notice tpa effets quantiques - epfl

Notice TPA

Effets quantiques

R. Sanjinés Date: 13.02.2010

NOTICE: Travaux pratiques avancés Effets quantiques Section de physique – FSB

2

2

I. Introduction

L’objectif de ce TP est de mieux comprendre la mécanique quantique en observant et en étudiant quelques phénomènes physiques simples qui sont interprétés dans le cadre de la mécanique quantique. Un des effets le plus connu est « la dualité onde-particule de la lumière » [1]. Lorsqu’on diminue les dimensions macroscopiques caractéristiques d’un système physique pour atteindre le domaine des nanomètres, souvent les propriétés physiques observées dans le domaine macroscopique ne sont plus valables. De nouvelles propriétés physiques, chimiques voir biologiques apparaissent.

A l’échelle atomique ou moléculaire, la physique classique n’est plus suffisante pour comprendre les phénomènes observés. On fait appel à la mécanique quantique. Lorsque les atomes se regroupent ils forment des molécules plus au moins complexes, des amas, et lorsque les amas contiennent quelques centaines voir de milliers d’atomes on forme de nanoparticules ou nanocristaux. Pour comprendre les propriétés physiques et chimiques de ces nanosystèmes on utilise l’approche dite « du bas en haut », c’est à dire on extrapole les résultats de la mécanique quantique [2].

II. Quelques effets quantiques

II.1. Conduction électrique quantique

Le premier effet quantique à étudier est la conduction électrique à 1 dimension (1D). Dans ce cas on peut montrer que la conductance électrique Gn est quantifiée et qu’elle est donnée

par la relation nheGN

22= où h est la constante de Planck et n un nombre entier [3]. Ou en

termes de résistance Ω=== knne

hG

Rn

n9.121

21* 2 .

La conductance quantifiée peut être observée au moyen d’un circuit électrique simple en mesurant la résistance R* d’un contact électrique entre deux conducteurs métalliques.


3

3

Donc à l’aide d’un oscilloscope on mesure la tension Uout=U4=R4I1. On peut facilement calculer la valeur de la tension de sortie Uout en fonction de la tension d’alimentation du circuit Uin=U1 et des résistances R1, R2 et R4. En appliquant les lois de Kirchhoff, on obtient

inout URRU ⎟⎟⎠

⎞⎜⎜⎝

⎛=

)R (R + R + Rn*) + R (Rn* +R 42421

42 (1)

Travail à effectuer :

Etudier le circuit montré schématiquement ci-dessous et calculer la tension de sortie U4 en fonction de la tension entrée U1 et des résistances Ri ainsi que de Rn*(vérifier la relation 1). Les valeurs caractéristiques du circuit électrique sont R1= 25 kΩ, R2=R4= 1kΩ et C=10 µF. En général il faut choisir des valeurs de Ri telles que la tension de sortie U4 pour n=1, c’est à dire pour R*= 12.9 kΩ, soit de quelques dizaines de mV.

Fig.1. Exemple des paliers de tensions observés au moment de la rupture du contact électrique et la formation d’un canal de conduction 1D

II.2 Fluorescence dans les puits quantiques (quantum dots)

Le deuxième phénomène physique traite d’un effet quantique dans l’optique: il s’agit de l’émission de lumière (fluorescence) observée dans le cas des nanocristaux de CdSe excités avec une source de lumière de longueur d’onde convenable (énergie de photons supérieur au seuil d’excitation). Les électrons et trous dans ces nanocristaux, contrairement à ceux dans un cristal macroscopique, se comportent comme des particules piégées dans une boîte quantique. Dans ce cas, la longueur d’onde de la lumière émise dépend de la dimension de la boîte quantique ou des nanocristaux. Lorsque le rayon de la boite quantique est plus grand que 20 nm, le seuil d’absorption optique correspond à celui du matériau massif. Aujourd’hui il est possible de fabriquer des nanoparticules de taille homogène comprise entre 2 et 20 nm qui émettent de la lumière dans le visible et le proche infrarouge [4].


4

4

Fig.2 Spectres typiques d’absorption et d’émission des nanocristaux de CdSe.

Travail à effectuer

Enregistrer les spectres d’émission (fluorescence) d’une série de quantums dots et déterminer l’énergie d’émission. A l’aide de l’équation (solution de l’eq. de Schrödinger pour une particule dans une boite à 3 Dimensions)

Re

mmRhEE

CdSeheg

0

2

2

2

48.111

8 επε−⎟⎟

⎠

⎞⎜⎜⎝

⎛++≈ (2)

Évaluer la taille des nanocrystaux de CdSe. Discuter les résultats.

danieloberli

Barrer

Vous allez faire la mesure et l’étude des spectres d'émission et d'absorption de 6 solutions colloïdales contenant des boites quantiques de tailles distinctes (nanocristaux de CdSe). Les solutions sont déjà prêtes à l'emploi.

Objectifs

L'objectif premier de cette expérience consiste à montrer l'existence d'un effet de taille sur la longueur d'onde d'émission d'une paire d'électron et de trou confinée dans un nanocristal. Vous serez amené à déterminer la taille moyenne et la distribution de taille des boites quantiques en solution. La comparaison entre le spectre d'absorption et d'émission vous permettra de mettre en évidence l'existence d'un décalage spectral, communément appelé "Stokes shift".

Répondre aux questions suivantes :

- Expliquez l'origine de ce décalage et quantifiez sa valeur- Comparez les largeurs spectrales des raies d'émission et d'absorption; sont-ellesidentiques? Sinon, pourquoi diffèrent-elles?- Identifiez les transitions optiques associées aux premiers états excités des boitesquantiques- Quantifiez la séparation en énergie entre l'état fondamental et le premier étatexcité- Etudiez la dépendance de cet écart d'énergie avec la taille des boites et expliquezson origine

Pour répondre à ces questions, vous pourrez consulter les articles scientifiques qui sont joints à cette notice.

Références :

[1] Tonomura A., Endo J., Matsuda T., Kawasaki T., and Ezawa H., Demonstration ofsingle-electron build-up of an interference pattern, Am. J. Phys. 57 (1989) 117.[2] M. Lahmani, C. Dupas, Ph. Houdy, « Les nano sciences : Nanotechnologies etnanophysique », Collection Echelles Belin Editions, 2004[3] F. Ott, J. Lunney, « Quatum conduction : a step-by-step guide », Trinity College,Dublin[4] J. Galloway, « Semiconductor quantum dots », Johns Hopkins University, 2007,[5] Adresse du site web du fabricant des solutions colloïdales :http://datasheets.globalspec.com/ds/3901/EvidentTechnologies

SPEC

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October 2006 - Quantum Dot Nanomaterials for Research Specifi cation Sheet© 2006 Evident Technologies, Inc. All specifi cations subject to change without notice

Page: 1

Quantum dots are a unique class of semiconductor particles, ranging from 2-10 nanometers (10-50 atoms) in diameter. At these small sizes, materials behave differently, giving quantum dots unprecedented tunability and enabling applications to exploit unique properties at the nanoscale.

The usefulness of quantum dots comes from their peak emission frequency’s extreme sensitivity to both the dot’s size and composition, which can be controlled using Evident Technologies’ proprietary manufacturing techniques, and control over surfaces.

Quantum Dots in Research QuantitiesEvident Technologies is the world’s premier supplier of quantum dot nanomaterials for the researcher and scientist. We supply small quantities of our proprietary nanomaterials that enable basic research to foster the understanding of material properties and the exploration of a wide variety of potential nanotech applications. Our semiconductor nanocrystals below are offered in a variety of wavelengths and composites materials systems.

Production quantities of a range of these and other quantum dot materials are available to our product development partners. To explore more with Evident, please see our Product Development Opportunities section.

The Advantages of Core-Shell EviDots Quantum DotsThe EviDots core-shell quantum dots have unique shells that stabilize the material, improve quantum yield and reduce photo-degradation. The properties of EviDot core-shell quantum dots include:

• Stability - The nanocrystal materials are very stable. They are composed of inert inorganic compounds and are further stabilized with our engineered shells that resist photochemical damage.

EviDots & EviComposites

For Research

QUANTUM DOTNANOMATERIALS

Tel. +82-31-478-2462 Fax. +82-31-478-2460

www.nanobest.co.kr [email protected]

SPEC

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Page: 2

• Emission Spectrum - The nanocrystals emission spectrum is very narrow compared to organic dyes due to a very monodisperse size distribution, as little as 5% variation in diameter. This results in the spectral width under 30nm FWHM, making the materials ideal for color multiplexing applications.

• Excitation - EviDot nanocrystals possess very broad excitation properties and can be excited with simple excitation sources at essentially any wavelength shorter than the emission peak. This facilitates simultaneous detection, imaging and quantifi cation.

• Intensity - EviDot nanocrystals fl uoresce intensely and exhibit high quantum yields. Brightness is comparable to or greater than traditional organic fl uor dyes.

• Fluorescence Lifetimes - Evident’s CdSe/ZnS nanocrystals have 15-20ns fl uorescence lifetimes which is an order of magnitude greater than conventional organic dyes and even greater than the auto-fl uorescence lifetime of organic dyes. This permits the quantum dot to be more easily distinguished from other sources of intrinsic fl uorescence when used in instruments that have fl uorescence lifetime capabilities, and improves the signal to noise ratio by orders of magnitude.

• Surface Chemistry - EviDot nanocrystals have a very fl exible surface chemistry that can be altered for various applications to allow solubility in organic and aqueous solutions.

• Engineered NanoMaterials - Explore material and product applications with EviDots, which can be engineered to meet cutting-edge needs for new fl uorescent or photonic materials in biotechnology, optical devices, optical computing, photovoltaics, light emitting diodes (LED), lasers or a fast-growing number of other applications.

EviComposites – Quantum Dots in CompositesEvident Technologies is offering a selection of resins composites to explore the use of quantum dots in matrix materials. EviComposites are an easy-to-use form of quantum dot nanomaterials. EviComposite matrix materials are available in a UV curable form and as a polymeric solution (PMMA) which provides versatility to explore the use quantum dots in plastics. The ready-to-use resin allows scientists and researchers to focus on the application of nanotechnology.

Quantum Dot in Other Composite SystemsEvident Technologies has expertise in a wide variety of nanocrystal materials and composites available to our partners. In addition to the research selection of resins, the company has experience in developing and using other insulating and conductive polymers. Evident Technologies has produced quantum dot composites in wavelengths from 465 nm to 2300 nm. We have experience in dispersing these nanocrystals into a variety of matrix materials including polymers such as epoxies, silicones, and other matrix materials including silica and titania. Composites compatible with spin coating, ink jetting, spraying and screen-printing are available to our partners and offer the ability to control fi lm thickness, uniformity and concentration. Additionally, wider variety of materials is available to our product development partners. For more information, please call our Product Development Specialist at 518-273-6266 x0 to arrange a consultation or write to us at [email protected].

October 2006 - Quantum Dot Nanomaterials for Research Specifi cation Sheet

© 2006 Evident Technologies, Inc. All specifi cations subject to change without noticePage: 3

Quantum Dot SpecificationsAp

prox

. Qu

antu

m

Yieldd

30%

-50%

Molec

ular

Weig

htc

[µg/

nmol]

3 10 15 23 44 86 200

Molar

Ex

tinct

ion

Coef

ficien

t at

1st

Excit

onb

[10^

5]

0.4

0.55

0.72

0.98 1.6

2.6

4.5

Crys

tal

Diam

eter

a [n

m-

appr

ox]

1.9

2.1

2.4

2.6

3.2 4 5.2

1st

Excit

ation

Pe

ak [n

m-

nom

inal]

470

505

525

545

570

590

610

Sugg

este

d Ex

citat

ion

<400

Typic

al FW

HM

[nm

]

<40

<35

<30

Emiss

ion

Peak

To

leran

ce

+/-1

0

Emiss

ion

Peak

[nm

]

490

520

540

560

580

600

620

Color

Lake

Pla

cid

Blu

e

Adi

rond

ack

Gre

en

Cat

skill

G

reen

Hop

s Ye

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Birc

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For

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Map

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Quan

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Do

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Cor

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Quan

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Do

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CdS

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S

Notes

on Q

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1.

Est

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Feb

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, 200

3.

2.

Mea

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(m

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SPEC

IFICA

TIONS

Page: 4

Quantum Dot Emission & Absorption Spectra

EviDot and EviComposites Quantum Dot Products

Quantum Dot Test Kits with EviDotsQuantum dot test kits are ideal for researcher and product developers who wish to experiment with small quantities of the nanomaterials. Each kit consists of standard colors, providing a selection of wavelengths in a convenient package to conduct research and characterization or for teaching purposes.

EviDot Quantum Dot Test Kits2 Color Test Kit – CdSe/ZnS Quantum Dots in Toluene – 490nm to 620nm (select colors)DK-C11-TOL-KIT2-1 Kit

4 Color Test Kit – CdSe/ZnS Quantum Dots in Toluene – 490nm to 620nm (select colors)DK-C11-TOL-KIT4-1 Kit

6 Color Test Kit– CdSe/ZnS Quantum Dots in Toluene – 490nm to 620nm (select colors)DK-C11-TOL-KIT6-1 Kit

EviDots Quantum Dot Test Kits are sold at concentrations where the optical density (absorbance) equals 0.5 through an optical path of 1mm, measured at the respective fi rst exciton peak.

Quantum Dot Research Products – EviDots and EviComposites

Item Description

EviDots ED-C11-TOL-0490 CdSe/ZnS Core Shell EviDots, Lake Placid Blue/ 490nm, 50mg or 200mg

ED-C11-TOL-0520 CdSe/ZnS Core Shell EviDots [Adirondack Green/ 520nm], 50mg or 200mg

ED-C11-TOL-0540 CdSe/ZnS Core Shell EviDots, Catskill Green/ 540nm, 50mg or 200mg

ED-C11-TOL-0560 CdSe/ZnS Core Shell EviDots, Hops Yellow/ 560nm, 50mg or 200mg

ED-C11-TOL-0580 CdSe/ZnS Core Shell EviDots, Birch Yellow/ 580nm, 50mg or 200mg

ED-C11-TOL-0600 CdSe/ZnS Core Shell EviDots/Fort Orange/ 600nm, 50mg or 200mg

ED-C11-TOL-0620 CdSe/ZnS Core Shell EviDots, Maple Red-Orange/ 620nm, 50mg or 200mg


Page: 5

Item Description

EviComposites EC-C11-R10-0490-50mg/10mL CdSe/ZnS EviComposite UV Clear Resin, Lake Placid Blue/ 490nm, 50mg/10mL or 100mg/mL

EC-C11-R10-0520-50mg/10mL CdSe/ZnS EviComposite UV Clear Resin, Adirondack Green/ 520nm, 50mg/10mL or 100mg/mL

EC-C11-R10-0540-50mg/10mL CdSe/ZnS EviComposite UV Clear Resin, Catskill Green/ 540nm, 50mg/10mL or 100mg/mL

EC-C11-R10-0560-50mg/10mL CdSe/ZnS EviComposite UV Clear Resin, Hops Yellow/ 560nm, 50mg/10mL or 100mg/mL

EC-C11-R10-0580-50mg/10mL CdSe/ZnS EviComposite UV Clear Resin, Birch Yellow/ 580nm, 50mg/10mL or 100mg/mL

EC-C11-R10-0600-50mg/10mL CdSe/ZnS EviComposite UV Clear Resin, Fort Orange/ 600nm, 50mg/10mL or 100mg/mL

EC-C11-R10-0620-50mg/10mL CdSe/ZnS EviComposite UV Clear Resin, Maple Red-Orange/ 620nm, 50mg/10mL or 100mg/mL

EC-C11-R40-0490-10mg/2g CdSe/ZnS EviDots in PMMA Solution, Lake Placid Blue/ 490nm, 10mg/2g

EC-C11-R40-0520-10mg/2g CdSe/ZnS EviDots in PMMA Solution, Adirondack Green/ 520nm, 10mg/2g

EC-C11-R40-0540-10mg/2g CdSe/ZnS EviDots in PMMA Solution, Catskill Green/ 540nm, 10mg/2g

EC-C11-R40-0560-10mg/2g CdSe/ZnS EviDots in PMMA Solution, Hops Yellow/ 560nm, 10mg/2g

EC-C11-R40-0580-10mg/2g CdSe/ZnS EviDots in PMMA Solution, Birch Yellow/ 580nm, 10mg/2g

EC-C11-R40-0600-10mg/2g CdSe/ZnS EviDots in PMMA Solution, Fort Orange/ 600nm, 10mg/2g

EC-C11-R40-0620-10mg/2g CdSe/ZnS EviDots in PMMA Solution, Maple Red-Orange/ 620nm, 10mg/2g

EC-000-R10-0000-1mL UV Clear Dilution Resin, 1ml

EviDots are sold by total mass with a minimum number of nanomoles per amount per wavelength as specifi ed in the table below. Included on the label is the number of nanomoles for a particular container. The number of nanomoles has been calculated from the molar extinction coeffi cients (at the exciton peak) for all amounts.

Ref: W. W. Yu. L. Qu, W. Guo. X. Peng, Chem. Mater.2003,15 (14), 2854-2860, corrected in volume 15, 2003.

Minimum Nanomoles of Core-Shell CdSe/ ZnS EviDots Delivered by Mass

Total Mass/ Approximate

Volume

Lake-Placid Blue

Adirondack Green

Catskill Green Hops Yellow Birch Yellow Fort Orange Maple Red- Orange

490nm 520nm 540nm 560nm 580nm 600nm 620nm

Minimum Nanomoles Delivered

50 mg / 5 mL

1,300 650 360 270 120 79 48

200 mg / 20 mL

5000 2600 1400 1100 480 310 190

Research and Development Use OnlyExcept as provided by written contracts, products ordered from Evident Technologies are sold for research and development use only. Cadmium and lead EviDots are not intended for food, drug, household, agricultural or cosmetic use.

SPEC

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Page: 6

Toxic Substances Control Act The buyer assumes responsibility to ensure that the products purchased are approved for use under the Toxic Substances Control Act (TSCA), if applicable. Use of Evident Technologies products shall be supervised by a technically qualifi ed individual. A Material Safety Data Sheet (MSDS) accompanies products defi ned as hazardous by the US Occupational Safety and Health Administration (OSHA) Hazard Communication Standard.

Fast-track Product Development OptionsQuantum dot nanomaterials can overcome the limitations of many current materials, providing product developer with endless opportunities for innovation, and can be used to create market-changing, life-changing products. But not every company has the necessary experience, expertise or intellectual property to harness the power of quantum dot technologies. That’s why we are an ideal partner for companies wanting to harness the power of quantum dots.

We have a product development approach that can combine our expertise with the capabilities of our industry partners.

• We have a team of scientists, researchers and product developers to fast-track the development of quantum dot-based products; and

• Offer a range of teaming and licensing arrangements that make our expertise, technology and IP available to partners.

This partnership formula offers companies a fast and cost-effi cient way to bridge from their established technologies to applications that harness the market power of quantum dot technology.

Call our product development specialists at 518-273-6266 to learn how fast-track teaming can work for your products.

The Next Step is EvidentIf you want to explore new quantum dot product development opportunities or accelerate the development of next generation products, we want to hear from you.

Tel. +82-31-478-2462 Fax. +82-31-478-2460

www.nanobest.co.kr [email protected]

Synthesis of Cadmium Selenide Quantum Dots from a Non-Coordinating Solvent: GrowthKinetics and Particle Size Distribution

Jeaho Park, Kwan Hyi Lee, Justin F. Galloway, and Peter C. Searson*,†

Department of Materials Science and Engineering, Johns Hopkins UniVersity, Baltimore, Maryland 21218

ReceiVed: April 29, 2008; ReVised Manuscript ReceiVed: June 19, 2008

Here we report on the synthesis of CdSe quantum dots from a noncoordinating solvent. We show that nucleationand growth is very fast and is completed within 100 s. The subsequent increase in average particle size is dueto diffusion limited coarsening. Growth and coarsening can be quenched by injection of dodecanethiol. Finally,we compare the size distribution obtained from analysis of the absorption edge with the size distributionobtained from analysis of transmission electron microscope images.

Introduction

Semiconductor quantum dots (QDs) have received consider-able attention due to their size dependent properties andapplications in fields such as solar cells,1 light emitting diodes,2

and biological imaging.3,4 For materials, such as quantum dots,that exhibit size dependent properties, it is essential to under-stand the details of nucleation and growth and how theyinfluence the evolution of the particle size and the particle sizedistribution.

CdSe is one of the most versatile quantum dot materials sinceits emission wavelength can be tuned across the visiblespectrum. CdSe has a band gap of about 1.72 eV5 correspondingto a band-to-band emission wavelength of about 730 nm. Asthe particle diameter decreases below about 10 nm, band gapenlargement becomes significant and blue emission can beachieved for particles about 3 nm in diameter.

The most common synthesis route for CdSe QDs involvesan organometallic precursor in a coordinating solvent.6-8

Typically dimethyl cadmium is reacted with selenium intrioctylphosphine oxide at elevated temperature, usually about300 °C. The addition of more strongly co-ordinating molecules,can lead to anisotropic growth and the synthesis of quantumrods.9-11 More recently, high quality CdSe QDs have beensynthesized from “greener” precursors such as CdO and Cdacetate, thereby avoiding many of the limitations associated withthe organometallic precursor.8,12-14 The CdO precursor is reactedwith Se in TOP, with the addition of another coordinating ligand,such as hexadecylamine, or a phosphonic acid.

The CdSe QDs studied here are synthesized from a nonco-ordinating solvent, octadecene, with oleic acid as a cappingligand. The reaction can be performed very simply by injectingthe precursor mixture into the solvent with the capping ligandat 290 °C. This is similar to the method reported for the synthesisof InP, InAs, and CdS QDs.15,16 We show that nucleation andgrowth are fast and that the subsequent increase in particle sizeis due to diffusion limited coarsening. Finally, we show thatgrowth can be quenched by injection of dodecanethiol. We alsoshow that the particle size distribution obtained from analysis

of the absorption edge is in good agreement with the distributionobtained from analysis of transmission electron microscopeimages.

Experimental Details

Synthesis. First, 0.1 M solutions of the cadmium (Cd) andselenium (Se) precursors were prepared separately. The Cdprecursor was prepared by mixing 0.3204 g CdO (Alfa Aesar,Puratronic, 99.998%), 6.94 mL oleic acid (OA, Aldrich,Technical grade 90%), and 18 mL 1-octadecene (ODE, Aldrich,tech. 90%) at 220 °C until the solution became transparent. TheSe precursor was prepared by dissolving 0.1579 g Se powder(Aldrich, 100mesh 99.5+ %) in 20 mL trioctylphosphine (TOP,Aldrich, Technical grade 90%) with sonication for at least 190min in a glovebox. One mL of each precursor was combinedwhile sonicating to make 2 mL of the Cd/Se precursor.

Next, 9 mL of ODE and 1 mL of OA were combined in a3-neck flask and heated to 100 - 120 °C while stirringvigorously under vacuum for 5 - 10 min. The temperature wasthen increased to 290 °C under inert gas (Ar) flow, with coldwater running through the condenser. The Cd/Se precursor(2 mL) was then rapidly injected into the hot solution. We notethat the temperature for this synthesis is easily achieved usinga heating mantle and does not require additional insulation.

Small aliquots (typically about 1 mL) were extracted fromthe mixture at different reaction times. The extracted aliquotswere immediately immersed in dry ice in order to quench furthergrowth of the nanocrystals. One mL of hexane (EMD, HPLCgrade) was added to each aliquot. Next, 2 mL methanol (EMD,HPLC grade) was added into the solution for washing. Whenthe separation between two layers was evident, the clear bottomlayer was discarded; this washing procedure was repeated threetimes. After washing the remaining organics and solvent, acetonewas added until the solution became opaque. The supernatantwas discarded after centrifugation at 8000 rpm for 10 min. Thesolid precipitate was redissolved in about 1 mL hexane.

Capping. For capping experiments, 8 mL 1-dodecanethiol(Aldrich,g98%) was injected into the QD suspensions resultingin a final concentration of 1.65 M. The addition of 0.033 moldodecanethiol corresponds to an excess of 1193 with respect tomonolayer coverage on 5.8 nm diameter particles assuming thatall precursor is reacted to form CdSe and all of the surfactantis adsorbed on the particles. The concentration of the CdSe QDsis 0.1 mmol (3.22 × 1016 CdSe particles) under this assumption

* To whom correspondence should be addressed. E-mail: [email protected].

† Institute for NanoBioTechnology, Johns Hopkins University, Baltimore,Maryland 21218.

J. Phys. Chem. C 2008, 112, 17849–17854 17849

10.1021/jp803746b CCC: $40.75 2008 American Chemical SocietyPublished on Web 10/29/2008

with a total CdSe surface area of 3.403 × 1018 nm2. Assumingan adsorption area of 20 Å2 for the thiols, then monolayercoverage corresponds to 0.028 mmol dodecanethiol (6.8 µLdodecanethiol).

Characterization. Photoluminescence measurements wereobtained using a fluorometer (Fluorolog-3 fluorometer, HoribaJobin Yvon). Absorbance spectra of the samples were obtainedusing a spectrophotometer (Cary 50 UV/vis). Suspensions ofthe CdSe QDs in hexane were placed in cuvettes with polishedsides (Starna Cells, Inc.). Transmission electron microscopeimages were obtained with a Philips EM 420 TEM and FEITecnai 12 TWIN. High resolution images were obtained usinga Philips CM 300 FEG TEM. Samples for transmission electronmicroscopy were prepared by placing a drop of the QDsuspension on a lacey-carbon grid.

The quantum yield (QY) was determined using Rhodamine6G as a standard. Rhodamine 6G has a QY of 95% at anexcitation wavelength of 488 nm.17 The calibration curve forRhodamine was obtained in DI water which has refractive index(nst) of 1.333. The integrated area under the fluorescence curves(excitation at 488 nm) was plotted versus the absorbance at 488nm (after subtraction of the solvent absorbance) for differentconcentrations. The same procedure was repeated for the QDsamples suspended in hexane (n ) 1.3749). The excitationintensity and slit width were held constant for all measurements.The QY of the QDs was obtained from QYQD ) QYst [(dI/dA)QD/(dI/dA)st] [nQD

2/nst2] where I is the area under the PL

curves and A is the corresponding absorbance.18

Results and Discussions

Particle Synthesis. Figure 1 shows a sequence of absorbancespectra as a function of reaction time after injection of the Cd/Se precursor mixture. The absorbance spectra show an excitonpeak close to the absorption onset, characteristic of manysemiconductor quantum dots. The overall absorbance at shortwavelengths (<420 nm) increases with time, saturating afterabout 100 s. The absorption edge is relatively sharp, indicatinga relatively narrow size distribution.19 With increasing reactiontime, the absorbance spectra red shift indicating a progressiveincrease in the average particle size.

Particle sizes were determined from analysis of TEM images.Figure 2 shows typical TEM images for CdSe QDs prepared atdifferent reaction times. In all cases the particles appearedspherical and relatively monodisperse. Figure 2a shows an imageof particles synthesized for 180 s where the average radius is3.3 nm. Large areas of the grid revealed particles self-assembledinto ordered close packed arrays, although no attempt was madeto optimize conditions for self-assembly. The inset in Figure2a shows a high resolution TEM image of a single QD wherethe lattice fringes show that the particle is a single crystal.

Figures 2b and 2c show particles after 60 and 10 s reactiontime. The selected area electron diffraction pattern (Figure 2d)confirms that wurtzite crystal structure.

The size distributions of the particles, obtained from analysisof the TEM images are shown in Figure 3. The distributionsappear symmetrical and the most-probable radius increases from2.5 nm at 5 s to 3.6 nm after 480 s. The size distributions arerelatively sharp even though no size selection was performedin the synthesis.

Figure 4 shows PL spectra for a series of CdSe QDs withreaction times from 2 to 480 s, corresponding at average particleradii of 2.0-3.6 nm. For all experiments with this reactionscheme, spectra obtained for reaction times less than 5 s

Figure 1. Absorbance spectra for CdSe QDs after reaction times from5 to 2536 s. The inset shows a photograph of suspensions of CdSeQDs after 2, 10, 30, 120, 480 s under UV excitation.

Figure 2. TEM images of CdSe QDs. Reaction time (a) 180, (b) 60,(c) 10 s. Average particle size (radius) (a) 3.3, (b) 3.2, (c) 2.5 nm. Theinset shows a high resolution TEM image of a CdSe after 180 s. (d)The selected area electron diffraction pattern confirms the wurtzitecrystal structure.

Figure 3. Size distribution histograms obtained from TEM images.Reaction time: (a) 5 (N ) 88), (b) 10 (N ) 110), (c) 15 (N ) 97), (d)60 (N ) 106), (e) 180 (N ) 111), (f) 480 s (N ) 205). (Solid lines)Size distributions calculated from absorbance spectra based.

17850 J. Phys. Chem. C, Vol. 112, No. 46, 2008 Park et al.

exhibited a small shoulder at longer wavelengths, whereas thespectra at longer times were symmetrical. The PL peak and fullwidth at half-maximum (fwhm) are shown in Figure 5. The PLpeak red shifts significantly in the first 50 s and then increasesmore slowly at longer times. The fwhm decreases slightly overthe first 200 s and then increases sharply at longer times. Forreaction times from 20 to 200 s, the fwhm is less than 30 nm,equivalent to the best results reported using other synthesismethods.6,13,14,20 The quantum yield for the QDs decreasedslightly from 11.5% (t ) 15 s) to 7.8% (t ) 200 s). These valuesare in good agreement with reports in the literature for uncappedCdSe QDs.7,21

An estimate of the average particle diameter can be obtainedfrom the PL peak. Assuming that the PL is due to band-to-band emission, then the wavelength at the PL peak can be usedto obtain the band gap. The average particle size can then beestimated from the band gap using a suitable model forconfinement.22 Figure 6 shows the band gap for CdSe QDsversus particle radius compared to the effective mass model. Inthe effective mass model the band gap is related to the particlesize by23

E * )Egbulk + p

2π2

2er2 ( 1memo

+ 1mhmo

)- 1.8e4πεεor

(1)

where E* is the band gap in eV, Egbulk is the bulk band gap, r

is the particle radius, me is the effective mass of the electrons,mh is the effective mass of the holes, mo is the mass of a freeelectron, ε is the relative permittivity, εo is the permittivity of

free space, p is Planck’s constant, and e is the charge on theelectron. The second term on the right is a quantum confinementterm and the third term is an electrostatic attraction betweenthe electron and hole. For CdSe we take Eg

bulk ) 1.72 eV, me

) 0.13, mh ) 0.45, and ε ) 10.5

From Figure 6 we see that there is good agreement betweenthe average particle size obtained from analysis of TEM imagesand the effective mass model down to a radius of about 2 nm,corresponding to an emission wavelength of 520 nm. We didnot analyze TEM images for smaller particles due to thedifficulty in obtaining precise measurements on a sufficientlylarge number of particles. Murray et al.6 reported good agree-ment between the average size of CdSe QDs obtained fromanalysis of TEM images and the effective mass model, but onlyfor particle radii down to about 3 nm. For smaller particles, theband gap enlargement was smaller than predicted by the model.Nonetheless, this correlation provides a convenient tool forestimating the average particle diameter from PL spectra withoutthe need for direct measurement using transmission electronmicroscopy, at least for radii from 2 to 4 nm.

An estimate of the average diameter of QDs in suspensioncan also be obtained from absorbance spectra. Figure 7 showsthat the wavelength at the inflection point in the absorbanceedge (i.e., the point where d2A/dλ2 ) 0) is in excellent agreementwith the wavelength of the PL peak, illustrating that the sameparticle size would be obtained from absorbance or PL spectra.

Figure 4. Photoluminescence spectra of CdSe QDs. The reaction timein seconds is indicated in the figure.

Figure 5. (a) PL peak wavelength and (b) fwhm versus the logarithmof the reaction time.

Figure 6. (a) Average band gap obtained from analysis of TEM imagesand (b) corresponding wavelength plotted versus average particle radius.The solid line corresponds to the effective mass model using Eg(bulk)) 1.72 eV, me ) 0.13 and mh ) 0.45, and ε ) 10.

Figure 7. Plot of the PL peak wavelength (Figure 5a) versus thewavelength at the inflection point in the absorbance edge of theabsorbance spectra (Figure 1).

Synthesis of Cadmium Selenide Quantum Dots J. Phys. Chem. C, Vol. 112, No. 46, 2008 17851

The time dependence of the average particle size obtained fromabsorbance spectra has been used to study the kinetics of growthand coarsening of QDs during synthesis.24-26

Growth and Coarsening. During synthesis, the Cd and Seprecursors combine to form stable nuclei that subsequently growas the reaction proceeds. The synthesis reported here is a simpleone-pot synthesis where the precursor mixture is injected intothe solvent/capping ligand solution at the reaction temperaturefor reaction times up to 2500 s. For the short reaction times,QDs that emit in the blue are easily obtained. At longer times,particle growth results in a progressive red shift, resulting inlarger particles that emit in the red, approaching the emissionwavelength of 730 nm for band-to-band emission from bulkCdSe. These results are consistent with previous reports usingdimethyl cadmium precursor, a co-ordinating solvent, and morecomplex thermal management schemes.7

The increase in particle size with reaction time during solutionphase synthesis involves nucleation, growth, and other processessuch as aggregation and coarsening. Bullen and Mulvaney haveshown that nucleation of CdSe from octadecene is very fastand stops almost immediately after precursor injection.27

Examination of the absorption spectra from several experimentsreveals that the absorbance at short wavelengths reaches amaximum very quickly, within the first 100 s. For an inorganicsemiconductor, the absorbance at short wavelengths is relatedto the total volume of the material and hence this result indicatesthat the precursor concentration has decreased to the saturationconcentration and that growth is completed within 100 s.

Subsequent changes in particle size can, therefore, only occurby processes such as coarsening or aggregation. Figure 8 showsa plot of particle radius versus time for a typical CdSe synthesis.The average particle radius was obtained from the inflectionpoint in the absorption edge using the effective mass model, asdescribed above. The particle radius increases very quickly inthe first 50 s, after which the particle size increases much moreslowly. The inset shows the particle radius replotted as r3 versustime. The linear region from about 200 s until 2500 s isconsistent with diffusion limited coarsening which has a ratelaw given by rav

3 - r03 ) kt.28,29

From the slope of the linear region we obtain the rate constantfor coarsening k ) 0.010 ( 0.0047 nm3 s-1 (N ) 6). This issomewhat larger than the rate constant for coarsening of ZnOin alcohol which in the range 10-4-10-3 nm3 s-1, dependingon anion, solvent, and temperature (25 - 65 °C).30-33 The rateconstant is given by31-33

k)8γVm

2 cr)∞

54πηaNA(2)

where γ is the surface energy, Vm is the molar volume, cr)∞ isthe equilibrium concentration at a flat surface (i.e., the bulk

solubility), η is the viscosity of the solvent, a is the solvatedion radius, and NA is Avogadro’s number. Since γ, Vm, η, anda are known or can be reasonably estimated, we can obtain anestimate for the bulk solubility of CdSe in octadecene at 290°C. The surface energy of solids at a solid-liquid interface isexpected to be in the range 0.1-0.5 J m-2,34 and a value of0.17 J m-2 has been reported for CdSe QDs in octadecene.27

The molar volume Vm for CdSe at 300 K is 32.9 cm3 mol-1

and since the linear expansion coefficient is about 7.4 × 10-6

K-1, 35 we take the same value for Vm at the reactiontemperature. Since octadecene is a noncoordinating solvent, weuse the ionic radius a ) 0.1 nm for the solvated ion radius.36

The viscosity of octadecene at 290 °C is estimated by extrapola-tion from lower temperatures (using ln η ) -4.986 +1879/T),37 giving a value of 0.19 cP. Taking k ) 0.01, we estimatea bulk solubility of 0.02 nM for CdSe in the reaction mixture.This value is about 8 orders of magnitude lower than theprecursor concentration (8.33 mM), providing a broad windowfor particle growth and confirming that most of the precursorreacts to form the solid phase. The bulk solubility is also withinthe range 0.01-1 nM, estimated for ZnO during synthesis inalcohol.31,32

These results reveal two important features of CdSe synthesis.First, nucleation and growth is very fast and is completed withinthe first 100 s. Second, the subsequent increase in averageparticle size is due to diffusion limited coarsening. Figure 5ashows that the PL peak red shifts significantly in the first 50 s,corresponding to the nucleation and growth phase. After 50 s,the rate at which the PL peak red shifts decreases significantlyin the regime where coarsening controls the increase in particlesize. The fwhm of the PL peak decreases slightly during thefirst 100 s and then increases sharply to about 50 nm after severalhundred seconds (Figure 5b).

The importance of coarsening in QDs transferred to cleansolvent after synthesis has been highlighted by experimentswhere CdSe QDs synthesized from hexadecylamine weretransferred to toluene.38 Both CdSe and CdSe/ZnSe QDs showedan increase in the average particle size due to diffusion limitedcoarsening for times up to 8 h after transferring to the cleansolvent.

Coarsening requires thermodynamic equilibrium at the particlesurface between the solid and liquid phases, indicating that the oleicacid and trioctylphosphine are weakly physisorbed to the surface.Figure 9 shows the particle radius plotted versus time for CdSesynthesis where an excess of dodecanethiol were injected after 15 s.The particle radius does not increase after injection showing thatthe thiol is chemisorbed to the surface and thereby preventingcoarsening. Octadecanethiol has been shown to quench growthduring synthesis of ZnO.39

Particle Size Distributions. For a bulk semiconductor, theshape of the absorption edge is determined by the electronic

Figure 8. Particle radius versus reaction time. The particle size wasobtained from the inflection point in absorbance spectra using theeffective mass model. The inset shows a plot of r3 versus time.

Figure 9. Particle radius versus reaction time. At 15 s, dodecanethiolwas injected into the reaction mixture.


band-to-band transition.40 For an ensemble of QDs, the shapeof the absorption edge close to the absorbance onset, is alsodependent on the particle size distribution and hence thedistribution of band gaps. However, if the particle size distribu-tion is sufficiently large, then the shape of the absorbancespectrum near the absorbance onset is dominated by the particlesize distribution. Thus the particle size distribution can beobtained from analysis of the absorption edge.

For sufficiently dilute suspensions, the absorbance A at anywavelength in the quantum regime is related to the total volumeof particles with radius greater than or equal to the sizecorresponding to the absorption onset19

A(r) ∝ ∫r

∞ 43

πr3n(r)dr (3)

where n(r) is the particle size distribution. The particle sizedistribution can be obtained from the absorbance edge by takingthe derivative of A(r) with respect to the particle radius, andnoting that as r f ∞, n(r) ) 0:

n(r) ∝ -dA(r) ⁄ dr43

πr3(4)

Having established that the effective mass model can be usedto relate the band gap to the particle radius (Figure 6), we candetermine the particle size distribution from the absorption edge.As described above, the particle size distribution is simplyrelated to the local slope of the absorbance spectrum at theabsorption edge. The procedure is as follows. First the absorptionedge is extracted from the absorption spectrum. This is theregion from the absorption onset to the point where the slopeis zero (the exciton peak). Next the wavelength axis is convertedto energy and then to radius using the effective mass model. Asuitable fitting method is used to obtain the slope (dA(r)/dr)) ateach radius. Finally, dA(r)/dr) is divided by the particle volumeat that radius to obtain n(r).

The particle size distributions obtained from analysis of theabsorption edges are superimposed on the distributions obtainedfrom analysis of the TEM images in Figure 3. There is excellentagreement between the most probable size obtained fromanalysis of the absorption edge and the TEM images, as expectedfrom Figure 6. The maximum in the distribution obtained fromanalysis of the absorbance spectra corresponds to the inflectionpoint in the absorption edge and can be considered the averageparticle size.

The particle size distribution obtained from the absorbancespectra at 480 s (rav ) 3.6 nm) shows good agreement with thedistribution obtained from analysis of TEM images. For shorterreaction times, the distributions obtained from analysis of theabsorbance spectra are progressively sharper than the distribu-tions obtained from TEM images. This may be the result of thedifficulty in obtaining precise measurements of smaller particlesfrom TEM images. It is also evident that the distributionsobtained from the absorbance spectra are noticeably sharper tothe left of the maximum at small particle sizes. This effect islikely due to the contribution of the exciton peak and can beseen from the absorbance spectra in Figure 1. We note, however,that particles in this size regime comprise only a small volumefraction of the overall distribution and we make no attempt tosubtract the contribution of the exciton peak from the absorbancespectra.

These measurements show that a reasonable estimate of theparticle size distribution for CdSe QDs can be obtained from

analysis of the absorbance spectra, without the necessity oftransmission electron microscopy.

Summary

We report on the synthesis of CdSe QDs using a noncoor-dinating solvent at 290 °C. QDs with PL emission from blue tored can be easily obtained with a narrow full width at half-maximum. Nucleation and growth are fast and completed withinthe first 100 s. The subsequent slow increase in average particlesize is due to diffusion limited coarsening. Growth andcoarsening can be quenched by injection of dodecanethiol. Weshow that the average particle size and PL peak agree very wellwith the effective mass model at least for radii from 2 to 4 nm.This relationship has been used to compare the particle sizedistribution from analysis of the absorption edge and thedistribution obtained from transmission electron microscopy.

Acknowledgment. We gratefully acknowledge Dr. M. Mc-Caffrey and Dr. K. Livi for assistance with TEM imaging. Wealso acknowledge Dr. M. Merzlyakov for assistance with PLmeasurements. J.P. acknowledges a Summer Fellowship fromthe Johns Hopkins Institute of NanoBioTechnology. This workwas supported by NIH (1R21EB006890-01).

References and Notes

(1) Huynh, W. U.; Peng, X. G.; Alivisatos, A. P. AdV. Mater. 1999,11, 923–927.

(2) Mattoussi, H.; Radzilowski, L. H.; Dabbousi, B. O.; Thomas, E. L.;Bawendi, M. G.; Rubner, M. F. J. Appl. Phys. 1998, 83, 7965–7974.

(3) Gao, X. H.; Cui, Y. Y.; Levenson, R. M.; Chung, L. W. K.; Nie,S. M. Nat. Biotechnol. 2004, 22, 969–976.

(4) Michalet, X.; Pinaud, F. F.; Bentolila, L. A.; Tsay, J. M.; Doose,S.; Li, J. J.; Sundaresan, G.; Wu, A. M.; Gambhir, S. S.; Weiss, S. Science2005, 307, 538–544.

(5) Berger, L. I. Semiconductor materials; CRC Press: Boca Raton,FL, 1997.

(6) Murray, C. B.; Norris, D. J.; Bawendi, M. G. J. Am. Chem. Soc.1993, 115, 8706–8715.

(7) Dabbousi, B. O.; RodriguezViejo, J.; Mikulec, F. V.; Heine, J. R.;Mattoussi, H.; Ober, R.; Jensen, K. F.; Bawendi, M. G. J. Phys. Chem. B1997, 101, 9463–9475.

(8) Peng, Z. A.; Peng, X. G. J. Am. Chem. Soc. 2001, 123, 183–184.(9) Peng, Z. A.; Peng, X. G. J. Am. Chem. Soc. 2001, 123, 1389–

1395.(10) Peng, X. G.; Manna, L.; Yang, W. D.; Wickham, J.; Scher, E.;

Kadavanich, A.; Alivisatos, A. P. Nature 2000, 404, 59–61.(11) Yu, H.; Li, J. B.; Loomis, R. A.; Gibbons, P. C.; Wang, L. W.;

Buhro, W. E. J. Am. Chem. Soc. 2003, 125, 16168–16169.(12) Qu, L. H.; Peng, Z. A.; Peng, X. G. Nano Lett. 2001, 1, 333–337.(13) Qu, L. H.; Peng, X. G. J. Am. Chem. Soc. 2002, 124, 2049–2055.(14) Mekis, I.; Talapin, D. V.; Kornowski, A.; Haase, M.; Weller, H. J.

Phys. Chem. B 2003, 107, 7454–7462.(15) Battaglia, D.; Peng, X. G. Nano Lett. 2002, 2, 1027–1030.(16) Yu, W. W.; Peng, X. G. Angew. Chem., Int. Ed. 2002, 41, 2368–

2371.(17) Magde, D.; Rojas, G. E.; Seybold, P. G. Photochem. Photobiol.

1999, 70, 737–744.(18) Demas, J. N.; Crosby, G. A. J. Phys. Chem. 1971, 75, 991–1024.(19) Pesika, N. S.; Stebe, K. J.; Searson, P. C. J. Phys. Chem. B 2003,

107, 10412–10415.(20) Talapin, D. V.; Rogach, A. L.; Kornowski, A.; Haase, M.; Weller,

H. Nano Lett. 2001, 1, 207–211.(21) Murray, C. B.; Kagan, C. R.; Bawendi, M. G. Science 1995, 270,

1335–1338.(22) Efros, A. L.; Rosen, M. Annu. ReV. Mater. Sci. 2000, 30, 475–

521.(23) Brus, L. J. Phys. Chem. 1986, 90, 2555–2560.(24) Bahnemann, D. W.; Kormann, C.; Hoffmann, M. R. J. Phys. Chem.

1987, 91, 3789–3798.(25) Wong, E. M.; Bonevich, J. E.; Searson, P. C. J. Phys. Chem. B

1998, 102, 7770–7775.(26) van Dijken, A.; Meulenkamp, E. A.; Vanmaekelbergh, D.; Meijerink,

A. J. Phys. Chem. B 2000, 104, 1715–1723.(27) Bullen, C. R.; Mulvaney, P. Nano Letters 2004, 4, 2303–2307.

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(28) Lifshitz, I. M.; Slyozov, V. V. J. Phys. Chem. Solids 1961, 19,35–50.

(29) Wagner, C. Z. Elektrochem. 1961, 65, 581–591.(30) Hu, Z. S.; Ramirez, D. J. E.; Cervera, B. E. H.; Oskam, G.; Searson,

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JP803746B


Band-edge exciton in quantum dots of semiconductors with a degenerate valence band:Dark and bright exciton states

Al. L. Efros and M. RosenNanostructure Optics Section, Naval Research Laboratory, Washington D.C. 20375

M. Kuno, M. Nirmal, D. J. Norris, and M. BawendiMassachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139

~Received 12 March 1996!

We present a theoretical analysis of the band-edge exciton structure in nanometer-size crystallites of directsemiconductors with a cubic lattice structure or a hexagonal lattice structure which can be described within theframework of a quasicubic model. The lowest energy exciton, eightfold degenerate in spherically symmetricdots, is split into five levels by the crystal shape asymmetry, the intrinsic crystal field~in hexagonal latticestructures!, and the electron-hole exchange interaction. Transition oscillator strengths and the size dependenceof the splittings have been calculated. Two of the five states, including the ground state, are optically passive~dark excitons!. The oscillator strengths of the other three levels~bright excitons! depend strongly on crystalsize, shape, and energy band parameters. The relative ordering of the energy levels is also heavily influencedby these parameters. The distance between the first optically active state and the optically forbidden groundexciton state increases with decreasing size, leading to an increase of the Stokes shift in the luminescence. Ourresults are in good agreement with the size dependence of Stokes shifts obtained in fluorescence line narrowingand photoluminescence experiments in CdSe nanocrystals. Mixing of the dark and bright excitons in anexternal magnetic field allows the direct optical recombination of the dark exciton ground state. The observedshortening of the luminescence decay time in CdSe nanoncrystals in a magnetic field is also in excellentagreement with the theory, giving further support to the validity of our model.@S0163-1829~96!05831-6#

I. INTRODUCTION

Size dependent optical spectroscopy of semiconductorquantum dots has now reached the state held by magneto-optical spectroscopy during the mid-1960s, when the avail-ability of high quality semiconductor materials on the onehand, and the development of the multiband Landau leveltheory by Pidgeon and Brown1 on the other hand, enabledthe description of the magnetic field dependence of the verycomplicated absorption spectra of zinc-blende semiconduc-tors ~see, for example, the review by Aggarwal2!. The highquality of recently available nanosize CdSe crystals has al-lowed one to resolve and study the size dependence of up toeight excited states in their absorption spectra.3–5 These ex-citation spectra, obtained in the strong confinement regimewhere the nanocrystals are small compared to the excitonBohr radius, are the result of transitions between discretequantum size levels of electrons and holes.6,7 The smallvalue of the crystal field splitting in CdSe~25 meV! allowsone to consider this semiconductor as a zinc-blende materialas a first approximation.8 As a result, multiband effectivemass theory which takes the degenerate valence band struc-ture into account9 has successfully described excitation spec-tra obtained in absorption,3 hole burning,4 and photolumi-nescence excitation experiments.5,10,11

The data on CdSe quantum dots, however, also providedus with a number of puzzles. While the large scale structureof the absorption spectra is now fairly wellunderstood,3–5,10,11 the nature of the emitting state has re-mained controversial. The photoluminescence of high qualitysamples with high quantum yield is found to be redshifted

with respect to the excitation frequency and has an unusuallylong radiative lifetime12 (tR;1ms at 10 K! compared to thebulk exciton recombination time (tR;1 ns!. Simple para-bolic band theory cannot explain these data in terms of re-combination through internal states. Rather, band-edge emis-sion in II-VI quantum dots~QD’s! was explained as a surfaceeffect and attributed to the recombination of weakly overlap-ping, surface-localized carriers.12,13 These two effects canalso be explained if the ground exciton were the opticallyforbidden state split from the first optically active state bythe electron-hole exchange interaction.14–18 Another puzzleis the very large Stokes shift of the luminescence with re-spect to the absorption for excitation far from the band edge,whose magnitude reaches;100 meV in 16 Å CdSecrystals,12 while the Stokes shift of the resonant band-edgephotoluminescence is only;9 meV.

In this paper we present a realistic multiband calculationof the band-edge exciton fine structure in quantum dots ofsemiconductors having a degenerate valence band, whichtakes into account the effect of the electron-hole exchangeinteraction, nonsphericity of the crystal shape, and the intrin-sic hexagonal lattice asymmetry. We predict and describe asize dependent Stokes shift of both the resonant and nonreso-nant photoluminescence and the fine structure in absorptionand hole burning spectra, and predict the formation of along-lived dark exciton.20 We show below that fluorescenceline narrowing~FLN! and photoluminescence~PL! spectra inCdSe quantum dots support the picture of dark exciton for-mation via excitation of higher energy fine structure statesfollowed by rapid thermalization to the exciton ground state.Particularly strong confirmation of our model is found in the

PHYSICAL REVIEW B 15 AUGUST 1996-IVOLUME 54, NUMBER 7

540163-1829/96/54~7!/4843~14!/$10.00 4843 © 1996 The American Physical Society

magnetic field dependence of the dark exciton decay time.16

In Sec. II we calculate the energy structure of the band-edge exciton and obtain transition oscillator strengths. Wealso calculate the lifetime of the optically passive groundexciton state in an external magnetic field. In Sec. III wepresent data on the size dependence of the resonant and non-resonant photoluminescence Stokes shift and on the mag-netic field dependence of the dark exciton decay time inCdSe quantum dots. The experimental results are comparedwith theory in Sec. IV and conclusions are drawn from thiscomparison.

II. THEORY

In semiconductor crystals which are smaller than the bulkexciton Bohr radius, the energy spectrum and the wave func-tions of electron-hole pairs can be approximated using theindependent quantization of the electron and hole motion~the so-called strong confinement regime!. The electron andhole quantum confinement energies and their wave functionsare found in the framework of the multiband effective massapproximation.19 The formal procedure in deriving thismethod demands that the external potential be smoothenough. In the case of nanosize semiconductor crystals thisleads to the condition 2a@a0, wherea is the crystal radiusanda0 is the lattice constant. In addition, the effective massapproximation holds only if the typical energies of the elec-tron and hole are close enough to the bottom of the conduc-tion band and to the top of the valence band. In practice, thismeans that the quantization energy must be much smallerthan the distance in energy to the next higher~lower! energyextremum in the conduction~valence! band.

In the framework of the effective mass approximation, forspherically symmetric crystals, i.e., finite size spherical crys-tals having a cubic lattice structure, the first quantum sizelevel of electrons is a 1Se state doubly degenerate with re-spect to its spin projection and the first quantum size level ofholes is a 1S3/2 state which is fourfold degenerate with re-spect to the projection of its total angular momentumK(M53/2, 1/2,21/2, and23/2).9,3 The energies and wavefunctions of these quantum size levels are easily found in theparabolic approximation. For electrons they are

E1S5\2p2

2mea2 ,

ca~r !5j~r !uSa&5A2

a

sin~pr /a!

rY00~V!uSa&, ~1!

whereme is the electron effective mass,a is the radius of thecrystal,Ylm(V) are spherical harmonic functions,uSa& arethe Bloch functions of the conduction band, anda5↑ (↓) isthe projection of the electron spin,sz51(2)1/2. For holesin the fourfold degenerate valence band they can be written

E3/2~b!5\2w2~b!

2mhha2 , ~2!

cM~r !52 (l50,2

Rl~r !~21!M23/2

3 (m1m5M

S 3/2 l 3/2

m m 2M DYlm~V!um , ~3!

where b5mlh /mhh is the ratio of the light to heavy holeeffective masses,w(b) is the first root of the equation21–24,8

j 0~w! j 2~Abw!1 j 2~w! j 0~Abw!50, ~4!

where j n(x) are spherical Bessel functions, (mi

nkpl ) are

Wigner 3j symbols, andum (m561/2,63/2) are the Blochfunctions of the fourfold degenerate valence bandG8:

25

u3/251

A2~X1 iY!↑, u23/25

i

A2~X2 iY!↓,

u1/25i

A6@~X1 iY!↓22Z↑#, ~5!

u21/251

A6@~X2 iY!↑12Z↓#.

The radial functionsRl(r ) are21,23,8

R2~r !5A

a3/2F j 2~wr /a!1j 0~w!

j 0~wAb!j 2~wAbr /a!G ,

R0~r !5A

a3/2F j 0~wr /a!2j 0~w!

j 0~wAb!j 0~wAbr /a!G , ~6!

where the constantA is determined by the normalizationcondition

E drr 2@R02~r !1R2

2~r !#51. ~7!

The dependence ofw on b is presented in Fig. 1~a!.8

For spherical dots the exciton ground state (1S3/21Se) iseightfold degenerate. However, shape and internal crystalstructure anisotropy together with the electron-hole exchangeinteraction lift this degeneracy. The splitting and the transi-tion oscillator strengths of the states, as well as their order,are very sensitive to crystal size and shape, as shown below.We calculate this splitting neglecting the warping of the va-lence band and the nonparabolicity of the electron and lighthole energy spectra.

A. Energy spectrum and wave functions

Nanocrystal asymmetry lifts the hole state degeneracy.The asymmetry has two sources: the intrinsic asymmetry ofthe hexagonal lattice structure of the crystal8 and the non-spherical shape of the finite crystal.26 Both split the fourfolddegenerate hole state into two twofold degenerate states—aKramers doublet—havinguM u51/2 and 3/2, respectively.

The splitting due to the intrinsic hexagonal lattice struc-ture,D int , can be written8

D int5Dcrv~b!, ~8!

4844 54AL. L. EFROSet al.

whereDcr is the crystal field splitting equal to the distancebetween theA andB valence subbands in bulk semiconduc-tors having a hexagonal lattice structure~25 meV in CdSe!.Equation~8! is obtained within the framework of the qua-sicubic model for the case when the crystal field splitting canbe considered as a perturbation.8 The Kramers doublet split-ting does not depend on crystal size but only on the ratio ofthe light to heavy hole effective masses. The dimensionlessfunction v(b) describing this dependence,8 shown in Fig.1~b!, varies rapidly in the region 0,b,0.3. TheuM u53/2state is the ground state.

We model the nonsphericity of the crystal by taking it tobe an ellipsoid whose deviation from sphericity is character-ized by the ratioc/b511m of its major to minor axes. Herem is the ellipticity of the crystal and is positive~negative! forprolate ~oblate! crystals. The splitting arising from this de-viation has been calculated in first order perturbationtheory:26

Dsh52mu~b!E3/2~b!, ~9!

whereE3/2 is the 1S3/2 ground state hole energy for sphericalcrystals of radiusa5(b2c)1/3. E3/2 is inversly proportional toa2 @see Eq.~2!# and the shape splitting is therefore a sensi-tive function of the crystal size. The functionu(b) ~Ref. 26!decreases from a value of 4/15 atb50, changes sign atb50.14, and goes to zero atb51 @see Fig. 1~c!#.

The net splitting of the hole state,D(a,b,m), is the sumof the crystal field and shape splitting,

D~a,b,m!5Dsh1D int . ~10!

In crystals where the functionu(b) is negative, e.g., in CdSecrystals whereb50.28,5 the net splitting decreases with sizein prolate (m.0) crystals. Even the order of the hole levelscan change, with theuM u51/2 state becoming the holeground level for sufficiently small crystals.27 This can bequalitatively understood within a model of uncoupledA andB valence subbands. In prolate crystals the energy of thelowest hole quantum size level is determined by its motion inthe plane perpendicular to the hexagonal axis. In this planethe hole effective mass in the lowest subbandA is smallerthan that in the higherB subband.8 Decreasing the size of thecrystal causes a shift of the quantum size level inverselyproportional to both the effective mass and the square of thenanocrystal radius. The shift is therefore larger for theAsubband than for theB subband and, as a result, can changethe order of the levels in small crystals. In oblate (m,0)crystals, where the levels are determined by motion along thehexagonal axis, theB subband has the smaller mass. Hencethe net splitting increases with decreasing size and the statesmaintain their original order.

The eightfold degeneracy of the spherical band-edge ex-citon is also broken by the electron-hole exchange interac-tion which mixes different electron and hole spin states. Ithas the following form:28,25

Hexch52~2/3!«exch~a0!3d~re2rh!sJ, ~11!

wheres is the electron Pauli spin-1/2 matrix,J is the holespin-3/2 matrix,a0 is the lattice constant, and«exch is theexchange strength constant. In bulk crystals with cubic lat-tice structure this term splits the eightfold degenerate groundexciton state into a fivefold degenerate optically passive statewith total angular momentum 2 and a threefold degenerateoptically active state with total angular momentum 1. Thissplitting can be expressed in terms of the bulk exciton Bohrradiusaex:

\vST5~8/3p!~a0 /aex!3«exch. ~12!

In bulk crystals with hexagonal lattice structure this termsplits the exciton fourfold degenerate ground state into a trip-let and a singlet state, separated by

\vST5~2/p!~a0 /aex!3«exch. ~13!

Equations~12! and ~13! allow one to evaluate the exchangestrength constant. In CdSe crystals, where\vST50.13 meV,29 a value of«exch5450 meV is obtained usingaex556 Å.

Taken together, the hexagonal lattice structure, crystalshape asymmetry, and electron-hole exchange interactionsplit the original ‘‘spherical’’ eightfold degenerate excitoninto five levels. The levels are labeled by the magnitude ofthe exciton total angular momentum projection,F5M1sz :one level with F562, two with F561, and two withF50. The level energies« uFu are determined by solving thesecular equation det(E2« uFu)50, where the matrixE con-sists of matrix elements of the asymmetry perturbations andthe exchange interactionHexch, taken between the excitonwave functionsCa,M(re ,rh)5ca(re)cM(rh):

FIG. 1. ~a! The dependence of the hole ground state functionw(b) on the light to heavy hole effective mass ratiob; ~b! thedimensionless functionv(b) associated with hole level splittingdue to hexagonal lattice structure;~c! the dimensionless functionu(b) associated with hole level splitting due to crystal shape asym-metry; ~d! the dimensionless functionx(b) associated with excitonsplitting due to the electron-hole exchange interaction.

54 4845BAND-EDGE EXCITON IN QUANTUM DOTS OF . . .

↑,3/2 ↑,1/2 ↑,21/2 ↑,23/2 ↓,3/2 ↓,1/2 ↓,21/2 ↓,23/2

↑,3/2 23h2

2D

20 0 0 0 0 0 0

↑,1/2 0 2h2

1D

20 0 2 iA3h 0 0 0

↑,21/2 0 0 h2

1D

20 0 2 i2h 0 0

↑,23/2 0 0 0 3h2

2D

20 0 2 iA3h 0

↓,3/2 0 iA3h 0 0 3h2

2D

20 0 0

↓,1/2 0 0 i2h 0 0 h2

1D

20 0

↓,21/2 0 0 0 iA3h 0 0 2h2

1D

20

↓,23/2 0 0 0 0 0 0 0 23h2

2D

2~14!

h5(aex/a)3\vSTx(b), and the dimensionless function

x(b) is written in terms of the electron and hole radial wavefunctions,

x~b!5~1/6!a2E0

a

dr sin2~pr /a!@R02~r !10.2R2

2~r !#.

~15!

The dependence ofx on the parameterb is shown in Fig.1~d!.

Solution of the secular equation yields five exciton levels.The energy of the exciton with total angular momentum pro-jection uFu52 and its dependence on crystal size are givenby

«2523h/22D/2. ~16!

The respective wave functions are

C22~re ,rh!5C↓,23/2~re ,rh!,

C2~re ,rh!5C↑,3/2~re ,rh!. ~17!

The energies and size dependence of the two levels, eachwith total momentum projectionuFu51, are given by

«1U,L5h/26A~2h2D!2/413h2, ~18!

whereU andL correspond to the upper~‘‘ 1’’ in this equa-tion! or lower ~‘‘ 2 ’’ in this equation! sign, respectively. Wedenote these states by61U and61L, respectively; i.e., theupper and lower state with projectionF561. The corre-sponding wave functions for the states withF511 are

C1U,L~re ,rh!57 iC1C↑,1/2~re ,rh!1C2C↓,3/2~re ,rh!;

~19!

for the states withF521

C21U,L~re ,rh!57 iC2C↑,23/2~re ,rh!1C1C↓,21/2~re ,rh!,

~20!

where

C65AAf 21d6 f

2Af 21d, ~21!

f5(22h1D)/2, andd53h2. The energies and size depen-dence of the twoF50 exciton levels are given by

«0U,L5h/21D/262h ~22!

~we denote the twoF50 states by 0U and 0L), with corre-sponding wave functions

FIG. 2. The size dependence of the exciton band-edge structurein ellipsoidal hexagonal CdSe quantum dots with ellipticitym: ~a!spherical dots (m50); ~b! oblate dots (m520.28);~c! prolate dots(m50.28); ~d! dots having a size dependent ellipticity as deter-mined from SAXS and TEM measurements. Solid~dashed! linesindicate optically active~passive! levels.


C0U,L~re ,rh!5

1

A2@7 iC↑,21/2~re ,rh!1C↓,1/2~re ,rh!#,

~23!

The size dependence of the band-edge exciton splittingfor hexagonal CdSe crystals with different shapes is shownin Fig. 2. The calculation was done usingb50.28.5 Inspherical crystals@Fig. 2~a!# theF562 state is the excitonground state for all sizes, and is optically passive, as wasshown in Ref. 8. The separation between the ground stateand the lower optically activeF561 state initially increaseswith decreasing size as 1/a3, but tends to 3D/4 for very smallcrystals. In oblate crystals@Fig. 2~b!# the order of the excitonlevels is the same as in spherical ones. However, the splittingdoes not saturate, because in these crystalsD increases withdecreasing size. In prolate crystalsD becomes negative withdecreasing size and this changes the order of the excitonlevels at some value of the radius@Fig. 2~c!#; in small crys-tals the optically passive~as we show below! F50 statebecomes the ground exciton state. The crossing occurs whenD goes through 0. In nanocrystals of this size the shapeasymmetry exactly compensates the asymmetry connectedwith the hexagonal lattice structure,27 and hence the excitonlevels have ‘‘spherical’’ symmetry. As a result there is onefivefold degenerate exciton with total angular momentum 2~which is reflected in the crossing of the 0L, 61L, and62levels! and one threefold degenerate exciton state with totalangular momentum 1~reflected in the crossing of the 0U and61U levels!. In Fig. 2~d! the band-edge exciton fine struc-ture is shown for the case where the ellipticity varies withsize,30 corresponding to small-angle x-ray scattering~SAXS!and transmission electron microscopy~TEM! measurementsof our CdSe crystals.31 The level structure closely resemblesthat of spherical crystals.

The size dependence of the band-edge exciton splitting incubic CdTe crystals with different shapes is shown in Fig. 3.The calculation was done using the parametersb50.086 and\vST50.04 meV. One can see that in the spherical nano-crystals the electron-hole exchange interaction splits theeightfold degenrate band-edge exciton into a fivefold degen-erate exciton with total angular momentum 2 and a threefolddegenerate exciton with total angular momentum [email protected]~a!#. Crystal shape asymmetry lifts the degeneracy of thesestates and completely determines the relative order of theexciton states@see Fig. 3~b! and Fig. 3~c! for comparison#.

B. Selection rules and transition oscillator strengths

To describe the fine structure of the absorption spectraand photoluminescence we calculate transition oscillatorstrengths for these five exciton states. The mixing of theelectron and hole spin momentum states by the electron-holeexchange interaction strongly affects the optical transitionprobabilities. The wave functions of theuFu52 exciton state,however, are unaffected by this interaction@see Eq.~17!#; itis optically passive in the dipole approximation becauseemitted or absorbed photons cannot have an angular momen-tum projection of62. The probability of optical excitationor recombination of an exciton state with total angular mo-

mentum projectionF is proportional to the square of thematrix element of the momentum operatorep between thatstate and the vacuum state,

PF5 z^0uepuCF& z2, ~24!

where u0&5d(re2rh), e is the polarization vector of theemitted or absorbed light, the momentum operatorp actsonly on the valence band Bloch functions@see Eq.~5!# andthe exciton wave functionCF is written in the electron-electron representation. Exciton wave functions in theelectron-hole representation are transformed to the electron-electron representation by taking the complex conjugate ofEqs.~17!, ~19!, and~23! and flipping the spin projections inthe hole Bloch functions (↑ and↓ to ↓ and↑).

For the exciton state withF50 we obtain

P0U,L5 z^0uepuC0

U,L& z25~161!2

3KP2cos2~u!, ~25!

whereP5^Su pzuZ& is the Kane interband matix element,uis the angle between the polarization vector of the emitted orabsorbed light and the hexagonal axis of the crystal, andK isthe square of the overlap integral8

K52

a U E drr sin~pr /a!R0~r !u2. ~26!

Its value is independent of crystal size and depends only onb; hence the excitation probability of theF50 state does notdepend on crystal size. For the lower exciton state, 0L, it isidentically zero. At the same time the exchange interactionincreases the excitation probability of the upper 0U exciton

FIG. 3. The size dependence of the exciton band-edge structurein ellipsoidal cubic CdTe quantum dots with ellipticitym: ~a!spherical dots (m50); ~b! oblate dots (m520.28);~c! prolate dots(m50.28); Solid~dashed! lines indicate optically active~passive!levels.


state by a factor of 2. This result arises from the constructiveand destructive interference of the wave functions of the twoindistinguishable exciton statesu↑,21/2& and u↓,1/2& @seeEq. ~23!#.

For the exciton state withF51 we obtain

P1U,L5

1

3 S 2Af 21d7 f6A3d2Af 21d

DKP2sin2~u!. ~27!

The excitation probability of theF521 state is equal to thatof theF51 state. As a result, the total transition probabilityto the doubly degenerateuFu51 exciton states is equal to2P1

U,L .Equations~25! and ~27! show that theF50 and uFu51

state excitation probabilities differ in their dependence on theangle between the light polarization vector and the hexago-nal axis of the crystal. If the crystal hexagonal axes arealigned perpendicular to the light direction, only the activeF50 state can be excited. Alternatively, when the crystalsare aligned along the light propagation direction, only theupper and loweruFu51 states will participate in the absorp-tion. For the case of randomly oriented crystals, polarizedexcitation resonant with one of these exciton states selec-tively excites suitably oriented crystals, leading to polarizedluminescence.8 Observation of this effect has been reportedin several papers.12,32,15Furthermore, a large energy splittingbetween theF50 and uFu51 states can lead to differentStokes shifts in the polarized luminescence.

To find the probability of exciton excitation for a systemof randomly oriented nanocrystals, we average Eqs.~25! and

~27! over all possible solid angles. The respective excitationprobabilities are proportional to

P0U,L5

~161!2KP2

9,

P1U,L5P21

U,L52KP2

9 S 2Af 21d7 f6A3d2Af 21d

D . ~28!

There are three optically active states with relative oscillatorstrengthsP0

U, 2P1U, and 2P1

L. The dependence of thesestrengths on size for different shapes is shown in Fig. 4 forhexagonal CdSe crystals. It is seen that the crystal shapestrongly influences this dependence. For example, in prolatecrystals@Fig. 4~c!# the61L state oscillator strength goes tozero for those crystals for whichD50; i.e., where the crystalshape asymmetry exactly compensates the internal asymme-try connected with the hexagonal lattice structure. For thesecrystals the oscillator strengths of all the upper states (0U,1U, and21U) are equal. Nevertheless, one can see that forall nanocrystal shapes the excitation probability of the loweruFu51 (61L) exciton state, 2P1

L, decreases with size andthat the upperuFu51 (61U) gains its oscillator strength.

This can be understood by examining the spherically sym-metric limit. In spherical nanocrystals the exchange interac-tion leads to the formation of two exciton states—with totalangular momenta 2 and 1. The ground state is the opticallypassive state with total angular momentum 2. This state isfivefold degenerate with respect to the total angular momen-tum projection. For small nanocrystals the splitting of theexciton levels due to the nanocrystal asymmetry can be con-sidered as a perturbation to the exchange interaction, which

FIG. 4. The size dependence of the oscillator strengths, relativeto that of the 0U state, for the optically active states in hexagonalCdSe quantum dots with ellipticitym: ~a! spherical dots (m50);~b! oblate dots (m520.28); ~c! prolate dots (m50.28); ~d! dotshaving a size dependent ellipticity as determined from SAXS andTEM measurements.

FIG. 5. The size dependence of the oscillator strengths, relativeto that of the 0U state, for the optically active states in cubic CdTequantum dots with ellipticitym: ~a! spherical dots (m50); ~b! ob-late dots (m520.28); ~c! prolate dots (m50.28).


grows as 1/a3. In this situation the wave functions of the61L, 0L, and62 exciton states turn into the wave functionsof the optically passive exciton with total angular momentum2. The wave functions of the61U and 0U exciton statesbecome those of the optically active exciton states with totalangular momentum 1. These three states therefore carrynearly all the oscillator strength.

In large crystals, for all possible shapes, we can neglectthe exchange interaction~which decreases as 1/a3), and thusthere are only two fourfold degenerate exciton states~seeFig. 3!. The splitting here is determined by the shape asym-metry and the intrinsic crystal field. In a system of randomlyoriented crystals, the excitation probability of both thesestates is the same:P0

U12P1U52P1

L52KP2/3.8

In Fig. 5 we show these dependences for variously shapedcubic CdTe nanocrystals.

It is necessary to note here that despite the fact that theexchange interaction drastically changes the structure and theoscillator strengths of the band-edge exciton, the polarizationproperties of the nanocrystal are determined by the internaland crystal shape asymmetries. All polarization effects areproportional to the net splitting parameterD and go to zerowhenD50.

C. Recombination of the dark exciton in magnetic fields

Time decay measurements of the dark exciton in CdSenanocrystals in the presence of external magnetic fields

strongly support our model.16 Recombination of the dark ex-citon is allowed if the magnetic field is not directed along thehexagonal axis of the nanocrystal. In this caseF is no longera good quantum number and the62 dark exciton states areadmixed with the optically active61 bright exciton states.This now allows the direct optical recombination of the62 exciton ground state.

For nanosize quantum dots the effect of an external mag-netic fieldH is well described as a molecular Zeeman effect:

HH51

2gemBsH2ghmBKH, ~29!

wherege andgh are the electron and holeg factors, respec-tively, andmB is the Bohr magneton. For electrons in CdSege50.68.33 The value of the holeg factor is calculated in theAppendix using the results of Ref. 34, and isgh521.09. InEq. ~29! we neglect the diamagneticH2 terms because thedots are significantly smaller than the magnetic length(;115 Å at 10 T!.

Treating the magnetic interaction as a perturbation, wecan determine the influence of the magnetic field on the un-perturbed exciton state using the perturbation matrix

EH8 5^Ca,MumB

21HHuCa8,M8&:

↑,3/2 ↑,1/2 ↑,21/2 ↑,23/2 ↓,3/2 ↓,1/2 ↓,21/2 ↓,23/2

↑,3/2 Hz(ge23gh)2

2 iA3ghH2

2

0 0 geH2

20 0 0

↑,1/2 iA3ghH1

2

Hz(ge2gh)2

2 ighH2 0 0 geH2

20 0

↑,21/2 0 ighH1 Hz(ge1gh)2

2 iA3ghH2

2

0 0 geH2

20

↑,23/2 0 0 iA3ghH1

2

Hz(ge13gh)2

0 0 0 geH2

2↓,3/2 geH1

20 0 0 2Hz(ge13gh)

22 iA3ghH2

2

0 0

↓,1/2 0 geH1

20 0 iA3ghH1

2

2Hz(ge1gh)2

2 ighH2 0

↓,21/2 0 0 geH1

20 0 ighH1 2Hz(ge2gh)

22 iA3ghH2

2↓,23/2 0 0 0 geH1

20 0 iA3ghH1

2

2Hz(ge23gh)2

~30!

whereHz is the magnetic field projection along the crystalhexagonal axis andH65Hx6 iH y . One can see from Eq.~30! that components of the magnetic field perpendicular tothe hexagonal crystal axis mix theF562 dark excitonstates with the respective optically activeF561 bright ex-

citon states. In small nanocrystals, whereh is of the order of10 meV, the influence of even the strongest magnetic fieldcan be considered as a perturbation. The case of large crys-tals whereh is of the same order asmBgeH will be consid-ered later. The admixture in theF52 state is given by


DC25mBH2

2 FgeC22A3ghC1

«22«11 C1

1

1A3ghC21geC

1

«22«12 C1

2G , ~31!

where the constantsC6 are given in Eq.~21!. The admixturein the F522 exciton state of theF521 exciton state isdescribed similarly.

This admixture of the optically active bright exciton statesallows the optical recombination of the dark exciton. Theradiative recombination rate of an exciton state,F, can beobtained by summing Eq.~24! over all light polarizations:35

1

t uFu54e2vnr3m0

2c3\z^0u pmuCF& z2, ~32!

wherev andc are the light frequency and velocity,nr is therefractive index, andm0 is the free electron mass. Using Eqs.~25!–~27! we obtain the radiative decay time for the upperexciton state withF50,

1

t05

8vnrP2K

93137m02c2

; ~33!

for the upper and lower exciton states withuFu51,

1

t1U,L 5S 2Af 21d7 f6A3d

2Af 21dD 1

t0. ~34!

Using the admixture of theuFu51 states in theuFu52 exci-ton given in Eq.~31!, we calculate the recombination rate ofthe uFu52 exciton in a magnetic field,

1

t2~H !53mB

2H2sin2~u!

8D2 S 2gh2ge2h1D

3h D 2 1t0 . ~35!

The characteristic timet0 does not depend on the crystalradius. For CdSe, calculations using 2P2/m0517.5 eV~Ref.3! give t051.6 ns.

In large crystals the magnetic field splittingmBgeH is ofthe same order as the exchange interactionh and cannot beconsidered a perturbation. At the same time, both these en-ergies are much smaller than the splitting due to the crystalasymmetry. We consider here the admixture in theuFu52dark exciton of the lowestuFu51 exciton only. This can becalculated exactly. The magnetic field also lifts the degen-eracy of the exciton states with respect to the sign of the totalangular momentum projectionF. The energies of the formerF522 andF521 states are

«21,226 5

2D13mBghHz

2

6A~3h1mBgeHz!

21~mBge!2H'

2

2, ~36!

where1 (2) refers to theF521 state with anF522admixture (F522 state with anF521 admixture! andH'5AHx

21Hy2. The corresponding wave functions are

C21,226 5AAp21unu26p

2Ap21unu2C↑,23/2

7n

A2Ap21unu2~Ap21unu26p!C↓,23/2,

~37!

wheren5mBgeH1 andp53h1mBgeHz . The energies andwave functions of the formerF52,1 states are~using nota-tion similar to that used just above!

«1,26 5

2D23mBghHz

26

A~3h2mBgeHz!21~mBge!

2H'2

2,

~38!

C1,26 5AAp821un8u26p8

2Ap821un8u2C↓,3/2

7n8

A2Ap821un8u2~Ap821un8u26p8!C↑,3/2,

~39!

wheren85mBgeH2 andp853h2mBgeHz . As a result thedecay time of the dark exciton in an external magnetic fieldcan be written

1

t~H !5

A11z212zcosu212zcosu

2A11z212zcosu

3

2t0, ~40!

wherez5mBgeH/3h. The probabilty of exciton recombina-tion increases in weak magnetic fields (z!1) as@(0.5mBgeH)

2/(3h)2#@3sin2(u)/2t0#, and saturates in strongmagnetic fields (z@1), reaching @3(12cosu)/4t0#@12(3h/mBgeH)(11cosu)].

One can see from Eqs.~35! and ~40! that the recombina-tion lifetime depends on the angle between the crystal hex-agonal axis and the magnetic field. The recombination timeis different for different crystal orientations, which leads to anonexponetial time decay dependence for a system of ran-domly oriented crystals.

III. EXPERIMENT

The samples in our Stokes shift study were prepared usingthe synthetic technique described in Ref. 31. This methodproduces nearly monodisperse wurtzite crystallites of CdSe(s,5%! which are slightly prolate and have surfaces passi-vated by an organic layer of tri-n-octylphosphine/tri-n-octylphosphine oxide ligands. In total, 18 samples wereprepared for this study. Their effective radii, as determinedby small angle x-ray scattering and TEM measurements,range from 12 to 56 Å. The samples were isolated as a pow-der and redispersed into a mixture ofo-terphenyl in tri-n-butylphosphine~200 mg/ml! to form an optically clearglass at liquid helium temperatures. Each sample was loadedbetween sapphire flats separated by a 0.5 mm thick Teflon


spacer and mounted in a helium cold finger cryostat for lowtemperature optical work. The 12 Å sample used in the mag-netic field and luminescence decay experiments was left inits original growth solution without being redispersed in tri-n-butylphosphine ando-terphenyl. With the exception of the12 Å sample just described, all the samples in our Stokesshift study were freshly prepared specifically for this study.

Absorption, luminescence, and fluorescence line nar-rowed spectra were taken on the same optical setup. In theabsorption experiment, light from a 300 W Xe arc lamp waspassed through the sample and the transmitted light detectedwith an optical multichannel analyzer~OMA! coupled to a0.33 m single spectrometer. The full luminescence spectrumof each sample was obtained by passing light from a 300 WHg-Xe arc lamp through a 0.25 m spectrometer and excitingit above its band edge. Typically, the excitation light waskept spectrally broad@50 nm full width at half maximum~FWHM!# to prevent any possible size selection of the dots.The emitted light was then dispersed and detected with thespectrometer/OMA combination described in the absorptionexperiment.

Fluorescence line narrowed spectra were acquired by ex-citing the samples with the output of aQ-switchedneodymium-doped yttrium aluminum garnet~Nd:YAG!/dyelaser system (;7 ns pulses!. The laser power was attenuatedto ensure that the luminescence varied linearly with the ex-citation intensity. The measured luminescence was dispersedand detected with a 0.33 m spectrometer coupled to a 5 nsgated OMA.

Magnetic field studies were conducted by placing thesample within the bore of a variable field superconductingmagnet. The excitation source was the Nd:YAG/dye lasersystem described above. The resulting luminescence was dis-persed through a 0.66 m single spectrometer and detectedwith a 5 nstime gated OMA. Luminescence decays wererecorded with the following: a 500 Mhz digitizing oscillo-scope, a photomultiplier tube with 2 ns resolution, and a 0.75m subtractive double spectrometer to eliminate scattered la-ser light.

A. Stokes shift of the resonant photoluminescence

We observe strong evidence for the predicted band-edgefine structure in our fluorescence line narrowing~FLN! ex-periments. By exciting our samples on the red edge of theabsorption, we selectively excite the largest dots present inthe residual size distribution of each sample. This reducesthe inhomogeneous broadening of the luminescence and theresulting emission is spectrally narrow, displaying a well re-solved longitudinal optical~LO! phonon progression. Inpractice, we excite our samples at that point on their red edgewhere the absorption is roughly 1/3 of the peak of the band-edge absorption. Figure 6 shows the FLN spectra for the sizeseries considered in this paper. The peak of the zero LOphonon line~ZPL! is observed to be shifted with respect tothe excitation energy. This Stokes shift is size dependent andranges from;20 meV for small crystals to;2 meV forlarge crystals. Moving the excitation position does not no-ticeably affect the Stokes shift of the larger samples; how-ever, it does make a difference for the smaller sizes. Weattribute this difference to the excitation of different size dots

within the size distribution of a sample, causing the observedStokes shift to change. The effect is largest in the case ofsmall nanocrystallites because of the size dependence of theStokes shift~see Fig. 7!.

In terms of the proposed model, excitation on the red edgeof the absorption probes the lowestuFu51 bright excitonstate@see Fig. 2~d!#. The transition to this state is followedby thermal relaxation to the darkuFu52 state, from whererecombination occurs through a phonon assisted16,8 ornuclear/paramagnetic spin-flip assisted transition.16 The ob-served Stokes shift is the difference in energy between the61L state and the dark62 state and increases with decreas-ing size.

We find good agreement between the experimental valuesof the size dependent Stokes shift and the values derivedfrom theory. Figure 7 compares the two results. The onlyparameters used in the theoretical calculation are taken fromthe literature: aex556 Å,3 \vST50.13 meV,29 andb50.28.4,5 The comparison shows that there is good quanti-tative agreement between experiment and theory for largesizes. For small crystals, however, the theoretical splittingbased on the size dependent exchange interaction begins tounderestimate the observed Stokes shift. This discrepancymay be explained, in part, by an additional contribution tothe Stokes shift by phonons, which is not accounted for inthe present model.36,37

FIG. 6. Normalized FLN spectra for CdSe QD’s between 12 and56 Å in radius. The mean radii of the dots are determined fromSAXS and TEM measurements. A 10 HzQ-switched Nd:YAG/dyelaser system (;7 ns pulses! serves as the excitation source. Detec-tion of the FLN signal is accomplished using a time gated OMA.The laser line is included in the figure~dotted line! for referencepurposes. All FLN spectra are taken at 10 K.


B. Stokes shift of the nonresonant photoluminescence

We have also studied the Stokes shift of the nonresonantphotoluminescence. In this experiment we excite oursamples above their band-edge absorption. The resulting‘‘full’’ luminescence contains contributions from all crystal-lites in the sample residual size distribution and is inhomo-geneously broadened. It shows no distinct phonon structure.This is unlike the FLN experiment where we suppress theinhomogeneous broadening of the luminescence by selec-tively exciting a narrow subset of crystallites. Figure 8 pre-sents the full luminescence spectrum measured for nine sizesin the size series considered. As with the FLN data, the fullluminescence shows a strong size dependence of the Stokesshift, ranging from;100 meV for small sizes to;25 meVfor large sizes. Note that the full luminescence Stokes shift istaken to be the difference in energy between the lowest en-ergy peak of the band-edge absorption and the peak of thefull luminescence~the peak of the lowest energy absorptionis determined by fitting the absorption spectra with a seriesof Gaussians!. We denote the full luminescence Stokes shiftas the ‘‘nonresonant’’ Stokes shift. The nonresonant Stokesshift requires too large a Huang-RhysS parameter to bereadily explained by exciton-phonon coupling.43 There are,however, two other explanations which account for the largevalue of the nonresonant Stokes shift.

The first one is directly connected with the nanocrystalsize distribution. In the strong confinement regime the totaloscillator strength of transitions between the electron andhole quantum size levels does not depend on crystal size.However, the excitation probability is proportional to thenumber of participating states and, as a result, is proportionalto the crystal volume and therefore toa3, for excitations farfrom the band edge.6 Thus while the first absorption peak

generally follows the crystal size distribution, the position ofthe luminescence line for nonresonant excitation is deter-mined by the largest crystals within this distibution. Thiscauses a Stokes shift because the energy of the band-edgetransitions in larger crystals is less than in smaller ones.

This, however, is not the main source of the nonresonantStokes shift in our samples, which have a very narrow crystalsize distribution (,5%!. The nonresonant Stokes shift hereis connected with the band-edge exciton fine structure. Insmall nanocrystals the two upper states of the band-edge ex-citon fine structure possess nearly all the oscillator strengthof the band-edge transition@see Fig. 4~d!#, and the first ab-sorption maximum is therefore determined by the positionsof these two states. The nonresonant Stokes shift then is thedifference in energy between these upper states and the62dark exciton ground state. This accounts for the sizable mag-nitude of the Stokes shift.

To describe the band-edge absorption we convolute thethree optically active states with the intrinsic size distributionof the sample, weighted by their respective oscillatorstrengths. The peak of the band-edge absorption occurs at theweighted energetic mean of these states. We obtain the po-sition of the full luminescence line by convoluting the posi-tion of the62 dark exciton state with the size distributionweighted by the nanocrystal excitation probability (;a3).

Figure 9 compares the shifts predicted by theory for asample with a 5% size distribution to the experimental valuesof the nonresonant Stokes shift. The theory is shown as a

FIG. 7. The size dependence of the resonant Stokes shift. ThisStokes shift is the difference in energy between the pump energyand the peak of the ZPL in the FLN measurement. The pointslabeled3 are the experimental values. The solid line is the theo-retical size dependent splitting between the61L state and the62exciton ground state@see Fig. 2~d!#.

FIG. 8. Normalized absorption and full luminescence spectra forCdSe QD’s between 12 and 56 Å in radius. A 300 W Xe arc lampserves as the excitation source for both absorption and lumines-cence experiments. The excitation light is intentionally broad~50nm FWHM! to prevent possible size selection of the dots. Detectionof the transmission/luminescence signal is carried out with an OMAcoupled to a 0.33 m spectrometer. The absorption spectra are indi-cated by solid lines; the corresponding luminescence spectra bydotted lines.


dashed line in the figure. We find a reasonable correlationbetween the two, but the theory underestimates the Stokesshift for the sizes considered. However, we have not takeninto account phonons, whose role is seen experimentally inthe LO phonon progressions observed in FLN and PLEspectra.18,16,38,39We include the contribution of phonons tothe theoretical Stokes shift phenomenologically by includingin the convolution with the size distribution the phonon de-pendent absorption and emission line shapes for single CdSequantum dots. We assume the following forms for the ab-sorption and emission line shapes:

A~n,n8!5(l51

3

(m50

4~Sa!

m

A2pg l ,mm!

3expS 2@n2~n81D l1mvLO!#2

2g l ,m2 D , ~41!

E~n,n8!5 (n50

4~Se!

n

A2pgnn!expS 2@n2~n82mvLO!#2

2gn2 D .

~42!

We consider the first five LO phonon replicas associatedwith each of the three light exciton states~denoted byl ) inabsorption as well as the first five LO phonon replicas inemission. In Eqs.~42! and ~43!, n8 denotes the position ofthe zero phonon line,vLO is the LO phonon frequency sepa-rating the phonon replicas,D l is the offset of thel th bright

exciton from theuFu52 dark exciton ground state,gn is thelinewidth of thenth phonon replica in absorption and emis-sion, andg l ,m is the width of themth phonon replica forabsorption into thel th light exciton state. The valueSa(e) isthe absorption~emission! exciton–LO-phonon coupling con-stant. It is equivalent to the Huang-RhysS parameter assum-ing a displaced harmonic oscillator model.40,41The values weuse forSa(e) are derived from experimental results.41 Includ-ing the contribution of phonons moves the predicted band-edge absorption maximum to the blue and emission to thered, and results in the solid curve shown in Fig. 9. The modi-fied curve still underestimates the nonresonant Stokes shift,but is in reasonable agreement with the experimental data.

We note that the nonresonant Stokes shift depends on thesize distribution. This is illustrated in Fig. 9, where we showthe increase of the calculated Stokes shift for a sample with a10% size distribution. The good fit to the experimental datadoes not necessarily imply though that we have a 10% sizedistribution. Any other type of inhomogeneous broadeningleads to an additional Stokes shift at the nonresonant excita-tion conditions. The nonresonant Stokes shift may containcontributions from shape distributions, structural inhomoge-neities, and differences in chemical environment experiencedby the dots dispersed in the glassy matrix.

C. Dark exciton lifetime in a magnetic field

Strong evidence for the dark exciton state is found in thestudy of the FLN spectra and luminescence decays in exter-nal magnetic fields. In Fig. 10~a! we show the magnetic fielddependence of the FLN between 0 and 10 T for 12 Å radiusdots. Each spectrum is normalized to the zero field one-phonon line for clarity. In isolation the62 state would havean infinite lifetime within the electric dipole approximation,since the emitted photon cannot carry off an angular momen-tum of 2. However, the dark exciton can recombine via anLO phonon assisted momentum-conserving transition.42

Spherical LO phonons with orbital angular momenta of 1 or2 are expected to participate in these transitions; the selectionrules are determined by the coupling mechanism.8,43 Conse-quently, for zero field the LO phonon lines are strongly en-hanced relative to the ZPL. With increasing magnetic field,however, the62 level gains optically active61 character@Eq. ~30!#, diminishing the need for LO phonon assisted re-combination in dots whose hexagonal axis is not parallel tothe magnetic field. This explains the dramatic rise of the ZPLintensity relative to the higher LO phonon replicas with in-creasing field.

The magnetic field induced admixture of the optically ac-tive 61 states shortens the exciton radiative lifetime. Lumi-nescence decays for 12 Å radius crystallites between 0 and10 T at 1.7 K are shown in Fig. 10~b!. The sample wasexcited far to the blue of the first absorption maximum toavoid orientational selection in the excitation process sincethe transition dipole of theuFu51 states is perpendicular tothe c axis @see Eq.~27!#. Excitons rapidly thermalize to theground state through acoustic and optical phonon emission.The longms luminescence at zero field is consistent with LOphonon assisted recombination from this state. Although thelight emission occurs primarily from the62 state, its longradiative lifetime allows the thermally partially populated

FIG. 9. The size dependence of the nonresonant Stokes shift.This Stokes shift is considered as the difference in energy betweenthe peak of the band-edge absorption and the peak of the full lumi-nescence. Experimental values are represented by3. The dashedline is the theoretical Stokes shift calculated for a sample with as55% size distribution. It is the difference between the mean en-ergy of the three light exciton states and the mean position of the62 exciton ground states of the participating crystals. The solidline includes the contribution of phonons to the theoretical splitting.The dotted line shows the theoretical results for a sample with a10% size distribution


61L state to also contribute to the luminescence. With in-creasing magnetic field the luminescence lifetime decreases;since the quantum yield remains essentially constant, we in-terpret this as an enhancement of the radiative rate.

The magnetic field dependence of the luminescence de-cays can be reproduced using three-level kinetics with61L

and62 emitting states.16 The respective radiative rates fromthese states,G1(u,H) andG2(u,H), in a particular nanocrys-tal, depend on the angleu between the magnetic field and thecrystal hexagonal axis. The thermalization rateG th of the61L state to the62 level is determined independently frompicosecond time resolved measurements. The population ofthe 61L level is determined by microscopic reversibility.We assume that the magnetic field does not affect the zeromagnetic field recombination but rather opens an additionalchannel for ground state recombination via admixture in the62 state of the61 states:G2(u,H)5G2(0,0)11/t2(u,H).This also causes a slight decrease in the recombination rateof the61L state.

The decay at zero field is multiexponential, presumablydue to sample inhomogeneities~e.g., in shape and symmetrybreaking impurity contamimations!. We describe the decayusing three three-level systems, each having a different valueof G2(0,0) and each representing a class of dots within theinhomogeneous distribution. These three-level systems are

then weighted to reproduce the zero field decay@Fig. 8~c!#.We obtain average values of 1/G2(0,0)51.42ms and1/G1(0,0)510.0 ns, in good agreement with the theoreticalvalue of the radiative lifetime for the61L state,t1L513.3 ns, calculated for a 12 Å nanocrystal using Eq.

~33!.In a magnetic field the angle dependent decay rates

@G1(u,H), G2(u,H)# are determined from Eq.~35!. The fielddependent decay is then calculated, averaging over all anglesto account for the random orientation of the crystallitecaxes. The simulated decay at 10 T@Fig. 8~c!#, using the bulkvalue for ge50.68,33 and the calculated values forD~19.4 meV! and h ~10.3 meV! for 12 Å radius dots, is inexcellent agreement with experiment. The holeg factor istreated as a fitting parameter since reliable values are notavailable. We usedgh521.00 consistent with theoreticalestimates for this parameter,gh521.09 ~see the Appendix!.

IV. DISCUSSION AND CONCLUSIONS

The size dependence of the Stokes shift we obtained forresonant excitation of the absorption band edge is in excel-lent agreement with the size dependence of the splitting be-tween the lowest optically active bright exciton state and theoptically passive dark exciton ground state. The discrepancyat the smallest sizes may be related to problems with theeffective mass approximation at these sizes or to the increas-ing role played by phonons in the luminescence. The phononinteraction through the deformation potential was shown

FIG. 10. ~a! FLN spectra for 12 Å radius dots as a function ofexternal magnetic field. The spectra are normalized to their one-phonon lines~1PL!. A small fraction of the excitation laser, whichis included for reference, appears as the sharp feature at 2.467 eV tothe blue of the zero phonon line~ZPL!. ~b! Luminescence decaysfor 12 Å radius dots for magnetic fields between 0 and 10 T mea-sured at the peak of the ‘‘full’’ luminescence~2.436 eV! and apump energy of 2.736 eV. All experiments were done in the Fara-day configuration (Hik). ~c! Observed luminescence decays for 12Å radius dots at 0 and 10 T.~d! Calculated decays based on thethree-level model described in the text. Three weighted three-levelsystems were used to simulate the decay at zero field with differentvalues ofg2 ~0.033, 0.0033, and 0.000 56 ns21) and weightingfactors~1, 3.8, and 15.3!. g1 ~0.1 ns21) andg th ~0.026 ps21) wereheld fixed in all three systems.

FIG. 11. Dependence of the hole radial function integralsI 1 andI 2, which enter in the expression for the holeg factor, on the holeeffective mass ratiob.


both theoretically44 and experimentally45 to increase with de-creasing size. This can lead to the formation of an exciton-polaron and be a source of an additional Stokes shift of theluminescence.36

The large difference between the resonant excitationStokes shift and the Stokes shift of the nonresonant photolu-minescence is related to the difference in the oscillatorstrengths of the upper and lower optically active bright exci-ton states. In the smallest crystals, the upper bright excitonsgain the oscillator strength of the lower bright excitons andgive the main contribution to the absorption. The nonreso-nant Stokes shift in this case is the energy difference betweenthe upper bright excitons and the optically passive dark ex-citon ground state. The experimental size dependence of thenonresonant Stokes shift is in reasonable agreement with thetheory. What discrepancy there is may be attributed to pho-non participation in the luminescence, which is not well un-derstood theoretically for CdSe nanocrystals.46

While the surface effects previously invoked to explainthe photophysical behavior of CdSe quantum dots may stillplay a role, especially via nonradiative processes, the ener-getics and dynamics of the band-edge emission can be quan-titatively understood in terms of the intrinsic band-edge ex-citon structure. Exciton thermalization to a dipole forbidden62 dark exciton state resolves the issue of the long lifetimesof the band-edge luminescence. An external magnetic fieldmixes the dark exciton with the optically active bright exci-ton states and allows its recombination. The magnetic fielddependence of the emission decays and the observed LOphonon structure confirm the presence of this ‘‘dark’’ exci-tonic state.

In zero external magnetic field, recombination of the62 dark exciton can take place via a LO phonon assistedtransition. Phonons can take up part of the total angular mo-mentum projection,62, which cannot be taken up by a pho-ton alone. However, even in the absence of an external mag-netic field the zero LO phonon line is weakly allowed,suggesting an alternate recombination pathway, for example,through coupling to acoustic phonons. Interaction with para-magnetic defects in the lattice can also provide an importantadditional mechanism for recombination. The spins of thesedefects potentially generate strong effective internal mag-netic fields, which, depending on the strength of the spin-spin exchange interaction with the carriers and on the crystalradius, can reach several tens of teslas, and induce spin-flipassisted transitions of the62 state, enabling zero phononrecombination to occur. Preliminary electron paramagneticresonance~EPR! spectra do in fact indicate a small concen-tration of paramagnetic centers in our samples, and efforts

under way to introduce an electrically neutral magnetic im-purity such as Mn21 into the lattice may confirm this mecha-nism.

In spite of the obvious success of our model in describingthe electronic structure of the band-edge excitation in CdSenanocrystals, several questions remain unanswered. Theseinclude the increase with decreasing size of the homoge-neous exciton absorption line width47 and the underestima-tion of the Stokes shift at small sizes. Both issues may beconnected to our lack of knowledge about the main mecha-nism of exciton interaction with polar optical phonons.46

In conclusion, we have described the band-edge excitonfine structure, and have explained in a self-consistent waymost of the complex and controversial experimental data innanosize CdSe quantum dots, e.g., the small Stokes shift ofthe resonant photoluminescence, the large Stokes shift of thenonresonant PL relative to the band-edge absorptionmaxima, the long radiative lifetime, its decrease in magneticfields, and the fine structure of the photoluminescence linenarrowing and photoluminescence excitation spectra.18

ACKNOWLEDGMENTS

M.N. and D.J.N. benefited from AT&T and NSF financialsupport, respectively. M.G.B. thanks the Lucille and DavidPackard Foundation and the Alfred P. Sloan Foundation forfinancial support. This work was funded in part by the MITCenter for Materials Science and Engineering~NSF GrantNo. DMR-90-22933! and by NSF ~DMR-91-57491!. Wealso thank the MIT Harrison Spectroscopy Laboratory~NSFGrant No. CHE-93-04251! for the use of their facilities.

APPENDIX: CALCULATION OF THE HOLE g FACTOR

The expression for theg factor of a hole localized in aspherically symmetric potential was obtained by Gel’montand D’yakonov:34

gh54

5g1I 21

8

5g~ I 12I 2!12kS 12

4

5I 2D , ~A1!

whereg1, g, andk are the Luttinger parameters48 and I 1,2are integrals of the hole radial wave functions@see Eq.~6!#:

I 15E0

a

drr 3R2

dR0dr

, I 25E0

a

drr 2R22 . ~A2!

These integrals depend only on the parameterb, and theirvariation with b is shown in Fig. 11. Usingg152.04 andg50.58,5 and the relationshipk522/315g/32g1/3,

49 onecalculates thatgh521.09.

1C. R. Pidgeon and R. N. Brown, Phys. Rev.146, 575 ~1966!.2R. L. Aggarwal, inModulation Techniques, edited by by R. K.Willardson and A. C. Beer, Semiconductors and SemimetalsVol. 9 ~Academic Press, New York, 1974!, p. 151.

3A. I. Ekimov, F. Hache, M. C. Schanne-Klein, D. Ricard, C.Flytzanis, I. A. Kudryavtsev, T. V. Yazeva, A. V. Rodina, and

Al. L. Efros, J. Opt. Soc. Am. B10, 100 ~1993!.4D. J. Norris, A. Sacra, C.B. Murray, and M. G. Bawendi, Phys.Rev. Lett.72, 2612~1994!.

5D. J. Norris and M. G. Bawendi, Phys. Rev. B53, 16 338~1996!.6Al. L. Efros and A. L. Efros, Fiz. Tekh. Poluprovodn.16, 1209

~1982! @Sov. Phys. Semicond.16, 772 ~1982!#.


7L. E. Brus, J. Chem. Phys.80, 4403~1984!.8Al. L. Efros, Phys. Rev. B 46, 7448~1992!.9G. B. Grigoryan, E. M. Kazaryan, Al. L. Efros, and T. V. Yazeva,Fiz. Tverd. Tela32, 1772 ~1990! @Sov. Phys. Solid State32,1031 ~1990!#.

10A. I. Ekimov, Phys. Scr.39, 217 ~1991!.11P. A. M. Rodrigues, G. Tamulaitis, P. Y. Yu, and S. H. Risbud,

Solid State Commun.94, 583 ~1995!.12M. G. Bawendi, W. L. Wilson, L. Rothberg, P. J. Carroll, T. M.

Jedju, M. L. Stegerwald, and L. E. Brus, Phys. Rev. Lett.65,1623 ~1990!.

13M. O’Neil, J. Marohn, and G. McLendon, J. Phys. Chem.94,4356 ~1990!; A. Hasselbarth, A. Eychmuller, and H. Weller,Chem. Phys. Lett.203, 271 ~1993!.

14P. D. J. Calcott, K. L. Nash, L. T. Canham, M. J. Kane, and D.Brumhead, J. Lumin.57, 257 ~1993!.

15M. Chamarro, C. Gourdon, P. Lavallard, and A. I. Ekimov, Jpn. J.Appl. Phys.34, Suppl. 34-1, 12~1995!.

16M. Nirmal, D. J. Norris, M. Kuno, M. G. Bawendi, Al. L. Efros,and M. Rosen, Phys. Rev. Lett.75, 3728~1995!.

17M. Chamarro, C. Gourdon, P. Lavallard, O. Lublinskaya, and A.I. Ekimov, Phys. Rev. B53, 1 ~1996!.

18D. J. Norris, Al. L. Efros, M. Rosen, and M. G. Bawendi, Phys.Rev. B53, 16 347~1996!.

19J. M. Luttinger and W. Kohn, Phys. Rev.97, 869 ~1955!.20Such optically passive exciton states were first observed in mag-

netic fields by J. B. Stark, W. H. Knox, and D. S. Chemla, Phys.Rev. B46, 7919~1992!.

21A. I. Ekimov, A. A. Onushchenko, A. G. Plukhin, and Al. L.Efros, Zh. Eksp. Teor. Fiz.88, 1490 ~1985! @Sov. Phys. JETP61, 891 ~1985!#.

22J. B. Xia, Phys. Rev. B40, 8500~1989!.23K. J. Vahala and P. C. Sercel, Phys. Rev. Lett.65, 239~1990!; P.

C. Sercel and K. J. Vahala, Phys. Rev. B42, 3690~1990!.24Al. L. Efros and A. V. Rodina, Solid State Commun.72, 645

~1989!.25G. L. Bir and G. E. Pikus,Symmetry and Strain-Induced Effects

in Semiconductors~Wiley, New York, 1975!.26Al. L. Efros and A. V. Rodina, Phys. Rev. B47, 10 005~1993!27L. E. Brus ~unpublished!.28E. I. Rashba, Zh. E´ksp. Teor. Fiz.36, 1703 ~1959! @Sov. Phys.

JETP9, 1213~1959!#.29V. P. Kochereshko, G. V. Mikhailov, and I. N. Ural’tsev, Fiz.

Tverd. Tela25, 759 ~1983! @Sov. Phys. Solid State25, 439~1983!#.

30In accordance with SAXS and TEM measurements the ellipticitywas approximated by the polynomialm(a)50.10120.034a13.5073 1023a221.1773 1024a311.86331026a421.41831028a514.196310211a6.

31C. B. Murray, D. J. Norris, and M. G. Bawendi, J. Am. Chem.Soc.115, 8706~1993!.

32M. G. Bawendi, P. J. Carroll, W. L. Wilson, and L. E. Brus, J.Chem. Phys.96, 946 ~1992!.

33W. W. Piper, inProceedings of the 7th International Conferenceon II-VI Semiconductor Compounds, Providence, RI, edited byD. G. Thomas~Benjamin, New York, 1967!, p. 839.

34B. L. Gel’mont and M. I. D’yakonov, Fiz. Tekh. Poluprovodn.7,2013 ~1973! @Sov. Phys. Semicond.7, 1345~1973!#.

35L. D. Landau and E. M. Lifshitz,Relativistic Quantum Theory,2nd ed.~Pergamon Press, Oxford, 1965!.

36T. Itoh, M. Nishijima, A. I. Ekimov, C. Gourdon, Al. L. Efros,and M. Rosen, Phys. Rev. Lett.74, 1645~1995!.

37S. Nomura and T. Kobayashi, Solid State Commun.82, 335~1992!.

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41The values ofSe50.45 andSa50.14 are consistent with experi-mental measurements~Ref. 18!.

42P. D. J. Calcott, K. J. Nash, L. T. Canham, M. J. Kane, and D.Brumhead, J. Phys. Condens. Matter5, L91 ~1993!.

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98, 8432~1993!; J. J. Shiang, R. K. Grubbs, and A. P. Alivisatos~unpublished!; J. J. Shiang, A. P. Alivisatos, and K. B. Whaley~unpublished!.

46Al. L. Efros, inPhonons in Semiconductor Nanostructures,editedby J.-P. Leburton, J. Pascual, and C. Sotomayor-Torres~KluwerAcademic Publishers, Boston, 1993!, p. 299.

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~1955!.


VOLUME 60, NUMBER 9 PHYSICAL REVIEW LETTERS 29 FEBRUARY 1988

Quantized Conductance of Point Contacts in a Two-Dimensional Electron Gas

B. J. van WeesDepartment of Applied Physics, Delft University of Technology, 2628 CJ Delft, The Netherlands

H. van Houten, C. W. J. Beenakker, and J. G. Williamson,

Philips Research Laboratories, 5600 JA Eindhoven, The Netherlands

L. P. Kouwenhoven and D. van der MarelDepartment of Applied Physics, Delft University of Technology, 2628 CJ Delft, The Netherlands

and

C. T. FoxonPhilips Research Laboratories, Redhill, Surrey RHJ 5', United Kingdom

(Received 31 December 1987)

Ballistic point contacts, defined in the two-dimensional electron gas of a GaAs-AlGaAs heterostruc-ture, have been studied in zero magnetic field. The conductance changes in quantized steps of e2/nh

when the width, controlled by a gate on top of the heterojunction, is varied. Up to sixteen steps are ob-served when the point contact is widened from 0 to 360 nm. An explanation is proposed, which assumesquantized transverse momentum in the point-contact region.

PACS numbers: 72.20.JV, 73.40.Cg, 73.40.Lq

As a result of the high mobility attainable in the two-dimensional electron gas (2DEG) in GaAs-A1GaAs het-erostructures it is now becoming feasible to study ballis-tic transport in small devices. ' In metals ideal tools forsuch studies are constrictions having a width W and

length L much smaller than the mean free path l, .These are known as Sharvin point contacts. Because ofthe ballistic transport through these constrictions, theresistance is determined by the point-contact geometryonly. Point contacts have been used extensively for thestudy of elastic and inelastic electron scattering. Withuse of biased point contacts, electrons can be injectedinto metals at energies above the Fermi level. This al-lows the study of the energy dependence of the scatteringmechanisms. With the use of a geometry containingtwo point contacts, with separation smaller than l„elec-trons injected by a point contact can be focused into theother contact, by the application of a magnetic field.This technique (transverse electron focusing) has been

applied to the detailed study of Fermi surfaces.In this Letter we report the first experimental study of

the resistance of ballistic point contacts in the 2DEG ofhigh-mobility GaAs-A1Ga As heterostructures. Thesingle-point contacts discussed in this paper are part of adouble-point-contact device. The results of transverseelectron focusing in these devices will be published else-where. ' The point contacts are defined by electrostaticdepletion of the 2DEG underneath a gate. This method,which has been used by several authors for the study of1D conduction, ' oA'ers the possibility to control thewidth of the point contact by the gate voltage. Controlof the width is not feasible in metal point contacts.

The classical expression for the conductance of a pointcontact in two dimensions (see below) is

G =(e /trh)kFW/tr

in which kF is the Fermi wave vector and W'is the widthof the contact. This expression is valid if l, ))W and theFermi wavelength XF« W. The first condition is satisfiedin our devices, which have a maximum width W,„= 250 nm and l, 8.5 pm. The second condition shouldalso hold when the devices have the maximum width.We expect quantum effects to become important when

the width becomes comparable to A, F, which is 42 nm in

our devices. In this way we are able to study the transi-tion from classical to quantum ballistic transportthrough the point contact.

The point contacts are made on high-mobilitymolecular-beam-epitaxy-grown GaAs-A1GaAs hetero-structures. The electron density of the material is

3.56X10' /m and the mobility 85 m /V s (at 0.6 K).These values are obtained from the devices containingthe studied point contacts. A standard Hall bar geome-try is defined by wet etching. Using electron-beamlithography, a metal gate is made on top of the hetero-structure, with an opening 250 nm wide (inset in Fig. 1).The point contacts are defined by the application of anegative voltage to the gate. At Vg = —0.6 V the elec-tron gas underneath the gate is depleted, the conductiontaking place through the point contact only. At this volt-

age the point contacts have their maximum width 8',„,about equal to the opening between the gates. By a fur-ther decrease of the gate voltage, the width of the pointcontacts can gradually be reduced, until they are fully

848 1988 The American Physical Society

Var UME 60, NUMBER 9 PHYSICAL REVIEW LETTERS 29 FEaRU~RV 1988

15

1P

LLj

I—5-

U3LLJCC

-2 —1.8 —1 6 -1.4 -$. 2 -i -0.8 -0 6

—1 4

GATE VOLTAGE IV)

—1 2

GATE VOLTAGE (V)

FIG. 1. Point-contact resistance as a function of gate volt-age at 0.6 K. Inset: Point-contact layout.

FIG. 2. Point-contact conductance as a function of gatevoltage, obtained from the data of Fig. 1 after subtraction ofthe lead resistance. The conductance shows plateaus at multi-ples of e /xh.

pinched off at Vg = —2.2 V.We measured the resistance of several point contacts

as a function of gate voltage. The measurements wereperformed in zero magnetic field, at 0.6 K. An ac lockintechnique was used, with voltages across the sample keptbelow kT/e, to prevent electron heating. In Fig. 1 themeasured resistance of a point contact as a function ofgate voltage is shown. Unexpectedly, plateaus are foundin the resistance. In total, sixteen plateaus are observedwhen the gate voltage is varied from —0.6 to —2.2 V.The measured resistance consists of the resistance of thepoint contact, which changes with gate voltage, and aconstant series resistance from the 2DEG leads to thepoint contact. As demonstrated in Fig. 2, a plot of theconductance, calculated from the measured resistanceafter subtraction of a lead resistance of 400 0, showsclear plateaus at integer multiples of e /&A. The abovevalue for the lead resistance is consistent with an es-timated value based on the lead geometry and the resis-tivity of the 2DEG. We do not know how accurate thequantization is. In this experiment the deviations frominteger multiples of e /zh might be caused by the uncer-tainty in the resistance of the 2DEG leads. Inserting thepoint-contact resistance at V~= —0.6 V (750 0) intoEq. (1) we find for the width W,„=360nm, in reason-

able agreement with the lithographically defined widthbetween the gate electrodes.

The average conductance increases almost linearlywith gate voltage. This indicates that the relation be-tween the width and the gate voltage is also almostlinear. From the maximum width W,„(360 nm) andthe total number of observed steps (16) we estimate theincrease in width between two consecutive steps to be 22nm.

We propose an explanation of the observed quantiza-tion of the conductance, based on the assumption ofquantized transverse momentum in the contact constric-tion. In principle this assumption requires a constrictionmuch longer than wide, but presumably the quantizationis conserved in the short and narrow constriction of theexperiment. The point-contact conductance G for ballis-tic transport is given by "

G =e NpW(It/2m)( [ k„~ ).

The brackets denote an average of the longitudinal wavevector k, over directions on the Fermi circle, N p

=m/eh 2 is the density of states in the two-dimensionalelectron gas, and W is the width of the constriction. TheFermi-circle average is taken over discrete transversewave vectors k» = ~ nz/W (n =1,2, . . . ), so that we canwrite

T

&Ik. l&= J d'krak, )&(k —kF) g 6' k»—7C F 8', - ) 8' (3)

Carrying out the integration and substituting into Eq. (2), one obtains the result

N,

(4)

where the number of channels (or one-dimensional subbands) N, is the largest integer smaller than kFW/x. For

849

VOLUME 60, NUMBER 9 PHYSICAL REVIEW LETTERS 29 FEBRUARY 1988

kFR'»1 this expression reduces to the classical formula[Eq. (1)]. Equation (4) tells us that G is quantized in

units of e /trh in agreement with the experimental ob-servation. With the increase of W by an amount of 1I.F/2,an extra channel is added to the conductance. This com-pares well with the increase in width between two con-secutive steps, determined from the experiment. Equa-tion (4) may also be viewed as a special case of the mul-

tichannel Landauer formula, '

N,

Zn, m 1

for transmission coefficientsi t„ i =b„corresponding

to ballistic transport with no channel mixing.It is interesting to note that this multichannel Lan-

dauer formula has been developed to describe the ideal-ized case of the resistance of a quantum wire, connectedto massive reservoirs, in which the inelastic-scatteringevents are thought to take place exclusively. As dis-cussed by Imry, ' it„ i =B„corresponds to the casethat elastic scattering is absent in the wire also. The factthat the conductance G =N, e /nh of such an ideal wireis finite' is a consequence of the inevitable contact resis-tances associated with the connection to the thermalizingreservoirs. The findings described in this Letter may im-

ply that we have realized an experimental system whichclosely approximates the behavior of idealized mesocopicsystems.

In summary we have reported the first measurementsof the conductance of single ballistic point contacts in atwo-dimensional electron gas. A novel quantum effect isfound: The conductance is quantized in units of e /xh.

We would like to thank J. M. Lagemaat, C. E. Tim-mering, and L. W. Lander for technological support andL. J. Geerligs for assistance with the experiments. Wethank the Delft Center for Submicron Technology forthe facilities offered and the Stichting voor Fundamen-teel Onderzoek der Materie (FOM) for financial sup-port.

'T. J. Thornton, M. Pepper, H. Ahmed, D. Andrews, andG. J. Davies, Phys. Rev. Lett. 56, 1198 (1986).

2H. Z. Zheng, H. P. Wei, D. C. Tsui, and G. Weimann,Phys. Rev. B 34, 5635 (1986).

K. K. Choi, D. C. Tsui, and S. C. Palmateer, Phys. Rev. B32, 5540 (1985).

4H. van Houten, C. W. J. Beenakker, B. J. van Wees, andJ. E. Mooij, in Proceedings of the Seventh InternationalConference on the Physics of Two-Dimensional Systems, SantaFe, 1987, Surf. Sci. (to be published).

5G. Timp, A. M. Chang, J. E. Cunningham, T. Y. Chang,P. Mankiewich, R. Behringer, and R. E. Howard, Phys. Rev.Lett. 58, 2814 (1987).

G. Timp, A. M. Chang, P. Mankiewich, R. Behringer, J. E.Cunningham, T. Y. Chang, and R. E. Howard, Phys. Rev.Lett. 59, 732 (1987).

7Yu. V. Sharvin, Zh. Eksp. Teor. Fiz. 48, 984 (1965) [Sov.Phys. JETP 21, 655 (1965)].

For a review, see I. K. Yanson and 0. I. Shklyarevskii, Fiz.Nizk. Temp. 12, 899 (1986) [Sov. J. Low Temp. Phys. 12, 509(1986)].

P. C. van Son, H. van Kempen, and P. Wyder, Phys. Rev.Lett. 58, 1567 (1987).

' H. van Houten, B. J. van Wees, J. E. Mooij, C. W. J.Beenakker, J. G. Williamson, and C. T. Foxon (to be pub-lished)."I. B. Levinson, E. V. Sukhorukov, and A. V. Khaetskii,

Pis'ma Zh. Eksp. Teor. Fiz. 45, 384 (1987) [JETP Lett. 45,488 (1987)].

i2R. Landauer, IBM J. Res. Dev. 1, 223 (1957); R. Lan-dauer, Phys. Lett. 85A, 91 (1981).

' M. Buttiker, Y. Imry, R. Landauer, and S. Pinhas, Phys.Rev. B 31, 6207 (1985). For a survey, see Y. Imry, in Directions in Condensed Matter Physics, edited by G. Grinstein andG. Mazenko (World Scientific, Singapore, 1986), Vol. 1, p.102.

i4D. S. Fishe~ and P. A. Lee, Phys. Rev. B 23, 6851 (1981).'SThe origina& Landauer formula (Ref. 12) containing the ra-

tio of transmission and reflection coefficients does give aninfinite conductance for a perfect system. However, this for-mula excludes contributions from the contact resistances.

850

notice tpa effets quantiques - epfl

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