idea is that the ions being about 2×103 to 105 times heavier,move much more slowly than the electrons. At any given instant, therefore, theelectrons “see” the ions fixed in some instantaneous positions. On the contrary,the ions at any given instant “see” the electron system relaxed and arrived toequilibrium, corresponding to instantaneous ionic arrangement. We say thatthe electrons follow the ionic motion adiabatically. In an adiabatic motion, anelectron does not make abrupt transitions from one state to others; instead, anelectronic state itself is deformed progressively by the ionic displacements. Iwill briefly sketch some results of the adiabatic approximation; for a detaileddiscussion please refer to Born and Huang [1954]; Born and Oppenheimer [1927];Brownman and Kagan [1967]; Chester [1961]; Maradudin [1974]; Migdal [1958];Ziman [1964, 1967]. The hamiltonian for the crystal is

H = Te + Tj + Vjj(R) + Vee(r) + Vje(r,R) , (2.12)

where r is a collective symbol for the coordinates of the valence electrons andR similarly for the ionic coordinates. The subscripts j and e denote ions and

48

2.3. Adiabatic approximation

valence electrons, correspondingly. Te and Tj are the kinetic energy operators ofthe electrons and ions, Vee(r) and Vjj are the electron-electron and direct ion-ion interactions, respectively, while Vij(r,R) is the electron-ion interaction. LetΨ(r,R) be the eigenfunction of the hamiltonian (2.12), so that

HΨ(r,R) = EΨ(r,R) , (2.13)

where E is the total energy of the entire system. In the adiabatic approximation,we imagine the ions to be fixed in some configuration, and for this particular Rwe can construct the Shrodinger equation for the electrons:

[Te + Vee(r) + Vje(r,R)] ψR(r) = Ee(R)ψR(r) . (2.14)

Here ψR(r) is a wavefunction for the entire system of electrons, depending on allvariables represented by r explicitly, and on ionic coordinates R – as parameters.Likewise, the electronic energy Ee(R) depends on the ionic coordinates. Ee(R) isnot just a potential energy; it includes, beside the electron-electron and the ion-electron interactions, the kinetic energy of the electronic motion which changesas the coordinates R are changed. If we use the following ansatz for Ψ(r,R):

Ψ(r,R) = ψR(r)χ(R) , (2.15)

substitute Eq. (2.15) in (2.13) and make use of (2.14), it may be shown thatEq. (2.15) is indeed a solution of Eq. (2.13), provided that certain terms can beneglected and the ionic wavefunction χ(R) is chosen to satisfy

[Tj + Vjj(R) + Ee(R)] χ(R) = Eχ(R) . (2.16)

An elementary discussion of the neglected terms was given by Ziman [1964].Eq. (2.16) is an equation for a wavefunction of the ions alone. The essential pointis that for the ionic motion, an effective potential energy function

U(R) = Vjj(R) + Ee(R) (2.17)

can be introduced, to which the electrons contribute through Ee(R). U(R) canbe used as a potential energy for discussing the motion of the ions.

The validity of the adiabatic approximation for lattice dynamics has beendiscussed by several authors, see Brownman and Kagan [1967]; Chester [1961];Migdal [1958]; Ziman [1964, 1967]. The main result is that the approximationis valid for the lattice dynamics of insulators as well as of semiconductors andmetals. In insulators and large bandgap semiconductors, the approximation isvalid because the excited states of the electrons are separated from the groundstate by energies large compared with phonon energies �ω. This is certainly not

49

Chapter 2. Basics of theory and approximations used

the case in small gap semiconductors and metals. Indeed, in the vicinity of theFermi surface of a metal, electronic transitions are possible with arbitrarily smallexcitation energies, and for electrons with energies within �ω of the Fermi energyEF, the above-mentioned abrupt transitions associated with the motion of theions can not be competely excluded. However, the number of electrons in thisthin “thermal layer” of thickness �ω is very small compared to the total numberof electrons in the conduction band. For the latter, the excitation energy is ofthe order of EF, and therefore they must follow the vibrating ions adiabatically.In my lattice dynamics study of the Zn-Be-Se and Cd-S-Se alloys, the validity ofthe adiabatic approximation is assumed. I should mention, however, here thatthe adiabatic approximation breaks down for phenomena where the dynamics ofthe electrons are directly involved, such as the resistance due to scattering ofelectrons by the phonons and superconductivity.

The adiabatic approximation makes it possible to work with an effective po-tential U(R) given by Eq. (2.17) if we are dealing with the motion of the ions.The evaluation of Ee(R) on the basis of Eq. (2.14), however, is a very difficultproblem in general.

As a substitution to first-principles schemes, it was common in the past, andstill is, for unconveniently complex systems, to assume some phenomenologicalpotential for U(R) depending only on few parameters. Phenomenological po-tentials for atoms in solids are discussed in chapter 4 of the book by Bruesch[1982].

The harmonic approximation assumes U(R) to be a quadratic form in allatomic displacements from equilibrium. Higher-order corrections give rise toanharmonic terms, considerably complicating the formalism. Such anharmonicterms are essential for a discussion of such properties as thermal expansion andthermal conductivity, but not of immediate importance for phase transition (apartfrom certain special issues); therefore any systematic study of anharmonicity willbe spared in the present work.

2.4 An introductory word about phonon

Phonons are quanta of lattice vibrations. They stand in the same relation tonormal vibration modes as a quantum-mechanical harmonic oscillator stands toa classical one: a more intense vibration, in the language of quantum mechanics,means creation of more phonons (which are subject to Bose-Einstein statistics).Thus, a given phonon is by definition harmonic; the anharmonicity is traditionallydescribed in terms of phonon-phonon interactions. A phonon is characterized byits wavevector value q, frequency ω and branch number, exactly as the normalvibration modes are. A phonon-phonon interaction may mix up, in general case,

50

2.4. An introductory word about phonon

phonons with different frequencies, characterized by different wavevectors, thatcomplicates any practical calculation immensely. Staying within the harmonicapproximation, when phonons exist as independent harmonic oscillators, is somuch simpler. Fortunately this is sufficient for treating a big number of problemsfrom the materials science, in which the temperature is not very elevated. Theharmonic approximation is a limitation which is admitted throughout the presentwork.

Each phonon corresponds to a one-dimensional harmonic oscillator; a wholesystem of such independent oscillators (whose total number 3N , N being thenumber of atoms in crystal) has their classical counterparts in the solutions ofthe system of coupled classical harmonic oscillators. The harmonic approximationallows to decouple the originally mutually dependent system into independentnormal modes, as is explained in detail in section 2.5. The calculated modes, withtheir frequencies and eigenvectors, provide, in case of need, a basis for quantum-mechanical treatment. In the present work, the quantum-mechanical aspect ofvibrations is addressed only once, in calculating the phonon contribution to theentropy of alloys, in chapter 7. For the rest of work, harmonic lattice vibrationsare treated in entirely classical way.

This classical treatment bases on the Born-Oppenheimer approximation, oradiabatic approximation, explained earlier in section 2.3. Within it, the (Newto-nian) forces acting on atoms follow from quantum-mechanical calculations doneon the electronic system, for a given configuration of fixed ions. What is neededfor solving dynamical equations are force constants, i.e., second derivatives ofenergy in atomic displacements from equilibrium. The ways to get them fromfirst-principles calculations are the following: (i) by numerical 2nd order differ-entiation of calculated total energy; (ii) by numerical 1st order differentiation offorces directly available from a first-principles code; (iii) directly from a codewhich implements a programmed analytical calculation of force constants, in alinear response scheme. The way (i) is prohibitively inefficient for typical phononcalculations (too many different calculations for various displacements have to bedone), apart from the simplest cases with just few atoms per unit cell. The way(ii) is within grasp of many calculation methods which provide accurate forces;in the present work, such calculations have been done for supercells with sev-eral tens of atoms, e.g., in chapter 6 and 7, using the Siesta code. The way(iii) is implemented, e.g., in the abinit code, and used for calculating phonondispersions in chapter 5.

Before entering a more detailed discussion about how vibration modes arecalculated and analyzed, we briefly address an important, in the following, issueof how different vibration modes are attributed with respect to their wavevectorsq.

A N -atomic molecule or finite cluster do not know anything about wavevec-

51

Chapter 2. Basics of theory and approximations used

tors; their 3N normal vibration modes are simply numbered throughout. Inperiodic crystals, a possible simplification appears. If we assume a periodicity of,say, a n-atomic unit cell and postulate that, moreover, the Born–von Karman pe-riodic boundary conditions are imposed on a crystal containing M (presumably avery big number) unit cells, the total of 3nM modes in such crystal are separatedinto M groups by 3n modes, each (now independent) group being characterizedby its different wave vector, nested somewhere in the Brillouin zone. This offersa tremendous simplification in practical calculations. Moreover, as the choice ofperiodic unit cell in a crystal is not unique, it is at our convenience to switchbetween the (minimal) primitive cell and arbitrarily enlarged multiple cells (su-percells). While describing the same physics, the mathematical language allowsto include more modes (as n increases in a multiple cell) with less allowed q valuesfor each (as the corresponding Brillouin zone schrinks). In the present work, manyresults refer to ZC q=(0 0 0), i.e. Γ modes, of different supercells (constructed asto incorporate impurities, or different substitution motifs in alloys). Not all ofthese modes, however, are genuine ZC with respect to the Brillouin zone of theunderlying crystal lattice and its primitive cell; some modes come in fact fromother wavevector values, as the unit cell is multiplied into a supercell, and itsphonon branches are folded onto a correspondingly smaller Brillouin zone. It isimportant to distinguish between “genuine” and “folded” Γ modes, because onlythe former markedly contribute “in phase” to experimentally observed Raman orinfrared lines of the crystal, whereas the effect of the latter sums up to virtuallyzero. In perfect crystals, the folding/unfolding of phonon branches in multiple(super)cells is clear-cut and straightforward; in systems with defects, where theperiodicity of the underlying lattice is “slightly” distorted, this analysis of the q“reduction” can be done only approximatively. The details of such treatment areaddressed in chapter 6.

2.5 Harmonic approximation formalism

Quantum functions of atomic waves explore the region of space near equilibriumpositions, where the probability of finding atoms is maximal. This quantumpicture is behind the idea of treating classically the vibration of the system, as-suming that the nuclei do small oscillations around their equilibrium positions. Inthese conditions of small displacements, the harmonic approach is often employed,that namely permits to separate coupled vibrations into a superposition of (non-interacting) normal modes. Once these modes with their frequencies identified,their corresponding quantized energies are given by:

En = �ωs(n +12) , (2.18)

52

2.5. Harmonic approximation formalism

where s = q is the oscillator in the state with quantum number ns (integer).The main characteristic of a given mode of quantum oscillator is therefore

its frequency, which (in the harmonic approximation) is exactly the same asin classical case, and can be obtained from solving the equations of motion, atextbook example in classical mechanics. A certain complication comes onlyfrom allowing couplings between different degrees of freedom in a crystal.

If the equilibrium position of ion α is denoted by R0α, the dynamics of this

ion can be characterized by its small displacements with respect to this position:Rα = R0

α + τα. The classical kinetic energy is then given by:

T =1

2

N∑α=1

∑m=x,y,z

Mα τ 2α,m , (2.19)

where the dot represents time derivative and τα,m is the cartesian component ofthe τα. The effective potential energy, introduced in Eq. (2.17), can be expressedas a Taylor series in the atomic displacements:

U = U0 +∑αm

[∂U

∂τα,m

]0

τα,m +1

2

∑αm, α′m′

[∂2U

∂τα,m ∂τα′,m′

]0

τα,mτα′,m′ + . . . (2.20)

The bracketed expressions are evaluated for atomic equilibrium positions. The

term linear in τα is canceled, since in these positions the forces are null,[

∂U∂τα,m

]0=0.

Therefore, in the harmonic approximation the Hamiltonian for the ionic systembecomes:

H =1

2

∑αm

Mατ 2α,m + U0 +

1

2

∑αm,α′m′

[∂2U

∂τα,m ∂τ ′α, m′

]0

τα,mτα′,m′ . (2.21)

In the harmonic approximation, the force on an atom is strictly proportional to itsdisplacement relative to its neighbours (which is nothing other than the Hooke’slaw), and the classical equation of motion reads:

Mατα,m = −∑α′m′

[∂2U

∂τα,m ∂τ ′α, m′

]0

τα′,m′ . (2.22)

Therefore the knowledge of second derivatives of the potential that acts on ionswith respect to shifts from their equilibrium positions (i.e., the force constants),fully determines the dynamics of the lattice in the harmonic approximation.Searching for solutions, for crystalline solids, of the general form

τα = M− 1

2α uαei(q·Rα−wt) . (2.23)

53

Chapter 2. Basics of theory and approximations used

reduces the system of differential equations (2.22) to an eigenvalue problem ofdimension 3N × 3N , where N is the number of atoms:

ω2(q)uα(q) = Dmm′αα′ (q) uα′(q) , (2.24)

and the matrix D(q), known as the dynamic matrix, is defined as:

D(q) ≡ Dmm′αα′ (q) = (MαMα′)−

12

∑Rα−Rα′

∂2U

∂τα,m∂τα′,m′e−iq·(Rα−Rα′ ) . (2.25)

Its eigenvalues and eigenvectors give the frequencies and vibration modes of thesystem. The previous formulation is the most general one equally applicable tonon-periodic materials, such as molecules.

The harmonic approximation is the starting point for all the theories of latticedynamics (except, perhaps, that of solid helium). Further corrections to U(R)expansion in Eq. (2.20), especially those with third and fourth derivatives indisplacements, are known as anharmonic terms. The theory of lattice dynamicsis much more complicated if anharmonic effects are considered. The additionalterms in U3 and U4, can often be treated as a perturbation.

2.5.1 Nonperturbative methods

Essential for solving the coupled dynamical equations from the previous sectionis to know force constants, i.e., the second derivatives of the total energy. Thiscan be achieved either by numerical differentiation, or analytically, if the algo-rithm/code in question allows it.

Numerical differentiation can be performed directly on total energy; this ishowever not much practical, because too many calculations are typically needed.A given element of the force constants matrix is a function of two variables,U(ταm, τα′m′); a 2nd order polynomial in two variables has 6 terms; hence (atleast) 5 calculations are needed for each (α, α′) pair (in addition to the undisplacedconfiguration, common for each pair) to complete its 2nd order fit and extract thesearched for second derivatives. With N atoms and N(N+1)/2 interactions, thismakes 5N(N +1)/2 + 1 different calculations – the number which can probablybe reduced, taking into account symmetry relations in the system.

If the calculation method allows to directly calculate forces on all atoms, theconstruction of force constants amounts to single differentiation of them:

∂2U

∂ταm ∂τα′m′= − ∂Fαm

∂τα′m′= −∂Fα′m′

∂ταm

, (2.26)

that can be achieved from 2 calculations (two different displacements, in additionto the result for the undisplaced equilibrium position). As all forces are available

54

2.5. Harmonic approximation formalism

Figure 2.4: A schematic view of a supercell representing a frozen phonon of com-mensurate wavelength with a periodic crystal. Clear circles represent equilibriumpositions of atoms. Dark circles represent the position of displaced atoms, withtheir directions of movement shown by arrows. Figure taken from PhD thesis ofPruneda [2001].

from each calculation, it is in principle sufficient to try the total of 6N + 1calculations (two tries in each Cartesian displacement of each atom), for anyinterestingly large system much less than without having the forces available.This approach is used in calculating phonons with the Siesta code.

This approach is sometimes referred to as finite-displacements one; or elsethe frozen phonons scheme. In a more precise perspective, one can say thatthe “frozen phonon” (see Fig. 2.4) technique rather refers not to accumulatingall force constants, but to scanning specifically some their symmetry-preparedcombinations, corresponding to a given symmetry of vibration.

The magnitude of displacement used for calculation of numerical derivativesmay somehow affect the results if the harmonicity hypothesis is badly justified,and the neglected terms (of the higher order than 2) in the effective potentialenergy expansion in Eq. (2.20) are in fact not small. Therefore, the magnitudeof trial displacements should be chosen with care – not too big (to reduce theambiguity due to higher-order terms) and not too small (to prevent numericaldivision error); Siesta calculations typically used ∼0.04 Bohr.

The above numerical-differentiation, or sampling, scheme is straightforwardonly if applied to “molecule” or cluster without periodicity, where the numberingof force constants is done directly by the indices of atoms, all being unique. Asubtlety arises if one turns to a periodic system in view of finding the dynamicalmatrix for a given q value, as in Eq. (2.25). There are two ways to achievethis. The first is to collect force constants from Eq. (2.26) in real space up tosufficiently large interatomic distances, at which the force induced on one atom

55

Chapter 2. Basics of theory and approximations used

due to displacement of the other is sufficiently small, and perform its Fouriertransform for arbitrary q. The other is, to organize a trial displacement as acollective one, by construction corresponding to a wave of given q, and analyze theforces induced on atoms in periodic unit cell as function of such wave’s magnitude.

The direct force method [Kunc and Martin, 1982; Parlinski et al., 1997; Yinand Cohen, 1982] proceeds by calculating, from first principles, the forces experi-enced by the atoms in response to various imposed displacements and by deter-mining the value of the force constant tensors that match these forces through aleast-squares fit. Note that such calculation is typically organized in a supercellgeomery, i.e., a trial displacement of a single atom is in fact accompanied by adisplacement of all its translated replicas on the infinite lattice. However in prac-tice, taking a suffucient number of different trial displacements allows to extractrelevant “locally numbered” force constants up to interatomic distances beyondwhich these force constants can be safely neglected.

Another possibility to study vibrational properties, circumventing the aboveformalism and, moreover, not restricted by the harmonic approximation, is the abinitio molecular dynamics. The simulation time step being a priori chosen, theatomic trajectories are generated from classical equations of motion, discretizedwith this step, as each atom is promoted from its instantaneous position to thenext one, depending on the atom’s initial velocity and instantaneous force actingon it. Essential for the success of such analysis are sufficiently large numberof simulation steps and sufficiently large system size, in order to guarantee agood enough thermodynamical sampling. Useful information, including vibrationfrequencies and correlations in the movement of different atoms, can be thenextracted from the velocity autocorrelation function – see Allen and Tildesley[1987]. A big advantage of the molecular dynamics scheme is that simulations canbe run either at fixed energy, or at fixed (or, even smoothly varying) temperature,imposed by some algorithmical realization of a thermostat. As this method wasnot used in the present work, its details are not discussed here. They can befound in a review by Marx and Hutter [2000] and in the mentioned book by Allenand Tildesley [1987].

2.5.2 Perturbative approach

In the late 80’s, Baroni et al. [1987] proposed a method to obtain the change indensity without having to calculate the dielectric matrix. The method in ques-tion employs the perturbation theory based on the dielectric matrix. However,the response of the system is not obtained by inversion of the dielectric matrix,but iteratively, till arriving to self-consistency. The method notably avoids thesummations over empty conduction bands (whose number has to be taken, inreasonably accurate calculations, quite large) and implements, instead, a Green’s

56

2.6. Pseudobinary alloy phase diagram

function technique. As a result, one arrives at estimating changes in the electrondensity with about the same computational effort as needed to calculate a fewrows of the dielectric matrix. Knowing the related variation of the density, δρ(q),one can get the second derivatives of the energy (force constants matrix), andhence the dynamic matrix (2.25). The first step, in the presence of perturbingexternal “bare” potential whose Fourier transform is φext(q), goes as follows:

δρ(q) = χ(q) φ(q) = q2 [1 − ε(q)] φ(q) . (2.27)

where ε(q) is the dielectric function, χ(q) electric susceptibility and φ is thescreened potential given by:

φ(q) =1

ε(q)φext(q) . (2.28)

A review by Baroni et al. [2001] outlined the state of art of the time it appearedin what regards applying DFT-based perturbative methods for the evaluation ofvibrational properties in crystals.

This linear response theory [Baroni et al., 2001; Gonze, 1997; Gonze and Lee,1997] can be used to calculate the dynamical matrix D(q) of Eq. (2.25) directlyusing second-order perturbation theory, thus circumventing a need for supercellcalculations. A big advantage of the linear response theory is that it allows to getresults for any q vector, even that incommensurate with crystal lattice, and thusscan the dispersions throughout the Brillouin zone. Moreover, it allows to includethe effects of (long-ranged) external electric fields with relative straightforward-ness. The linear response theory is also particularly useful when a system possesslong-range force constants, as in the presence of Fermi surface instabilities.

2.6 Pseudobinary alloy phase diagram

Mixing free energy, miscibility gap,and order-disorder transitions

The following brief introduction in construction and analysis of concentration–temperature alloy phase diagrams is quite general. As I wont’t apply it to any-thing else than pseudobinary semiconductors, I assume from the beginning that,for an A1−xBxC alloy, the A and B atoms more or less randomly occupy the sitesof one fcc sublattice, whereas the C atoms are fixed as spectators at the other fccsublattice.

The equilibrium state at fixed temperature T and pressure p is the one withthe minimum Gibbs free energy (2.8). In order to identify such state, one should

57

Chapter 2. Basics of theory and approximations used

be able to provide, within whatever approximation or model, the internal energy(e.g., the zero-temperature electronic total energy) U throughout the concentra-tion range, 0≤x≤1. This might be necessary to do for several competing phases.Namely, the Gibbs’s phase rule [e.g., Landau and Lifshitz, 1980] states that a so-lution containing n species can have up to n+2 phases coexisting in equilibrium.

What is of importance for numerical analysis is not the energy at a givenconcentration as such (its absolute value can strongly depend on the calculationmethod, level of approximation, e.g., whether the deep core electronic states areincluded in the calculation or not), but the energy gain (or loss) relatively tonon-alloyed parent constituents. The latter, obviously for their non-interactingmixture, should be taken linearly weighted with concentration. For the following,we’ll argument in such relative, e.g. mixing thermodynamical variables (U , F ,G, etc.). In ambient conditions, the pV term is zero or, for practical purposes,negligible, so that the property of interest is the Helmholtz free energy, whosemixing value explicitly is:

ΔF (x, T ) = F (x, T ) − (1 − x)FAC − xFBC . (2.29)

Separating it into total energy and entropy differences:

ΔF = ΔU − TΔS . (2.30)

For elevated pressures, the mixing property of reference will be the mixing en-thalpy ΔH, with a straightforward generalization of Eqs. (2.29, 2.30) towards theGibbs free energy. It must eventually include the change of equilibrium volumebetween the alloy and the parent phases for a given pressure.

The internal energy of mixing, ΔU at T = 0, can be either positive or negativefor different systems; notably for most pseudobinary semiconductor alloys it ispositive, hence being a concave function of x, as in the upper curve of Fig. 2.5. Incase of alloying two binary compounds of similar structure, its end points are ob-viously fixed, by definition of Eq. (2.29), as ΔF (0, T ) = ΔF (1, T ) = 0. However,if one proceeds from different structures, as in our case of Cd(S,Se) addressed inchapter 7, then ΔF (0, T ) = ΔF (1, T ) = 0. The other concentration dependencein Eq. (2.29) comes from the mixing entropy. The latter is, primarily, the config-uration entropy, due to a variety of ways to place (A,B) atoms at their sites in analloy. The other temperature-dependent part of entropy, due to latice vibration,which is typically less important, was briefly mentioned in section 2.1.8; its moredetailed discussion is postponed till section 2.8. The evaluation of configurationalentropy is a standard exercise in statistical physics, yielding

ΔS = −kB [xlnx + (1 − x)ln(1 − x)] , (2.31)

per site of the (A,B) sublattice. Eq. (2.31) describes also a concave function ofx, pinned to zero at both limiting values of x. A difference of concave (ΔU) and

58

2.6. Pseudobinary alloy phase diagram

Figure 2.5: Upper panel:schematic of mixing free energiesΔF as a function of position xof a pseudobinary alloy A1−xBxCwith a positive mixing enthalpyat three temperatures: T1 = 0,T2 < Tc and T3 > Tc. Lowerpanel: schematic miscibility gapand spinodal curves. Regions I,II, and III are stable, metastable,and unstable, respectively. Fig-ure taken from [Chen and Sher,1995] with the permission of theauthors.

convex (−TΔS) functions may have a complicated shape (see Fig. 2.5, upperpanel, regime T2 < Tc), until at high enough temperature the TΔS term startsto dominate, enforcing the convex character of ΔF (x, T ) at T3 > Tc in the abovefigure.

Such globally convex character at sufficiently high temperatures indicates thatfor all x values, the disordered alloy will be stable. At lower temperatures, thereis a miscibility gap for x within a certain interval x1 < x < x2. Here x1 and x2

are the points at which the tangents constructed to the ΔF (x) curve merge intoa common one. This situation means that the alloy is thermally stable againstdecomposition for x < x1 and x > x2, but for x inside this gap, the alloy tends todecompose into two fractions with minimal and maximal allowed concentrations

59

Chapter 2. Basics of theory and approximations used

x1 and x2 in relative proportions P1 and P2, governed by the “lever rule”:

P1 =x2 − x

x2 − x1

; P2 =x − x1

x2 − x1

, (2.32)

because notably this decomposition minimizes the free energy. Such points, takenover a range of temperatures up to Tc, trace in the (x, T ) plane the binodal curve,called also coexistence curve, which is shown as the solid line in Fig. 2.5, bottompanel.

For temperatures below the critical one, one marks moreover two other specialconcentration values, x′

1 and x′2, called spinodal points. They are inflection points

of ΔF , given by the condition

∂2ΔF

∂x2

∣∣∣∣x′1

= 0 ;∂2ΔF

∂x2

∣∣∣∣x′2

= 0 . (2.33)

For varying temperatures, the resulting spinodal curves x′1(T ), x′

2(T ) separate the(x, T ) plane into unstable and metastable regions, as shown in Fig. 2.5, bottompanel. For x1 < x < x′

1 or x′2 < x < x2, the alloy is metastable against local

decomposition, because the ΔF value for any x in these regions is lower than thelever-rule average value of ΔF with any two compositions in the neighborhoodof x. This increase in ΔF acts as a temporary energy barrier against alloydecomposition into its final equilibrium concentrations x1 and x2. A uniform alloywith x′

1 <x<x′2 is inherently unstable because of the lack of such decompostion

barrier. Tracing the binodal and spinodal curves by continuously varying thetemperatures and tracing the values of these gaps, as indicated in Fig. 2.5, bottompanel, shows that the both gaps gradually shrink at increasing temperature untileventually closing at Tc, when all the values x1, x′

1, x2, x′2 merge together. Above

Tc, up to melting, the disordered solid phase is stable for all alloy concentrations.

In case if, contrarily to as assumed above, ΔU < 0 at T = 0, there is atendency to form a long-range ordered alloy. Whether the alloy is ordered ordisordered depends on the temperature. Fig. 2.6 schematically compares themixing free energy of the disordered alloy against that of an ordered compoundABC2. With a negative mixing enthalpy, the energy of the ordered system ΔU0

is lower than the disordered alloy (i.e., ΔU0 < ΔU at T = 0), so that the orderedstate is the equilibrium one.

As temperature increases, ΔF of the disordered alloy becomes more negative,mainly because the entropy term in Eq. (2.30) decreases roughly linearly in T .The system remains in the ordered phase, until a transition temperature T0 isreached when ΔF is equal to ΔU0. The disordered alloy then becomes the stablestate for temperatures greater than T0.

60

2.7. Phase diagram analytical models

Figure 2.6: Schematic picture of mixing free energies ΔF as a function of alloycomposition x in a pseudobinary alloy A1−xBxC with a negative mixing enthalpyat three temperatures: T1 = 0, T2 = Tc and T3 > Tc. The narrow wedge-likeregion in the middle represents the ABC2 ordered phase. Figure taken from[Chen and Sher, 1995] with the permission of the authors.

2.7 Phase diagram analytical models

This section reviews some analytical models, which are useful for illustrating basicconcepts and for semiemperical phase diagram evaluation. I focus on disorderedalloys with positive mixing energies, as this the most relevant case for most bulksemiconductors, and in particular our case, CdSxSe1−x, full treatment of which isoutlined in Chapter 7.

2.7.1 Ideal-solution model

The ideal-solution model is equivalent to the case where the mixing energy van-ishes, i.e., ΔU = 0 in Eq. (2.30). Thus, in this approximation, the mixing free

61

Chapter 2. Basics of theory and approximations used

energy ΔF results totally from the additional entropy arising from a randomarrangement of NA A atoms and NB B atoms on their N lattice sites, withN = NA + NB and x = NB/N . The number of different arrangements is

Φ0 =N !

NA!NB!; (2.34)

the corresponding mixing entropy is ΔS = kBlnΦ0. Using the Stirling’s formulafor large N , lnN !approxN lnN − N , the mixing entropy for the random alloyreads:

ΔS = −NkB [(1 − x) ln(1 − x) + x lnx] (2.35)

This model always yields negative values of ΔF = −TΔS for all x and T . As afunction of x, ΔF in Eq. (2.30) is always convex upward, so no miscibility gapexists at any T .

2.7.2 Zeroth approximation

The situation when ΔU is not zero, but the mixing entropy is still set equal to therandom alloy result, was called by Guggenheim [1952] the “strict-regular solution”model, or the zeroth approximation. If, as often experimentally observed inpseudobinary alloys, ΔU has an x dependence given by

ΔU = Nx(1 − x)ω , (2.36)

then the miscibility gaps exist at low temperatures for positive interaction pa-rameter ω. At this point ω is being treated as an empirical fitting parameter.

This strict-regular model is the simplest one that contains some aspect ofreality, because it relates a positive mixing enthalpy to a miscibility gap andcritical temperature Tc. For example, Tc can be obtained explicitly by setting∂2ΔF/∂x2 equal to zero at x = 1

2,1where ΔF = ΔU −TcΔS with ΔU in the form

of Eq. (2.36), and ΔS adopts the random alloy expression of Eq. (2.35). Theresult is Tc = ω/(2kB).

2.8 First-principles calculations of thermodynamic

properties

In a solid, thermal fluctuations take the form of electron excitations and latticevibrations, and, accordingly, the free energy can be written as

F = U − TS + Fe + Fvib , (2.37)

1ΔF has its second derivative vanish at x = 12 in this case because of the assumed symmetric

forms of Eqs. (2.36). In more general cases that extend beyond pair interactions, the secondderivative can vanish at different x values.

62

2.8. First-principles calculations of thermodynamic properties

where U is the absolute zero total energy while Fe and Fvib denote electronicand vibrational free energy contributions, respectively. This section describesthe calculation of the electronic and vibrational contributions most commonlyconsidered in phase-diagram calculations under the assumption that electron-phonon interactions are negligible (i.e., Fe and Fvib are simply additive).

To account for electronic excitations, electronic DFT can be extended tononzero temperatures by allowing for partial occupations of the electronic states[Mermin, 1965]. Within this framework, and assuming that both the electroniccharge density and the electronic density of states can be considered temperatureindependent, the electronic contribution to the free energy Fe(T ) at temperatureT can be decomposed as

Fe(T ) = Ee(T ) − Ee(0) − TSe(T ) , (2.38)

where the electronic band energy Ee(T ) and the electron entropy Se(T ) are re-spectively given by

Ee(T ) =

∫fμ,T (ε) g(ε)dε , (2.39)

Se(T ) = −kB

∫{fμ,T (ε) ln fμ,T (ε)

+ [1 − fμ,T (ε)] ln[1 − fμ,T (ε)]} g(ε)dε , (2.40)

where g(ε) is the electronic density of states obtained from a density functionalcalculation, and fμ,T (ε) is the Fermi distribution with the electronic chemicalpotential μ:

fμ,T (ε) =

[1 + exp

(ε − μ

KBT

)]−1

. (2.41)

The chemical potential μ is the solution to fμ,T (ε)g(ε)dε = ne, ne being the totalnumber of electrons. Under the assumption that the electronic density of statesnear the Fermi level is slowly varying relative to fμ,T (ε), the equations for theelectronic free energy reduce to the well-known Sommerfeld model, an expansionin powers of T whose lowest order term is

Fe = −π2

6k2

B T 2g(εF) , (2.42)

where g(εF) is the zero-temperature value of the electronic density of states atthe Fermi level (εF).

The quantum treatment of lattice vibrations in the harmonic approximationprovides a reliable description of thermal vibrations in many solids for low to

63

Chapter 2. Basics of theory and approximations used

moderately high temperatures. In Appendix C, I provide a complete derivationof the vibrational free energy:1

Fvib = kBT∑

α

∞∫0

ln

(2 sinh

�ω

2kBT

)δ(ω − ωα) dω . (2.43)

The associated vibrational entropy Svib of the system can be obtained from thewell-known thermodynamic relationship Svib = −∂Fvib/∂T . The high tempera-ture limit (which is also the classical limit) of Eq. (2.43) is often a good approx-imation over the range of temperature of interest in solid-state phase diagramcalculations:

Fvib = kBT∑

α

∞∫0

ln

(�ω

kBT

)δ(ω − ωα) dω . (2.44)

The high temperature limit of the vibrational entropy difference between twophases is often used as convenient measure of the magnitude of the effect oflattice vibrations on phase stability. It has the advantage of being temperature-independent, thus allowing a unique number to be reported as a measure ofvibrational effects. Fig. 2.7 from van de Walle et al. [2007] illustrates the use ofthe above formalism to assess the relative phase stability of the θ and θ′ phasesresponsible for precipitation hardening in the Al-Cu system. Interestingly, ac-counting for lattice vibrations is crucial in order for the calculations to agree withthe experimentally observed fact that the θ phase is stable at typical processingtemperatures (T > 475 K).

A simple improvement over the harmonic approximation, called the quasi-harmonic approximation, is obtained by using volume-dependent force constanttensors. This approach maintains all the computational advantages of the har-monic approximation while permitting the modeling of thermal expansion. Thevolume dependence of the phonon frequencies induced by the volume-dependenceof the force constants is traditionally described by the Gruneisen parameter,γ = −∂ ln ωb(q)/∂ ln V . However, for the purpose of modeling thermal expan-sion, it is more convenient to directly parameterize the volume dependence of thefree energy itself. This dependence has two sources: the change in entropy dueto the change in the phonon frequencies and the elastic energy change due to theexpansion of the lattice:

F (T, V ) = U(V ) − TS + Fvib(T, V ) , (2.45)

where U(V ) is the energy of a motionless lattice whose unit cell is constrained toremain at volume V , while Fvib(T, V ) is the vibrational free energy of a harmonic

1Fvib is nothing but Gν encountered in Eqs. 2.8 and 2.10.

64

2.8. First-principles calculations of thermodynamic properties

Figure 2.7: Temperature dependence of the free energy of the θ and θ′ phases ofthe Al2Cu compound. Insets show the crystal structures of each phase and thecorresponding phonon density of states. Dashed lines indicate region of metasta-bility and the θ phase is seen to become stable above about 475 K. (Figure from[van de Walle et al., 2007] with the permission of the authors).

system constrained to remain with a unit cell volume V at temperature T . Theequilibrium volume V ∗(T ) at temperature T is obtained by minimising F (T, V )with respect to V . The resulting free energy F (T ) at temperature T is then givenby F (T, V ∗(T )). The quasiharmonic approximation has been shown to provide areliable description of thermal expansion of numerous elements up to their meltingpoints, as shown in Fig. 2.8.

First-principles calculations, either frozen phonon or linear response ones, canbe used to provide the necessary input parameters for the above formalism.

Above we have our discussion centered around the application of harmonic(or quasi-harmonic) approximations to the statistical modeling of vibrationalcontributions to free energies of solids. While harmonic theory is known to behighly accurate for a wide class of materials, important cases exist where thisapproximation breaks down due to large anharmonic effects. Examples includethe modeling of ferroelectric and martensitic phase transformations, where the

65

Chapter 2. Basics of theory and approximations used

Figure 2.8: Thermal expansion of selected metals calculated within the quasi-harmonic approximation (diamonds denote experimental measurements). From[van de Walle et al., 2007] with the permission of the authors.

high-temperature phases are often dynamically unstable at zero temperature,i.e., their phonon spectra are characterized by unstable modes. In such cases,effective-Hamiltonian methods have been developed to model structural phasetransitions from first principles [Zhong et al., 1994]. Alternatively, direct appli-cation of ab initio molecular-dynamics offers a general framework for modelingthermodynamic properties of anharmonic solids [Alfe et al., 2000, 2002].

66

Chapter 3

First-principles study of gapbowing in InAs1−xPx alloys

3.1 Introduction

This chapter outlines early results of the present work [Breidi et al., 2009], ob-tained on the electronic structure of semiconductor alloys, without yet addressingpressure effects and assuming, as a simplest model, a sequence of ordered super-structures. The main point of interest was the study of band gap bowing. Evenas “conventional” DFT calculations are known to largely underestimate the bandgap in semiconductors and insulators, certain hope exists, sometimes confirmedby comparison with experiment, that the rough concentration-dependent trendsin the band gap variation can be grasped from “naive” supercell calculations. Theinterest for studying band gaps is mostly driven by hopes of tuning propertiesof semiconductors devices, and by the need to have reasonable predictions. It isknown that many pseudobinary semiconductors allow variation of concentrationin quite broad limits, accompanied by substantial changes in lattice constantsand optical properties. By tuning the concentration of appropriately chosen con-stituents, one can hopefully adjust the lattice constant (to the desired substrate)and band gap (to desired optical absorption threshold).

Whereas the lattice constant often follows the Vegard’s law [Vegard, 1921] oflinear interpolation between the end components, the variation of the band gapis usually less predictable. Deviations from the linear trend are characterizedby band gap bowing, defined below in Sec. 3.3.2. The magnitude of the bandgap bowing vary between systems, so that even crude predictions may be useful.Concerning comparison with experiment, one notes that the band gap and bandgap bowing quite depend on the quality of sample and, for example, can bedifferent for bulk or thin films.

67

Chapter 3. First-principles study of gap bowing in InAs1−xPx alloys

Specifically, the system of study under discussion is the InAs1−xPx alloy, whichreceived a considerable attention since its energy gap varies at room temperature(R.T.) from 0.36 eV for InAs to 1.35 eV for InP, and hence covers the infrared(IR) spectrum. This makes the alloy suitable for emitters and detectors for the1.3 – 3.0 μm region of the spectrum [Kalvoda et al., 1997; Tsang, 1984]. Al-though a number of experimental and theoretical studies of the electronic struc-ture and some of related properties of the cationic InAs1−xPx semiconductor alloyhave been published [Adachi, 1987; Bouarissa, 2001; Huang and Wessels, 1988;Theodorou and Tsegas, 2000], no known data are available to date to the best ofour knowledge for basic electronic properties, i.e., the bandgap and gap bowingparameters.

3.2 Method and results

We employ the FP-LAPW method [Andersen, 1975] within the framework ofthe density functional theory (DFT) [Hohenberg and Kohn, 1964; Kohn andSham, 1965] as implemented in the WIEN2k code [Blaha et al., 2001] that hasbeen shown to yield reliable results for the electronic and structural properties ofvarious solids. The exchange-correlation potential for structural properties wascalculated using GGA after Perdew et al. [1996], while for electronic properties, inaddition, the Engel and Vosko GGA (EV-GGA, Engel and Vosko [1993]) schemewas applied, to enhance the prediction of the band gap.

In the muffin-tin spheres of radius RMT , the l-expansion of the non-sphericalpotential was carried out up to lmax=9, whereas the charge density was Fourier-expanded up to Gmax=14 Ry1/2. In order to achieve energy eigenvalue conver-gence, the wavefunctions in the interstitial region were expanded in PW with acutoff of Kmax = 9/RMT. Both the muffin-tin radius and the number of k-pointswere varied to ensure total energy convergence. The core states that are com-pletely confined inside the corresponding muffin-tin spheres were treated fullyrelativistically, while for the valence states that are rather non-localized we usedthe scalar-relativistic approach that explicitly includes the mass-velocity and Dar-win terms, but omits spin-obit coupling.

3.3 Results and discussion

3.3.1 Structural properties

Along with benchmark calculations done for binary end systems (in ZB structure),intermediate (25%; 75%) concentrations have been approximated by 8-atom su-percells, constructed, in their anion sublattice, as corresponding to Cu3Au struc-

68

3.3. Results and discussion

ture. The Cu3Au phase has no free parameters or internal coordinates; however,decorated with the cation sublattice, it makes the structure of luzonite. Thereis a free coordinate, fixing the placement of, say, indium inside the As3P (orAsP3) tetrahedron. For 50% concentration, the simplest ordered structure wouldbe that of the CuAu type in anionic sublattice; however, being an extremelyanisotropic one, it was rejected in favour of two times larger (hence 16-atom),chalcopyrite-type one. It has a free tetragonality parameter and an internal co-ordinate governing the placement of indium ions inside the As2P2 tetrahedra.Excused by utter simplicity of the then utilized alloy “model”, I neglected inthus early work deviations from perfect underlying ZB structure, and only variedlattice parameter, uniformly scaling the atomic positions.

The calculated total energies at many different volumes around equilibriumwere fitted by the Murnaghan equation of state [Murnaghan, 1944] in order toobtain the equilibrium lattice constant and the bulk modulus for the binary com-pounds and their alloy. Table 3.1 summarizes the results of our calculations andcompares them with other experimental and theoretical predictions. It is clearthat our calculated lattice constants using the GGA scheme are in reasonableagreement with experimental values.

The results obtained for the composition dependence of the calculated equi-librium lattice parameter for InAs1−xPx alloy are shown in Fig. 3.1, left panel. Asmall deviation from Vegard’s law [Vegard, 1921] with a marginal upward bow-ing of ∼0.02 A in the middle is clearly visible. However, violation of Vegard’s

Table 3.1: Calculated equilibrium lattice constant a and bulk modulus B forInAs, InP binary compounds in comparison with experimental data and previouscalculations, and original calculation results for InAs1−xPx alloys at intermediateconcentrations. Both previous calculations cited are done using PP method.

Lattice constant a (A) Bulk modulus B (GPa)

x Our work Exp. Other calc. Our work Exp. Other calc.

0 6.189 6.036a 5.921b 49.446 59d 61.7b

0.25 6.139 50.7570.5 6.081 52.8680.75 6.012 55.8521 5.964 5.861a 5.789c; 5.968e 59.958 74a 57.037e

aLin et al. [2009]; bWang and Ye [2002] (PW, LDA); cDebernardi [1998] (PW,LDA); dVohra et al. [1985]; eBrik et al. [2010] (PW, GGA).

69

Chapter 3. First-principles study of gap bowing in InAs1−xPx alloys

Figure 3.1: Composition dependence of the calculated lattice constant (leftpanel, filled squares) and calculated bulk modulus (right panel, filled squares)in InAs1−xPx alloy. Linear interpolation is shown by dotted line in both cases.

law has been both reported experimentally [Jobst et al., 1996] and theoretically[El Haj Hassan, 2005] in semiconductor alloys. The physical origin of this devi-ation should be mainly due to the small mismatches of the lattice constants ofInAs and InP compounds.

Fig. 3.1, right panel shows the bulk modulus as a function of x. A deviationof the bulk modulus from the linear concentration dependence with downwardbowing equal to 7.38 GPa was observed. The large value of the bulk modulusbowing for alloy roots in the significant mismatch of the bulk modulus of thebinary compounds.

3.3.2 Gap bowing and its origins

To compute band gaps of InAs, InP binary compounds, and their alloy InAs1−xPx

self consistently, GGA and EV-GGA are used within DFT. Calculated results fordirect band gaps are given in Table 3.2 ans Fig. 3.2. It is clearly seen that theband gap values obtained by EV-GGA are in reasonable agreement with exper-iment, while GGA gives zero band gap for x=0, 0.25 and 0.5, hence the GGAapproximation failed to predict the right band gap. In fact,the GGA functionalshave simple forms that are not sufficiently flexible to accurately reproduce bothexchange correlation energy and its charge derivative. Engel and Vosko, consid-ering this shortcoming, constructed a new functional form of GGA which is ableto better reproduce exchange potential at the expense of worse agreement in ex-change energy. This EV-GGA approach yields a better band splitting and someother properties which mainly depend on the accuracy of exchange-correlation

70

3.3. Results and discussion

Table 3.2: Direct band gap (in eV) in InAs1−xPx alloy at different concentrations.

Our work

x GGA EV-GGA Exp. Other calc.

0 0.000 0.000 0.360f 0.360h

0.25 0.000 0.180 0.544g 0.538h

0.5 0.000 0.448 0.773g 0.766h

0.75 0.179 0.736 1.042g 1.034h

1 0.413 1.003 1.350f 1.350h

fAdachi [1987] and references therein; gThomas [1997] (exp. data reported at300 K); hBouarissa [2001] (semiempirical).

potential. On the other hand, in this method, the quantities which depend onan accurate description of exchange energy Ex such as equilibrium volumes andbulk modulus are far from experiment [El Haj Hassan et al., 2006].

One speaks of band gap bowing if the variation of band gap, as function ofconcentration, between two end points x=0 and x=1 deviates from linear. Theband gap bowing is quantitatively characterized by a parameter b, measured in

Figure 3.2: Composition dependence of the calculated band gap, using GGA(solid squares) and EV-GGA (solid circles) for the InAs1−xPx alloy.

71

Chapter 3. First-principles study of gap bowing in InAs1−xPx alloys

the units of energy, as the band gap Eg itself, and which is defined as follows:

b = 2[Eg(x = 0) + Eg(x = 1)] − 4Eg(x = 1/2) , (3.1)

i.e., four times the deviation of the bad gap downwards from the linear interpola-tion. If the band gap nonlinearity is parabolic one, one sees that b exactly equalsthe coefficient at x2. The second-order polynomial fit to the band gap resultsshown in Fig. 3.2 is as follows:

EGGAg (x) = 0.0098 − 0.33717x + 0.73939x2 . (3.2)

EEV−GGAg (x) = −0.0113 + 0.80576x + 0.21942x2. (3.3)

An analysis suggested long ago by Bernard and Zunger [1986]; Wei et al. [1990]helps to get some insight into different contributions behind the gap bowing.According to its very definition, the gap bowing coefficient (b) is assumed to beindependent of composition x and is decomposed into three components. So tomeasure bowing b at x=0.5, the following conjecture is applied:

AB(aAB) + AC(aAC) → AB0.5C0.5(aeq) , (3.4)

where aAB and aAC are the equilibrium lattice constants of the binary compoundsAB and AC, respectively, and aeq is the alloy equilibrium lattice constant. We nowdecompose Eq. (3.4) into three components, corresponding to volume deformation(VD), charge-exchange (CE), and structural relaxation (SR):

VD : AB(aAB) + AC(aAC) → AB(a) + AC(a) ; (3.5)

CE : AB(a) + AC(a) → AB0.5C0.5(a) ; (3.6)

SR : AB0.5C0.5(a) → AB0.5C0.5(aeq) . (3.7)

The first step measures therefore the effect of volume deformation on the energygap bowing. The corresponding contribution to the bowing bVD, to the total gapbowing parameter represents relative response of the band structure of the binarycompounds AB and AC to hydrostatic pressure. This is due to the deviation oftheir individual equilibrium lattice constants from the alloy value a as wouldfollow from Vegard’s rule.

The second step, due to charge-exchange contribution bCE, reflects a charge-transfer effect that is due to the different (averaged) bonding behavior at thelattice constant a. The final step measures changes due to the structural relax-ation, in passing from the unrelaxed to the relaxed alloy by bSR. Consequently,the total gap bowing parameter is defined as:

b = bVD + bCE + bSR , (3.8)

bVD = 2 [εAB(aAB) − εAB(a) + εAC(aAC) − εAC(a)] , (3.9)

bCE = 2 [εAB(a) + εAC(a) − 2εABC(a)] , (3.10)

bSR = 4 [εABC(a) − εABC(aeq)] , (3.11)

72

3.4. Conclusion

Table 3.3: Decomposition of optical band gap bowing into volume deforma-tion (VD), charge exchange (CE) and structural relaxation (SR) contributionscompared with that obtained by a quadratic interpolation.

Procedure afterBernard and Zunger [1986] Quadratic equation

GGA EV-GGA GGA EV-GGA

bVD 0.644 −0.125bCE −1.497 0.291bSR 1.679 0.046b 0.826 0.212 0.739 0.219

where ε is the energy gap which has been calculated for the indicated atomicstructures and lattice constants. Energy gap terms in Eqs. (3.9), (3.10) and(3.11) are calculated separately with self-consistent band structure approach FP-LAPW. Different contributions to the direct gap bowing were calculated usingGGA and EV-GGA schemes and the results are given in Table 3.3.

3.4 Conclusion

In the course of studying electronic, structural and properties of InAs1−xPx

ternary alloy using the FP-LAPW method, a quite small deviation from theVegard’s law for the lattice constant was found, contrasting with a large devia-tion of bulk modulus from linear trend. The physical origin of this effect shouldbe mainly due to the significant mismatch between bulk modulus of InAs andInP compounds. It was found that the GGA approximation failed to predict theright gap bowing for InAs1−xPx alloy. The EV-GGA band gap exhibits nonlinearbehavior versus the composition x.

73

Chapter 4

Pressure-induced phase diagramof the II-VI semiconductorcompounds: ZnSe and ZnS

4.1 Introduction

Two compounds addressed in this chapter have ZB phase as their ground-state,and are known to transform, under sufficiently high pressure, into metallic RSphase. An interesting question is that about intermediate phases, of which namelycinnabar-type and SC16 are brought into discussion. Despite many calculationsdone by various methods, and a number by experiments by different groups,certain controversies existed, that was the motivation for the present study. Itsresults on the ZnSe system were published in a paper by Breidi et al. [2010].

For a long time it was thought that the existence of a low-enthalpy cinnabar-type phase at low and moderately high pressures was a characteristic of mercurychalcogenides exclusively – see, e.g., Sun and Dong [2005] and references therein.However, the last decade of the last century has seen the discovery of cinnabar-type phases in ZnTe [San-Miguel et al., 1993], CdTe [Nelmes et al., 1993], andGaAs [McMahon and Nelmes, 1997]. Lee and Ihm [1996] confirmed the stability ofthe cinnabar phase of ZnTe, using a first-principles PP-PW technique, whereasthat of CdTe has been confirmed by the full-potential linear muffin-tin orbitalcalculations of Ahuja et al. [1997]. However, Kelsey et al. and Mujica et al.from their respective PP-PW calculations concluded that GaAs [Kelsey et al.,1998; Mujica et al., 1998] and GaP [Mujica et al., 1998] might have cinnabar asa metastable phase only.

As concerns the SC16 structure, it has been tried in many early theoreticalcalculations and found by Crain et al. [1994b, 1995] to be a stable high-pressure

75

Chapter 4. Pressure-induced phase diagram of ZnSe and ZnS

phase in GaAs, AlSb and InAs. This was followed by calculations by Mujica etal. who showed SC16 to be an instable phase in Al-based [Mujica et al., 1995,1999; Serrano et al., 2000] and In-based [Mujica and Needs, 1997; Serrano et al.,2000] semiconductors and moreover in GaN [Serrano et al., 2000], but stable inGaAs [Mujica et al., 1995] and GaP [Mujica and Needs, 1997]. Interestingly, ithas been argued that the route to BC8, the monoatomic analog to SC16, in, say,silicon via the so-called R8 structure [Crain et al., 1994a] is such that its “gen-eralization” over binary semiconductors would involve the formation of “wrong”bonds between the like atomic species, and thus is probably forbidden. Never-theless, the SC16 phase has been observed experimentally in GaAs [McMahonet al., 1998] at high pressure. It was obtained by heating the high-pressure Cmcmphase to above ∼400 K at ∼14 GPa, and cooling back to room temperature – seeMcMahon et al. [1998]. Later on, McMahon et al. [2001] detected SC16 as an in-termediate phase between the ZB and the high-pressure Cmcm, on both increase(from about 15 GPa onwards) and decrease (from about 18 GPa downwards) ofpressure in GaAs heated to 400◦. Moreover, this structure has been detected insome I-VII compounds, namely CuCl and CuBr [Hull and Keen, 1994].

Cote et al. [1997] have studied the stability of the cinnabar and Cmcm phasesin ZnSe, ZnTe, CdSe and CdTe, using a PP-PW method, and predicted a sta-ble cinnabar phase to emerge in ZnSe, before reaching the RS structure. Theyreported the pressures “window” for the stability of the cinnabar phase to be3.2 GPa. Qteish and Munoz [2000] carried out another PP-PW calculation forZnSe, making use of the internal parameters of the cinnabar structure deter-mined by Cote et al. [1997], which are u=v=0.5. Surprisingly, they found thestable phase to be SC16, while the cinnabar being not able to win in any pressurerange over the direct ZB to RS transition. This is in an apparent disagreementwith Cote et al. [1997]. From the side of experiment, Pellicer-Porres et al. [2001]performed an energy-dispersive XRS on ZnSe at room temperature, and wereable to obtain a cinnabar phase similar to that observed in GaAs [McMahonand Nelmes, 1997] and ZnTe [McMahon and Nelmes, 1996a; Nelmes et al., 1994,1995]. The cinnabar phase was detected within a very small pressure interval10.4 – 9.9 GPa while slowly relieving the pressure from the RS phase.

Quite recently, a detection of the cinnabar phase was implied by Ovsyannikovet al. [2009] from their resistivity study of ZnSe (among other semiconductors)under varying pressure. While not accompanied by crystallographic characteri-zation, their identification of an “additional phase” ruled out its possibility to beSC16, on the basis of presumably predicted semimetallicity of the latter.1

1In fact, we failed to find in the citation given by Ovsyannikov et al. [2009] in supportof semimetallicity of ZnSe in the SC16 phase (their Ref. 31) any statements to this end, andanyway the calculation method used in the paper in question is not suited to providing anaccurate study of electronic structure in loosely packed semiconductor phases.

76

4.3. Methods and technical details

4.2 Methods and technical details

The calculations in relation with the present problem should were supposed toachieve an ultimate accuracy of the total energy for different phases, possibleat the DFT level for solids, that included also optimization of internal coordi-nates. The method of choice was, then, the FP-APW+lo method (Schwarz andBlaha [2003]; Schwarz et al. [2002]), implemented in the WIEN2K code [Blahaet al., 2001]. The method uses the “muffin-tin (MT) geometry”, separating thespace into atom-centered MT spheres and the interstitial. The XC energy ofelectrons is described both in the LDA and in the GGA using the functionalparametrization of Perdew–Burke–Ernzerhof [Perdew et al., 1996]. After neces-sary tests (to control the stability of energy differences between phases of ourinterest, the following values for crucial parameters of calculation have been ac-cepted: RMTKmax = 9, with RMT denoting the smallest MT radius and Kmax

the magnitude of the largest reciprocal space vector in the PW expansion. TheMT radii used in the calculations were 1.3 a.u. for Zn, and 1.65 a.u. for Se and1.50 a.u. for S – i.e., necessarily small to allow the scan, without overlappingthe MT spheres, of the range of volumes including those corresponding to quitehigh compression. The initial step of the radial mesh, a crucial parameter for theaccuracy of calculated forces, was set at 5·10−5 Bohr, acceptedly small enoughfor medium-weight atoms. The angular expansion of wavefunctions within thespheres was confined to lmax=10. The charge density in the interstitial region wasFourier expanded up to Gmax=20 Ry1/2. A mesh of 47, 58, 45 and 47 special k-points for ZB, cinnabar, SC16 and RS, respectively, were taken in the irreduciblewedge of the Brillouin zone for the total energy calculation. We emphasize thatconvergence tests for the PW cutoff and the number of k-points were essentialto assure reliable total energy differences. The variation of c/a parameter (in thecinnabar phase) was done “by hand” for each given volume, whereas the internalcorrdinates (for each given volume in SC16; for each volume and c/a in cinnabar)were optimized using the MINI script integrated into the WIEN2k package, whichtakes into account the forces acting on atoms [Kohler et al., 1996; Madsen et al.,2001; Schwarz and Blaha, 2003; Yu et al., 1991]. The space group was imposedand kept fixed in WIEN2k throughout the relaxation of each particular system.In this way, a spontaneous lowering of symmetry from a high-symmetric phasewas not possible, but we’ll see that, other way around, the cinnabar phase atcertain conditions finds a way to spontaneously arrange itself in a high-symmetryphase.

77

Chapter 4. Pressure-induced phase diagram of ZnSe and ZnS

4.3 ZnSe: preliminary considerations and total-

energy results

Two of considered phases, cinnabar and SC16, are characterized by internalcoordinates of atoms which may vary under pressure. The hexagonal cinnabarphase (B9), moreover, allows a variation of the c/a ratio. The space group of thisphase is either P3221 (Nr 154) or P3121 (Nr 152), the two being enantiomorphic.The Wyckoff positions (3a), say for cation and (3b), say for anion are characterizedby u and v internal coordinates, correspondingly. u = v = 1

3or 2

3, with c/a=

√6,

recovers the RS phase. A continuous transformation exists also between thecinnabar and the RS phase. The relation between the three has been describedby Sowa [2003, 2005]. What is of importance in the present context is that theZB phase can undergo, at least hypothetically, a transformation into RS on twodifferent ways, the one passing by cinnabar and the other not – see details inAppendix A.

The cubic phase SC16 has the space group Pa3 (Nr 205), in which both thecations and the anions are in the (8c) positions, each one characterized by a singleinternal coordinate. While trying the effect of pressure and varying the volume,we performed the relaxation of all the free parameters involved – c/a, u(Zn) andv(Se) for cinnabar, and the cation and anion internal coordinates in the SC16.

The variation of the total energy with volume for all considered structuresis presented in Fig. 4.1. As it is generally known that LDA and GGA do notalways arrive at (even qualitatively) consistent conclusions, the comparison isgiven for the results obtained with both these approximations for the XC. Themost marked difference is, as could be expected, in smaller equilibrium volumeper atom (145.6 Bohr3, for ZB) in LDA than in GGA (160.8 Bohr3), which bothvalues bracket the experimental value (153.8 Bohr3), and the resulting overall shiftof the whole system of energy-volume curves. The differences in the curves’ shapeare noticeable (in favor of generally “softer” GGA prediction), but not excessive:the bulk moduli of RS/cinnabar phases (89/79 GPa in LDA, 68/61 GPa in GGA)stay markedly higher than in ZB/SC16 (71/69 GPa in LDA, 55/48 GPa in GGA).With both XC schemes, the SC16 phase is slightly but definitely lower in energythan the cinnabar one, notably in the region of interest between the stabilitydomains of ZB and RS phases. The difference is that, according to the GGAcalculation, the two intermediate phases – cinnabar and SC16 – protrude moredownwards between the RS and ZB than in the LDA case. When transposed intothe Gibbs free energy vs. pressure diagram (Fig. 4.2), this yields a small stabilityregion of both SC16 and cinnabar phases underway between ZB to RS after theGGA calculation, but the presumed stability of SC16 only, and not of cinnabar,according to LDA.

78

4.3. ZnSe: preliminary considerations and total-energy results

100 120 140 160Volume per atom (a.u.3)

-0.58

-0.56

-0.54

-0.52

Tot

al e

nerg

y pe

r at

om +

4220

(R

y) ZnSe: LDA

100 120 140 160 180Volume per atom (a.u.3)

-0.4

-0.38

-0.36

-0.34

Tot

al e

nerg

y pe

r at

om +

4226

(R

y)ZnSe: GGA

zincblendeSC16cinnabarrocksalt

Figure 4.1: Total energy vs. volume for ZnSe in different phases, as calculatedwithin the LDA (left panel) and GGA (right panel). Continuous lines show fit tothe Murnaghan equation of state for each phase. In cinnabar and SC16 phases,the internal coordinates have been relaxed in each point.

10 12 14 16Pressure (GPa)

-4

-2

0

2

4

Ent

halp

y pe

r at

om r

elat

ive

to B

3 (m

Ry)

10 12 14 16 18Pressure (GPa)

-4

-2

0

2

4

SC16cinnabarB1B3

LDA GGA

ZnSe

Figure 4.2: Static enthalpy per atom in different phases of ZnSe, relative to theZB phase, as calculated with the LDA (left panel) and GGA (right panel).

79

Chapter 4. Pressure-induced phase diagram of ZnSe and ZnS

The spread of LDA/GGA prediction, in view of how they often “bracket”the reality, gives a kind of “tolerance margin” for the present and other ab initioresults. It can be implied, however, on the basis of some experimental evidenceaddressed below, that the GGA provides somehow more adequate description.

Let’s now look at earlier ab initio calculations in this context. In what con-cerns the stability of the SC16 phase, Qteish and Munoz [2000] reported that ittakes over ZB starting from 9.2 GPa and has a stability range of ΔP=7.2 GPa;in the present LDA calculation, the pressure range of SC16 is much more narrow,2.5 GPa, starting from 10.8 GPa. Remarkably, the present GGA prediction ofthe SC16 stability is not that different – it goes over ΔP=4.3 GPa, setting onat 11.8 GPa. So at least qualitative agreement should be pointed out betweendifferent calculation schemes in what concerns the presumed placement of thisphase on the pressure-driven phase diagram.

The issue of stability of the cinnabar phase is more subtle. The LDA calcu-lation by Cote et al. [1997], predicting a stability region for the cinnabar phasebetween ZB and RS, is at variance with LDA calculations by Qteish and Munoz[2000] as well as with the present one (according to which both the cinnabar re-mains thermodynamically “hidden” in the ZB to RS transition), and hence seemsfortuitous. An emergence of cinnabar as a metastable phase seems to be a qual-itatively new result delivered with the GGA. Interestingly, our predicted regionof metastability for the cinnabar phase, albeit narrow (ΔP=0.55 GPa, between13.35 and 13.9 GPa), falls in remarkable agreement with experimental results(0.5 GPa) of Pellicer-Porres et al. [2001].

Turning to experiment, I note that the data are not decisive; at least, tobest of my knowledge, no experimental study that would combine the effects ofhigh pressure and high temperature, in analogy with the above case of GaAs[McMahon and Nelmes, 1997], has yet been done.

Pellicer-Porres et al. [2001] performed a careful search, under pressure at roomtemperature, for an intermediate phase of Te-doped ZnSe between the ZB andRS, but none was found in the upstroke. This is at variance with the resultsof Kobayashi [2001] who claim the existence of a new phase between the ZBand RS, observed in the upstroke, however without giving any structural char-acterization. Moreover, the “additional phase” referred to by Ovsyannikov et al.[2009] as cinnabar, albeit without structural characterization, appears on bothupstroke and downstroke, from about 14 GPa onwards and 9 GPa downwards,correspondingly, the existency range in each case being (quite smeared) around4 GPa.

In the downstroke, Pellicer-Porres et al. [2001] managed to obtain the diffrac-tion pattern of the new high-pressure phase, best pronounced in the spectraof samples with the highest Te content, ZnSe0.8Te0.2. The observed existencePellicer-Porres et al. [2001] of this phase is from 10.4 down to 9.9 GPa. The anal-

80

4.3. ZnSe: preliminary considerations and total-energy results

ysis of the diffraction pattern revealed the intermediate phase to be of hexagonalsymmetry, without further elaborating. By analogy with ZnTe [McMahon andNelmes, 1996a; Nelmes et al., 1994, 1995], however, a suggestion has been madeabout the phase in question being cinnabar. Pellicer-Porres et al. [2001] stressedthat, in the composition range of Zn(Se,Te) alloys studied, the cinnabar range ofexistence diminishes as the Te content is reduced.

Assuming this hypothesis demands to explain two peculiarities: i) why thecinnabar phase appears in the downstroke more pronouncedly than in the up-stroke, and ii) why does the computationally favorable, according to all calcula-tions, SC16 phase escape experimental detection. A plauisble opinion, formulatedin [Breidi et al., 2010], on the former issue is that the ZB–RS conversion mightnot be necessarily reversible. As mentioned above, the ZB structure might bedriven to RS, namely by a non-trivial twist in each xy plane, accompanied byslight opposite z-displacements of consecutive atomic planes (see Sowa [2003]for details). This scenario roughly maintains the coordination of each atom, untilsuddenly each one gets six nearest neighbours of the opposite kind, at the place offour. Apparently, no experimental evidence of such transformation has yet beenreported. Important is that this transformation does not pass via the cinnabarstructure. At the same time, a different structural transformations exist whichrelate RS structure to cinnabar, and then cinnabar to ZB. In other words, thethree phases ZB, cinnabar and RS are connected by a “triangular” path (whichis moreover almost “equilateral”, judging by average atom displacements in eachtransformation). An important assumption is that for whatever reason, whichmight be clarified from modeling the kinetics of the processes, the ZB to RStransformation (in the upstroke) chooses the direct way, whereas the back trans-formation (in the downstroke) passes, at least in part of the sample, via thecinnabar.1 The second peculiarity, that of experimental absence (at least in roomtemperature measurements) of computationally predicted SC16 phase could beso understood that the formation of SC16 is kinetically hindered, presumably dueto high energy barriers [Crain et al., 1994b, 1995].

As was suggested by Crain et al. [1994b, 1995], SC16 may be formed under con-ditions of high pressure and temperature and could then persist as a metastablestate to ambient pressure in analogy to the case of SC16-GaAs [McMahon et al.,1998] where SC16 phase appears upon combining the role of high pressure andtemperature.

It is worth noting that the present calculations, as many of those done before,do only probe the zero-temperature energy relations between different phases,whereas molecular dynamics or other calculations involving high temperatures

1In Appendix A, we discuss the relations between the three crystal structures and possiblepaths between them.

81

Chapter 4. Pressure-induced phase diagram of ZnSe and ZnS

might be useful to shed the light on the mechanisms of how barriers betweendifferent phases [Blanco et al., 2000] are overcome.

In the following, the discussion is given of how do lattice parameters changewith pressure, according to our calculations. Apart from the values of pressureat which the phase transition takes place, the trends are almost the same in LDAand GGA calculations, therefore only the latter ones are covered.

4.3.1 Cinnabar structure

Note for the following that the cinnabar structure is described by the P3121space group, and remind that we assume Zn in the (3a) positions with internalcoordinate u and Se in the (3b) with internal coordinate v. A schematic view ofnearest neighbourhood to cation and anion sites, for realistic (i.e., between 1

3and

12) but exaggeratedly different u and v values, is shown in Fig. 4.3. The “genuine”

internal coordinates, as optimized from total energy calculations for a range ofpressures, are depicted in Fig. 4.4, and the c, a parameters (along with their ratio)in Fig. 4.5. At low pressures, c/a increases markedly over-linearly, that conteststhe validity of previous calculations [Qteish and Munoz, 2000] which found c/ato change sub-linearly throughout the cinnabar phase. Eventually at 18 GPa itjumps to

√6 while u and v set to 1

3, the values corresponding to the RS phase.

Apparently at this jump the c parameter grows with pressure (see Fig. 4.5, upper

032

32

61

61

3

3

12

56

65

v u

vu1

1

cationanion

Figure 4.3: Scheme of distorted tetrahedric coordinations around anion andcation sites in the cinnabar structure, in the projection onto the (x, y) plane.Numbers in the circles indicate z-coordinates of corresponding atoms.

82

4.3. ZnSe: preliminary considerations and total-energy results

-10 0 10 20 30 40 50Pressure (GPa)

0.3

0.35

0.4

0.45

0.5

inte

rnal

coo

rdin

ates

u,v

v (Se)u (Zn)

3

13 3

2

32

2 12

31

13

anion

Figure 4.4: Internal coordinates in the cinnabar phase of ZnSe as function ofpressure. Note the collapse into the RS phase at 18 GPa. The crystal structure(around an anion site, as in the left panel of Fig. 4.3) corresponding to thissituation is shown in the inset.

panel). Cote et al. [1997] computed c/a to increase smoothly without exhibitingany discontinuity during the whole range of pressure, apparently due to the factthat they fix u and v to constants throughout the pressure range, thus preventinga transition into the RS phase.

We note that ZnSe remains semiconducting in the cinnabar phase, until be-coming a metal upon a transition into RS (see Fig. 4.6). The band gap nearlylogarithmically increases with pressure (as was studied over many zinc-blende bi-nary semiconductors by Wei and Zunger [1999]), from 0.5 eV at zero pressure to1.25 eV at 15 Gpa. We remind that our calculation scheme involved a straightfor-ward GGA functional, without special provisions to increase, or otherwise tune,the band gap. The electronic structure of cinnabar-ZnSe under pressure is shownin Fig. 4.7. The LDA calculation, done at the comparable volume (Fig. 4.7, lowerpanel), yields a very similar electronic structure: the Se4s and Zn3d bands shiftupwards by 0.35 and 0.18 eV, correspondingly, and the band gap squeezes by onethird, whereas the band dispersions remain indistinguishable.

It is obvious from Fig. 4.3 that the four cation-anion bond lengths split intotwo distinct, for u =v, pairs. At 13.35 GPa these are two of 2.416 A and two of2.384 A. This can be compared with four bonds lengths of 2.356 A in ZB ZnSe at

83

Chapter 4. Pressure-induced phase diagram of ZnSe and ZnS

0 10 20 30 40 50 60 70Pressure (GPa)

2.2

2.3

2.4

2.5

c/a

0 10 20 30 40 50 60 70

8.5

9

9.5

c(Å

)

c

3.5

4

4.5

a(Å

)

a

6

Figure 4.5: Variationof a, c lattice para-meters in the cinnabarphase of ZnSe as func-tion of pressure. Theideal c/a =

√6 value

indicated is for the RSstructure representedas hexagonal one.

the same pressure, indicating a certain relieve of strain on escaping from the ZBgeometry. On the other end of our calculated “cinnabar window” (see Fig. 4.2),at 13.90 GPa, the bond lengths in ZnSe-cinnabar are two at 2.412 A and two at2.384 A, to be compared to six bonds of 2.561 A in ZnSe-RS at the same pressure– again a relieve of stress at the expense of acquiring a higher coordination.

The six bond angles within a given twisted tetrahedron of the cinnabar struc-ture (Fig. 4.3) split as 2+2+1+1, whereby Zn-Se-Zn and Se-Zn-Se angles are(slightly1) different. Fig. 4.8 shows variation of bond angles and bond lengthswith pressure. Due to a small but noticeable difference between u and v (seeFig. 4.4), the bond lengths remain split by about 0.04 A throughout the rangeof pressures from zero to the collapse of the cinnabar phase (at 18 GPa), ex-periencing underway a gradual contraction by ∼0.2 A. On collapse into the RSstructure, the bond lengths become equal and roughly recover their initial (larger)magnitude at zero pressure, as the structure packing becomes more dense. Onfurther increase of pressure, the bond lengths decrease with pressure at a smaller

1One angle pair is exactly the same for cation-anion-cation and anion-cation-anion groups.

84

4.3. ZnSe: preliminary considerations and total-energy results

Figure 4.6: Band gap evolution with pressure in four ZnSe phases (GGA).

slope than in the cinnabar phase, preserving the constance of bond angles (90and 180◦).

Each of four inequivalent bond angles undergoes a smooth yet non-trivialvariation within about 2◦ throughout the interval of pressures before the collapse.The doublet angles are the lowest ones, close to 94◦ and 104◦. On the collapse toRS, they drop to 90◦, followed by the smallest (∼128◦) of two singlet angles. Thelargest bond angle (∼138◦), on the collapse, opens to 180◦. Simultaneously, twopreviously more distant atoms (from the adjacent unit cells) approach the atomin the middle of a twisted tetrahedron, completing a neighbourhood of a givenatom to an octahedron – see inset in Fig. 4.4.

These variations of bond lengths and angles can be expressed in terms offour independent parameters – a, c, u and v, – as explained in Appendix B. Theircombined adjustment is essential to allow the structure to gradually accommodateat high pressure. It is seen from Fig. 4.4 that, as the variation of u, v underpressure is not large (prior to the collapse into RS), it is, at least, different for Znand Se. The internal parameters describing both Zn and Se positions remain, atlow pressures, close to 1

2. Even if this value is not set by symmetry, it was used as

fixed in some previous calculations, irrespectively of pressure [Cote et al., 1997;Qteish and Munoz, 2000]. We think that intentionally imposed constraints, or

85

Chapter 4. Pressure-induced phase diagram of ZnSe and ZnS

Γ M K Γ A

-15

-10

-5

0

5

Ene

rgy

(eV

)

ZnSe: cinnabar at 13.74 GPa (GGA)

10 20states/eV

Figure 4.7: Band structure and total density of states as calculated in thecinnabar phase of ZnSe within the GGA (upper panel, volume per atom123.12 Bohr3), and LDA (lower panel). Both calculations are done at roughlythe same volume, corresponding to the stability of cinnabar phase in GGA.

86

4.3. ZnSe: preliminary considerations and total-energy results

0 10 20 30 40 50 60 70Pressure (GPa)

90

100

110

120

130

140

150

160

170

180B

ond

angl

es (

degr

ees)

Zn-Se-ZnSe-Zn-Se

0 10 20 30 40 50 60 70Pressure (GPa)

2.4

2.5

2.6

Bon

d le

ngth

(Å

)

ZnSe (B9)

Figure 4.8: Bond lengths and bond angles as calculated under different pres-sures in the cinnabar structure of ZnSe. The opened circles and crosses symbolscorrespond to two distinct length values of the all-four bonds.

insufficient precision, in some earlier calculations were responsible for inaccurateresults.

Cote et al. [1997] reported their u and v to yield the energy minimum atu=v=0.5 throughout the whole applied pressure range, which implies that thePP method underestimated the difference between the anionic and cationic freeparameters. Pellicer-Porres et al. [2001] simulated the diffraction pattern corre-sponding to their observed (presumably cinnabar) structure by accepting u=v=0.5,that gave a qualitative agreement with the experimental diffraction intensities.In an alternative try, they [Pellicer-Porres et al., 2001] assumed the evolution ofnearest-neighbor distances in ZnSe throughout the phase transitions to be likein HgTe [San-Miguel et al., 1995] and CdTe [McMahon et al., 1993], i.e., almostcontinuously on both ZB–cinabar and cinnabar–RS transitions. This demandedthe internal parameters to be u=0.63 and v=0.55 (to compare with u∼0.64 andv∼0.56 for CdTe and HgTe [McMahon et al., 1993; San-Miguel et al., 1995]. Set-ting these values in the diffraction pattern simulation has not provided intensitieswhich would agree with the experimental spectra.

87

Chapter 4. Pressure-induced phase diagram of ZnSe and ZnS

Pellicer-Porres et al. [2001] concluded therefore that their presumably cinnabarstructure of ZnSe must be close to that associated with GaAs [McMahon andNelmes, 1997] and ZnTe [McMahon and Nelmes, 1996a; Nelmes et al., 1994, 1995],i.e. having both internal parameters close to 0.5. The difference from the othersituation tested is not simply numerical, because the two inspected distinct pos-sibilities to arrange (u, v) give rise either to “2 close + 4 distant”, or to “4 close+ 2 distant” neighbors coordinations [McMahon et al., 1993; Wright et al., 1993].The review by Mujica et al. [2003] well explains the varieties of cinnabar phases.

The result by Pellicer-Porres et al. [2001] concerning the lattice parameters oftheir hexagonal phase is a=3.829 A and c=8.996 A, at 10.4 GPa. This compareswell to what we have calculated, a=3.777 A and c=8.840 A, at 13.90 GPa – thefirst theoretical appearance of cinnabar coming from RS. The experimental c/aaxial ratio Pellicer-Porres et al. [2001] falls in the interval 2.31–2.35 in all thestudied samples, which is in good agreement with our calculations: we foundc/a to vary within 2.335–2.340 in our regime of stability of cinnabar between13.35–13.90 GPa, whereas the reported c/a ratio from PP calculations [Coteet al., 1997] is ∼2.26, too much off. As in ZnTe and CdTe [Cote et al., 1997],the PP method slightly underestimates the c/a ratio. We have computed thevolume collapse at the ZB–cinnabar and the cinnabar–RS phase transitions to be9.03% and 7.57%, respectively, in good qualitative agreement with experimentalindications [Pellicer-Porres et al., 2001] (9.8% and 6.3%, respectively).

It should be mentioned that the phase transition from cinnabar to RS phase inHgS has been simulated by Sun and Dong [2005], who used the same calculationmethod as we do in the present work. They traced the variations of all latticeparameters, as we do now. The difference is that the native cinnabar structure ofHgS is characterized by very different values of u and v (∼0.5 for cation, ∼0.75 foranion – see the above discussion), which converge to 2/3 on the transition. Thevariation of c/a with pressure is sub-linear in HgS and merges into the

√6 value

without singularity; the bond lengths have no discontinuity. In total, the citedstudy confirms the experimental indications for the second-order phase transitionin HgS, in contrast to presumably first-order character in ZnSe.

4.3.2 SC16 structure

The SC16 structure also has one internal parameter to define cation position andone for anions; we retain for them the notations u and v, although their meaningis of course quite different from that in the cinnabar structure (see Appendix B).The space group is now Pa3 (Nr 205), and the sites in question are (8c). Sincethere is an ambiguity in defining the internal coordinate so as to generate the sameset of eight equivalent positions, we specify that the cation at (u u u) and theanion at (v v v) are first neighbours, whose connecting bond length is a(v−u)

√3

88

4.3. ZnSe: preliminary considerations and total-energy results

uv

Figure 4.9: A side view of the cubicprimitive cell of the SC16 structure un-derlying its trigonal symmetry and in-dicating the definition of internal coor-dinates u (for cations) and v (for aions).

(Fig. 4.9). Differently from the twist as in the cinnabar phase, the “elementary”tetrahedron undergoes a pyramidal distortion, in which a singular cation-anionbond (from the central atom to the pyramid’s summit, along the spatial diagonalof the cube in Fig. 4.9) differs from three equal bonds to the ions in the pyramid’sbasal plane. Correspondingly, six bond angles in a given tetrahedron split intothree involving the singular bond and three others, excluding it. Finally, a pairof cation-anion-cation bond angles is, in general, different from the anion-cation-anion pair.

The lattice parameter and internal coordinates optimized in the SC16 phaseof ZnSe as functions of pressure are shown in Fig. 4.10, and corresponding bondlengths and bond angles – in Fig. 4.11. The relation between them is explainedin Appendix B. Throughout the range of pressures studied, the system remainssemiconducting, although the band gap variation under pressure differs from thatin the cinnabar phase. Starting from 0.85 eV at zero pressure, the band gap (seeFig. 4.6) rises to 1.04 eV at 14 GPa, from where a less steep descent brings itsback to 0.83 eV at 45 GPa. Qualitatively, the variation of the band gap resemblesthat of the v(Se) parameter in Fig. 4.10.

We see that the distance from a cation to its second-neighbor (not bonded)counterpart, i.e. between the atoms at (u u u) and (u u u), is gradually decreasingwith pressure, whereas the corresponding anion-anion distance (across the centerof the cubic cell in Fig. 4.9), although being roughly of the same value, stronglyresists compression. Till about 12 GPa, the relative anion-cation separation (v−u)

89

Chapter 4. Pressure-induced phase diagram of ZnSe and ZnS

0 10 20 30 40

0.14

0.15

0.16

u(Z

n)

0 10 20 30 40Pressure (GPa)

0.35

0.36

v(S

e)

0 10 20 30 40Pressure (GPa)

6.5

7

latti

ce p

aram

eter

(Å

)

ZnSe (SC16)

Figure 4.10: In-ternal coordinatesu, v (left panels)and lattice param-eter (right panel)in the SC16 phaseof ZnSe as opti-mized at differentapplied pressures.

in fact grows with pressure, that is however (over)compensated by the volumiccompression, so that the both bond species get shortened (at different rate; thetriplet bond compresses faster). Near 12 GPa, the change of behavior occurs, as

0 10 20 30 40Pressure (GPa)

95

100

105

110

115

120

Bon

d an

gles

(de

gree

s)

Zn-Se-ZnSe-Zn-Se

0 10 20 30 40Pressure (GPa)

2.3

2.4

2.5

2.6

Bon

d le

ngth

(Å

)

singlet bondtriplet bond

ZnSe (SC16)

Figure 4.11: Bond lengths and bondangles in the SC16 phase of ZnSeoptimized at different applied pres-sures. Open circles mark degeneratelength values of three bonds, filledcircles – unique value of the fourthbond to the central atom in cationicor anionic tetrahedron.

90

4.3. ZnSe: preliminary considerations and total-energy results

the two bond length values cross:1 from here on, the anion-centered tetrahedrabecome “rigid”, as the bond angles get stabilized at about 97◦ and 118◦. Onthe contrary, the cation-centered tetrahedra do smoothly “flow” with pressure,the cation being gradually pressed into the basal plane of the pyramide: the Zn-centered bond angles go away towards ∼90◦ and ∼120◦, the single bond contractsonly slightly whereas the “planar” triple bonds get considerably shorter.

We traced the structure modifications in the SC16 phase deep into the regionof pressures where the RS phase must definitely win (according to calculations).At the low-pressure end of the “SC16 window” (Fig. 4.2), at 11.8 GPa, we inter-polate the triple bonds to be 2.402 A and the single one 2.383 A. This compareswell with four bonds of 2.367 A in ZnSe-ZB at the same pressure. On the oppositeend, at 16.1 GPa, we find triple bonds of 2.367 A and the single one of 2.378 A,to be compared to six bonds of 2.550 A in the ZnSe-RS at the same pressure.The calculated volume collapse on ZB–SC16 and SC16–RS phase transitions are8.7% and 7.56%, respectively. The experimental data to compare with are, asmentioned above, so far not available.

4.3.3 Open questions

An important observation from the present study is that when considering thecinnabar phase as intermediate one between the ZB and RS, the optimization ofinternal coordinates u, v along with the c, a lattice parameters is essential forgetting, at the same time, all the following properties right: (i) a preference inenergy over both ZB and RS in a certain window of pressures; (ii) the widthof this window; (iii) the ultimate collapse, at a sufficiently high pressure, of thecinnabar phase into the RS. An unresolved question remains, why is the SC16phase, apparently (i.e. consistently over results of many calculations) havinglower enthalpy than the B9, not observed experimentally. In order to shed alight on this issue, the study of the energy barrier connecting ZB to SC16, incomparison with that from ZB to cinnabar, has to be done. Such simulationwould already be quite helpful when done in a static model, assuming a plausiblepath for the structure transformation. However, dynamical aspects may also playan important role. So far, the argumentation concerning the absence of the SC16phase in experiment included that “the formation of SC16 is kinetically hindered,which is extremely likely given the high energy of the intermediate R16 structure”[Kelsey et al., 1998]. Moreover, as suggested by Crain et al. [1994b, 1995], SC16may be formed under conditions of high pressure and temperature and could thenpersist as a metastable state to ambient pressure. An analogy is known for GaAs,where the SC16 phase was detected to appear under a combined effect of high

1A similar crossing was reported by Crain et al. [1994b] in their simulation of SC16-GaAs.

91

Chapter 4. Pressure-induced phase diagram of ZnSe and ZnS

pressure and temperature [McMahon et al., 1998].

4.3.4 Conclusion on ZnSe

In [Breidi et al., 2010], it was found that the cinnabar phase, a fourfold-coordinatedstructure formed from twisted tetrahedra, could exist as a high pressure thermo-dynamic metastable phase between the semiconducting ZB and the RS metallicphases. The observation of cinnabar phase in the downstroke reveals that a bigenergy barrier exists between RS and the ZB and SC16 phases. A suspicion isstrong that upon pressure increase from the ZB phase, transition to covalentlybonded fourfolded-coordinates are inhibited by big energy barriers at room tem-perature. While upon pressure decrease from the RS phase, a transition to SC16remains hindered, a transition to the cinnabar phase remains possible, becausethere is a low energy path between RS and cinnabar. There are striking similar-ities in the calculated high pressure phase diagrams of ZnSe and GaAs [Kelseyet al., 1998; Mujica et al., 1998]. For both systems, the cinnabar occurs as alow-pressure metastable phase, and were found experimentally, at room tempera-ture, only upon release of pressure from the metallic high-pressure RS (for ZnSe:Pellicer-Porres et al. [2001]) and Cmcm (for GaAs: McMahon and Nelmes [1997])phases; also u∼v∼0.5 gave an excellent fit to the diffraction pattern in both cases.The striking similarity in behavior between ZnSe and GaAs adds to the develop-ing links between the II-VI and the III-V semiconductors (McMahon and Nelmes[1996b]). It is of high interest to investigate the relationship further by lookingfor the possibility to arrive at SC16 of ZnSe along a similar path as was discussedfor GaAs [McMahon et al., 2001; McMahon et al., 1998].

4.4 Zinc Sulfide

It is well known that under applied pressure, ZnS, as many other binary semi-conductors, departs from its ground-state ZB phase and eventually arrives intothe RS metallic phase. Along with the ZB, the WZ phase is competitive asthe ground-state one (it was shown to be metastable at ambient conditions).As “usual suspects” for intermediate high-pressure phases underway from ZB toRRS, cubic SC16 and hexagonal cinnabar-type are typically brought into discus-sion, again as a common observation with a number of other semiconductors. Amotivation for the present study was a hope to shed light on some existing contro-versies between earlier calculations and experiment. Moreover, certain predictedtrends for ZnS seem to be unexpectedly different from those for ZnSe, a prioriquite similar system. The present contribution is hoped to be useful for under-standing strained semiconductor systems, notably the ways to force them into a

92

4.4. Zinc Sulfide

desired crystal structure.

4.4.1 Experimental and theoretical evidence;relation to ZnSe and other materials

Smith and Martin [1965] observed a ZB–RS transition at 11.7 GPa; later studiesfound that the transition takes place at higher pressure – 14.5(5) GPa [Nelmesand McMahon, 1998], or 15.5 Gpa [Uchino et al., 1999]. Structural, optical, andelectronic properties of cubic ZnS were studied by Ves et al. [1990] and Zhou et al.[1991] where the RS phase was found to be an indirect-bandgap semiconductor.This was, later on, confirmed by ab initio calculations [Jaffe et al., 1993].

The WZ-type ZnS has been reported to transform to the ZB structure priorto a transition at higher pressures into RS [Desgreniers et al., 2000]. The effectof the decrease of grain size on the increase of the onset of transition has beenstudied by Jiang et al. [1999] and Qadri et al. [2001].

At even higher pressures, the RS phase was reported to transform into theorthorhombic Cmcm one, without a change in volume. The transition pressurewas estimated at ∼65 GPa by Nelmes and McMahon [1998]. Desgreniers et al.[2000] reported ∼65 GPa as transition pressure from RS-ZnS to Cmcm-ZnS withno significant change in volume. Lopez-Solano et al. [2003] confirmed the RS-Cmcm transition at 65 GPa in their calculation using the PW-PP method.Whereas III-V and II-VI binary compounds are since long well studied, the lastdecade brought about a breakthrough in systematization of their phase transitionsunder pressure. New low-symmetry phases were shown to exist, according to abinitio calculations, between ZB and RS, on one side, and Cmcm, on the other side– see, e.g., Mujica et al. [2003]. McMahon et al. [1998] and McMahon et al. [2001]carried a high temperature / high pressure experiment on the III-V thoroughlyriched material GaAs, where they observed the SC16 as intermediate stable phasebetween ZB and the high pressure Cmcm phase, both on pressure increase fromabout 15 GPa onwards and decrease from about 18 GPa downwards. McMahonet al. [1998] concluded that the the more ionic bonding in the II-VI systemsshould facilitate the formation of SC16. Interestingly, Qteish and Parrinello [2000]carried out a PW-PP calculation on the II-VI ZnS semiconductor and found theSC16 phase to be a stable intermediate one between the (semiconducting) ZB andthe (metallic) RS. Contrary to predictions of many PP calculations, notably forAlSb, InAs [Crain et al., 1994b, 1995], GaP [Mujica and Needs, 1997], ZnS [Qteishand Parrinello, 2000] and ZnSe [Qteish and Munoz, 2000], the SC16 phase hasonly been observed in the GaAs semiconductor [McMahon et al., 2001; McMahonet al., 1998].

Regarding cinnabar, its stability at ambient pressure in mercury chalcogenides

93

Chapter 4. Pressure-induced phase diagram of ZnSe and ZnS

HgO [Aurivillius and Carlsson, 1958], HgS [Aurivillius and Malmros, 1961], and atvery low “high pressure” HgSe [Kafalas et al., 1962], HgTe [Mariano and Warekois,1963] was experimentaly observed since long. However, the last decade has shownthat other binary compounds adopt cinnabar as intermediate stable phase atmoderate and high pressures: CdTe [McMahon et al., 1993; Nelmes et al., 1993];ZnTe [McMahon and Nelmes, 1996a; Nelmes et al., 1995; San-Miguel et al., 1993];GaAs [McMahon and Nelmes, 1997], and finally ZnSe [Pellicer-Porres et al., 2001].[Lee and Ihm, 1996] confirmed the stability of the cinnabar phase in ZnTe, usingPP-PW technique, whereas that in CdTe has been confirmed by full-potentialLinear muffin-tin orital calculations of Ahuja et al. [1997]. However, Kelsey et al.[1998] and Mujica et al. [1998] have shown from their PP-PW calculations thatGaAs might have cinnabar as a metastable phase only. Nazzal and Qteish [1996]have calculated the existence of the stability for a ZnS-cinnabar phase, interme-diate between the ZB and the RS phases, but this was not confirmed by latercalculations by Qteish et al. [1998]. However, a rigorous search was made by Des-greniers et al. [2000] along the 300 K-isotherm, between 11.4 and 14.5 GPa, tolook for indication of the ZnS-cinnabar phase, with negative results. The lackedappearance, to the best of our knowledge, of the SC16 phase in ZnS, despite itstheoretical prediction [Qteish and Parrinello, 2000] as stable intermediate phasebetween ZB and RS, motivated me to reconsider its stability issue. As with ZnSe,the calculations were done with both LDA and GGA, and the same four crys-talline phases taken into account. In principle, the calculation is a pendant toearlier study on ZnSe, but its results are not yet published.

4.4.2 Calculation results

As for ZnSe, the optimization of the c/a and minimization of the cation and anioninternal parameters in cinnabar and SC16 structures, was done throughout fordifferent values of hydrostatic pressure.

The total energies of the four different phases considered, calculated with bothXC schemes at T=0, are depicted in Fig. 4.12; the “Gibbs energy versus pres-sure” diagram is shown in Fig. 4.13. The left panel (LDA calculations) of thisfigure shows that the SC16 phase is definitley low in enthalpy and, consequently,is thermodynamically stable. My finding that SC16 is thermodynamically sta-ble in LDA is in agreement with that of Qteish and Parrinello [2000]. I foundmoreover that ZB is unstable against the SC16 phase at 14.45 GPa and that thestability range of the latter phase is 1.65 GPa. Qteish and Parinnello reportedstability of SC16 to start from 12.8 GPa, with the stability range being 3.4 GPa.Moreover, in agreement with Qteish and Parinnello I found the cinnabar phasefar from being stable. Turning to GGA calculations, one finds that the resultsare at variance with those of LDA. The construction of the Gibbs free energy

94

4.4. Zinc Sulfide

80 100 120 140 160Volume per atom (a.u.3)

-0.52

-0.48

-0.44

-0.4

Tot

al e

nerg

y pe

r at

om +

2191

(R

y) ZnS: LDA

80 100 120 140 160 180Volume per atom (a.u.3)

-0.36

-0.32

-0.28

Tot

al e

nerg

y pe

r at

om +

2195

(R

y)ZnS: GGA

zincblendeSC16cinnabarrocksalt

Figure 4.12: Total energy vs. volume for ZnS in different phases, as calculatedwithin the LDA (left panel) and GGA (right panel). Continuous lines show fit tothe Murnaghan equation of state for each phase. In cinnabar and SC16 phases,the internal coordinates have been relaxed in each point.

vs. pressure diagram (see Fig. 4.13, right panel) shows that the SC16 is definit-ley not a thermodynamically stable phase, as its enthalpy is higher than that ofZB and RS (notably in the critical pressure range corresponding to the ZB–RStransition). As for cinnabar, it is unstable everywhere. Fitting total energies

12 14 16Pressure (GPa)

-2

0

2

4

Ent

halp

y pe

r at

om r

elat

ive

to B

3 (m

Ry)

12 14 16 18Pressure (GPa)

-2

0

2

4

SC16cinnabarRSZB

LDA GGA

ZnS

Figure 4.13: Static enthalpy per atom in different phases of ZnS, relative to theZB phase, as calculated with the LDA (left panel) and GGA (right panel)

95

Chapter 4. Pressure-induced phase diagram of ZnSe and ZnS

and volumes into Murnaghan equation of state for each phase made it possible toextract their corresponding structural parameters. The LDA, as expected, pro-duces lower equilibrium volume (125.4 Bohr3, for ZB) than GGA (136.86 Bohr3).The bulk modulus of ZB/SC16 phase (87/82 in LDA, 68/64 in GGA) stay lowerthan RS/cinnabar (107/94 in LDA, 84/73 in GGA).

Let us turn to experimental data. Desgreniers et al. [2000] carried XRD study,using synchrotron radiation. A rigorous search along the 300 K isotherm as afunction of pressure gave no indication of existence of any intermediate phase. Inthis experiment the pressure was increased and decreased by steps of ∼0.5 GPaon most samples and by ∼0.1 GPa on one sample to ensure that a new phase witha restricted pressure stability range would not be overlooked. Desgreniers et al.[2000] concluded that, starting from the ZB structure, ZnS transforms unambu-gously to the RS structure upon pressure increase at room temperature. Also, theback transformation (RS to ZB) has been observed upon pressure decrease, andfound to proceed without an intermediate phase. Our predicted GGA-instabilityof SC16 and cinnabar phases in ZnS confirms, theoretically, the above observationby Desgreniers et al. [2000] of non-existence of any phase between ZB and RSupon pressure increase or decrease.

There is a quite big difference between ZnS and ZnSe regarding the staticenthalpy phase diagram, even if the both systems share the same cation. InZnSe, according to GGA, the SC16 phase is predicted to be stable while cinnabarmetastable (Breidi et al. [2010]), whereas in ZnS I found both SC16 and cinnabarunstable. However this seems to be in contradiction with the previous claims(Qteish and Parrinello [2000]) that the relative stability between SC16 and cinnabarstructures of II-VI semiconductors depends strongly on the cation.

96

Chapter 5

Dynamical instability ofzincblende phase in ZnSe

The static approach to phase transformations under pressure, outlined in theprevious chapter, presumes a simple comparison of enthalpies of different phases,but tell us nothing about how the phase becomes inherently instable, nor givesa clue about a possible transformation path. Such conclusions can be driven onthe basis of lattice-dynamical simulations. Even if done on the same systemsas above, they demanded a use of completely different calculation approach –a linear-response analysis of phonon dispersions. Correspondingly, this part ofstudy is organized into a separate chapter. After a brief introduction, the methodis outlined, and the results on convergence tests are followed by substantial resultson ZnSe.

5.1 Introduction

The standard method for the calculation of relative stability of high-pressurephases is usually based on the examination of enthalpy as a function of pressurefor different phases. Such procedure does usually impose symmetry constraintson different phases, exactly in order to keep them distinguishable. However, noguarantee is given that a given phase would remain stable once such symmetryconstraint is removed.

The general stability of crystal lattice is summarized in three stability condi-tions according to Peierls [2001], which all must be fulfilled simultaneously for agiven crystal. The first condition (a) is that the crystal lattice is free from forces,i.e., that the total force on each atom is zero. The second condition (b) is thatthe crystal lattice is stable against macroscopic displacement such as compression(expansion) or shear. The third stability condition (c) is that the crystal lattice

97

Chapter 5. Dynamical instability of zincblende phase in ZnSe

is stable against any small displacement. This means that for any not too largedisplacement, there is a restoring force bringing atoms back to the equilibrium.This condition is equivalent to the statement that all phonon frequencies are real[Born and Huang, 1954]. The opposite would mean that the total energy surfacehas negative curvature along at least one direction in the space of all possibleatomic displacements, and the system will find its way to lower its energy movingaway from what was presumed to be an equilibrium structure. Since phononfrequencies do, in some sense, reveal the curvature of global total energy surfacealong its “main axes” (revealed by normal vibration modes), a considerable soft-ening of a given phonon is a good indication of a tendency for displacive phasetransition. Pressure is a convenient external parameter (along with say electricor magnetic field) that can be changed smoothly, thus enabling the study of thetransition.

5.2 Method, parameters, accuracy tests

The softening of phonon indicating a tendency towards phase transition mayoccur not necessarily at the center of the Brillouin zone. This is a potentiallyvery important information: the knowledge of the instability wavevector qinst willthen give a hint as to which superstructure wants to develop under transition,that is otherwise difficult to guess: in such superstructure, a spatial wave withwavevector qinst will make a full period along each of supercell vectors. “In theideal case” each of the superstructure vectors Ti must make with qinst the scalarproduct of 2π×(integer number), so that the supercell (that corresponding tothe phase after transition) will be commensurate with the initial underlying unitcell. In case of such commensurability, qinst spans the reciprocal space (alongwith “conventional” reciprocal vectors), imposing a new “folded structure”. Itsinversion will give the real-space “superstructure” searched for. This will beexplained in the section 5.3. For better understanding such unfolding - supercelltransformation, the work by Moreno and Soler [1992] is very enlightening.

As we a priori do not know the instability wavevector, we would like to haveaccess to full phonon dispersion, in order to scan for instability throughout theBrillouin zone. Therefore, a linear response method offers a more convenientworking tool than a method of “frozen phonon” category.

The calculations in the present chapter have been done with abinit package[Gonze et al., 2002], which is (apart from many features and extensions specificfor the package, irrelevant in the present context) a realization of PW PP method.Basically it is a solver of Kohn-Sham equations (2.3) along with, in what regards

linear response, the perturbative method [Baroni et al., 2001; Gonze, 1997; Gonzeand Lee, 1997]. XC energy and potentials can be treated according to different

98

5.3. Method, parameters, accuracy tests

-5 0 5 10 15 20 25Pressure (GPa)

120

140

160

180

Vol

ume

per

ato

m (

Boh

r3 )

WIEN2kabinit

Figure 5.1: Volume-pressure curves for ZB ZnSe, calculated with abinit andWIEN2k.

schemes; in the present calculations, the GGA was used, specifically with thePerdew-Burke-Ernzerhof parametrization [Perdew et al., 1996]. Norm-conservingPP of the Troullier-Martins type [Troullier and Martins, 1991] were taken asavailable in the abinit database, corresponding to 12 and 6 valence electrons forZn and Se, correspondingly, without further adjustment.

Apart from this, the usual convergence / accuracy tests had to be done, withrespect to two calculation parameters which are usually essential for quality offorces and hence phonon calculations by almost any method: the planewave cutoffand the k-space mesh. Both can be gradually and independently increased in aplanewave method; the aim was to suppress the corresponding fluctuations oftotal energy (and its derivatives), in the course of structure optimization andphonon calculations. Following these tests, the safe values of ecut=32 Ha and8×8×8 (reducible) k-points were accepted.

Fig. 5.1 gives the comparison of volume-pressure curves calculated by abinitmethod in the present setup and by all-electron WIEN2k, as discussed in theprevious chapter. Both calculations have been done with (in fact, the sameflavour of) GGA. Besides the obvious closeness of the equation of state curves,one can mark an excellent numerical agreement between both calculation methodsin what regards the equilibrium volume per atom (162.3 Bohr3 with abinit vs.160.8 Bohr3 with WIEN2k) and bulk modulus (55 GPa in both cases).

99

Chapter 5. Dynamical instability of zincblende phase in ZnSe

5.3 Results and discussion

Phonon dispersion curves have been calculated along different high-symmetrydirections in the Brillouin zone, for some values of imposed pressure. Resultingdispersion plots are shown in Fig. 5.2. One can see that two transversal acoustic(TA) modes get considerably softened, particularly in the vicinity of the X point,and (less) at L. This is an unusual trend; more typical under pressure is thatthe total energy surface gets a steeper profile, and the vibration frequencies gethardened. This is indeed seen for other branches in Fig. 5.2. As the pressure

Freq

uenc

y (c

m-1

)

0

50

100

150

200

250

300

350

400

Γ K X Γ L X W L

p=12 GPa

Freq

uenc

y (c

m-1

)

0

50

100

150

200

250

300

350

400

Γ K X Γ L X W L

p=19 GPa

Figure 5.2: Phonon dispersion curves of ZnSe crystal in cubic ZB structure, attwo pressures, before and after the ultimate softening at X.

100

5.3. Results and discussion

130 132 134 136 138 140

Cell volume per atom (Bohr3)

-20

0

20

40

X

TA

Fre

quen

cy (

cm-1

)

12 14 16 18 20 22Pressure (GPa)

-20

0

20

40

X

TA

freq

uenc

y (c

m -1)

Figure 5.3: TA phonon frequency at X point of the BZ for ZnSe in the ZB struc-ture, depending on volume (left panel) and on pressure (right panel). Imaginaryfrequency values are shown as negative ones.

increases, the TA phonon frequency in X decreases, eventually passes throughzero and turns imaginary; it is customary to represent the corresponding valuesas negative ones, ω ← −√−ω2, as is done in Figs. 5.2 and 5.3. A steep drop ofcalculated frequency towards, and across, the “root” in Fig. 5.3 makes it easy toestimate the critical volume and hence critical pressure for structure instability;it turns to be about 18 GPa, much larger than transition pressure estimated fromstatic calculations in the previous chapter.

Let’s now discuss what the softening at X would really imply. The latticevectors of ZB structure are

a1 = a(0

1

2

1

2

); a2 = a

(1

20

1

2

); a3 = a

(1

2

1

20)

; (5.1)

and the corresponding reciprocal lattice vectors

b1 =2π

a(−1 1 1) ; b2 =

2π

a(1 −1 1) ; b3 =

2π

a(1 1 −1) . (5.2)

The coordinate of X is 2πa

(1 0 0). The softening at X means that this point nowbecomes a node of the reciprocal lattice which corresponds to the superlattice, inwhose unit cell the spatial period of the corresponding lattice vibration would befully confined. Therefore the above X coordinates must be added to the vectorsin (5.2), to search for a “more dense” reciprocal lattice. Easy to see that thevectors which span its primitve cell can be chosen as, e.g.,

B1 =2π

a(1 0 0) B2 =

2π

a(0 1 −1) ; B3 =

2π

a(0 1 1) , (5.3)

101

Chapter 5. Dynamical instability of zincblende phase in ZnSe

and their inversion in the real space would be

A1 = a(1 0 0

); A2 = a

(0

1

2− 1

2

); A3 = a

(0

1

2

1

2

), (5.4)

or (producing the same lattice)

A1 = a(1 0 0

); A2 = a

(0 1 0

); A3 = a

(0

1

2

1

2

), (5.5)

The resulting unit cell is therefore tetragonal, with a square base a√2× a√

2in the

yz plane and of the length a along (1 0 0). Its volume is doubled, with respect tothe underlying ZB cell.

This cell doubling have something in common with a transformation into theCmcm structure shown in Fig. 1.5. Even if this figure describes transition fromthe RS structure, the lattice vectors are the same as in ZB. The transformationinvolves gliding of alternating planes in the cubic setting, reducing the symmetryto orthorhombic. A more attentive analysis of transformation trend could bededuced from the analysis of the phonon eigenvector at X (not yet done at thismoment).

It should be noted that the transformation potentially resulting from the soft-ening of the X (or any other) acoustical phonon is a displacive one, as it impliesthe group/subgroup relation between the two structures. However, the transitionsdiscussed in the previous chapter, ZB to cinnabar etc., were reconstructive onesand could not have been hinted for by a particular phonon softening. Thereforethe transformation like that indicated by the present phonon anomaly is plausi-ble, but demands a too high pressure, and in reality would be probably preventedby some reconstructive transition.

102

Chapter 6

Pressure effect on latticedynamics in (ZnBe)Se

6.1 Introduction

The study of phonon spectra in mixed (mostly, pseudobinary) semiconductors, viacareful analysis of published data, own Raman measurements, phenomenologicalmodelling, all this being assisted by first-principles calculations, is an importantdomain of specialization in Laboratoire de Physique des Milieux Denses (LPMD)in Metz, my host laboratory at the French side in the course of my present PhDwork. The percolation model emerged as a main paradigm, primarily due to effortsby Olivier Pages,1 which were systematized in a number of publications, aimed atdifferent semiconductor mixed systems and situations [Pages et al., 2006, 2008].

The main message is that, in going beyond the simplest VCA (see Section1.1.2) and concentrating on details of local environment for this or that individualcation-anion bonds, one can get insight into effects of disorder on a mesoscopicscale. Fractal-like behaviour and percolation effects over given type of bonds,for a given concentration) may offer an elegant interpretation for some unusualeffects in vibration spectra. The most concise overview of the related ideas is dueto Pages et al. [2009], and a detailed analysis for particular systems – in recentPhD works done in the laboratory [Chafi, 2008], notably the latest one by my co-worker and co-author in this field, Jihane Souhabi [2010]. The aspect of pressureas an additional and useful tool, able to affect internal strains and modify thevibration modes, is quite new in this context.

In the percolation model, a random AxB1−xC alloy is viewed as a composite of

1Director of Laboratoire de Physique des Milieux Denses, Universite Paul Verlaine de Metz-France, http://www.univ-metz.fr/recherche/labos/lpmd/alliages-semiconducteurs/pages.html

103

Chapter 6. Pressure effect on lattice dynamics in (ZnBe)Se

“AC-like” and “BC-like” regions, in which the A–C and B–C bonds are highly self-connected, respectively, each region providing one phonon per bond (1-bond →2-mode) – see Pages et al. [2009]. This reveals that phonons give a natural insightinto the alloy disorder at the unusual mesoscopic scale. O. Pages introduced aterminology that the percolation doublet (with splitting denoted by Δ acts asa ‘mesoscope’ into the alloy disorder. In principle, the change in frequency andintensity of each 1-bond → 2-mode (AC-like or BC-like) “mesoscope” may bestudied under the influence of any stimulus. This opens the way for a basicunderstanding beyond that given by the usual 1-bond → 1-mode ‘macroscope’ ofChang and Mitra [1971], based on a description of an alloy as a uniform continuumaccording to the VCA (see Sec. 1.1.2).

In this work, we explore the phonon behavior of semiconductor alloys underpressure. The aim is to decide whether the AB- and AC-like regions are affectedin the same way by hydrostatic pressure or whether one region behaves in aspecific manner. The most suitable alloy to address such issue is Zn1−xBexSe.While ZnSe adopts the usual sixfold-coordinated (metallic) cubic RS phase underpressure, BeSe is one of the few exceptional systems that transforms to the sixfold-coordinated hexagonal NiAs phase (Mujica et al. [2003] and Luo et al. [1995]).Furthermore, the transition pressure is much higher for BeSe (∼56 GPa) thanfor ZnSe (∼13 GPa). Luo et al. [1995] observed that the volume collapse forBeSe (∼ 11%) is small as compared with ZnSe (∼17%); in fact the smallest everachieved at a first-order fourfold-to-sixfold phase transition [Luo et al., 1995].They advanced an idea that due to the small size of the Be cations there existsa strong Pauling repulsive interaction between the large-size Se anions, whichprevents drastic volume reduction.

An interesting question then is what happens to the Be-Se bonds of ZnBeSecrystals with moderate Be content when the ZnSe-like medium begins its naturalZB→RS transition, an unnatural one for the Be-Se bond. This we investigate bytaking advantage of the well-resolved Be-Se mesoscope (Δ ∼40 cm−1). Ramanmeasurements were performed, by our colleagues1 and co-authors of [Pradhanet al., 2010], at increasing pressure up to the ZB→RS transition using a seriesof alloys with moderate Be content that allows to cover all possible topologies ofthe BeSe-like region as determined by two critical x values. The first is the Be-Sebond percolation threshold (xBe = 0.19), at which limit the spatial organizationof the BeSe-like region turns from a dispersion of finite-size clusters (x < xBe)into a pseudoinfinite treelike continuum (x > xBe), and the second is x = 0.5,beyond which limit the latter continuum becomes dominant over the coexisting

1Gopal K. Pradhan and Chandrabhas Narayana of: Light Scattering Laboratory, Chem-istry and Physics of Materials Unit, Jawaharlal Nehru Centre for Advanced Scientific Research(JNCASR), Jakkur P.O., Bangalore 560064, India

104

6.3. Experimental details and ab initio methods

ZnSe-like one, see Stauffer [1985]. We then attempt to achieve consistent under-standing of the observed effects at both the microscopic (ab initio calculations)and mesoscopic (percolation scheme) scales.

6.2 Experimental details and ab initio methods

Experiments, the details of which are given in [Pradhan et al., 2010] but are notof immediate importance here, were done on Zn1−xBexSe single crystals with xvalues of 0.11, 0.16, 0.24, and 0.55 (the largest achievable limit so far). The pres-sure was applied in DAC, using the Methanol/ethanol/water (16/3/1) mixtureas transmitting medium. The pressure environment is assured to be hydrostaticuntil ∼14 GPa, beyond which it can be considered as quasihydrostatic up to∼30 GPa. Unpolarized Raman spectra were recorded in backscattering geometryon the (111) crystal face with nonresonant 532.0 nm excitation. In such geometryboth the transversal optical (TO) and longitudinal optical (LO) modes are Ra-man active. We recall that these modes consist of out-of-phase vibrations of thequasirigid anion (Se) and cation (Zn,Be) sublattices, either perpendicular to (TO)or along (LO) the direction of propagation. Ab initio ZC TO-DOS (zone-centertransverse optical density of states) calculations were done with the Siesta code[Soler et al., 2002]. I used 64 atom supercells containing the ‘ultimate’ motif(referred to as the 2-imp. motif), i.e., a pair of next-nearest-neighbor impurities(say A) immersed in the environment of the other species (BC-like). Both theZnBeSe and GaAsP alloys were considered (the latter for reference purpose), us-ing the basis function and PP settings as explained by Pages et al. [2009]) andPostnikov et al. [2005]. After performing the full relaxation of lattice parametersand atomic positions within each supercell, the dynamical matrix was constructedby probing small individual displacements of atom from their equilibrium posi-tions and collecting the forces induced on all sites. The diagonalization of thedynamical matrix yielded the frequencies of TO phonons and the correspondingeigenvectors. A subsequent projection of eigenvectors onto the uniform transla-tion pattern of each species throughout the crystal, as explained by Postnikovet al. [2005], permits to single out the vibrations of genuinely zone-center char-acter. The correspondingly weighted (and artificially broadened) spectrum ofdiscrete vibration frequencies can then be compared with the TO Raman spec-trum. It should be noted that the matrix elements of Raman scattering were nottaken into account in the calculation, so that the relative magnitudes of differentpeaks do not always mimic those in the experiment.

105

Chapter 6. Pressure effect on lattice dynamics in (ZnBe)Se

6.3 Results and discussion

Before entering the body of the discussion we briefly specify the general contextof the study, as derived from preliminary XRD measurements performed over theinvestigated pressure domain (not shown). The basic trend is that the ZB→RStransition is regularly shifted to higher pressure when the incorporation of Beincreases. This is due to the reinforcement of the highly ionic (soft) ZnSe latticeby the strongly covalent (stiff) Be-Se bonds. Besides, the x-ray diffraction dataindicate that the ZB→RS transition is very sharp, and that the crystal adoptsthe RS structure as a whole. Indeed no signal due to any residual ZB phase couldbe detected beyond the transition pressure.1

In the discussion of the Raman data we focus on TO modes, because the LOones are affected by their coupling to electric field, and otherwise clear trends invibration spectra are smeared out and masked.

In the following, the notation of modes is used, in which the subscript refersto a bond species and the superscript – to a predominant cation in the region inquestion, e.g. TOBe

Zn−Se stands for a vibration of predominantly Zn–Se bond inpredominantly BeSe-rich region.

For the presentation and discussion of the data we focus on the intermediatex value (x=0.24), which corresponds to well-resolved Zn-Se and Be-Se Ramansignals, but the effects are general. Representative pressure-dependent Ramanspectra are shown in Fig. 6.1. The frequency versus pressure variations of themain features are shown in the inset. A 2TO+1LO signal shows up clearly in theBe-Se spectral range (∼550 cm−1).

With increasing pressure the intensity of the LO modes decreases, a well-known effect of the progressive metallization of the sample resulting in the ZB→RStransition. The total LO extinction occurs at ∼23 GPa, coinciding with the actualZB→RS transition observed by x-ray diffraction (not shown1). More fascinatingeffects relate to TO modes as marked by capitals, i.e., (C) the convergence of thelower Be-Se TO branch onto the upper one.

Be-Se spectral range

We emphasize that the feature (C) is not composition dependent, i.e., it occursat the same critical pressure of 14±1 GPa for all samples. Apparently no effect isseen on whether the sample consists of a dispersion of finite-size clusters (x < xBe),of the as-formed pseudoinfinite continuum (x just above xBe), or of the dominantpercolation cluster (x just above 0.5). This hints for a local nature of the bondstate. Fortunately, first-principle calculations allow direct insight in the pattern of

1A. Polian, T. Ganguli and S. K. Deb, private communication.

106

6.3. Results and discussion

Figure 6.1: Pressure-dependent Raman spectra of Zn0.76Be0.24Se. The frequencyversus pressure variations of the main features are shown in the inset. There, theplain and open circles refer to the LO and TO modes, respectively. Remarkabletrends (A,B,C) are marked by circles. Note: In my dissertation I discuss only thetrend labeled (C).

this or other mode, an advantage not accessible in experiment. The visualisationof individual vibration modes, on the basis of their corresponding eigenvectors,can be done with the XCrySDen code [Kokalj, 1999, 2003].

The singularity (C) was observed in all samples, corresponding to differentBe concentrations. Therefore it was sufficient to investigate theoretically thissingularity in the simplest model of alloy beyond the dilution limit, i.e., two

107

Chapter 6. Pressure effect on lattice dynamics in (ZnBe)Se

interacting impurities (at the nearest cation sites) in possibly large supercell (64atoms, in our case). Indeed, the ab initio ZC TO-DOS per Be atom with the2-imp. (Be) motif in ZnSe, calculated at both pressures, 0 and 10 GPa, dowell exhibit the (C) singularity. The curves are shown in the body of Fig. 6.2,specifying the vibration pattern per mode. In view of these, we identify themicroscopic mechanism behind (C) as the transformation of TOBe

Be−Se (vibrationalong the Be-Se-Be chain, noted ‖ chain) from the bond-stretching type to thebond-bending type under pressure. The nature of the doubly degenerate TOZn

Be−Se

(vibrations perpendicular to the Be-Se-Be chain, noted ⊥ chain, in-plane and out-of-plane) remains unchanged, i.e., of the bond-bending type.

For more insight we performed “symmetric” ab initio calculations with the2-imp. (Zn) motif in BeSe (a heavy impurity dilution limit). The longer Zn-Sebonds create a local compression in surrounding BeSe, driving a local Be-Se modeclose to Zn, i.e., TOZn

Be−Se (see Fig. 6.2, right panel), at a higher frequency thanthe bulk Be-Se mode, i.e., TOBe

Be−Se. This gives a clear Be-Se doublet, though thesplitting is less than in the Be-dilute limit, of ∼11 cm−1. We observed the samesplitting at 0 and 10 GPa, indicating that singularity (C) disappears when theBe-Se bonds are dominant in the alloy, thus not influenced by the Zn-Se bonds.

It is interesting to compare the situation in (Zn,Be)Se alloys with that inGa(As,P). The substitution now being at anion sites, the primary effect is similar:the shorter bond (Ga-P) gives rise to a distinct percolation doublet of Δ∼12 cm−1

[Pages et al., 2009]. This happens because the length and the stiffness of thebond in question depend on whether it finds itself in more GaAs-rich or GaP-richenvironment. A further similarity to (Zn,Be)Se is the sensitivity to pressure: bothparent binary systems GaAs and GaP, in ZB structure at ambient conditions,adopt the same phase (Cmcm) under pressure, notably at almost the same criticalpressure (∼15±3 GPa) as (Zn,Be)Se transforming into the RS phase.

The ZC TO-DOS per P atom obtained1 at 0 and 10 GPa with the 2-imp. (P)motif in GaAs are shown in the inset of Fig. 6.2. The Ga-P percolation doublet isshifted upwards as a whole when approaching the phase transition, and no C-typeconvergence is observed. Hence, it emerges that the C singularity is not intrinsicto short bonds in alloys, but specific to the Be-Se bonds in ZnBeSe, when theirfraction is such (low or moderate) that they are forced to adopt the ZnSe-like RSphase under pressure.

For an intuitive discussion of the physics behind C we turn to the percolationmodel.2 A parallel between the percolation model (mesoscopic scale) and ab ini-

1The ZC phonon density of states of Ga(As,P) alloy was calculated by my co-student, JihaneSouhabi [2010].

2Note that such phenomenological model relies on a scalar description of the alloy (linear-chain approximation) so that everything comes down to a question of bond-stretching forcesonly, by construction.

108

6.3. Results and discussion

Figure 6.2: Left panel: ab initio ZC TO-DOS per impurity of the 2-imp. Be(main curves) and P (inset) motifs in ZnSe- and GaAs-like supercells, respectively.The Be-Se vibration patterns are indicated, emphasizing the bond-stretchingmodes (dashed areas) in reference to feature C (circle). Right panel: schematicvibration of Be, sharing the same anion with the Zn impurity. Be and Se atomsmove in the Be-Se-Be plane, normal to the Zn-Zn line.

tio calculations (microscopic scale) can be drawn by realizing that bond-bendingwithin a given impurity motif corresponds to bond stretching of the (like) sur-rounding bonds from the host matrix, and vice versa. If we refer to the 2-imp.(Be) motif in ZnSe, this comes down to discuss the low- and high-frequency Be-Semodes in terms of Be-Se stretching within the BeSe- and ZnSe-like regions, respec-

109

Chapter 6. Pressure effect on lattice dynamics in (ZnBe)Se

tively (see the shaded areas in the vibration patterns, Fig. 6.2, consistently withthe terminology of the percolation model. Within such stretching-type model, thefeature C can thus be understood as due to the progressive ‘freezing’ of the Be-Sebonds from the minor BeSe-like region when this is forced to adopt the unnaturalRS phase of the host ZnSe-like region. The oscillator strength is transferred tothe close Be-Se bonds of the latter region.

6.4 Conclusion

In brief, the percolation mesoscope reveals that the lattice dynamics of theZnBeSe crystal basically changes when approaching the ZB→RS ZnSe-like transi-tion under pressure, which is an unnatural transition for the Be-Se bonds. Whilethe application of pressure reveals a percolation-type fine structure of the Zn-Se mode, that remains basically unaffected by pressure, the highly self-connectedBe-Se bonds from the BeSe-like region ‘freeze’, the oscillator strength being chan-neled to the less self-connected Be-Se bonds of the surrounding ZnSe-like region.This process, transparent for the Zn-Se dynamics, is completed (∼14 GPa) beforethe actual transition to RS occurs (∼23 GPa at 24% Be). Generally this workshows how the percolation mesoscope may help to achieve further understandingof the pressure dependence of phonons in alloys beyond the usual VCA paradigm.

110

Chapter 7

Computational modelling ofstructural, electronic andthermodynamic properties ofCd(S,Se) alloy

7.1 Introduction

Among the group II-VI semiconductors whose energy bands cover the visible spec-tral range, there are many promising candidates for use in opto-electronic devices,see, e.g., Gu et al. [2005] and Pillai et al. [1983]. CdS and CdSe are such importantwide gap semiconductors, already broadly used in opto-electronics, notably non-linear optics, visible-light emitting diodes and lasers [Pan et al., 2006b]. However,some opto-electronic applications demand precisely tuned emission wavelength.The desired tunability can be achieved through composition modulation, as manyexamples of pseudobinary compounds with continuously varied band gap, e.g.CdSxSe1−x, ZnxMg1−xO, and CdxMn1−xS, had readily shown [Chen et al., 2002;Hsu et al., 2005; Ku et al., 2005; Liang et al., 2005; Pan et al., 2005]. CdSxSe1−x

can be grown over the whole composition range x = 0− 1, yielding the band gapfrom ≈ 2.44 eV for CdS to ≈ 1.72 eV for CdSe, that corresponds to wavelengthvalues spanning from 509 to 720 nm, hence almost covering the entire visiblerange. In what regards the opto-electronic device efficiency, the mixed materialis usually halfway between CdS in which very high sensitivity is possible but re-sponse time is high and CdSe in which the response time is shorter at the cost ofsomehow lower sensitivity [Hsu et al., 2005].

Due to its excellent properties, including high quantum efficiency, large non-linear susceptibility, narrow band edge, good photoconduction, and fast response

111

Chapter 7. Computational modelling of properties of Cd(S,Se) alloy

times, Cd(S,Se) has been widely used as a photoconductor, with much interestfocused on applications in photovoltaic and photoelectrochemical devices [Maneand Lokhande, 1997; Mei, 1992; Nogami et al., 1993; Pan et al., 2006a; Pernaet al., 1999; Premaratne et al., 2004; Roussignol et al., 1990; Vaynberg et al.,1996].

Furthermore, Cd(S,Se) is a good laser medium to produce lasing in the visiblespectral range region [Hurwitz, 1966; Johnston, 1971; Mane and Lokhande, 1997;Roxlo et al., 1971]. Briefly concerning the preparation issues, Cd(S,Se) withvariable composition can be produced from binary end-member compounds, eachhaving ZB structure, by CdS/CdSe co-evaporation, laser ablation, solvothermalreactions, electrodeposition, solid-state diffusion, and vapor-growth [Dutta et al.,1993; Gupta et al., 1993; Handelman and Kaiser, 1964; Korostelin and Kozlovsky,2004; Liu et al., 2006; Mane and Lokhande, 1997; Nogami et al., 1993; Perna et al.,2000].

7.2 Methods

Total energy calculations have been done using the PW-PP ab initio simulationpackage [VASP: Kresse and Furthmuller, 1996a,b] and non-normconserving PP[Vanderbilt, 1990]. Semicore 4d states of Cd were included as valence ones in thegeneration of the corresponding PP. A relatively high cutoff energy of 36.75 Ry isused throughout this work. A Monkhrost-Pack mesh of 11×11×6 is used for thefour-atom WZ unit cell of CdS and CdSe systems; a 11×11×11 mesh was used forthe two-atom ZB CdS and CdSe unit cells (finer meshes resulted in total-energychanges of <0.01 meV/atom). In all calculations involved, the total energy issupposed to be minimized with respect to structure variables (lattice parameters,internal coordinates). In the intermediate (alloy) part, some search / optimization/ sampling over possible alloy configurations must be, in principle, done. In thefollowing, such configurations are replaced by SQS, of arbitrarily chosen size of32 atoms, constructed for three intermediate concentrations x=0.25, 0.50, 0.75.

7.3 Structural characterizations

Lattice constants of most of the II-VI and III-V ternary compounds are known toclosely follow the Vegard’s law [Vegard, 1921]. In what regards the microscopicstructure, e.g. bond lengths, the simplest “naive” picture can be that of VCA,according to which the “averaged” effective atomic species make a perfect crystalwith all equal and gradually scaled bonds. The opposite, but also simple, conceptis due to Pauling, presuming the conservation of covalent radii, according towhich the individual cation-anion bond lengths in an alloy would be as inherited

112

7.3. Structural characterizations

from parent binary compounds. This can be to some extent verified with thehelp of EXAFS, a powerful structural tool for the determination of bond lengthsbetween selected atomic pairs. The earliest EXAFS experiments dealing withstudies of the local structure of ternary semiconductor systems were performedin Mikkelsen Jr. and Boyce [1983], as mentioned in Chapter 1. Further EXAFSstudies of ternary II-VI and III-V compounds with the ZB structure have givensimilar results [Balzarotti et al., 1985; Boyce and Mikkelsen Jr., 1989; Letardiet al., 1987; Marbeuf et al., 1989; Motta et al., 1985; Pong et al., 1989].

Using a valence force field (VFF) approach which takes into account bond-stretching and bond-bending forces, Martins and Zunger [1984] introduced amodel where the relaxation parameters for the A-B and A-C bonds in ABxC1−x

compounds can be calculated as follows:

εAC = εAC|BC:A| =1

1 + (d0AC/d0

BC)3 (2

3− 1

2f i

BC

) , (7.1)

where d0AC and d0

BC are the bond lengths in binary AC and BC crystals. Theequation above is for the case where A is an isolated impurity in the BC crystal,and f i

BC is the ionicity of the host system BC. Therefore, ε describes the relaxationof the A-C bond in the ternary compound: ε = 0 corresponds to the VCA limit,ε = 1 – to the Pauling’s limit. Moreover, Eq. (7.1) was established for the ZBstructure; ε is obtained by minimizing the elastic deformation energy with respectto the breathing mode displacements of the first two shells around the impurity.Martins and Zunger [1984] predicted the value of ε for an isolated impurity in a ZBsystem (for many II-VI, III-V and IV-IV ternary systems). Typical ε values rangefrom 0.6 to 0.8. Eq. (7.1), using ionicities values f i

CdS=0.794 and f iCdSe=0.841

calculated by Christensen et al. [1987], yields εCdS= 0.82 and εCdSe= 0.77. Thesecalculations showed a satisfying agreement with experimental values measuredby EXAFS [Levelut et al., 1991].

From our calculations we found that in the whole composition range, thedistances dCdS and dCdSe are different and very close to respective values in thebinary compounds (Fig. 7.1). The behaviour of CdSxSe1−x compounds is there-fore closer to Pauling’s limit than to the VCA model. Still, the relaxation is notperfect. A small decrease in the dCdSe distance is predicted when going from pureCdSe to sulphur-rich compounds. Moreover, a small increase in the NN distancedCdS is observed when going from CdS to selenium-rich compounds. Experimen-tal values for the relaxation parameters εCdS and εCdS obtained by Levelut et al.[1991] are, respectively, εCdS = 0.87±0.04 and εCdSe = 0.88±0.08. For relaxationparameters ΓAC(x) and ΓBC(x), introduced earlier in Eq. (1.3), the actual relaxedbond lengths yield ΓCdS(x) = 0.86; 0.88; 0.94 (for x=0.25; 0.5; 0.75, respectively),and ΓCdSe(x) = 0.94; 0.89; 0.85 (for x=0.25; 0.5; 0.75), see Fig. 7.2.

113

Chapter 7. Computational modelling of properties of Cd(S,Se) alloy

0 0.2 0.4 0.6 0.8 1Sulfur composition (x)

2.55

2.6

2.65

2.7

NN

dis

tanc

e (Å

)

Pauling’s limit: full relaxation of dCdS bond

Pauling’s limit: full relaxation of dCdSe bond

Cd-SeCd-S

VCA; null relaxation limits

Figure 7.1: Variation of the bond lengths dCdS and dCdSe in CdSxSe1−x alloy withcomposition, from calcultions in ZB supercells. The VCA and Pauling’s modellimits are shown.

Among the NNN distances (see Fig. 7.3), those between cations clearly showtwo-mode behavior, exhibiting a clear separation of magnitudes, depending on

0 0.25 0.5 0.75 1Sulfur composition (x)

0.85

0.90

0.95

1.00

Rel

axat

ion

para

met

er (

unitl

ess)

Cd-SeCd-S

Figure 7.2: Variation of the relaxation parameters ΓCdS(x) and ΓCdSe(x), definedaccording to Eq. (1.3), in ZB CdSxSe1−x alloy with composition. Γ=0 correspondsto the VCA limit and Γ=1 to the Pauling’s model.

114

7.4. Composition-induced phase transition and bandgap bowing

0 0.25 0.50 0.75 1Sulfur composition (x)

4.2

4.25

4.3

4.35

4.4

next

-nea

rest

nei

ghbo

urs

dist

ance

(Å

)

Cd-S-CdS-SSe-SeS-SeCd-Se-Cd

Figure 7.3: Schematic variation of the anionic and cationic NNN distances inCdSxSe1−x alloy with alloying composition x, in ZB supercells.

whether S or Se connects the two Cd atoms in question. In contrast, all threedifferent species of (anion)-Cd-(Anion) distances roughly follow the Vegard’s law.Thus, the anionic sub-lattice behaves almost as a rigid one that shrinks or dilatesas a whole. This is a quasi standard picture for all ZB-structure alloys.

7.4 Composition-induced phase transition and

bandgap bowing

Comparing calculated total energies in pure compounds, once they are relaxedin different structures, one finds that the ground state of CdSe (x=0) is ZB andthat of CdS (x=1) – WZ. Consequently, a crossover from one structure to theother must happen at some inermediate concentration. In the following only fivespecial cases are presented, namely: x=0, 0.25, 0.5, 0.75 and 1. For low S con-centrations, the ZB phase is favorable (Fig. 7.4), and a direct ZB↔WZ transitionis possible at 0.25 < x < 0.5. This finding is consistent with the experimentalwork of Kumar and Sharma [1998] and Kainthla et al. [1982]. Using screen print-ing method, Kumar and Sharma [1998] prepared sintered CdSxSe1−x films, andcharacterized their structure by XRD. They reported that the CdSxSe1−x filmshave a ZB structure up to x=0.4, and the WZ structure for x >0.4. Even earlier,Kainthla et al. [1982] have studied structural and optical properties of solution-grown CdSxSe1−x films. This study found the CdS films to have predominantly

115

Chapter 7. Computational modelling of properties of Cd(S,Se) alloy

0 0.25 0.5 0.75 1S composition (x)

-1

0

1

EW

Zto

tal -

EZ

Bto

tal (

meV

per

ion)

ZincblendeWurtzite

ZB WZ

0.25<x<0.5?

x(exp.)ZB-WZ [Kumar] = 0.4

x(exp.)ZB-WZ [Kainthla] = 0.6

Figure 7.4: Total energies of CdSxSe1−x WZ supercells relative to those in corre-sponding ZB supercells. The energies are taken per ion (i.e. par half of formulaunit), in each case at its corresponding equilibrium volume.

cubic (ZB) structure with traces of hexagonal(WZ) one, whereas the CdSe filmswere purely cubic. In alloys films, Kainthla et al. [1982] found the cubic structureto persist up to x ∼ 0.6, while for x > 0.6; the films were predominantly cubicwith small amount of hexagonal phase. The authors observed that the relativeamount of hexagonal modification increases with x.

The electronic structure of alloys is characterized by a small band gap bowingparameter b (Fig. 7.5, see also Chapter 3), defined by Eq. (3.1), to yield b=0.32 eV(in ZB) and 0.25 eV (in WZ). Our results are consistent with theoretical work ofWei et al. [2000]. Altough the stable crystal structure of CdS is WZ, the latterauthors assumed the ZB structure throughout. They employed the FP-LAPWmethod and LDA to study the band structure of Cd-based alloys. Wei et al. [2000]obtained a bowing parameter of 0.28 eV. Experimentally, as was demonstratedby Adachi [2009], the quadratic fit of Bondarenko et al. [1972] (solid circles)and Perna et al. [2000] gives a value of 0.7 eV. Such (both experimentally andtheoretically) low value of the gap bowing can be argued to due to small latticemismatch between CdS and CdSe (∼ 4%) and similarity of their band structures.

116

7.5. Thermodynamic properties

0 0.25 0.5 0.75 1Sulfur composition

0.6

0.8

1

Ban

d ga

p (e

V)

ZincblendeWurtzite

optical bowing = 0.251

optical bowing = 0.3196

Figure 7.5: Variation of optical bandgap of CdSxSe1−x as function of x.

7.5 Thermodynamic properties

In spite of numerous investigations on preparation and optical properties in theCdSxSe1−x system, little, experimentally, has been published on their thermo-dynamic behaviour, and no theoretical study has been reported up to our day.Investigations of mixing in solid solutions is important in determining solid solu-bility, physical properties, and the distribution of components among coexistingsolid and liquid phases.

In this work, I employ DFT to examine the thermodynamic properties ofboth WZ and ZB CdSxSe1−x. For both structures, a direct comparison is doneof the formation enthalpies and free energies, on the basis of calculations usingexactly the same PP and energy cutoff for both structures. These results allowto predict binodal and spinodal lines on the binary phase diagram, delineatingthe miscibility gap and the region where the solid solution is unstable. The sim-ulation method uses the SQS formalism [Wei et al., 1990; Zunger et al., 1990]to efficiently represent the structure of the random alloy. The effect of latticevibrations is included in the harmonic approximation, whereby the dynamicalmatrix is determined using DFT methods. Even though lattice vibration effectshave been shown to be significant for the thermodynamic properties of some in-termetallics [van de Walle and Ceder, 2002], the effects of vibration have beencommonly neglected for semiconductor alloys. However I should mention herethat a recent study [Gan et al., 2006] done on InxGa1−xN has shown that inclu-sion of the vibrational contribution to the free energy of formation of compound

117

Chapter 7. Computational modelling of properties of Cd(S,Se) alloy

semiconductor alloys can have a profound effect on the critical temperature forthe order-disorder transition on the cation sub-lattice. Not only does the vibra-tional entropy shift the critical temperature, it also modifies the solubility limits.Such effects were not previously included, in part, because an accurate treat-ment of lattice vibrations requires substantial computing resources, above thatneeded for modeling the effects of alloying on the heat of formation. Advancesin first-principles computational methods and hardware make routine inclusionof vibrational effects in alloy phase diagram calculations possible.

Earlier work on solid solution semiconductor alloys Si-Ge and Ga1−xInxP [Sil-verman et al., 1995] suggested that lattice vibration effects are negligible. Onthe other hand, an inclusion of these effects for intermetallic compounds havebeen shown to be important in determining phase stability, notably in the Cd-Mg [Asta et al., 1993], Al-Sc [Ozolins and Asta, 2001], Cu-Au [Ozolins et al.,1998] and Al-Cu [Wolverton and Ozolins, 2001] systems. Further on, vibrationeffects have been shown to be relatively unimportant in other metallic systemsthat exhibit compound formation, e.g., Ni-Al [van de Walle et al., 1998] and Pd-V [van de Walle and Ceder, 2000]. Previous works on phase stability of metalssuggests that vibrational effects may or may not be important in those systems(intermetallic ones).

7.5.1 Enthalpy of formation

The enthalpy of formation, first estimated at zero temperature, results from com-paring calculated total energies of intermediate (alloy) configurations with thoseof end members, weighted according to a given concentration. The peculiarity ofthe present system in question is that pure CdS and CdSe have different ground-state phases, WZ and ZB correspondingly; therefore the alloy decomposition, ifit happens, would proceed into two-phase mixture, to be taken as reference forthe entropy difference. However, the structure of alloy for any concentration is apriori not known, and two sequences of difference entropies must be calculated,for phases σ being either WZ or ZB. Taking into consideration that under normalpressure of about 1 atmosphere, the difference between the enthalpy H and thetotal energy U , H − U = PV , is insignificant for a solid or a liquid, we assume:

ΔHσ = Eσ (CdSxSe1−x) − xEWZ(CdS) − (1 − x)EZB(CdSe) (7.2)

The total energies are in the following always given per “formula unit” (equiva-lently refered to as “per cation” or “per anion”). The ZB CdSe has lower totalenergy than WZ CdSe by 2.03 meV/anion, while WZ CdS structure has lowertotal energies than ZB CdS by 2.54 meV/anion.

The formation enthalpy results are shown in Fig. 7.6. The numbers extractedfrom first-principles calculations of WZ and ZB structures were fitted to the

118

7.5. Thermodynamic properties

Form

atio

n e

nth

alpy

per

an

ion

(eV

)

0

0.01

Sulfur concentration (x)0 0.2 0.4 0.6 0.8 1

SQS-16, WZ

Fit to WZ results

SQS-16, ZB

Fit to ZB results

VFF (Marbeuf)

CdSe CdS

Figure 7.6: Formation enthalpy of CdSxSe1−x in WZ and ZB structures as func-tion of composition x. Solid curves are a fit to calculated values (open circles andsquares). The VFF results of Marbeuf et al. [1994] are included for comparison.

polynomial forms ΔH = (α + βx)x(1 − x) + 0.0020325(1 − x) and ΔH = (α +βx)x(1 − x) + 0.002541(x) respectively, beyond the strict regular solution model[Chen and Sher, 1995]. For the WZ structure, we obtain α=0.03714 eV/anionand β=0.00339 eV/anion. For the ZB structure, we find α=0.03666 eV/anionand β=0.00404 eV/cation. Focusing on the WZ results, we see that the VFFpredictions for the heat of formation [Marbeuf et al., 1994] are a bit smallerthan those obtained here (by as much as 2.48 meV/anion at x=0.50). ΔH isasymmetric, leading to a slight deviation toward the highest (lowest) x ZB (WZ)side. The WZ version of the curve lies higher on the lower-x (CdSe) side and ZBbegin higher on the high-x (CdS) side, with the crossing point near x=0.38. Theeffect of this asymmetry will be pronounced in the phase diagram shown later.

Because of the big volume mismatch between CdS and CdSe (around 11 %),the energetics of mixing are expected to be strongly positive if no factors otherthan the size effect are important. But, as seen, we have very small values of ΔHindicating presence of important effects for mixing other than size effect. Actually,S and Se possess high electro-negativities (S: 2.5; Se: 2.4, according to Pauling[1960]), and consequently high atomic covalency (covalent radii: 1.05 A for S and

119

Chapter 7. Computational modelling of properties of Cd(S,Se) alloy

1.2- A for Se)1, and high anion polarizability (2.9 A3 for S and 3.8 A3 for Se)2,yielding an enhancement in the covalency of the bonds and reduction of anion-cation interatomic distance. Thus, the size effect can be partly compensated bythe polarization of anions being mixed, resulting in a thermodynamically idealsolid solution at low growth temperature. Physically, the positivity of ΔH atT=0 K means that unlike-anions dislike neighboring each other and this unfavor-able mixing on the substitutional sublattice is highly pronounced at x=0.5. Ourfindings are in perfect agreement with direct calorimetric measurements, done onthe Cd(S,Se) solid solutions by Xu et al. [2009], who found ΔH of CdSxSe1−x

alloy to be zero (within experimental errors of ∼40 K) at 298 K.

7.5.2 Lattice vibration calculations

As was discussed in Chapter 2, the construction of phase diagrams needs theknowledge of free energy and hence, implicitly or explicitly, calculation of entropy.One (“static”) contribution to entropy comes from configurational disorder, plac-ing S and Se over the anion sites. Another source of disorder is due to latticevibrations. The latter do also affect the energy at a given temperature, accord-ing to population of levels in harmonic oscillators associated to different normalmodes of vibration. In fact it seems easier not to separate both, but directlyconstruct the phonon free energy (see details in Appendix C). What is needed forthis is the phonon density of states g(ω), calculated by any convenient method.In principle, g(ω) depends on phonon dispersions throughout the Brillouin zone.With respect to alloys, two remarks are due: i) the dispersions are not as cleanlydefined as in perfect periodic crystal; ii) as we deal with supercells simulatingdifferent alloy configurations, the Brillouin zone shrinks anyway, and ultimatelya good sampling over phonon modes, approximating g(ω), can be achieved bytaking just a sum over (hopefully numerous and dense) Γ-vibration modes of thesupercell. With 3N such modes, ωα, for a N -atom supercell known, g(ω) becomesapproximated as

g(ω) =3N∑α

δ(ω − ωα) ,

∞∫0

g(ω)dω = 3N , (7.3)

where δ(ω) are not necessarily discrete peaks, but, optionally, some convenientlybroadened functions. The working formula for the phonon free energy then:

Fν = kBT

3N∑α

∞∫0

ln

(2 sinh

�ω

2kBT

)δ(ω − ωα) dω . (7.4)

1http://www.webelements.com2http://www.chemicool.com

120

7.5. Thermodynamic properties

The calculated phonon densities of states of both WZ and ZB CdSxSe1−x areshown in Fig. 7.7 for several alloy compositions. The broadening of the phononspectra on alloying is associated with disorder reducing the coherence of thephonon modes.

Although the phonon DOS are done, the phonon free energies have not yetbeen calculated by the completion of this thesis, and the subsequent analysis ofthe Gibbs free energy of mixing is restricted to the enthalpy of mixing (ΔH) andTΔS terms.

7.5.3 Alloy phase diagram

Helmholtz free energy of formation ΔF of the CdSxSe1−x is given by:

ΔFm = ΔU − TΔS + ΔFν (7.5)

0.5

1

1.5

2

Pho

non

DO

S

x=0

Zincblende supercells

0

0.25

0.5

0.75

1

Pho

non

DO

S x=1/2

0.25

0.5

0.75

1

Pho

non

DO

S

x=3/4

0

0.25

0.5

0.75

1

Pho

non

DO

S

x=1/4

0 50 100 150 200 250 300 350

Frequency (cm -1

)

0.2

0.4

0.6

0.8

1

Pho

non

DO

S x=1

0.5

1

1.5

2

Pho

non

DO

S

x=0

Wurtzite supercells

0

0.2

0.4

0.6

Pho

non

DO

S x=1/2

0 50 100 150 200 250 300 350

Frequency (cm -1

)

0

0.2

0.4

0.6

Pho

non

DO

S

x=3/4

0

0.2

0.4

0.6

0.8

1

Pho

non

DO

S

x=1/4

0 100 200 300 400 500

Frequency(cm -1

)

0

0.1

0.2

0.3

0.4

0.5

0.6

Pho

non

DO

S

x=1

Figure 7.7: Phonon DOS of ZB (left panel) and WZ (right panel) CdSxSe1−x, forx=0, 1/4, 1/2, 3/4 and 1, calculated along Eq. (7.3) for the zone center of 32-atomSQS and artificially broadened with halfwidth parameter of δ(ω) = 5 cm−1.

121

Chapter 7. Computational modelling of properties of Cd(S,Se) alloy

As described above, ΔU and ΔFν in this equation are known from fitting the DFTand phonon calculations to particular functional forms. ΔS corresponds to theentropy of mixing and, in the present analysis, is described in the Bragg-Williamsapproximation as ΔS = −kB[xlogx+(1−x)log(1−x)] (taken per anion). As is wellknown, the Bragg-Williams approximation can be severe. However, the resultingerrors are expected to be important predominantly in frustrated systems andsystems where the phase diagram has a complex topology [De Fontaine, 1979].Neither of these situations applies to the CdSxSe1−x system. We employ theWZ and ZB reference states for the unmixed CdS and CdSe in order to make ameaningful comparison of the free energies of CdSxSe1−x.

From the variation of ΔFm as a function of x, we deduce the temperature-composition phase diagram which shows the stable, metastable, and unstablecomposition regions of a mixed crystal for a given growth temperature. Below acritical temperature TC , the free energy of mixing, remaining negative, acquiresa shape with two minima at two composition points (i.e., binodal1 points x1 andx2) and one maximum. In other way, x1 and x2 are the points where commontangent touches the free-energy curve. With the definition of the chemical po-tential μ(x, T ) = (∂/∂x)Fm(x, T ), it therefore holds μ(x1, T ) = μ(x2, T ). For acomposition in the range 0 < x < x1 or x2 < x < 1, the solid solution ABxC1−x

phase is thermally stable against decomposition. However, in the compositionrange of x1 < x < x2, the total Helmholtz free energy of the system is lowestif the system remains as a mixture of two immiscible solutions ABx1C1−x1 andABx2C1−x2 . The composition region between two binodal points is the miscibilitygap. Thermodynamically unstable phases may exist as metastable in cases wherethe decomposition kinetics is slow, combined with rapid quenching. Within themiscibility gap, there also exists two inflection points x3 and x4 (spinodal points),where ∂2Fm/∂x2 = 0. For a composition x between the binodal and spinodalpoints, the Helmholtz free energy increases when the solid solution decomposeslocally into a mixture of two compositions in the neighborhood of x. In this sense,there exist local decomposition barriers, and the ABxC1−x phase might exist asmetastable one. In Fig. 7.8 and 7.9, the resulting phase diagram of CdSxSe1−x

is shown. The critical alloy formation temperature occurs at a point where thebinodal and spinodal curves meet. The miscibility gap disappears at Tc= 228 K(ZB) and Tc= 225 K (WZ), and it is slightly asymmetric about x = 0.5 due tothe asymmetry of ΔHm. The estimated critical composition is xc = 0.55.

It is evident that the form of the free energy curve changes with temperature.At low temperature, the enthalpy contribution to the Helmholtz free energy asa function of concentration introduces two minima, one at the sulfur-rich side

1In Appendix D, I provide the needed analysis to construct the binodal and spinodal curves,that have required us to solve system of non-linear equations using Newton’s method.

122

7.5. Thermodynamic properties

and the other at the selenium-rich side, hence favoring clustering of like anions.However, the contribution from the enthalpy term can largely be neglected at hightemperatures, where the atoms become randomly mixed by thermal agitation, sothat the free energy curve has a single minimum only.

Figure 7.8: Upper panel: free energy variation in CdSxSe1−x in the ZB struc-ture with alloy composition at T=180 K. Lower panel: x − T phase diagram ofCdSxSe1−x-ZB. Tc, is the critical temperature, marking the solubility limits belowwhich phase separation will occur, is 228 K for the ZB structure in the absenceof vibrational contributions. “I” designates stability, “II” – metastability, “III” -instability region.

123

Chapter 7. Computational modelling of properties of Cd(S,Se) alloy

Figure 7.9: Same as Fig 7.8, for WZ structure.

124

General conclusions

This work lasted over three years, offering me various challenges related to thestudy of semiconductor alloys. As a result, something could have been learnedabout properties of different classes of semiconductor materials.

The structural and electronic properties of the ZB-type pseudobinary solidsolution In(As,P) were examined using a very simple approximation for alloying.The end compounds equilibrium lattice parameters excellently matched the ex-perimental ones; those for the alloy were at that time not known. Too small bandgaps due to GGA have been compared with larger ones obtained with the Engel-Vosko XC potential, and together the trends in the concentration dependence ofband gap were explored and discussed.

The ambiguities associated with experimentally detected (or not yet) andtheoretically reported (or not yet reported) low-symmetric intermediate-pressurephases of zinc chalcogenides have been elucidated using the “enthalpy compar-ison method” at zero temperature. The use of all-electron methods helped meto cope with the experimental techniques which acceptedly matches the reality,allowing me to justify the experimentally recognized non-traditional phases inthe usual ZB-RS hierarchy. My findings regarding the metastability of cinnabarphase (in a very narrow pressure-window) in ZnSe and instability of SC16 phasein ZnS (at any pressure) support the experimental observations and contest pre-vious theoretical reports. My prediction for a small stability pressure-range of aZnSe-SC16 phase is not yet validated by an experimental observation. This isunderstood on the ground that the experimental investigations done on this com-pound have been performed at room temperature, while the SC16 phase seems tobe hindered by large kinetic barrier. The latter needs to be overcome for the ZB– SC16 transition to occur; to this end, experimental techniques should involvethe effect of high temperature simultaneously with compressing / relieving thesample, in analogy with what was already reported in relation with GaAs.

In parallel, I probed structural instabilities for only one compound (ZnSe)and one (ZB) structure, by employing the density functional perturbation theorywhich uniquely gives me access to the whole Brillouin zone, opening new horizonsin the study of phonons. I monitored the transverse acoustical frequency depen-

125

General conclusions

dence on pressure and observed a softening at the zone edge. I concluded thatthe softening corresponds to a doubling of the unit cell, resembling the transfor-mation into the Cmcm structure. However, it would be of great interest for mein the near future to scrutinize the phonon eigenvectors in order to allocate thenew stable phase.

The high-pressure Raman measurements results on (Zn,Be)Se alloy lattice dy-namics were successfully reproduced via first-principles calculations. My ab initiocalculation manifested the convergence of the ZC-TO Be-Se doublet upon apply-ing pressure, particularly near the ZnSe ZB-RS transition. I could get a directinsight into the vibrations mechanism, and I could analyze each vibrational modeand particularly monitor the changing vibration pattern of the Be-Se doublet.The experimentally observed and theoretically determined ultimate convergenceof the doublet is explained and correlated to the “freezing” of the Be-Se bondsfrom the Be-Se region.

Finally, the thermodynamic stability of II-VI alloy Cd(S,Se) upon substitutingions on the anionic sublattice is demonstrated. I found a low value of the criticaltemperature, marking the solubility limits and order-disorder transition on theanionic sublattice, in accordance with a very recent calorimetric measurements,suggesting that Cd(S,Se) WZ-CdS and ZB-CdSe form an ideal solution at lowgrowth temperature, despite a substantial difference in molar volume and anionradius. The ionic thermal contribution to the free energy is determined, but itseffect on the solubility curve is yet to be done. On the other hand, and againsupported by experimental evidence, I found that the alloy structure is ZB whenthe anionic sublattice is Se-rich, yet transits to WZ when the S/Se ratio is equalor greater than 2/3. Full structural characterization is given for the alloy in theZB phase, illustrating that the NN and NNN distances vary typically as found inprevious studied II-VI and III-V ZB alloys.

126

Appendix A

Relation between zincblende,rocksalt and cinnabar structures

A.1 Preliminaries

The three discussed structures are ZB, RS and cinnabar. The latter exists intwo settings, related by mirror symmetry. All three can be described within thehexagonal system, each one with three formula units (cation+anion pairs) perprimitive lattice. All transformations occur within the same hexagonal unit cell(probably changing c/a slightly, but – most important – displacing the atoms).

A.1.1 ZB structure

In the cubic setting, space group F 43m (Nr 216),lattice vectors a1 = (1 0 0), b1 = (0 1 0), c1 = (0 0 1), lattice parameter say A,Wyckoff positions: 4(a) = (0 0 0) and 4(c) = (1

414

14)

with multiplicities (0 0 0)+ (0 12

12)+ (1

20 1

2)+ (1

2120)+.

In the hexagonal setting, space group R3m (Nr 160),lattice vectors t1 = −1

2a1 + 1

2b1, t2 = −1

2b1 + 1

2c1, t3 = 1

2a1 + 1

2b1 + 1

2c1, hence

in terms of the cubic A: t1 = 12(−1 1 0); t2 = 1

2(0 −1 1); t1 = 1

2(1 1 1). As

expected,

the cell volume V =1

4

∣∣∣∣∣∣−1 1 0

0 −1 11 1 1

∣∣∣∣∣∣ =3

4;

(t1 · t2) = −1

4; (t1 · t3) = (t2 · t3) = 0 ;

|t1| = |t2| =1√2

, |t3| =√

3 ; cos (t1, t2) = −1

2.

127

Appendix A. Relation between ZB, RS and cinnabar structures

The hexagonal lattice parameters are a = A/√

2; c = A√

3.

Wyckoff positions in the R3m group: 3(a) = (0 0 z), with z, say, = −18

forcation and 1

8for anion (note that z is not fixed by symmetry, only its difference

between two sublattices must be 14).

The multiplicity of Wyckoff positions in R3m is (0 0 0)+ (23

13

13)+ (1

323

23)+.

From R3m, Sowa [2003, 2005] introduces a further symmetry change intoR3 (Nr 146), with different site symmetry but otherwise the same Wyckoff po-sitions and multiplicity. From R3, we’ll further discuss a transition to the B9structure.

A.1.2 RS structure

In the cubic setting, space group Fm3m (Nr 255), lattice vectors a1, b1, c1 asabove,Wyckoff positions: 4(a) = (0 0 0) and 4(b) = (1

212

12),

with multiplicities as above, (0 0 0)+ (0 12

12)+ (1

20 1

2)+ (1

2120)+.

In the hexagonal setting (with the lattice vectors as described above for ZB), thesequence of space groups considered is (see Fig. 2 of Sowa [2003]):R3m (Nr 166); 3(a) = (0 0 0) and 3(b) = (0 0 1

2) , multiplicity (0 0 0)+ (2

313

13)+

(13

23

23)+ ;

R32 (Nr 155) – the same as above, only with different site symmetry;R3 (Nr 146) – still the same multiplicity, but the Wyckoff position now get avariable internal coordinate, 3(a) = (0 0 z). z may be 0 for cation and 1

2for anion

(or different, maintaining the difference of 12

between the two sublattices). As inthe previous subsection on ZB, the R3 space group will be our starting point ina transition to cinnabar.

128

A.1. Preliminaries

A.1.3 Relation between ZB and RS

As argued by Sowa Sowa [2005], the trans-formation within the same hexagonal setting,varying from z = 1

4to z = 1

2, is not prac-

tical and cannot be realized because it willinvolve a too large atomic displacement (ofA/4

√3≈ 0.433A), breaking all four bonds and

creating six new ones. A more probable tran-sition must involve a rearrangement in thexy plane, accompanied by a small adjustmentalong z. The latter adjustment is inavoidablebecause the cation/anion interplane distancesare different in ZB and in RS, following as3:1:3:1 in the former and 2:2:2:2 in the latter –see Figure on the right. The numbers along thevertical axis denote the z coordinates of differ-ent layers, in the units of 1

24.

11

13

19

21

5

3

22

18

14

10

6

2

A.1.4 B9 structure

A peculiarity of cinnabar phase is that it may exist, and is routinely referredto, in two space groups, P3121 (Nr 152) and P3221 (Nr 154). The two groupsare enantiomorphic, i.e. possess different chirality with respect to the hexagonalaxis; they are related by mirror symmetry. It is seen from the fact that the twogroups have different stacking of hexagonal planes, ABCABC or ACBACB. Inthe following analysis, we’ll make distinction between two cases:

• “Our” cinnabar, space group P3221 (Nr 154), Wyckoff positions:

(u 0 23)

(0 u 13)

(u u 0)

⎫⎪⎬⎪⎭ (3a) ;

(v 0 16)

(0 v 56)

(v v 12)

⎫⎪⎬⎪⎭ (3b) . (A.1)

Let’s choose, for further reference, that we have cations in (3a) and anions in(3b). The u, v coordinates are free but they tend to be u≈ 1

2, v≈ 1

2in the

calculated cinnabar phase (although slightly varying under pressure). Settingthem to u = v = 1

3, as we’ll see, recovers the RS structure. The P3221 group

is a special (high symmetry) case of P32 (Nr 145), which has the following (3a)positions: (x y z), (y x−y z+ 2

3), (−x+y x z+ 1

3). The above positions in the

P3121 group correspond to x=y=u, z=0 for cations and x=y=v, z=12

for anions.

129

Appendix A. Relation between ZB, RS and cinnabar structures

• “Genuine” cinnabar, space group P3121 (Nr 152), Wyckoff positions:

(u 0 13)

(0 u 23)

(u u 0)

⎫⎪⎬⎪⎭ (3a) ;

(v 0 56)

(0 v 16)

(v v 12)

⎫⎪⎬⎪⎭ (3b) . (A.2)

Again, we assume, for further reference, to have cations in (3a) and anions in(3b). The P3121 group is a special (high symmetry) case of P31 (Nr 144), whichhas the following (3a) positions: (x y z), (y x−y z+ 1

3), (−x+y x z+ 2

3).

It is obvious that the only difference between the P31 and P32 groups is theplane stacking sequence, z+ 1

3/ z+ 2

3.

A.2 Path from ZB via cinnabar to RS

Having already brought the zincblende into the hexagonal setting in the R3m(Nr 160) and further R3 (Nr 146) group, with z=−1

8for cations and 1

8for anions,

we have explicitly the following positions:

(0 0 2124

)

(23

13

524

)

(13

23

1324

)

⎫⎪⎬⎪⎭ (3a) ;

(0 0 324

)

(23

13

1124

)

(13

23

1924

)

⎫⎪⎬⎪⎭ (3b) .

Now we can break the mirror symmetry, introducing an uniform translation inthe (x, y) plane, which will bring us into either P31 or P32 group.

• Translation (13

13

0), into P32:

(13

13

2124

)

(0 23

524

)

(23

0 1324

)

⎫⎪⎬⎪⎭ (3a)– cations ;

(13

13

324

)

(0 23

1124

)

(23

0 1924

)

⎫⎪⎬⎪⎭ (3b)– anions . (A.3)

This has to be transformed into “our” cinnabar of (A.1), to which a translation(0 0 18

24) has to be added:

(u 0 1024

)

(0 u 224

)

(u u 1824

)

⎫⎪⎬⎪⎭ (3a)– cations ;

(v 0 2224

)

(0 v 1424

)

(v v 624

)

⎫⎪⎬⎪⎭ (3b)– anions . (A.4)

For “ideal cinnabar” values u = v = 12, the top view of the unit cell is shown

in the following figure. The circles mark positions of atoms in the ZB structure,squares – in cinnabar. The numbers indicate the z-coordinate in the units of 1

24.

130

A.2. Path from ZB via cinnabar to RS

1

2

t

t

10

1821

5

10

182

2

10

10

2

5

13

2

13

(cations)1

2

t

t

22

14

14

11

36

6

2222

14

22

19

11

14

19

(anions)

Cations move by 16

in the plane and by 324

downwards. The diplacement of eachatom is (assuming c, a to be as in the ideal cubic ZB with lattice constant A):

d =

√(a

6

)2

+( c

8

)2

=A√

35

24≈ 0.2465 A . (A.5)

As this (x, y)-movement continues, keeping however the z coordinate, the atomsarrive at the next symmetric positions, marked in the figure by pentagons:

(13

0 1024

)

(0 13

224

)

(23

23

1824

)

⎫⎪⎬⎪⎭ (3a)– cations ;

(13

0 2224

)

(0 13

1424

)

(23

23

624

)

⎫⎪⎬⎪⎭ (3b)– anions . (A.6)

This is the RS structure, by the force of the following. Introducing in the RSphase, as grasped by the R3 group, a uniform displacement (0 1

30) and hence a

symmetry lowering from R3, we arrive in the P32 (Nr 145), hence with Wyckoffpositions (x y z) (y x−y z+ 2

3) (−x +y x z+ 1

3). Writing it down with x=0,

y= 13

(as the abovementioned translation) and z= 224

(cations), z= 1424

(anions), wearrive exactly at the coordinates (A.6). The atomic displacement from cinnabarto rocksalt is

d =a

6=

A√

2

12≈ 0.118 A .

131

Appendix A. Relation between ZB, RS and cinnabar structures

• Translation (23

23

0), into P31:

(23

23

2124

)

(13

0 524

)

(0 13

1324

)

⎫⎪⎬⎪⎭ (3a)– cations ;

(23

23

324

)

(13

0 1124

)

(0 13

1924

)

⎫⎪⎬⎪⎭ (3b)– anions . (A.7)

This has to be transformed into “genuine” cinnabar (A.2), to which a translation(0 0 18

24) has to be added:

(u 0 224

)

(0 u 1024

)

(u u 1824

)

⎫⎪⎬⎪⎭ (3a)– cations ;

(v 0 1424

)

(0 v 2224

)

(v v 624

)

⎫⎪⎬⎪⎭ (3b)– anions . (A.8)

For “ideal cinnabar” values u=v= 12, the top view of the unit cell is shown in the

following figure. The circles mark positions of atoms in the ZB structure, squares– in the cinnabar. The numbers indicate the z-coordinate in the units of 1

24.

1

2

t

t

2

13

10

10

21

18

22

5

10

2

13

5

10

18

(cations)1

2

t

t

14

22

22

36

6

1414

11 19

11

22

14

22

19

(anions)

As this movement continues, keeping the z coordinate, we arrive at

(23

0 224

)

(0 23

1024

)

(13

13

1824

)

⎫⎪⎬⎪⎭ (3a)– cations ;

(23

0 1424

)

(0 23

2224

)

(13

13

624

)

⎫⎪⎬⎪⎭ (3b)– anions , (A.9)

marked in the figure by pentagons. As before for the mirror-equivalent structuretransformation, this one brings us to rocksalt. The RS structure written down in

132

A.3. Path from ZB directly to RS

the R3 space group must undergo a symmetry lowering: a (23

0 0) displacement,accompanied for some (0 0 z) shift. The first displacement brings the system intothe P31 (Nr 144) group, with Wyckoff positions: (x y z) (y x−y z+ 1

3) (−x +

y x z+23). Writing it down with x= 2

3, y=0 (as the abovementioned translation)

and z= 224

(cations), z= 1424

(anions), we arrive exactly at the coordinates (A.9).

A.3 Path from ZB directly to RS

There is another transformation from ZB to RS which do not pass throughcinnabar; [Sowa, 2003] is entirely devoted to its analysis. Below, only certainissues are discussed (in a bit more of detail) which are of our immediate concern.

• Within the P32 (Nr 145) space group, this is the movement from the coor-dinates as in (A.3) to

(13

0 2224

)

(23

23

624

)

(0 13

1424

)

⎫⎪⎬⎪⎭ (3a)– cations ;

(0 13

224

)

(13

0 1024

)

(23

23

1824

)

⎫⎪⎬⎪⎭ (3b)– anions , (A.10)

which are positions within P32 for x= 23, y= 2

3, z= 6

24(cations), z= 18

24(anions) and

– hence two sublattices do only differ by Δ z= 12

– a valid RS. The transformationis shown in the Figure (circles: atoms in ZB, pentagons: atoms in RS; numbers– values of z for each atom i the units of 1

24). A similar drawing is given in Fig. 3

of Sowa [2003].

1

2

t

t

13

21

5

5

14 6

1422

22

13

(cations)

2

t1

t

11

10

19

3

2

11 19

18

2

10

(anions)

133

Appendix A. Relation between ZB, RS and cinnabar structures

• Within the P31 (Nr 144) space group, this is the movement from the coor-dinates as in (A.7) to

(23

0 1424

)

(0 23

2224

)

(13

13

624

)

⎫⎪⎬⎪⎭ (3a)– cations ;

(23

0 224

)

(0 23

1024

)

(13

13

1824

)

⎫⎪⎬⎪⎭ (3b)– anions , (A.11)

which are positions within P31 for x= 13, y = 1

3, z = 6

24(cations), z = 18

24(anions).

The transformation (mirror to the previous one) is shown in the Figure.

1

2

t

t5

14

22

6

5

21

22

13

14

13

(cations)1

2

t

t

19

2

11

10

3

11

18

2

10

19

(anions)

These direct transformations involve atomic displacements of the magnitude

d =

√(a

3

)2

+( c

24

)2

=A√

35

24≈ 0.2465 A

and hence miraculously enough – for the c/a ratio being still that as in the perfectZB lattice – these displacements are equal to the ZB–cinnabar displacements of(A.5).

A worthy question might be – whether the structure, half-way of this trans-formation, indeed passes through the cinnabar phase. Let’s construct the corre-sponding coordinates, with (x, y) indeed half-way of the path, but z coordinatealready as in the RS (because in cinnabar it must be the same). The coordinates

134

A.4. Conclusions

will be (within the P31 group):

(56

23

624

)

(56

16

1424

)

(13

16

2224

)

⎫⎪⎬⎪⎭ (3a)– cations ;

(23

56

1824

)

(16

13

224

)

(16

56

1024

)

⎫⎪⎬⎪⎭ (3b)– anions , (A.12)

Now the game is to find such uniform translation that would recover from thesecoordinates the standard form (A.1) for cinnabar – if not for general u and v thenat least for u=v=1

2. Apparently it does not work (or, a more tricky transformation

is needed).

A.4 Conclusions

The transition from ZB may occur into both cinnabar and RS structures. Themagnitudes of these transitions (measured by the individual displacement of eachatom in the course of transformation) are equal (under the assumption that c/adoes not change), even if two paths are completely different. The magnitude ofthe displacement involved is nearly 1/4 of the cubic (ZB) lattice constant, e.g.,≈0.6 of the cation–anion bond length – not small but probable (not that theatomic displacements are probably mostly “collateral” so that the bond anglesbut not lengths are primarily concerned).

Moreover, a transition cinnabar – rocksalt is possible and apparently easy,because two structures are more than two times close, in terms of the said dis-placement. What follows from this, in terms of research, is that three metastablephases – ZB, cinnabar, RS – make a triangle; transformations between them hasto studied independently, in order to estimate the corresponding barrier (pairwisebetween phases).

In view of our present research, the “triangularity” of path seems enlighten-ing for understanding the “backstroke” phenomenon: the pressure, for whateverreason, promotes the structure from ZB directly to RS, from where it finds con-venient, as the pressure is removed, to take a different path and go visit the B9on the return.

135

Appendix B

Bonds and angles in cinnabarand SC16 phases:crystallographic considerationsand explicit formulas

B.1 Cinnabar phase

The cinnabar phase can be described by either of two enantiomorphic (mirror-symmetric) space groups:• “Our” cinnabar, space group P3221 (Nr 154), Wyckoff positions:

(u 0 23)

(0 u 13)

(u u 0)

⎫⎪⎬⎪⎭ (3a) ;

(v 0 16)

(0 v 56)

(v v 12)

⎫⎪⎬⎪⎭ (3b) . (B.1)

Assume cations in (3a) and anions in (3b); u≈ v≈ 0.5;• “Genuine” cinnabar, space group P3121 (Nr 152), Wyckoff positions:

(u 0 13)

(0 u 23)

(u u 0)

⎫⎪⎬⎪⎭ (3a) ;

(v 0 56)

(0 v 16)

(v v 12)

⎫⎪⎬⎪⎭ (3b) . (B.2)

It suffices to discuss “genuine” cinnabar only, since “our” one is related to itby mirror symmetry. Around a cation C0 at (1−u 1−u 0) one finds four anions:

A1(v 0 − 1

6

); A2

(v 1 − 1

6

); A3

(0 v 1

6

); A4

(1 v 1

6

).

137

Appendix B. Bonds and angles in cinnabar and SC16 phases

We’ll calculate distances and angles using the metric tensor,

G =

⎛⎜⎝ a2 −a2

20

−a2

2a2 0

0 0 c2

⎞⎟⎠ for the hexagonal lattice.

With this, the scalar product in terms of crystallographic coordinates of twovectors is:

(r1| r2) = (x1 y1 z1) (G)

⎛⎝x2

y2

z2

⎞⎠ .

The bond lengths split into two pairs:

|A1 − C0||A3 − C0|

}=

√a2(u2 + v2 + uv − 2u − v + 1) +

c2

36;

|A2 − C0||A4 − C0|

}=

√a2(u2 + v2 + uv − u − 2v + 1) +

c2

36. (B.3)

The bond angles do apparently split as 1+2+2+1:

(A1 − C0)(A3 − C0) = a2

(u2 − v2

2+ uv − 2u − v + 1

)− c2

36;

(A1 − C0)(A2 − C0)(A3 − C0)(A4 − C0)

}= a2

(u2 + v2 + uv − 3u

2− 3v

2+

1

2

)+

c2

36;

(A1 − C0)(A4 − C0)(A2 − C0)(A3 − C0)

}= a2

(u2 − v2

2+ uv − 3u

2+

1

2

)− c2

36;

(A2 − C0)(A4 − C0) = a2

(u2 − v2

2+ uv − u + v − 1

2

)− c2

36. (B.4)

Inversely, around an anion A0 at (1−v 1−v 12) one finds four cations:

C1(u 0 1

3

); C2

(u 1 1

3

); C3

(0 u 2

3

); C4

(1 u 2

3

).

The bond lengths are expectedly like those of Eq. (B.3):

|C1 − A0||C3 − A0|

}=

√a2(u2 + v2 + uv − u − 2v + 1) +

c2

36;

|C2 − A0||C4 − A0|

}=

√a2(u2 + v2 + uv − 2u − v + 1) +

c2

36. (B.5)

The grouping of bond angles is like in Eqs. (B.4):

(C1 − A0)(C3 − A0) = a2

(−u2

2+ v2 + uv − u − 2v + 1

)− c2

36;

138

B.1. Cinnabar phase

(C1 − A0)(C2 − A0)(C3 − A0)(C4 − A0)

}= a2

(u2 + v2 + uv − 3u

2− 3v

2+

1

2

)+

c2

36;

(C1 − A0)(C4 − A0)(C2 − A0)(C3 − A0)

}= a2

(−u2

2+ v2 + uv − 3v

2+

1

2

)− c2

36;

(C2 − A0)(C4 − A0) = a2

(−u2

2+ v2 + uv + u − v − 1

2

)− c2

36. (B.6)

Note that whereas these formulae are symmetric to (B.4) in the sense of inter-changing u↔ v, the C-A-C and A-C-A angles as such, for a given general (u, v)pair, are not identical (with the exception of the second angle in each group,which is, indeed, symmetric with respect to u and v). Hence there are, in total,seven different bond angles in the cinnabar structure.

Consider now special cases. For “ideal” cinnabar with u = v = 12, all bond

lengths become identical,

|A? − C0| =

√a2

4+

c2

36,

whereas the angles split into three pairs:

(A1 − C0)(A3 − C0)(A2 − C0)(A4 − C0)

}= −

(a2

8+

c2

36

);

(A1 − C0)(A2 − C0)(A3 − C0)(A4 − C0)

}= −

(a2

4− c2

36

);

(A1 − C0)(A4 − C0)(A2 − C0)(A3 − C0)

}=

a2

8− c2

36. (B.7)

Take now c/a = 2.28 (about as calculated for zero pressure) Then bond lengthssquared are 0.3944 a2, the three cosines from scalar products of Eq. (B.7) become−0.2694, −0.1056 and −0.0194, and the corresponding angles - 133◦, 105◦ and93◦. This is indeed as was “directly” yielded by ab initio calculations.

For “cinnabar like RS” with u = v = 13, all bond lengths are

|A? − C0| =

√a2

3+

c2

36,

and the angles make two groups of three:

(A1 − C0)(A3 − C0)(A1 − C0)(A4 − C0)(A2 − C0)(A3 − C0)

⎫⎬⎭ =

a2

6− c2

36;

(A1 − C0)(A2 − C0)(A3 − C0)(A4 − C0)(A2 − C0)(A4 − C0)

⎫⎬⎭ = −

(a2

6− c2

36

).

139

Appendix B. Bonds and angles in cinnabar and SC16 phases

With an additional condition c = a√

6, all these scalar products become zero.

B.2 SC16 structure

The SC16 structure (space group Pa3, Nr 205) has 8c sites (the following num-bering of sites differs from that in International Tables)

(1) ( u u u ) (2) ( u u u )

(3) ( 12−u u 1

2+u ) (4) ( 1

2+u u 1

2−u )

(5) ( 12+u 1

2−u u ) (6) ( 1

2−u 1

2+u u )

(7) ( u 12+u 1

2− u ) (8) ( u 1

2−u 1

2+ u )

(B.8)

for cations, and other 8c (with v parameter) for anions. In the literature, (x1, x2)are often used instead of (u, v); we’ll come back to their meaning later. Withthis definition, cation at (u u u) and anion at (v v v) will be first neighbours,connected by a bond of length a0(v − u)

√3. For u≈ 0.16 and v≈ 0.36, the local

neighbourhood of a given cation (C1) (u u u) is:

(A1) ( v v v )

(A4) ( v− 12

v 12−v )

(A6) ( 12−v v− 1

2v )

(A8) ( v 12−v v− 1

2)

(B.9)

Stewart Clark in his thesis of 1996 available chapterwise on the web1 uses twodifferent parameters, x1 and x2, to characterize cation and anion positions corre-spondingly, whose relation to u and v is recovered below. There are two differ-ent (nearest neighbours) cation-anion distances, both (differently) depending onx1 + x2 only. From (B.8) and (B.9), the bond lengths (in terms of cubic latticeconstant) are:

|C1 − A1| : (v − u)√

3 ; (B.10)

|C1 − A4||C1 − A6||C1 − A8|

⎫⎬⎭ :

√1

2+ 3v2 + 3u2 − 2uv − 2v . (B.11)

They are referred to in the following as ‘single’ and ‘triple’ bond lengths, dS

and dT, correspondingly. From this it is clear that the distorted coordinating

1http://cmt.dur.ac.uk/sjc/thesis/thesis/node44.html

140

B.2. SC16 structure

tetrahedra do (locally) maintain trigonal symmetry. The bond angles will belabelled as ‘single-triple’ and ‘triple-triple’, derived from scalar products:

(A4 − C1)(A1 − C1)(A6 − C1)(A1 − C1)(A8 − C1)(A1 − C1)

⎫⎬⎭ = (v − u)(v − 3u) ;

(A6 − C1)(A4 − C1)(A8 − C1)(A4 − C1)(A8 − C1)(A6 − C1)

⎫⎬⎭ = v − (v − u)(v + 3u) − 1

4.

Bond angles:

cos(A1 − C1 − A4)cos(A1 − C1 − A6)cos(A1 − C1 − A8)

⎫⎬⎭ = − 3u − v√

3(12

+ 3v2 + 3u2 − 2uv − 2v)≡ cos αST;

cos(A4 − C1 − A6)cos(A4 − C1 − A8)cos(A6 − C1 − A8)

⎫⎬⎭ = − 1 + 4v2 − 12u2 + 8uv − 4v

2(1 + 6v2 + 6u2 − 4uv − 4v)≡ cos αTT.

The local neighbourhood of the anion (A1) (v v v) is:

(C1) ( u u u )

(C4) ( u+ 12

u 12−u )

(C6) ( 12−u u+ 1

2u )

(C8) ( u 12−u u+ 1

2)

It is seen that the formula of anion positions around a cation and cations aroundan anion are not symmetric with respect to (u↔v). This is so because someions need to be translated, to make a compact first shell. The bond lengths areexpectedly as before:

|A1 − C1| : (v − u)√

3 ;

|A1 − C4||A1 − C6||A1 − C8|

⎫⎬⎭ :

√1

2+ 3v2 + 3u2 − 2uv − 2v

However, the bond angles are different. The scalar products are now:

(C4 − A1)(C1 − A1)(C6 − A1)(C1 − A1)(C8 − A1)(C1 − A1)

⎫⎬⎭ = (v − u)(3v − u − 1) ;

(C6 − A1)(C4 − A1)(C8 − A1)(C4 − A1)(C8 − A1)(C6 − A1)

⎫⎬⎭ =

1

4− v + (v − u)(u + 3v − 1) .

141

Appendix B. Bonds and angles in cinnabar and SC16 phases

Consider the shell of next nearest neighbours. For each of four anions (A1, A4,A6, A8) connected to the C1 cation, we list three further cations connected. Thenumbering of cations is according to the list (B.8), eventually adding translations.

— neighbours to A1 — — neighbours to A6 —

(C4) ( 12+u u 1

2−u ) d1 (C7) ( u u− 1

212−u ) d2

(C6) ( 12−u 1

2+u u ) d1 (C6) ( 1

2−u u− 1

2u ) d1

(C8) ( u 12−u 1

2+u ) d1 (C3) ( 1

2−u u 1

2+u ) d2

— neighbours to A4 — — neighbours to A8 —

(C7) ( u 12+u 1

2−u ) d2 (C5) ( 1

2+u 1

2−u u ) d2

(C5) ( u− 12

12−u u ) d2 (C8) ( u 1

2−u u− 1

2) d1

(C4) ( u− 12

u 12−u ) d1 (C3) ( 1

2−u u u− 1

2) d2

The two different cation-cation distances (of twelve next-nearest neighbours tothe central C1), marked in the above list, are:

d1 =

√1

2− 2u + 4u2 ; d2 =

√1

2− 2u + 8u2 , (B.12)

which make, for u = 0.153 (ZnS at zero pressure) d1 ≈ 0.536 and d2 ≈ 0.617. Tobe noted that the (C2) at ( u u u ) is at 2u

√3≈ 0.530 from (C1), although

not connected to it via an anion. Stewart Clark gives in his thesis a formulafor the distance to the nearest non-bonded neighbour, R1

5/a0 =√

3(1/2 − 2x1),to be compared with the above 2u

√3, which yields u = 1

4− x1. Therefore,

apparently the two internal parameters after Clark are mearured from (14

14

14)

in the opposite directions, i.e., our v is related to his x2 as v = 14

+ x2. Thenv−u = x1+x2, and we recover the formula for the shortest bond length (B.10),RA/a0 =

√3(x1 + x2) after Clark. However, the other bond length according

to Clark, RB/a0 =√

2(x1 + x2)2 − (x1 + x2) + 1/4, in this convention should be√1/4 + 2v2 + 2u2 − 4uv, that does not match our result (B.11). The controversy

is important, because Clark claims that only relative positions (his x1+x2, ourv−u) matter for bond lengths whereas it seems from the above that they bothenter independently.

Let’s estimate bond lengths and angles from the above formulae, for the pa-rameters given in the WIEN2k’s struct (from ab initio ) calculation file: a=12.55446Bohr = 6.6435314 A, u=0.15331443, v=0.35997171. Then the ‘single’ bond lengthis 2.37799 A, the ‘triple’ one 2.385515 A. This agrees well with ab initio simu-lation, but it is not clear which branch shows what (since they about cross inthis point). Concerning the angles, cos αST = −0.16074, cos αTT = −0.46124;αST = 99◦; αTT = 117◦ –somehow a bit different.

142

Appendix C

Phonon energy expression

In the following, we stay in the harmonic approximation, but take into accountthe quantum (Bose) statistics for phonons. The mean number of phonons ωq (weskip writing q in the following, as the statistics does not depend on it), excitedat temperature T , is:

nω =1

e�ω

kBT − 1. (C.1)

The internal energy comes from summing up the energies �ω(nω + 12) over (non-

interacting) phonons whose density is given by g(ω):

U =

ωmax∫0

�ω

(1

e�ω

kBT − 1+

1

2

)g(ω)dω =

ωmax∫0

�ω

2

e�ω

kBT + 1

e�ω

kBT − 1g(ω)dω . (C.2)

We assumed here that the density of modes is normalized “correctly”, i.e., toyield

∫ ωmax

0g(ω)dω = 3N . The constant-volume specific heat is

CV =∂U

∂T; assume x ≡ �ω

kBTand f(x) =

ex + 1

ex − 1;

∂f

∂T=

∂f

∂x

∂x

∂T=

2ex

(ex − 1)2

x

T;

CV =

ωmax∫0

�ω

kBT︸︷︷︸=x

kBT

2

2ex

(ex − 1)2

x

Tg(ω)dω = kB

ωmax∫0

(x

ex2 − e−

x2

)2

g(ω)dω . (C.3)

Now we turn to calculation of the (Helmholtz) free energy F = −kBT lnZ andentropy S = −∂F/∂T . The non-zero entropy comes from the fact that each oscil-lator, independently of all others, can be excited from its ground state in differentways (characterized by the number of phonons created). F and S will eventuallyappear as sums over free energis and entropies of individual oscillators. This isan approximation in the sense that it neglects interactions between oscillators,but this a is exactly the essence of the harmonic approximation.

143

Appendix C. Phonon energy expression

We start from evaluation of the partition function Z:

Z =∑

i(states)

e− Ei

kBT ; Ei =∑k

(oscillators)

�ωk

(nk +

1

2

). (C.4)

The oscillators k are in fact the normal modes of vibration characterized bydifferent ω(q); the sum over them will be at the end replaced by an integral overg(ω)dω. A “state” of a system consistinng of many oscillators means a vectori ≡ {n1, . . . , nk . . .} describing the excitation number of each oscillator. Sincephonons with a given ω(q) are all indistinguishable, the state of a given oscillatoris uniquely characterized by the phonon number, nk. For each oscillator, the nk

value numbering its states goes from zero to infinity. Different oscillators are alldistinguishable, even in the case the corresponding modes are degenerate.

Z =∑

{... nk...}e− 1

kBT

P

k�ωk(nk+ 1

2)=

∑{... nk...}

∏k

e− �ωk

kBT(nk+ 1

2). (C.5)

Now we note that the sum over all states means scanning all nk values indepen-dently, i.e.,

∑{... nk...}(· · ·) =

∏k

∑nk

(· · ·); it is like running through all cells of ak-dimensional cube. With this, the expression for Z factorizes as follows:

Z =∏

k

e− �ωk

2kBT

∞∑n=0

(e− �ωk

kBT

)n

=∏

k

e− �ωk

2kBT

(1 − e

− �ωkkBT

)−1

, (C.6)

so that

F = −kBT ln Z = −kBT∑

k

[− �ωk

2kBT+ ln

(1 − e

− �ωkkBT

)−1]

=∑

k

[�ωk

2+ kBT ln

(1 − e

− �ωkkBT

)]. (C.7)

Replacing the sum over individual oscillators by an integral over continuous fre-quencies with density of modes and introducing, as in Eq. (C.3), x = �ω/(kBT )yields, in several different equivalent forms:

F = kBT

ωmax∫0

[x

2+ ln

(1 − e−x

)]g(ω)dω = kBT

ωmax∫0

{x

2+ ln

[e−x (ex − 1)

]}g(ω)dω

= kBT

ωmax∫0

[ln(ex − 1) − x

2

]g(ω)dω (C.8)

144

Appendix C. Phonon energy expression

= kBT

ωmax∫0

{ln[e

x2

(e

x2 −e−

x2

)]−x

2

}g(ω)dω = kBT

ωmax∫0

[x

2+ ln

(e

x2 −e−

x2

)−x

2

]g(ω)dω

= kBT

ωmax∫0

ln(2 sinh

x

2

)g(ω)dω . (C.9)

S = −∂F

∂T= kB

ωmax∫0

[xex

ex − 1− ln(ex − 1)

]g(ω)dω (C.10)

= kB

ωmax∫0

[x

ex − 1− ln

(1 − e−x

)]g(ω)dω . (C.11)

For S, an equivalent formula is sometimes used, in terms of average number ofphonons in a given mode, Eq. (C.1), which expresses

ex − 1 =1

nω

; ex =nω + 1

nω

. (C.12)

With this, e.g. the integrand of Eq. (C.11) becomes:

x

ex − 1− ln(1 − e−x) = nω x − ln[e−x (ex − 1)︸ ︷︷ ︸

1nω

] = nω ln ex︸︷︷︸x

+ ln(nω ex)

= nω lnnω + 1

nω

+ ln

(nω

nω + 1

nω

)= nω ln(nω + 1) − nω ln nω + ln(nω + 1)

= (nω + 1) ln(nω + 1) − nω ln nω , (C.13)

hence

S = kB

ωmax∫0

[(nω + 1) ln(nω + 1) − nω ln nω

]g(ω)dω , (C.14)

an equivalent expression, bearing the parametric dependence on T .Let’s discuss now the practical aspects of calculating the above integrals, tak-

ing Eq. (C.9) as an example. The integrand of F , ln(2 sinh x

2

), goes asymptoti-

cally as ∼ ln x for x→0+ and as ∼ x2

for x→∞ (see Figure). Note that we don’thave an exact density of modes g(ω) but are obliged to mimic it by sampling,having (hopefully many) discrete peaks from a supercell calculation:

g(ω) ≈∑

α

δ(ω − ωα) . (C.15)

145

Appendix C. Phonon energy expression

1 2 3 4x

-2

-1

0

1

2

xln

(ex -1

) -

x/2

ln(x)

x/2Table. x = �ω

kBTfor some values

of ω and T .

ω T (K)(cm−1) 50 300 400

50 1.44 0.24 0.18250 7.20 1.20 0.90400 11.51 1.92 1.44

In practical sense, δ(ω) can be taken either as true δ-peaks, in which case theintegrals reduce to the sum over modes,

F = kBT∑

α

ln

(2 sinh

�ωα

2kBT

), (C.16)

or we introduce some artificial smearing to Lorentz or other function, with somead hoc width parameter

F = kBT∑

α

∞∫0

ln

(2 sinh

�ω

2kBT

)δ(ω − ωα) dω . (C.17)

In either case, the three acoustic modes ωα = 0 must be removed before takingthe sum (so I think, because they will result in divergence if included, and theirimportance tends to zero as the supercell size increases). As for other modes, theTable shows that their corresponding x values will be typically spread around theroot, strongly drifting with temperature, that may lead to an interesting tem-perature dependence of the resulting F . However, as the multiplying function is“well-behaving”, one may hope that the fine details of the spectrum, also whetherit is artificially broadened or taken as δ-peaks, will not drastically influence theresult. Still, preliminary tests are advisable.

146

Appendix D

Calculation of binodal andspinodal linesin the alloy phase diagram

The change of the Helmholtz free energy on alloying (x)AC+(1−x)BC → AxB1−xCis

ΔF (x, T ) = ΔU(x) − TΔS(x), (D.1)

where ΔU is the change in total energy (of an alloy, with respect to weightedvalues of pure constituent systems), which can be obtained from zero-temperaturetotal energy calculations, and ΔS(x) is configurational entropy,

ΔS = −kB [x ln x + (1 − x) ln(1 − x)] . (D.2)

The energy is parametrized according to convenience, typically by a polynomial.We accept a cubic interpolation with three parameters,

ΔU = (α + βx)x(1 − x) + (1 − x)γ . (D.3)

The function ΔF (x, T ) typically passes, on increase of T , through several regimes:first a single smooth maximum at some intermediate x value, then two localminima emerging and drifting inwards from the edges x = 0 and x = 1; finallythe two minima merge into a single one. In the two-minima regime, the commontangent gives the concentrations x1, x2 which mark the binodal lines. We’ll searchfor these two values under a parametrized dependence on T . In the following, weskip Δ in front of F (x, T ), for brevity. The common tangent condition reads:

∂F

∂x

∣∣∣∣x1

=∂F

∂x

∣∣∣∣x2

=F (x2) − F (x1)

x2 − x1

. (D.4)

147

Appendix D. Calculation of binodal and spinodal lines

This amounts to a system of two nonlinear equations:⎧⎨⎩

∂F∂x

∣∣x1

− F (x2)−F (x1)x2−x1

≡ f1(x1, x2) = 0 ;

∂F∂x

∣∣x2

− F (x2)−F (x1)x2−x1

≡ f2(x1, x2) = 0 .(D.5)

Its numerical solution can be attempted iteratively, using the Newton method:⎛⎝ ∂f1(x1,x2)

∂x1

∂f1(x1,x2)∂x2

∂f2(x1,x2)∂x1

∂f2(x1,x2)∂x2

⎞⎠(

δx1

δx2

)=

( −f1(x1, x2)

−f2(x1, x2)

), (D.6)

correcting in each step

⎛⎝ x

(i)1 + δx1

x(i)2 + δx2

⎞⎠ →

⎛⎝ x

(i+1)1

x(i+1)2

⎞⎠ ,

having started from a reasonable guess of the trial vector and iterating to suffi-

ciently small

(δx1

δx2

). The solution of the 2×2 system (D.6),

(w11 w12

w21 w22

)(δx1

δx2

)=

(b1

b2

), reads:

δx1 =b1w22 − b2w12

w11w22 − w12w21

, δx2 =b2w11 − b1w21

w11w22 − w12w21

. (D.7)

The matrix elements of the Wronskian are as follows:

w11 ≡ ∂f1

∂x1

= F ′′(x1) +F ′(x1)

x2 − x1

− F (x2) − F (x1)

(x2 − x1)2;

w12 ≡ ∂f1

∂x2

= − F ′(x2)

x2 − x1

+F (x2) − F (x1)

(x2 − x1)2;

w21 ≡ ∂f2

∂x1

=F ′(x1)

x2 − x1

− F (x2) − F (x1)

(x2 − x1)2;

w22 ≡ ∂f2

∂x2

= F ′′(x2) − F ′(x2)

x2 − x1

+F (x2) − F (x1)

(x2 − x1)2.

The explicit form of F (x, T ) with its first and second derivatives in x is:

F = (α + βx)x(1 − x) + (1 − x)γ + kBT [x ln x + (1 − x) ln(1 − x)] ;

∂F

∂x= (α − δ) + 2(β − α)x − 3βx2 + kBT ln

x

1 − x;

∂2F

∂x2= 2(β − α) − 6βx +

kBT

x(1 − x).

148

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176

Index

abinit , 42, 51, 98, 99α-Sn (structure), 11anharmonicity, 50, 54annealing, 18

band gap, 49, 67, 70, 72, 83, 111, 116,125

β-Sn (structure), 23, 24binodal (points, curves), 60, 122Bloch theorem, 34Born–Oppenheimer, 48, 51Born–von Karman, 34, 52Brillouin zone, 52, 57, 98bulk modulus, 9, 20, 69–71, 73, 78, 96

CALPHAD, 26, 28chalcopyrite (structure), 18, 69charge transfer, 72cinnabar (structure), 9, 23–25, 75–78,

80, 81, 127, 130cluster expansion, 27Cmcm (structure), 24, 25, 102, 108correlation effects, 31CPA, 15Cu2Au (structure), 69CuAu (structure), 18, 69CuPt (structure), 18

DAC, 19, 23DFT, 7, 29–31, 33, 37, 39, 43, 63, 67,

70, 77, 117, 122diagonalization, 40diamond, 19, 20, 22, 23

direct method (in lattice dynamics), 9,56

displacive transition, 21, 45, 102downstroke (pressure studies), 21, 80,

81, 92

effective crystal, 15elastic parameters, 26electron density, 20electronegativity, 119energy barriers, 18, 21, 22, 30, 45, 81,

82, 91, 92energy cutoff, 39, 117enthalpy, 8, 20, 43, 45, 46, 58, 96, 97,

118entropy, 143, 147

vibrational, 27, 64EXAFS, 11, 15–17, 113exchange energy, 30

Fermi distribution function, 63finite displacements (for phonon calcu-

lation), 55first-order transitions, 20, 21, 25, 44,

104force constants, 51, 53free energy, 20, 27, 28, 65, 117frozen phonon, 8, 9, 43, 55, 65, 98

GaAs, 91–93, 108GaP, 108GGA, 32, 33, 43, 68–71, 73, 77–80, 82,

83, 94, 96, 99, 125

177

Index

Gibbs free energy, 20, 46, 57, 78, 94Gruneisen parameter, 64graphite, 19, 22, 23Green’s function, 57group-subgroup relationship, 21, 102

harmonic approximation, 8, 9, 50, 53,117, 143

harmonic oscillator, 50Hartree–Fock theory, 30, 31, 38Helmholtz free energy, 8, 121, 122, 143,

147HgS, 88Hooke’s law, 53hysteresis, 21, 22

ionicity, 113

Kohn–Shamequations, 31, 34, 37, 98functions, 30, 35, 40, 42

Landau phase transitions theory, 21LDA, 31, 32, 43, 69, 77–80, 82, 83, 94,

96, 116LDA+U , 33lever rule (in phase diagrams), 60linear response, 8, 9, 42, 51, 57, 65, 97,

98local orbitals (in FP-LAPW), 41long-range ordering, 18luzonite (structure), 69

metastable phase, 19miscibility gap, 59, 62, 122molecular dynamics, 33, 56, 66, 81MREI, 15muffin-tin

geometry, 40spheres, 68

Murnaghan equation of state, 46, 69,79, 95, 96

octet rule, 13optical band gap, 8, 9opto-electronics, 111order parameter, 21

PAW, 39percolation model, 103percolation threshold, 104phase diagrams, 18phonon

dispersion, 8, 97, 98eigenvectors, 7, 126softening, 21, 98, 100–102, 126zone-center modes, 52

photoluminescence, 22planewave cutoff, 68PP, 34, 37–40, 42, 69, 75, 93, 94, 98, 99,

105, 117pseudobinary alloy, 8, 13, 17, 42, 58, 62,

67PW, 37, 38, 41, 42, 69, 75, 93, 94, 98

quasiharmonic approximation, 9, 47, 64,65

Raman scattering, 22reconstructive transition, 21RS (structure), 24, 25, 75–77, 81–84,

91–96, 102, 104, 108, 127–131,133, 139

SC16 (structure), 9, 23, 24, 75, 78, 92–94, 140

second-order transitions, 20, 21self-interaction, 31semicore states, 40, 112Si, 38Siesta, 42, 43, 51, 55Slater determinant, 31Sommerfeld model, 63spinodal (points, curves), 122SQS, 8, 9, 27, 112, 117

178

Index

static model (for phase transitions), 9stress, 18

thermal expansion, 50, 64Thomes–Fermi theory, 30

upstroke (pressure studies), 21, 80, 81

van der Waals forces, 22VASP, 42, 112VCA, 14, 15, 17, 103, 104, 110, 112–114Vegard’s law, 14, 16, 67, 69, 72, 73, 112,

115VFF, 113

WIEN2k, 40, 42, 68, 77, 142WZ (structure), 11, 12, 24, 25, 92, 93,

112, 116

X-ray emission; absorption, 40XC potential, 36, 98XRD, 14, 15, 22, 23, 96, 106, 115

ZB (structure), 11, 12, 24, 25, 68, 69,78, 92–96, 101, 102, 108, 113,116, 127, 129, 130, 133

zero-point motion, 20ZnS, 45

179