analytical modeling of silicon based resonant tunneling diodes for

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N° d’ordre : 2010-ISAL-0076 Année 2010 THESE présentée devant L’INSTITUT NATIONAL DES SCIENCES APPLIQUÉES DE LYON Pour obtenir le grade de Docteur ÉCOLE DOCTORALE : Électronique, Électrotechnique, Automatique SPÉCIALITÉ : Dispositifs de l’Electronique Intégrée par Emanuela BUCCAFURRI Analytical modeling of silicon based resonant tunneling diode for RF oscillator application (Modélisation analytique et simulation de diode tunnel résonante sur silicium pour application oscillateur radiofréquence) soutenue le 5 octobre 2010 devant la commission d’examen : Pr. Olivier VANBESIEN Univ. des Sciences et Technologies de Lille Rapporteur & Président Pr. Fabrizio PIRRI Politecnico di Torino Rapporteur M. Thomas ERNST CEA - LETI Examinateur Pr. Xavier ORIOLS Universitat Autònoma de Barcelona Examinateur M. Raphaël CLERC INP Grenoble PHELMA Co-encadrant de thèse Pr. Alain PONCET INSA Lyon Co-directeur de thèse M. Francis CALMON INSA Lyon Directeur de thèse

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N° d’ordre : 2010-ISAL-0076 Année 2010

THESE

présentée devant

L’INSTITUT NATIONAL DES SCIENCES APPLIQUÉES DE LYON

Pour obtenir le grade de

Docteur

ÉCOLE DOCTORALE : Électronique, Électrotechnique, Automatique

SPÉCIALITÉ : Dispositifs de l’Electronique Intégrée

par

Emanuela BUCCAFURRI

Analytical modeling of silicon based resonant tunneling diode for RF oscillator application

(Modélisation analytique et simulation de diode tunnel résonante sur silicium pour application oscillateur radiofréquence)

soutenue le 5 octobre 2010 devant la commission d’examen :

Pr. Olivier VANBESIEN Univ. des Sciences et Technologies de Lille Rapporteur & Président

Pr. Fabrizio PIRRI Politecnico di Torino Rapporteur

M. Thomas ERNST CEA - LETI Examinateur

Pr. Xavier ORIOLS Universitat Autònoma de Barcelona Examinateur

M. Raphaël CLERC INP Grenoble PHELMA Co-encadrant de thèse

Pr. Alain PONCET INSA Lyon Co-directeur de thèse

M. Francis CALMON INSA Lyon Directeur de thèse

3

INSA Direction de la Recherche - Ecoles Doctorales – Quadriennal 2007-2010

SIGLE ECOLE DOCTORALE NOM ET COORDONNEES DU RESPONSABLE

CHIMIE

CHIMIE DE LYON http://sakura.cpe.fr/ED206

M. Jean Marc LANCELIN

Insa : R. GOURDON

M. Jean Marc LANCELIN Université Claude Bernard Lyon 1 Bât CPE 43 bd du 11 novembre 1918 69622 VILLEURBANNE Cedex Tél : 04.72.43 13 95 Fax : [email protected]

E.E.A.

ELECTRONIQUE, ELECTROTECHNIQUE, AUTOMATIQUE http://www.insa-lyon.fr/eea M. Alain NICOLAS Insa : C. PLOSSU [email protected] Secrétariat : M. LABOUNE AM. 64.43 – Fax : 64.54

M. Alain NICOLAS Ecole Centrale de Lyon Bâtiment H9 36 avenue Guy de Collongue 69134 ECULLY Tél : 04.72.18 60 97 Fax : 04 78 43 37 17 [email protected] Secrétariat : M.C. HAVGOUDOUKIAN

E2M2

EVOLUTION, ECOSYSTEME, MICROBIOLOGIE, MODELISATION http://biomserv.univ-lyon1.fr/E2M2 M. Jean-Pierre FLANDROIS Insa : H. CHARLES

M. Jean-Pierre FLANDROIS CNRS UMR 5558 Université Claude Bernard Lyon 1 Bât G. Mendel 43 bd du 11 novembre 1918 69622 VILLEURBANNE Cédex Tél : 04.26 23 59 50 Fax 04 26 23 59 49 06 07 53 89 13 [email protected]

EDISS

INTERDISCIPLINAIRE SCIENCES-SANTE Sec : Safia Boudjema M. Didier REVEL Insa : M. LAGARDE

M. Didier REVEL Hôpital Cardiologique de Lyon Bâtiment Central 28 Avenue Doyen Lépine 69500 BRON Tél : 04.72.68 49 09 Fax :04 72 35 49 16 [email protected]

INFOMATHS

INFORMATIQUE ET MATHEMATIQUES http://infomaths.univ-lyon1.fr M. Alain MILLE Secrétariat : C. DAYEYAN

M. Alain MILLE Université Claude Bernard Lyon 1 LIRIS - INFOMATHS Bâtiment Nautibus 43 bd du 11 novembre 1918 69622 VILLEURBANNE Cedex Tél : 04.72. 44 82 94 Fax 04 72 43 13 10 [email protected] - [email protected]

Matériaux

MATERIAUX DE LYON M. Jean Marc PELLETIER Secrétariat : C. BERNAVON 83.85

M. Jean Marc PELLETIER INSA de Lyon MATEIS Bâtiment Blaise Pascal 7 avenue Jean Capelle 69621 VILLEURBANNE Cédex Tél : 04.72.43 83 18 Fax 04 72 43 85 28 [email protected]

MEGA

MECANIQUE, ENERGETIQUE, GENIE CIVIL, ACOUSTIQUE M. Jean Louis GUYADER Secrétariat : M. LABOUNE PM : 71.70 –Fax : 87.12

M. Jean Louis GUYADER INSA de Lyon Laboratoire de Vibrations et Acoustique Bâtiment Antoine de Saint Exupéry 25 bis avenue Jean Capelle 69621 VILLEURBANNE Cedex Tél :04.72.18.71.70 Fax : 04 72 43 72 37 [email protected]

ScSo

ScSo* M. OBADIA Lionel Insa : J.Y. TOUSSAINT

M. OBADIA Lionel Université Lyon 2 86 rue Pasteur 69365 LYON Cedex 07 Tél : 04.78.69.72.76 Fax : 04.37.28.04.48 [email protected]

*ScSo : Histoire, Geographie, Aménagement, Urbanisme, Archéologie, Science politique, Sociologie, Anthropologie

4

Remerciements - Acknowledgements

5

REMERCIEMENTS - ACKNOWLEDGEMENTS

Voilà, c’est bientôt la fin de cette aventure qui a duré 3 ans ! Cette page de remerciements ne sera pas en anglais comme le reste du manuscrit, mais en au moins en 3 autres langues: français, espagnol et italien.

Je commence par remercier le gens sans lesquels cette expérience n’aurait pas eu lieu.

Mon directeur de thèse Francis Calmon et mon co-directeur de thèse Alain Poncet, que je remercie pour m’avoir donné la possibilité de faire un doctorat, ce qui a toujours été mon projet mais aussi pour l’indépendance qu’ils m’ont accordée dans ce travail. Enfin, je les remercie pour leurs qualités humaines et pour avoir partagé avec moi leur expérience.

Mon co-encadrant Raphaël Clerc, qui m’a offert son aide scientifique et, plus important encore, son amitié pendant ces trois ans. Merci beaucoup pour ses conseils, pour sa patience, pour son immense disponibilité et son appui dans les moments difficiles de cette thèse.

Merci aux membres du jury d’avoir accepté de prendre part à ma thèse. Je dois remercier tous les gens qui sont passés à l’INL au cours de ces 3 ans (4 si l’on considère mon stage !). Merci à tous les permanents, aux doctorants, aux post-docs, aux secrétaires d’avoir été pour moi une grande famille durant tout ce temps. Plus particulièrement, je dois remercier mes amis les plus proches du laboratoire, avec qui j’ai partagé les hauts et les bas de cette expérience et qui ont été un indispensable soutien au quotidien: Sylvain, Andrea et Guillaume.

Avant de changer de langue, je sors du labo et je remercie ma famille lyonnaise, les meilleurs amis que j’aurais pu rencontrer et qui ont été chaque jour présents pour m’encourager : mon cher Bruno, ma grande amie Maria, pour la deuxième fois tout à fait méritée Sylvain Pellok, mon inséparable Pepa, mes anciennes colocs Nat, Isa et Tere, Nico mon super coloc’, il mio Ste sempre pronto per un abbraccio et toute la colocation de la rue Molière.

Encore plus en dehors, hors de France du moins, je remercie mes chères amies Nora et Rasa. Qu’aurais-je fait sans vos mails et votre estime ?! Un grand merci pour avoir toujours cru – souvent plus que moi-même – en mes capacités.

No podría olvidarme de toda la gente que conocí en Barcelona.

Tengo que empezar por el Professor Xavier Oriols sin el cual mi experiencia a la UAB no habría sido posible: Muchas gracias Xavier por aceptarme en tu laboratorio, gracias por todo lo que me has enseñado y por tu disponibilidad durante todo el período en tu equipo.

He pasado 9 meses en Barcelona y fueron suficientes para guardar un buen recuerdo de todos los doctorantes del departamento de electrónica (Gonzalo, Ferrán, Nuria, Alfonso, Guillerm, Fito,

Remerciements - Acknowledgements

6

Alberto, Gonzalo, Fran, Marta, Vanessa, Siso, Gerard, Ender, Jordi, Simone... ¡no se si me olvido de alguien!) que me han acogido enseguida como una de ellos y que me han hecho sentir integrada desde el primer día. Gracias chicos por todos los buenos momentos, ¡me acordaré siempre de vosotros!

E ancora all’Autonoma, meritano un ricordo speciale Fabio e Nuria. Fabio un collega eccellente e un solido appoggio su cui poter contare. Nuria, dolcissima e disponibilissima dal primo giorno, una grande amica.

Y fuera de la Autónoma, mis compañeros de piso de Can Bruixa: Albert, Olga y Goeske. ¡Qué bien se está en este piso! Cuantos buenos momentos, cuantas risas y cuanto os echo de menos.

Otra línea en español la merece Quique, que estuvo presente desde el primer día y en los momentos más críticos, gracias por aguantar mi histeria en algunos momentos.

Passiamo alla mia lingua adesso, perché non ci dimentichiamo che c’è tanta gente dall’altra parte delle Alpi che mi ha sostenuta e incoraggiata dal primo fino all’ultimo giorno. I miei migliori amici, Mario, Andrea, Adriana e Ciccio che da Milano, Roma, Torino e Reggio Cal. hanno fatto il tifo per me, da quando ho espresso questa idea strana di fare un dottorato! E che hanno varcato più volte le Alpi per riabbracciarmi e farmi sentire che mi stavano vicini. Grazie ragazzi.

Grazie alla mia seconda famiglia italiana, Antonella, Franco e Giuliana per avermi fatto sentire il loro affetto e per essere stati accanto alla mia famiglia durante la mia lontananza.

E per chiudere, i ringraziamenti più forti, vanno ai miei genitori che sempre mi sono stati accanto, nella quotidianità, ogni giorno, sopportando la distanza, incoraggiandomi anche nei momenti più duri e per non avermi mai fatto pesare la lontananza soprattutto nei momenti più provanti che abbiamo attraversato quest’ultimo anno. E’ a loro che questa tesi è dedicata.

Ça y est je suis docteur!

List of abbreviations

7

LIST OF ABBREVIATIONS: MOS Metal – Oxide- Semiconductor RST Real Space transfer injection Transistor LITT Lateral Interband Tunnel Transistor VTT Vertical Tunnel Transistor RITD Resonant Interband Tunnel Diodes FET Field Effect Transistor NDR Negative Differential Resistance RTD Resonant Tunneling Diode ADC Analog-Digital Converter SOI Silicon On Insulator MBE Molecular Beam Epitaxial PVCR Peak to Valley Current Ratio DB Double Barrier TB Triple Barrier RT Room Temperature LT-MBE Low Temperature Molecular Beam Epitaxial HFET Heterostructure Field Effect Transistor HEMT High Electron Mobility Transistor HBT Heterojunction bipolar transistor VCO Voltage Controlled Oscillator TMM Transfer Matrix Method NEMO NanoElectronic MOdeling WFM Wigner Function Method BTE Boltzmann Transport Equation BT Bohme Trajectories SSEC Small Signal Equivalent Circuit EMA Effective Mass Approximation

List of abbreviations

8

Table of contents

9

Table of contents

Remerciements - Acknowledgements .......................................................................... 5

List of Abbreviations: ................................................................................................... 7

General Introduction .................................................................................................. 13

Chapter I: Introduction and state of the art ............................................................ 17

I.1.Introduction ................................................................................................................................. 17

I.2. Historical background: From tunneling to resonant tunneling .................................................. 18

I.3. Resonant tunneling: Principles .................................................................................................. 19

I.4. Figures of merit .......................................................................................................................... 21

I.5. State of the Art: Materials for RTD ........................................................................................... 22

I.5.1. III –V based RTD ................................................................................................................ 22

I.5.1.1. From GaAs/AlxGa1-XAs To GaAs/AlAs RTDs .......................................................................... 22

I.5.1.2.From InGaAs/InAlAs To InAs/AlSb RTDs ................................................................................ 23

I.5.1.3. III-V RTD on Silicon .................................................................................................................. 24

I.5.1.4. Technological research on RTD: from III-V materials to the silicon compatible solutions ...... 25

I.5.2. Silicon based RTD .............................................................................................................. 25

I.5.2.1. Si/Si1-XGex .................................................................................................................................. 26

I.5.2.2. Resonant Interband Tunnel Diodes (RITD) ............................................................................... 28

I.5.2.3. Si/CaF2 ....................................................................................................................................... 29

Table of contents

10

I.5.2.4. Si/SiO2 ........................................................................................................................................ 29

I.5.2.5. High-K oxides for barriers ......................................................................................................... 30

I.5.3. RTD: Which materials? ....................................................................................................... 33

I.6. Applications of RTD .................................................................................................................. 34

I.6.1 Digital applications .............................................................................................................. 35

I.6.2. Analog applications ............................................................................................................. 38

I.7. The aim of this thesis: ................................................................................................................ 42

References of Chapter I .................................................................................................................... 43

Chapter II: Modeling of the RTD .............................................................................. 47

II.1. Introduction .............................................................................................................................. 47

II.2. Numerical methods for current calculation in RTD: an overview ........................................... 47

II.2.1. Transfer matrix method ...................................................................................................... 48

II.2.1.1 Analytical approach based on transfer matrix method ............................................................... 51

II.2.2 Coherent tunneling .............................................................................................................. 59

II.2.3 Green function .................................................................................................................... 62

II.2.3.1. Introduction ............................................................................................................................... 62

II.2.3.2. NEMO (klimeck) ...................................................................................................................... 63

II.2.4. Wigner function ................................................................................................................. 65

II.2.5. Bohm trajectory ................................................................................................................. 67

II.2.6. Numerical approaches for RTD simulations: conclusion .................................................. 69

II.3. Analytical modeling of RTD .................................................................................................... 70

II.3.1. Why do we need compact models? .................................................................................... 70

II.3.2. Previous analytical model of RTD ..................................................................................... 71

II.3.1.1 Model For Pspice of Yan et al. [Yan95] .................................................................................... 71

II.3.1.2 Brown et al. model [Chang93] ................................................................................................... 73

II.3.1.3 Schulman et al. model [Schulman96] ........................................................................................ 73

Table of contents

11

II.3.3. Analytical models: conclusion ........................................................................................... 74

II.4. Our analytical DC model of RTD ............................................................................................. 75

II.4.1. Potential profile .................................................................................................................. 75

II.4.2. Resonant levels in the well ................................................................................................ 77

II.4.3. Transmission coefficient calculation ................................................................................. 78

II.4.4. Current calculation ............................................................................................................. 81

II.4.5. Extension of the model to the multi-valley semiconductor: .............................................. 84

II.4.6. Analytical model validation ............................................................................................... 87

II.5. DC Analytical model: conclusion ............................................................................................. 89

References of Chapter II .................................................................................................................. 89

Chapter III: Transient behavior of the RTD ........................................................... 91

III.1. Introduction ............................................................................................................................. 91

III.2. Numerical approaches ............................................................................................................. 91

III.3. Analytical approaches ............................................................................................................. 95

III.4. Our AC analytical modeling of RTD ...................................................................................... 99

III.5 Interpretation of the small signal analytical model by numerical simulations ....................... 103

III.5.1 Conduction and displacement current: the current conservation ..................................... 104

III.5.2 Charge neutrality ............................................................................................................. 106

III.5.3 Numerical complete simulations and small signal equivalent circuit ............................. 108

III.5.3.1 Non self consistent simulation and charge neutrality not included ........................................ 108

III.5.3.2 Current conservation included and charge neutrality neglected ............................................. 111

III.5.3.3 Qualitative explanation of impact of Coulomb interaction in dynamic responds .................. 113

III.5.3.4 Current conservation and Charge neutrality (leads resistance) .............................................. 113

III.6. Conclusion of chapter III ....................................................................................................... 116

References of Chapter III ............................................................................................................... 116

Table of contents

12

Chapter IV: Results & Discussion ........................................................................... 119

IV.1. Introduction ........................................................................................................................... 119

IV.2. Impact of physical properties: comparison of different materials ........................................ 120

IV.3.1. Impact of geometrical parameters on I-V characteristics ............................................... 123

IV.3.2. Impact of geometrical parameters on intrinsic cut off frequency .................................. 124

IV.3.3. Conclusion on geometrical parameters impact .............................................................. 125

IV.4. Toward silicon based RTD ................................................................................................... 125

IV.5. Strained silicon based RTD ................................................................................................... 127

IV.6. Silicon RTD: Conclusion ...................................................................................................... 131

IV.7. RF oscillator based on strained-silicon RTD devices ........................................................... 132

IV.7.1 Basic differential oscillator ............................................................................................. 132

IV.7.2 Muramatsu oscillator ....................................................................................................... 135

IV.7.2.1 Muramatsu oscillator: Variability ........................................................................................... 136

IV.7.3. Oscillator based on strained-silicon RTD devices: conclusion ...................................... 139

References of Chapter IV ............................................................................................................... 140

General conclusion .................................................................................................... 143

Annex: Discussion on the effective mass approximation ...................................... 147

Résumé en français.................................................................................................... 151

List of publications .................................................................................................... 161

Folio administratif ..................................................................................................... 163

General introduction

13

GENERAL INTRODUCTION

Introduction

Since the introduction in microelectronics of the MOS transistor, the trend has been to reduce the size of each device, in order to increase density and performance, and to reduce costs [Mollick06]. Device nowadays are in a range of dimensions governed by Quantum Physics. In the context of MOS transistor, quantum effects are usually parasitic effects, penalizing the ideal transistor operation. Indeed, quantum confinement is responsible for a degradation of the coupling between the gate and the channel [Takagi95], quantum tunneling through the gate responsible of an increase of power consumption [Chia03] and source to drain tunneling is probably one of the main physical limits to the transistor operation [Lundstrom02]. In consequence, at the nano scale, it may be smarter to integrate new devices operating according to the law of quantum physics rather than to classical Physics.

Many devices based on quantum tunneling, hot electrons injections and charge quantization, have been investigated in the last two decades, mostly using III-V materials [Chang74] .Indeed, the molecular beam epitaxy technique, invented in the late 1960s, has made possible to realize well defined nanostructures, with large mean free path. However, nowadays, semiconductor technology continue to be dominated by silicon CMOS. Thus, it should be interesting to combine both quantum devices (usually investigate in III V materials) and nanoscaled silicon technology. This may be done using SOI substrates. Indeed, the continuous miniaturization on SOI substrates with very thin silicon films and buried oxide makes possible to envisage several devices based on quantum effect, CMOS compatible, before only conceivable using III-V materials.

In particular, Tunneling devices in combination with silicon transistors are particularly interesting, as they offer a way to extend the performance of existent technologies by increasing circuit speed and decreasing static power dissipation [Luryi04]. They thus could represent serious possibility to complete with the CMOS technology [Luryi07].

In this context, resonant tunneling diodes (RTDs) present interesting characteristics. Its maximum operating frequency is in the Terahertz range and offers a wide range of applications, in analog (analog-digital converter ADC, frequency divider or multiplier, oscillator [DelosSantos01]) as well as digital (“multi-value” logic [Lin94]) circuits. Its I-V characteristic presents an unusual negative differential resistance (NDR). Such negative differential resistance is usually achieved by a circuit involving more devices, and significant power consumption.

Like most of the other tunnel devices, the first RTD was realized on III-V materials [Chang74]. Nevertheless, difficulty to integrate III-V materials on silicon, pushed to find silicon compatible solution for RTD. In an attempt to realize these devices in a Silicon compatible technology, SiGe/Si heterostructures have been also investigated, leading to encouraging results [Paul04].

General introduction

14

In this work, an alternative option to integrate these innovative devices in a silicon process has been considered. In fact, today progress on SOI technology makes possible to envisage silicon based RTD compatible with CMOS technology, exploiting the vertical transport (gate to gate) in a double gate MOSFET. In such device, the thin body of silicon (today thinner than 10 nm) [Vinet05] can ensure quantization in a well and electrons can tunnel through the two thin high k barriers.

The aim of this work is thus to estimate by the means of an original analytical models, the theoretical expected performances of silicon based RTDs (in static and dynamic regime) and to compare them with conventional heterostructures. Finally, these models have been introduced in a circuit simulator (CADENCE), and have been used to estimate silicon RTD for RF oscillator applications.

This document is organized as follows:

The Chapter I of this thesis presents the state of the art of Resonant Tunneling Diode and circuits. In a first part, an overview of the main realizations of RTD and technological difficulties is reported. In the second part, the main analog and digital applications of RTD are listed.

The Chapter II of this work is dedicated to the development of the static analytical model. In the first part of this chapter, an overview of the main numerical and analytical approaches to model the transport in RTD is presented. In the second part, the analytical model developed in this thesis is detailed. At the end of this chapter, the analytical model is compared with numerical simulations realized with self-consistent ballistic simulations, performed using QUANTIX, a code elaborated in INL laboratory by Pr. Alain Poncet and numerical simulations based on Bohm Trajectories, performed with the code elaborated in UAB by Pr. Xavier Oriols.

In the Chapter III, the transient behavior of RTD is investigated. In the first part, the main numerical approaches to address this issue and corresponding small signal equivalent circuit, are presented. In the second part, two different techniques are carried out to analyze the frequency behavior of the RTDs. The first one is based on the Quantum Monte Carlo (QMC) technique exploiting many-particle Bohm trajectories, simulator elaborated in Universidad Autonoma de Barcelona in the Xavier Oriols’s team. Furthermore, this rigorous approach has been used to understand many of the dynamics involved in the time-dependent current. The second approach consists in an analytical Small Signal Equivalent Circuit (SSEC) derived from the DC physics-based model presented in the chapter II. Finally, this compact model has been tested comparing with the cut-off frequency sough for from numerical simulations.

The Chapter IV summarizes the main results of this thesis work. This chapter is focused on the comparison of III-V, SiGe and double gate like silicon RTD by the means of this complete DC-AC analysis. This comparison allows computing I-V characteristics and intrinsic cut off frequency as function of geometrical parameters and materials properties in order to optimize the silicon RTD structure. Finally, circuit simulations are presented: two different architectures of RF oscillator based on silicon RTD will be investigated and compared with conventional and III-V RTD based oscillators.

[Chang74] L. L. Chang, L. Esaki, and R. Tsu, “Resonant tunneling in semiconductor double barriers”, Appl. Phys. Lett. 24, 593 (1974).

General introduction

15

[Chia03] Yee-Chia Yeo Tsu-Jae King, and Chenming Hu, “MOSFET Gate Leakage Modeling and Selection Guide for Alternative Gate Dielectrics Based on Leakage Considerations” IEEE Transactions on Electron Devices, Vol. 50, n° 4, April 2003 p. 1027.

[DelosSantos01] H. J. De Los Santos, K. K. Chui, D. H. Chow, H. L. Dunlap, “An Efficient HBT/RTD Oscillator for Wireless Applications” IEEE Microwave and Wireless Components Letters, Vol. 11, pp. 193-195 (2001).

[Lin94] H.C. Lin, “Resonant Tunneling Diodes For Multi-Valued Digital Applications” in: Proceedings IEEE Int. Symp. Multiple Valued Logic, pp. 188-195 (1994).

[Lundstrom02] Jing Wang, Lundstrom M., “Does source-to-drain tunneling limit the ultimate scaling of MOSFETs?” Electron Devices Meeting, 2002. IEDM '02.

[Luryi04] S. Luryi, A. Zaslavsky “Blue sky in SOI: new opportunities for quantum and hot-electron devices” Solid-State Electronics 48 (2004) 877–885.

[Luryi07] S. Luryi, A. Zaslavsky “Nonclassical devices in SOI: Genuine or copyright from III–V” Solid-State Electronics 51 (2007) 212–218.

[Mollick06] Mollick, E. “Establishing Moore's Law”, Annals of the History of Computing, IEEE Volume: 28, Issue: 3, Page(s): 62 – 75 (2006)

[Paul04] D. J Paul, “Si/SiGe heterostructures: from material and physics to devices and circuits”, Semicond. Sci. Technol. 19 (2004) R75–R108.

[Takagi95] Shin-ichi Takagi, Akira Toriumi, “Quantitative Understanding of Inversion-Layer Capacitance in Si MOSFET' s”, IEEE Transactions on Electron Devices, Vol. 42, No. 12, December 1995, pp. 2125-2130.

Chapter I: Introduction and state of the art

16

Chapter I: Introduction and state of the art

17

CHAPTER I: INTRODUCTION AND STATE OF THE ART I.1.INTRODUCTION

Resonant Tunneling Diodes (RTDs) present very attractive characteristics, such as a high intrinsic cut-off frequency (theoretical value in the THz range) and current peaks associated with Negative Differential Resistance (NDR) regions. These RTD specificities are exploited in digital applications (“multi-value” logic [Lin94]) as well as analog applications [DelosSantos01] (ADC, frequency divider or multiplier, oscillator), leading to simpler circuits reducing the size of circuit as shown in the table I, with a large gain in power consumption and high frequency performance.

Requiring deposition of few nanometers thick layers, thanks to epitaxial technologies allowing an accurate control of device dimension at the nanometric scale, RTDs have been elaborated so far with III–V materials and technologies.

Nevertheless the difficulty to integrate these materials in a silicon process pushes to find possible solutions to realize them with silicon compatible materials. With the scaling down of CMOS technology (where double gate transistors with body thicknesses less than 10 nm have been demonstrated [Vinet05]), and considering progress on the epitaxy and high-k thin oxide deposition (using in double gate realization), it is now possible to realize a stack of nanometric dielectric layers on silicon.

Consequently, at present, it may be possible to think to realize Si based RTD for the same applications that were previously realized only with III-V materials. As a matter of fact, evidence of resonant tunneling effects has been recently observed in Si structures [Rommel88], [Ishikawa01]. However, the performances obtained in these works in term of current peak and negative resistance are still very poor (both in amorphous or epitaxial silicon).

In the first part of this chapter an overview of the main realizations of RTD are presented and in the second part the main applications found in literature are listed.

CIRCUIT TRANSISTOR-TRANSISTOR

LOGIC

CMOS LOGIC

COUPLED EMITTER

LOGIC

NEGATIVE DIFFERENTIAL

RESISTANCE (RTD)

Bistable Xor 33 16 11 4

Bistable Majority 36 18 29 5

MullerC-Element 45 8 44 4

9-State Memory 24 24 24 5

Nor2+Flipflop 14 12 33 4

Nand2+Flipflop 14 12 33 4

Table 1: Comparison of number of devices needed to realize a logic function for several technologies [Mazumd98].

Chapter I: Introduction and state of the art

18

I.2. HISTORICAL BACKGROUND: FROM TUNNELING TO RESONANT TUNNELING

Probably the most fascinating phenomenon associated to quantum physics is the possibility of a carrier to “tunnel” through a potential barrier of arbitrary shape.

This effect has been theorized by the Schrodinger equation. As the electron is characterized by a wave function and the square of the wave function represents the density of probability to find a particle in a point of the space, an electron incident in a barrier has finite probability to appear in the other side of the barrier.

One of the first studies about tunneling was driven by Fowler and Nordheim [Fowler28] that in the late 1920s proposed the theory of the electron field emission from bulk metals.

After, the interest was focused on tunneling between metals through thin insulator layer (MIM) and then between a metal and a semiconductor (MIS). In the basis of the development of band theory, Zener [Zener43] proposes the concept of interband tunneling, which consists in electron tunneling between conduction and valence band.

Later, in the 1940-1950, technology became mature enough to realize the p-n junction and finally this effect could be observed experimentally.

Shortly after, first NDR was observed in the so-called Esaki Diode [Esaki57], which consists of a heavily doped p-n junction, where Zener inter-band tunneling happens between the two bands.

In the middle of 1970s, progress in crystal growth and in molecular beam epitaxial (MBE) enabled the realization of very thin layers with high quality interfaces.

In 1973, the concept of resonant tunneling was proposed in a pioneering work of Tsu and Esaki at IBM [Tsu73]. The first experimental prototype of RTD was then realized by the same group (Chang et al. in 1974) on GaAs/AlxGa1-x [Chang74].

As RTD provides a large amount of information about quantum mechanicals aspect of electron transport, it therefore became of great interest. Several efforts have been also made by MIT group’s [Sollner83] to obtain negative differential resistance at room temperature and to understand fundamental the physics which govern this simple structure.

Technology improvement on growth of III-V materials and SiGe heterostructures combined with a better knowledge of physics and device design have led to the fabrication of RTD interesting for application in high speed microwave system and digital circuit.

Nowadays, efforts are focused to realize silicon compatible RTD, in order to introduce them in CMOS processing.

The principle and physics of resonant tunneling will be explained in the next paragraph.

Chapter I: Introduction and state of the art

19

I.3. RESONANT TUNNELING: PRINCIPLES In this section, the principle of resonant tunneling through a double barrier is explained in detail.

An RTD consists of an undoped quantum well sandwiched between two barriers and two doped reservoirs, emitter and collector. A simple scheme the band diagram in a RTD is provided in figure 1. The strong quantum confinement between the two barriers gives rise to quantized energy level in the well, as shown in Fig. I.1.

Fig. I.1: Schematic RTD structure.

By applying a voltage bias on the collector side, and maintaining the emitter grounded, the position of the different sub-band and Fermi levels are modified. The phenomenon of resonant tunneling occurs when the electron transmission coefficient results sharply peaked in correspondence of particular energy values.

Figure I.2 shows the conduction band of a double barrier structure for different applied voltage and the resulting conduction state of the device on the I-V characteristics. By progressively increasing the collector bias, the following phenomenon is occurring:

(a) When any voltage is not applied (V = 0 V), the Fermi levels in the emitter and collector are perfectly aligned, energy levels (Er1 and Er0) are still above the Fermi levels and current cannot flow through the double barrier.

(b) By applying a positive voltage (V > 0), the conduction band is tilted. Energy levels in the well are pulled down and when the first resonant level reach the Fermi level in the emitter side, carriers can flow from the emitter to the collector.

(c) Increasing the bias induces a current increase until the energy level Er0 reaches the conduction

band bottom in the emitter side. When the energy level Er0 is in front of the conduction band, the

current is in its maximum.

(d) Biasing further the collector lower the sub-band Er0 below the emitter conduction band, which turning off the electron available for tunneling through barrier. The current thus decreases with an increasing voltage, providing the negative differential resistance region (NDR).

(e)A last bias increase will lower the second level Er1, which will enable a second tunneling process,

and increase the thermionic current, which flow above the barrier.

Ef emitter

Er0

Er1

Barrier Barrier

Well

Ef collector

Chapter I: Introduction and state of the art

20

In other words, the phenomenon of resonant tunneling current through a double barrier structure basically depends on presence of a peak in the transmission coefficient. This concept will be introduced in Chapter II.

Fig. I.2: Conduction band profile of a RTD (typically of III-V heterostucture for example) and I-V characteristic at different bias: (a) zero bias; (b) threshold bias; (c) peak voltage; (d) resonance ; (e) valley current.

The origin of the particular I-V characteristics of RTD has thus been discussed. The next paragraph will present the key figure of merit to estimate the performances of RTD.

Chapter I: Introduction and state of the art

21

I.4. FIGURES OF MERIT

The I-V characteristic of a RTD fundamentally consists of a peak of current with a negative differential resistance region (NDR). Figure I.3 provides a schematic representation of an I-V characteristic of an RTD, showing of the key figures of merits of the performance of this device.

.

Fig. I.3: Typical I-V RTD characteristics.

• A high peak current is required for high speed applications and for many analog applications in order to ensure high output power.

• Large NDR is suitable for digital applications (memory, analog to digital converter, logic circuits…) and analog application (as oscillators) that exploit the natural instability of RTD in this region.

• The Peak to Valley Current Ratio (PVCR), which represents the distance between the current peak and the valley current, is an important figure of merit of RTD for both analog and digital application. In particular for memory and logic functions a large PVCR is suitable to choose operation points and in order to optimize noise margins. In addition, switching applications also require large PVCR, as a small valley current is needed to minimize the off-state power dissipation.

• The intrinsic cut off frequency [Brown89] is also an important figure of merit for RTD in the case of high frequency application. Several limitations come into play in the time response of RTDs: the intrinsic time to charge and discharge the well (i.e. the carrier life time in the well, which will be discussed in chapter 3), the transit time across the non-tunneling region (contact and spacer), and the constant time associated to the capacitance of the structure. The frequency performances of RTDs can be optimized by device design and material choice.

Chapter I: Introduction and state of the art

22

After explaining principle and the main features of RTD, the state of art of RTD fabrications will be detailed in the next section. In fact, depending on its intrinsic characteristics (band offset, effective masse, dielectric constant…), some materials can be more suitable than other for RTD applications.

I.5. STATE OF THE ART: MATERIALS FOR RTD Since the 70s, when molecular beam epitaxy was developed, RTDs were realized mainly on III-V materials. Nevertheless, difficulty to integrate III-V materials on silicon, pushed to find silicon compatible solution for RTD. The best performances have been achieved with SiGe/Si heterostructure and resonant intraband tunneling diodes (RITD). In this section, an overview of the main realizations of RTD with various materials and their performances has been detailed. At first, it is possible to list possible RTD realizations in two large categories: III- V heterostructures and silicon compatible technologies.

I.5.1. III –V BASED RTD

Small electron effective masse and low band-offset in III-V heterostructures (see table 2 and 3), make these materials interesting candidates for RTD fabrication. The first prototype of RTD has been realized on GaAs/AlxGa1-xAs and presented in [Chang74]. Then, several III-V alloys have been extensively explored in order to optimize static and frequency performances. In the next paragraphs, the most relevant realizations of RTD on III-V have been summarized.

I.5.1.1. FROM GaAs/AlXGa1-XAs TO GaAs/AlAs RTDs The first prototype of RTD, which consisted of a GaAs well between two Al0.7Ga0.3As barriers, with GaAs emitter and collector regions, has been realized in 1974 [Chang74] and presented a small NDR a 77 K.

By changing concentration of Al and Ga, Sollner et al. reported the observation of effect tunnel resonant for of GaAs wells between Al0.25Ga0.75As barrier at 200 K and 25 K, with larger current peak, a peak to valley current (PVCR) ratio of 6, and a theoretical frequency up 2.5 THz [Sollner83].

The first RTD with room-temperature NDR has been built with a similar structure in 1985 [She85].

Later, diodes with AlAs barriers also became common [Tsuchiya85] when it was discovered that the minimum usable thickness of AlxGa1-xAs barriers was considerably higher than the minimum thickness of AlAs barriers, due to the random alloy nature of the tertiary compound.

On the one hand, the thinner AlAs barriers provides higher current densities and higher speeds, but, on the other hand, the disadvantage of the GaAs/AlxGa1-xAs is the lower band offset between GaAs and AlGaAs, than for GaAs/AlAs structure, which are respectively only 0.23 eV [Mehdi90]

Chapter I: Introduction and state of the art

23

and 0.56 eV. This results presents relatively large thermionic current, and hence a rather poor PVCR. In fact, this effect limits the PVCR to 4.

Nevertheless, GaAs/AlGaAs systems for RTDs remain one of the best option, due to the experienced gained on the fabrication of this technology

However, in a frequency point of view, an important drawback of the GaAs is its large contact resistance [Mönch90], which is prejudicial for the frequency behavior of this device. One of the fastest GaAs/AlGaAs device reported has been used in a 412 GHz fundamental-mode oscillator [Brown89].

To answer this limitation, a modification of the standard GaAs/AlAs diode has been proposed in several papers [Allen93, Kon93, Smith94] that essentially replace the highly-doped collector region with a Schottky metal, significantly reducing the series parasitic resistance and improving the maximum cut off frequency.

An fmax of 900 GHz has been computed for a diode with this kind of contacts [Smith94], and of course, Schottky contacts could be also extended to other materials systems, possibly leading to better performances.

I.5.1.2.FROM InGaAs/InAlAs TO InAs/AlSb RTDs The InGaAs/InAlAs material systems were introduced to overcome the drawbacks of the GaAs/AlGaAs RTDs. For instance, the barrier in In0.53Ga0.47As/In0.52Al0.48As is about 0.53 eV, i.e. twice higher than one of the GaAs/AlGaAs system discussed above. In addition, the effective electron mass in In0.52As0.48As is 0.084 m0, compared to 0.092 m0 for Al0.30Ga0.70As [Mehdi90]. These diodes offered some of the highest PVCRs available at the time of their introduction [Sen87, Sug88]. However, they rapidly became obsolete after the introduction of the material system such as InAs/AlSb [Sod91]. The InAs/AlSb heterostructures present several advantages over GaAs/AlAs. First, the InAs/AlSb band offset (type II at the Γ point) allows an electron to tunnel through an AlSb barrier with a smaller attenuation coefficient than at the same energy in the AlAs barrier of a GaAs/AlAs structure (type-I band offset) [Brown91]. This advantage leads to a higher current density. Then, a shorter depletion-layer transit time and lower series resistance provide high frequency performance. Brown in [Brown91] provides a comparison between frequency performances of different material systems [Brown91]. The highest operating frequency corresponds to 712 GHz for InAs/AlSb double barrier RTD.

Chapter I: Introduction and state of the art

24

GaAs AlxGa1-xAs

m* 0.063m0 m* (0.063 + 0.083x) m0

(x<0.45) Dielectric constant 12.9 ε0 Dielectric constant (12.90-2.84x) ε0

Conduction Band Discontinuity (eV) : GaAs/AlxGa1-xAs: x<0.41 ΔEc = 0.79x (eV) x>0.41 ΔEc = 0.475-0.335x+0.143x2 (eV)

Table 2: Material parameters of GaAs/AlxGa1-xAs (From http://www.ioffe.ru).

InGaAs InAlAs

m*(In0.53Ga0.47As) 0.044 m0 m*(In0.52Al0.48As) 0.084m0

Dielectric constant (In0.53Ga0.47As):

13.9 ε0 Dielectric constant

(In0.52Al0.48As): 12.45 ε0

Conduction Band Discontinuity (eV) In0.53Ga0.47As / In0.52Al0.48As:

c g 0.52 0.48 g 0.53 0.47

g 0.53 0.47

g 0.52 0.48

E (E (In Al As)-E (In Ga As))

E (In Ga As) 0.71eV

E (In Al As) 1.51eV

0.667 Band parameter

Γ Γ

Γ

Γ

Δ = β

=

=

β =

Table 3: Material parameters of In0.53Ga0.47As / In0.52Al0.48As (From [Mizuta]).

I.5.1.3. III-V RTD ON SILICON Several attempts have been done to integrate RTD based in III-V material in CMOS technology. In this context III-V RTD bonding on silicon substrates merit to be listed. In 1996 [Envers96] has presented a AlAs/InGaAs/InAs grown on a InP substrate and then separated of substrate grown to be bonded on silicon substrates.

The obtained RTD did not present degradation due to the bonging procedure and peak current and PVCR remain constant before and after integration on silicon.

However, some variability was observed for several bonded devices, due to the condition of bonding and InP removal. In 1999, Bergman et al. [Bergman99] have presented the first resonant tunneling CMOS circuit, which was a 1-bit comparator. The RTD are fabricated in III-V technology on an InP substrate and transferred to a CMOS die using a thin-film transfer and bonding process. The main limitations of this RTD/CMOS integration are the high cost and the strong parasitic capacitances due to the integration process, which are prejudicial for high frequency applications.

Chapter I: Introduction and state of the art

25

I.5.1.4. TECHNOLOGICAL RESEARCH ON RTD: FROM III-V MATERIALS TO THE SILICON COMPATIBLE SOLUTIONS Thanks to epitaxial technology that allows realizing thin layers of good interfacial quality, III-V

materials became commonly used for RTD realizations. Due to the small effective mass and low band offset in III-V heterostructures, high current density, large NDR and high cut off frequency have been observed for these structures. For these reasons III-V based RTD have been extensively explored. However, because of the difficulty to integrate III-V materials on silicon, several attempts have been done to fabricate silicon compatible RTD.

In the next section the silicon compatible solution envisaged at present for RTD fabrication will be summarized.

I.5.2. SILICON BASED RTD

Because of the difficult integration of materials III-V on silicon, a possible solution to realize RTD with silicon or with another material compatible with silicon technologies is presently strongly studied.

Recently, resonant tunneling has been observed in silicon structure [Kubota06] [Rommel98], [Ishikawa01] even if poor NDR and peak current have been observed. Several solutions have been envisaged to realize the barriers, such as conventional SiO2, SiGe heterostructure, more complex CaF2 deposition, and recently high-k barriers [Osten07].

At present the significant progress of Silicon On Insulator (SOI) technology has made possible to process devices on ultra thin crystalline silicon with thickness lower than 10 nm [Vinet05], with good uniformity and roughness properties [Weber08], that may be used as a quantum well for RTD [Luryi07, Majkusia06]. In fact it is possible to think the vertical transport between the two gates as a resonant tunneling effect. Such RTD could easily be integrated using Double Gate (DG) or ultra thin SOI technology.

Fig. I.4: TEM cross section of DGMOS described in [Vinet05].

Chapter I: Introduction and state of the art

26

In this context, the first silicon compatible RTDs envisaged were Si/SiGe heterostructures. The most significant results using this technology will be summarized in the next section.

I.5.2.1. Si/Si1-XGeX SiGe RTDs have been intensively investigated as suitable integration on silicon circuits with good

performances in terms of peak to valley current ratio and negative differential resistance, comparable with the ones obtained with III-V heterostructures. Si-Ge based RTD have been realized by epitaxial processes and both holes and electrons tunneling have been explored.

• Holes tunneling

The majority of SiGe devices have been designed for holes tunneling, since a larger valence band discontinuity than conduction band discontinuity can be easily obtained. Si/Si1-xGex heterostuctures are grown by molecular beam epitaxy and there are two possibility to obtain a band offset (for holes) between SiGe well and Si barriers.

a) SiGe compressively strained well is sandwiched between two unstrained Si barriers [Liu88]

b) unstrained well between two tensile strained Si barriers. [Rhee88].

NDR behavior has been observed in these structures based on holes transport only at very low temperature (4.2k and 77k), which is obviously very prejudicial for circuit applications.

• Electrons tunneling:

SiGe RTD based on electron tunneling have been also fabricated, employing strain-relieved SiGe as the barrier layers surrounding pseudomorphic tensile strained Si, leading to a small conduction band offset. (150-170 meV) [Ismail91].

Peak current of 400 A/cm2 and a PVCR of 1.2 have been observed at room temperature. Higher PVCR results have been obtained for lower temperature. In fact, as the temperature increases, the energy distribution of incident electrons becomes broader and the electron-tunneling rate at higher energy levels increases, leading a weak NDR characteristic. To overcome this limitation Suda et al. [Suda01] has demonstrated that a Si0.7Ge0.3 /Si electron tunneling triple barriers (TB) RTD can provide larger PVCR.

• Triple barriers (TB) RTD

Figure 5 shows the conduction band of TB RTD. Based on the same operation as RTD, tunneling is possible only through resonant levels which are aligned in the double quantum well.

TB RTD exhibits a high peak-to-valley current ratio of more than 7.6 even at room temperature.

The larger NDR, compared to double barrier RTD, at room temperature, is explained by presence of a double well. In this structure, the second well can avoid electron tunneling from higher energy levels, thus providing a sharper NDR, as shown in figure I.6, taken from [Suda01].

Chapter I: Introduction and state of the art

27

Fig. I.5: Triple barrier SiGe RTD: Conduction band profile [Suda01].

Fig.I.6: I-V characteristic at 300°K [Suda01].

• Improved n-type SiGe RTD

Performances comparable to III-V RTD have been achieved with n-type SiGe RTD [Paul02], with Jp of 282 A/cm-2 and a PVCR de 2.4. Heating effects and thermal dissipations due to the poor thermal conductivity of SiGe substrates have been found to reduce the current density. Indeed, higher performances require appropriate heat sink designs and technological optimizations. To this aim, passivating the device using polyamide featuring a thermal conductivity of 30 m−1 K−1 at 298 K and reducing the device surface, as shown in figure I.8, has allowed to reduce and dissipate heat in the sample, thus improving its current density. [Paul02]

Fig. I.7: N type SiGe RTD structure [Paul02]. Fig. I.8: I-V characteristics for 3 different diode surface [Paul02].

Besides the conventional RTD, where transport occurs between conductions bands, good performances have been observed in resonant interband diodes (RITD) using SiGe or silicon only. The next section will be dedicated to this device.

Chapter I: Introduction and state of the art

28

I.5.2.2. RESONANT INTERBAND TUNNEL DIODES (RITD) Si-based resonant interband tunnel diode (RITD) is actually a hybrid of conventional RTD and

Esaki diode. RITD have been demonstrated in Si/Si0.5Ge0.5/Si heterostructures with good performances, comparable to those of conventional III-V heterostructure [Rommel98][Jin03].

In RITD, conduction occurs between conduction and valence band and not between conduction bands as in conventional RTDs as shown in fig.I.10.

The quantum well is fabricated, following Sweeny and Xu proceeding [Sweeny89], by introduction of δ-doping planes (B, P or Sb doping) to achieve confined states, while the tunnel junction is obtained by Si and SiGe layers. The structure is shown in fig.I.9.

Fig. I.9: RITD structures [Rommel98]. Fig. I.10: Band diagram of interband resonant tunnel diode.

These devices are realized by low temperature molecular beam epitaxy (LT-MBE). Therefore to activate doping, it is necessary to anneal at high temperature, typically 700-800 °C. In order to limit doping diffusion some SiGe spacers surrounding B δ doping layers have also been introduced.

By optimizing thickness and doping, higher performances can be achieved with this device. RITDs featuring peak current densities over 150 kA/cm-2 and PVCRs greater than 2, have been obtained [Jin03] and some circuit demonstrators have been realized with this technology [Chung03].

Epitaxial fabricated δ -doped RITD structures consisting purely of Si which exhibit NDR at room temperature has been presented by Rommel et al. [Rommel99]. Using only Si allows avoiding problem leaded by critical thickness and strain due to the SiGe alloys.

The performances have been shown to strongly depend on the δ-doping spacing and on the post growth annealing. Current peak of 2.7kA/cm2 and PVCR of 2 have been measured at room temperature.

Technological research on RTD: from SiGe RTD to the silicon RTD

RITD and the optimized SiGe RTD can achieve performances comparable to the III-V heterostructures. Although SiGe is a good solution to realize RTD compatible with silicon

Chapter I: Introduction and state of the art

29

technology, several successive works have been oriented to realize silicon based RTD, which can be competitive with III-V and SiGe heterostructure.

Several materials have been proposed for the barriers. The main challenges are:

• Good mismatch between the lattice parameters of silicon and barriers materials.

• Low band offset (but high enough to limit the thermionic current)

• Technology providing thin layers with adequate interface between layers.

• Obtaining crystalline well

In the next section, the main results obtained on silicon based RTD will be presented.

I.5.2.3. Si/CaF2 CaF2 is a good candidate as barrier materials for silicon based RTD, because of its lattice

parameter comparable with the silicon one (mismatch of 0.6%) and good band offset (2.9 eV).

However, the epitaxy of this material on silicon (001) can present some difficulties and a good stability is therefore hard to obtain. Room temperature NDR has been observed by [TsuiTsui99 ] in a structure with 1.8nm CaF2 barrier and 3.7 nm silicon amorphous well, fabricated by molecular beam epitaxy on n+ Silicon (111) substrate.

Another possibility is to grow a CdF2/CaF2/CdF2 on silicon substrate. Asada group [Kanasawa07] has observed negative differential resistance NDR characteristics at room temperature using this stack. The peak-to-valley current ratio was 13 at maximum and the peak current density was between 80 and 90 A/cm2. These approaches have been hampered by the difficulty of fabricating high quality silicon on single CaF2 crystal that lead to poor interface qualities.

I.5.2.4. Si/SiO2 Since first investigations of silicon based RTD, SiO2 was considered to be the best candidate for

the barriers oxide, as the Si/SiO2 structure is well controlled in CMOS technology and could allow an easy integration of RTD in MOS processing.

SiO2 is an amorphous material and disorder leads to some difficulties to fabricate a double barrier SiO2/Si/SiO2.with a crystalline well between two oxides barriers. However, quantization on SiO2/Si/SiO2 structure has been observed on [Lu95] for photonic applications, but to have a resonant effect and NDR, quantization is not sufficient, as conservation of both total energy and transverse momentum are needed [Lake98].

Disorder in amorphous barriers can lead the breaking of the transverse momentum conservation. Numerical calculations do not show promising performances from such materials, but have however demonstrated that amorphous barriers with a crystalline well appear to be sufficient to obtain a small PVCR. In addition, it has been shown that polycrystalline wells may be suitable if the domain sizes are large enough [Lake98].

Chapter I: Introduction and state of the art

30

In order to obtain a crystalline well, several attempts have been done by Tsybeskov et al. to realize silicon nanocrystals in a SiO2 matrix. The structure of the nc-Si:SiO2 interface features low defect density, and the Coulomb blockade effect has been observed in [Tsybeskov00]. Even if a low NDR has been measured, this structure promises rather good performances for memory applications.

Another way to realize a thin mono-crystalline silicon layer between two SiO2 barriers is to use the Silicon On Insulator (SOI) technology. A weak NDR has been observed at low temperature (15°K) for a SiO2/single-crystalline-Si double barrier diodes fabricated from a silicon-on-insulator (SOI) wafer, which featured an ultrathin SiO2 buried oxide (BOX) layer of 2 nm [Ishikawa01].

In this context, the progress made in the use of the SOI technology allows to process devices on thin crystalline silicon with good interface and roughness properties. In this context Double Gate MOSFET that may be used as a quantum well for RTD. Simulations of Choi et al 2003 have demonstrated that the gate tunnelling current of DG structure has a negative differential resistance like the one of conventional RTD.

Recently, silicon-based resonant tunneling diodes have been fabricated using an ultra-high-vacuum wafer bonding system by Lee et al.[Lee07], as shown in fig.I.11, but even in this case, the performances are poor: at 15°K, the PVCR observe was 1.3 and the current peak density of 1.6 10-6

A/cm2 and the NDR was lost with increasing temperature.

Fig. I.11: Process flow for the fabrication of Si-based RTD using ultra-high vacuum wafer bonding [Lee07].

I.5.2.5. HIGH-K OXIDES FOR BARRIERS In MOSFET technology epitaxial oxide on silicon is a good replacing of SiO2. For RTD, using

high-k oxide for barriers may allow obtaining higher current density, by reducing the band offset between Si and oxide and could be a good solution to integrate of RTD in new CMOS proceeding.

Two main results have to be reported in literature:

- An Al / γ-Al2O3 / Si / γ- Al2O3 / n-Si(111) structure [Shahj02]

- A Pt / Gd2O3 / Si / Gd2O3 /Si(111) structure [Osten07]

These structures are described in the following subsection.

Chapter I: Introduction and state of the art

31

• γ-Al2O3 / Si / γ-Al2O3

Double and triple barrier quantum well devices fabricated with thin epitaxial heterostructures formed with 3 nm of γ -Al2O3 barriers and 4 nm of Si well, with good roughness have been realized by [Shahj02]. The NDR effect with a PCVR of 3 for the double barrier structure and 4.5 for the triple barrier was observed at room temperature for the first time (I-V characteristics of double and triple barriers are respectively shown in figure I.12 and I.13).

Fig. I.12: I-V characteristics of γ-Al2O3 / Si / γ-Al2O3 double barrier for a surface of 100 × 100μm2

[Shahj02].

Fig. I.13: I-V characteristics of γ-Al2O3 / Si / γ-Al2O3 triple barrier for a surface of 100 × 100μm2

[Shahj02].

• Gd2O3 / Si / Gd2O3

Fabrication of silicon based double barrier is rather difficult because very thin barrier defect free with good interface are needed. In addition, it is also hard to obtain epitaxially grown silicon well on insulator.

Because of the difference in surface free energy between silicon and the oxide, silicon deposition at high temperature results in small islands. The Osten group [Osten07] has nevertheless managed to realize double barrier Gd2O3 / Si / Gd2O3 with an ultra thin single crystalline silicon well featuring good interfacial quality with rare earth oxide. Low NDR and low peak current have been observed by measurement of this RTD structure at low temperature (77K). I-V characteristic is shown in figure I.14.

Hysteresis has been observed during measurement, probably due to the charged and discharged traps in the structure or in the island. The poor performances obtained with this structure confirm the present difficulty led to realize silicon based RTD.

Chapter I: Introduction and state of the art

32

Fig. I.14: I-V and conductance versus voltage characteristics of Gd2O3 / Si / Gd2O3 diode [Osten07].

The main results provided in this overview of silicon based RTD have been summarized and compared in table 4.

Structure Barriers (nm)

Well (nm)

Type conduction

Temperature (°K)

PVCR Jp (A/cm2)

Reference

Si/Si1-xGexstrain/Si 3.3 6 holes 4.2 2.2 31.4 [Liu 88] Si strain/Si1-xGex/Si

strain 4 5 holes 4.2/77 2.1/1.6 152.8 [Rhee88]

Si1-xGe/Si/Si1-xGex 7.5 5 electrons 77/300 1.5/1.2 ~350 [Ismail 91] Triple barriers 3 6 electrons 300 7.6 ~400 [Suda01]

n-type SiGe/Si/SiGe 2 3 electrons 300 2.4 282 k [Paul02] Si-only RITD 3 5 Interband

conduction 300 1.4 2.3 k [Rommel99]

CaF2/Si/CaF2 1.8 2.8 electrons 300 3.1 71068.4 −⋅ [Tsui99]

CaF2/Si/CaF2 1.6 3.4e electrons 77 6.3 8104 −⋅ [Watan00]

CaF2/CdF2/CaF2(on Si substrate)

1.2 3.1 electrons 300 15.6 175 [Tsui06]

CaF2/CdF2/CaF2(on Si substrate)

1.24 1.86 electrons 300 2 to 13 80-90 [Kanaz07]

SiO2/Si/SiO2 2 2 electrons 15 (and 50) 1.8 5105 −⋅ [Ishik01]

γ-Al2O3/Si/γ-Al2O3 3 4 electrons 300 3 4105 −⋅ [Shahj02]

γ-Al2O3/Si/γ-Al2O3 2 3 electrons 300 248 1.53 [Kathun07] Gd2O3/Si/Gd2O3 1.8/2.3 12.3 electrons 77 (and 210) 1.3 ~10-9 [Osten07]

Table 4: Silicon based diode performances summary.

Chapter I: Introduction and state of the art

33

I.5.3. RTD: WHICH MATERIALS? An overview of RTD realization has been exposed in this section. III-V materials have been for

long time the more suitable for RTD fabrication, but the last prototypes of SiGe/Si heterostructures , (where section and contact have been improved for electron tunneling devices), are a good solution for silicon compatible technology. Instead, the silicon based RTDs presently achieve poor performances, not always measurable at room temperature, as reported in table 4. Fig I.15 summarizes the main results of the best Si-based RTD compared to III-V heterostructure by plotting the PVCR versus its corresponding Jp, showing an overview of the comparable performance at room temperature.

1 2 3 4 5 6100

1k

10k

100k

1M

Pe

ak c

urr

en

t (A

/cm

2 )

PVCR

InGaAs/AlAsInAs/AlSb

GaAs/AlAs

Esaki Diode (Dushl 2000)

RITD (Rommel 1998)Si-onlyRITD (Rommel 1999)

RITD SiGe (jin 2003)

SiGe-n (Paul 2002)

RITD (Dushl 2000)

Esaki Diode (Jorge 1993)

SiGe RITD (jin 2002)

Esaki Diode (Dushiell,2000)

Requirement for Memory and Logic Function (PVCR > 2)

Re

quirement fo

r High spe

ed applications

or mixed sign

al circuitry ( Jp >

10kA/cm

2)

Fig. I.15: The Peak current of several of SiGe (and Si)-based diodes and III-V heterostructures plotted versus PVCR. Open symbols represent Esaki diodes, solid blue symbols represent SiGe RITD and n-SiGe RTD. The only red solid symbol is the

only RITD silicon based measured at 300°K (InAs/AlSb [Sod91], InGaAs/AlAs [Broek90], GaAs/AlAs [Wolak91]).

Therefore, depending on applications, a material can be suitable than another for RTD fabrication.

On one hand, a higher current peak is needed for high speed applications, and a good PVCR for memory and logic functions. III-V double barriers present good requirement for these applications. n-SiGe optimized heterostructures and some RITDs also appear as good compromises.

On the other hand intrinsic operating frequency is an important parameter for high frequency application. Table 5 summarizes the main measured cut off frequencies found in literature.

Chapter I: Introduction and state of the art

34

Comparing table 4 and figure I.15, where the static performances have been reported, it turns out that, depending on application, a compromise between high cut off frequency, good PVCR and high current is needed in the choice of different materials. Altogether, improved SiGe and RITD present performances equivalent to III-V materials.

Table 5: Measured intrinsic cut off frequency for different material systems.

As explained later in this thesis, table 4 shows that the silicon realizations are not yet competitive with III-V and SiGe heterostructures and that any applications have not been demonstrated for this structure at present.

In the next chapters, a deep investigation of possible improvement in silicon structures and the analysis of the limitation of silicon devices will be presented.

The main digital and analog RTD application demonstrated in literature are listed in the next section.

I.6. APPLICATIONS OF RTD Very high speed operation and negative differential resistance make RTDs very interesting for

several digital and analog applications, enabling optimized circuit designs. For example signal processing circuit with reduced size and multi-valued memory cell based on RTD have been demonstrated. In addition, with their small size and simple structure, RTDs represent a good option to be integrated with conventional devices, to complement MOSFET and bipolar transistor in several applications, which should lead to drastic reduction in static power consumptions.

n-SiGe

[Paul02]

GaAs/AlAs

[Brown89]

Schottky GaAs/AlAs

[Smith94]

InAs/AlSb

[Brown91]

InGaAs/AlAs

[Brown91]

648 GHz

(2.3THz theoretical) 420 GHz 900 GHz

712 GHz

(Theoretical 1.24 THz) ~200GHz

Chapter I: Introduction and state of the art

35

I.6.1 DIGITAL APPLICATIONS

RTD, thanks to its particular I-V characteristics and high speed operation is suitable for multi-valued digital applications. In [Lin94], the authors have summarized promising multi valued applications based on RTD architecture.

Several applications are also possible by exploiting the natural instability of RTD in NDR region and the eventual multi peaks characteristic and hysteresis of RTD: memory application, counters and shift register, analog to digital converter, logic circuits (logic gates, flip flop…) and fuzzy logic.

Among them, we will focus on the memory cell and ADC applications, explaining the basic concept of multi-valued logic, from which other applications may also arise.

• Memory application

To increase the memory integration, two possibilities exist: memory cell scaling or multi-states memory cell implementation. RTD with series resistance can work like a memory element. In this configuration, the resistance is used to control the optimal operation point of the memory cell. [Wei92].

Fig. I.16: Simple memory cell, composed by a RTD and a resistor load.

Fig. I.17: Piecewise linear I-V characteristics for an RTD and a load line.

The principle of operation of this static memory is the following:

The characteristic of fig I.17 (RTD and load) presents 3 intersection points: 1 and 3 are stable point and 2 in unstable [Mizuta]. In the circuit of fig I.19, when Vin increase, Vout increases in the first linear part of the I-V diode characteristics from zero to the point 1, until the voltage peak. By increasing Vin , Vout pass switching in the third linear part of characteristics of ID When Vin decreases after switching, the diode voltage decreases from the point 3 to the valley voltage. With a further decreasing of Vin, the diode voltage passes to the first linear part of the characteristic. [Mizuta].

Reading the memory state is possible by sensing the voltage in output node. Writing is possible by forcing up or down the input voltage.

The memory element is placed in a matrix array and access is possible by the means of the transistors in the word line and bit line. As this memory element is very simple, it is suitable for integration with FET, in order to fabricate a memory integrated circuit, as in figure I.18.

Chapter I: Introduction and state of the art

36

Fig. I.18: Typical configuration of an RTD based static memory integrated circuit.

Compared to the conventional memory cell which needs 6 transistors or 4 transistors and two resistors for each memory cell, RTD based memory need only three devices: A RTD , a resistor and an FET, in the simplest case. Thus, RTD memory cell can lead to significant reduction in memory cell size.

Multi-peaks RTD is also a good option for a multi-valued logic [Lin94]. To bias the multi-peaks RTD in a multi-stable state mode, it is possible to use a resistance load or a current source. From N RTD peaks 2N+1 memory states are obtained. [Wei92].

Fig. I.19: Multi-peaks I-V characteristics of RTD with resistor loads.

Several configurations are possible to replace resistance load, such as a FET, which could use a lower power supply voltage, or another RTD in cascade, which is preferable in term of integration.

• Analog to Digital Converter (ADC) :

Conventional ADC converter consists of a bank of comparators in parallel to compare input signal with a reference voltage. By using multi threshold RTD digitizers, ADC does not need comparators, thus reducing the circuit complexity. This should simplify the circuit architecture and lead to a

Chapter I: Introduction and state of the art

37

reduction of the power dissipation and the chip area. In the multi peak RTD, the I-V characteristics can be controlled by connecting a resistance load. This should increase the positive resistance and reduce the negative resistances, as shown in figure I.20:

Fig. I.20: Multi-peaks I-V characteristics of RTD (curve a); with resistor loads (curve b) [Wei93].

The digitizer consists of two devices and two connected resistances [Wei93].

Fig. I.21: RTD digitizer. Fig. I.22: Four peaks I-V characteristics of RTD with resistor loads.

By increasing the node voltage, the RTD is biased in different part of its characteristics, which corresponds to 0 or 1 value. A multi-threshold digitizer is thus formed. Fig I.23 shows a schematic of a 4-bits ADC. The voltage Vin+Vq/2 is divided by 2z (Z = 0, 1, 2 and 3) using a resistor string to obtain Vin Vin/2, Vin/4, and Vin/8. The transfer curves for all four digital outputs are shown in Fig. I.24.

Chapter I: Introduction and state of the art

38

Fig. I.23: Schematic diagram of a 4-b AD converter using RTD’s.

Fig. I.24: Simulation results of a 4-bit ADC.

A flash-ADC composed of a HFET sample and-hold followed by 16 RTD/HFET clocked-comparators was proposed by [Broek98]. Also in this case, the RTD/HFET architecture reduces the number of circuit components, providing a lower power dissipation and higher speed compared to traditional circuit.

I.6.2. ANALOG APPLICATIONS

• Frequency divider:

A frequency divider based on a RTD chaos circuit has been fabricated by Kawanao et al. [Kawano01] for an input frequency of 50 GHz. This kind of frequency divider allows low power consumption and ultra high frequency operation. In [Kawano01] it has been demonstrated that, from a 50 GHz input frequency, output of 1/2,1/3 and ¼ frequencies can be obtained.

• Frequency multiplier:

The combination of RTD and transistor can avoid biasing RTD in the NDR region, assuming a stable operation at any frequency. A possible application of a HFET/RTD structure is the frequency multiplier. Connecting an RTD as a load at the drain of a HFET, the output voltage sharply swing at both peak and valley current, producing a rectangular output voltage rich in odd harmonics. In [Auer96], a frequency multiplier up to 26.5 GHz has been demonstrated.

Chapter I: Introduction and state of the art

39

Fig. I.27: HFET/RTD frequency multiplier.

Fig. I.28: Load characteristic HFET/RTD and a basic illustration for the generation of rectangular-like output

voltage [Auer96].

• Oscillator:

The standard differential oscillator circuit consists in a complex architecture, where the negative resistance, is obtained by means of a feedback loop (e.g. using a cross-coupled transistor pair [Bao04]), in order to compensate oscillator losses and to maintain oscillation

This circuit can be considerably simplified using NDR devices such as RTD, leading in principle

to the reduction of the oscillator circuit size (as it uses less devices), the reduction of the power consumption and to higher frequency performances. There are several demonstrations of RTD based oscillators.

Resonant tunneling diode/heterojunction bipolar transistor (RTD/HBT) VCO has been proposed

by De los Santos [DeLosSantos02], and before a BJT/RTD based oscillator was patent by the same author. The HBT/RTD structure, in figure I.29, introduces important advantages compared to conventional oscillators. For instance, it can consume almost 90% less power than actual feedback oscillators. As the supply voltage consists only in the voltage drop across the RTD (0.1- 0.5 V), the HBTs active region (i.e. the collector-emitter region ), and the inductor, it is strongly reduced, for example at 1.3 V. For a frequency of 5.78 GHz this circuit can produce an output power of +3.13 dBm.

Chapter I: Introduction and state of the art

40

Fig. I.29: HBT/RTD oscillator [DeLosSantos02].

Other architectures of tunneling diode monolithic microwave integrated circuit (MMIC) VCOs

have been reported in literature [Choi04] [Choi05].

Fig. I.30: Circuit architecture of the RTD MMIC VCO.

In the architecture shown in figure I.30, to begin the oscillation, the magnitude of the effective

negative resistance Rn generated by the RTD has to be smaller than the overall resistance, consisting of the effective input shunt resistance of the buffer (Rbuffer) and the equivalent resistance of the resonator (RRes), so that sBuffern RRR Re//< . The required voltage is hence determined by the NDR

voltage range of the RTD.

The NDR voltage range of the InP based RTD lies from 0.3 to 0.7 V. Thus, a very low voltage can be used in the VCO core circuit, which could reduce the dc power consumption compared to conventional-type VCO. For a frequency of 17GHz the output power obtained with this structure is -8dBm, and the dc power consumption is 1.42 mW.

Chapter I: Introduction and state of the art

41

In contrast to the previous detailed oscillator structures, Muramatsu et al.[Muram05] propose a new VCO architecture where oscillations are not based on small signal behavior of RTD around NDR, but on the large signal behavior of RTD.

In the circuit configuration of figure I.31, oscillations are obtained by associating the transistor load line with the switching originating from peak current to the valley current in the RTD. In this way, larger output voltage amplitude can be achieved, exceeding the output power limitation of conventional RTD based oscillator.

Fig. I.31: Circuit architecture of the Muramatsu oscillator [Muram05].

Recently, an InGaAs/AlAs RTD oscillator integrated with a slot antenna has been realized by Asada group [Hinata09]. High output power in the THz range have been achieved. Maximum output power have been found where the impedance matched between the RTD and the antenna. Experimentally an output power of 150 μW at 250 GHz has obtained with a RTD peak current of 7 mA/μm2.

Fig. I.32: Structure of RTD oscillator with offset-fed slot antenna.

Chapter I: Introduction and state of the art

42

As shown in this section, several applications have been proposed for RTD. The main expected advantages of RTD for analog and digital applications are:

• Low power consumption

• High frequency performances

• Reduction of circuit size and complexity

Most of the demonstrators have been made with III-V based RTD. Coupling RTD with FET and bipolar transistor provides several circuit advantages. In this context it would be interesting to investigate silicon based RTDs integrated with CMOS circuit and their possible applications. This question has motivated the work presented in this thesis.

I.7. THE AIM OF THIS THESIS:

The aim of this work is to give an estimation of the expected performances of silicon based RTDs and to compare them with conventional heterostructures. This thesis proposes to answer the following questions:

Will be silicon suitable for RTD? At which conditions and for which applications?

For this goal an original analytical and physically based model has been developed, which allows to compute I-V characteristics and intrinsic cutoff frequency of RTD in function of materials parameters and geometrical dimensions,. In addition, this model is compact enough to be introduced in a circuit simulator and has been used to estimate silicon RTD for RF oscillator applications.

The Chapter II of this work will dedicate to the development of the analytical model and to an overview of the main approaches to compute RTD I-V characteristics. A validation of the model will also be presented.

In the Chapter III, the DC analytical model will be extended to the AC regime, to compare the frequency limitation of several structures and materials. A time dependent many particles simulator based on Bohm trajectories will be used to analyze and validate the small signal equivalent circuit derived in AC regime.

The Chapter IV will focus at the comparison of III-V RTD performances and silicon based RTD by the means of this analytical model, in order to optimize the RTD structure and to obtain equivalent performances. Finally the model will be introduced in a circuit simulator and two different architectures of RF oscillator based on silicon RTD will be presented and compared with conventional oscillators and III-V RTD based oscillators.

The results of this work will finally be summarized in the conclusive chapter of this thesis.

Chapter I: Introduction and state of the art

43

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[Jin03] N. Jin, S. Chung, A. T. Rice, and P.R. Berger “151 kA/cm2 peak current densities in Si/SiGe resonant interband tunneling diodes for high-power mixed-signal applications” Appl. Phys. Lett. Vol. 83, N. 16 October 2003.

[Kanasawa07] T. Kanazawa, R. Fujii, T. Wada, Y. Suzuki, M. Watanabe, and M. Asadab “ Room temperature negative differential resistance of CdF2 /CaF2 double-barrier resonant tunneling diode structures grown on Si(100) substrates” Appl. Phys. Lett. 90, 2007.

[Kawano02] Y. Kawano, Y. Ohno, S. Kishimoto, K. Maezawa and T. Mizutani “50 GHz frequency divider using resonant tunnelling chaos circuit” Electronics Letters March 2002 Vol. 38 No. 7.

[Kawano98] Kawano, Y., Kishimoto, S., Maezawa, K., and Mizutani, T.: ‘Resonant tunneling chaos generator for high-speed/low-power frequency divider’, Jpn. 1 Appl. Phys., 1999, 38, pp. L1321-Ll322.

[Kon93] Y. Konishi, S. T. Allen, M. Reddy, and M. J. W. Rodwell, "AlAs/GaAs Schottky-Collector Resonant-Tunnel-Diodes", Solid-State Elec. 36, 1673 (1993).

[Kubota06] J. Kubota, A. Hashimoto, Y. Suda “Si1-x Gex sputter epitaxy technique and its application to RTD” Thin Solid Films 508 (2006) 20 – 23.

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[Majkusia06] B. Majkusia, "Resonant tunneling devices on SOI basis" NATO Advanced Research Workshop on Nanoscaled Semiconductor-on-Insulator Structures and Devices, Oct. 15-19, 2006 p. 341.

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[Mazumd98] P. Mazumder, S. Kulkarni, M. Bhattacharya, J.P. Sun, G.I. Haddad, “Digital circuit applications of resonant tunnelling devices”, Proceedings of the IEEE 86, 4, p664, (1998).

[Mehdi90] Imran Mehdi and George Haddad, “Lattice matched and pseudomorphic In0.53Ga0.47As/InxAl1-xAs resonant tunneling diodes with high current peak-to-valley ratio for millimeter-wave power generation”, J. Appl. Phys. 67, 2643 (1990).

[Mönch90] W. Mönch, “Schottky Barriers on GaAs”, in David K. Ferry, ed., Gallium Arsenide Technology, vol. II (Howard W. Sams & Co., Carmel, Indiana, 1990), p. 144.

[Muram05] N. Muramatsu, H. Okazaki, T. Waho, “A novel oscillation circuit using a resonant tunneling diode”, ISCAS 2005 proceedings, pp. 2341 – 2344.

[Osten07] H.J. Osten, D. Kuehne, E. Bugiel, A. Fissel, “Fabrication of single-crystalline insulator/Si/insulator double-barrier nanostructure using cooperative vapor–solid-phase epitaxy” Physica E 38 (2007) 6–10.

[Paul02] D.J. Paul, P. See, I.V. Zozoulenko , K.-F. Berggren, B. Holla¨nder , S. Mantl, N. Griffin, B.P. Coonan, G. Redmond, G.M. Crean “n-type Si/SiGe resonant tunnelling diodes” Materials Science and Engineering B89 (2002) 26–29.

[Rhee88] S.S. Rhee, J.S. Park, R.P.G. Karunasiri, Q. Ye, and K. L. Wang “Resonant tunnelling through a-Si/GexSi1-x on a GeSi buffer layer”. Appl. Phys. Lett. 53 (3)18 July 1988.

[Rommel88] Sean L. Rommel, Niu Jin, T. E. Dillon, Sandro J. Di Giacomo, Joel Banyai, Bryan hf. Card, C. D'hperio, D. J. Hancmk, N.Kirpalani. V. Emanuele. Paul R. Berger. Phillip E. Thompon Karl D. Hobart:a and Roger Lake “Development of δB/i-Si/δSb and δB/i-Si/δSb/i-Si/δB Resonant Interband Tunnel Diodes For Integrated Circuit Applications” IEDM 98.

[Rommel98] S. L. Rommel, T. E. Dillon, M. W. D., H. Feng, J. K., and P. R. Berger “Room temperature operation of epitaxially grown Si/Si0.5 Ge0.5 /Si resonant interband tunneling diodes” App. Phys. Lett. Vol. 73, No 15 October 1998.

[Schulman96] J. N. Schulman, H. J. De Los Santos , « Physics-Based RTD Current-Voltage Equation» IEEE Electron De-vice Letters, Vol 17, n°5, pp. 220- 222 (1996).

[Sen87] S. Sen, F. Capasso, A. L. Hutchinson, and A. Y. Cho, "Room-Temperature Operation of Ga0.47In0.53As/Al0.48In0.52As Resonant Tunneling Diodes", Elec. Lett. 23, 1229 (1987).

[Shahj02] M. Shahjahan, Y. Koji, K. Sawada, M. Ishida, “Fabrication of RTD by γ-Al2O3/Si multiple heterostructures”, Jap. J. Appl. Phys. 41, p2602, (2002).

[She85] T. J. Shewchuk, P. C. Chapin, P. D. Coleman, W. Kopp, R. Fischer, and H. Morkoç, "Resonant tunneling oscillations in a GaAs-AlxGa1-xAs heterostructure at room temperature", Appl. Phys. Lett. 46, 508 (1985).

[Smith94] R. P. Smith, S. T. Allen, M. Reddy, S. C. Martin, J. Liu, R. E. Muller, and M. J. W. Rodwell, "0.1 m Schottky-Collector AlAs/GaAs Resonant Tunneling Diodes", IEEE Elec. Dev. Lett. 15, 295 (1994).

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Chapter II: Modeling of the RTD

CHAPTER II: MODELING OF THE RTD

II.1. INTRODUCTION

Due to its simple 1-D structure and to the quantum phenomena (tunnel and confinement) that provide its particular characteristic I-V, RTD was widely investigated and modeled as vehicle to study transport in quantum devices.

Several approaches have been proposed for numerical and analytical simulation of RTD.

Among these methods, many approaches calculate first a transmission coefficient and then deduce from it the current characteristics (wave approach). There are also other approaches where the transmission coefficient is not explicitly calculated. In the first part of this chapter an overview of the most applied numerical methods is presented.

Numerical models provide a rigorous depiction of the physics that governs the static (and dynamic) behavior of RTD, but these models are not suitable for compact model application.

In literature, several analytical models can be found. In the second part of this chapter, we focus on the most relevant examples.

The aim of this thesis is developing an analytical model, physics based and compact enough to be introduced in a circuit simulator, in order to estimate the possible applications of silicon based RTD, compared with other materials.

In the last part of this chapter the DC analytical model developed during this PhD is presented. The key points of state of art in numerical and analytical modeling for RTD presented in this chapter will be useful to understand the originality and limits of this model.

II.2. NUMERICAL METHODS FOR CURRENT CALCULATION IN RTD: AN OVERVIEW

An overview of the most applied methods for current computation in RTD is provided in this section.

The first presented method is the transfer matrix method that allows computing transmission coefficient of an incident wave function and from this, deriving the current. For its relative simplicity and relatively low computation requirement, it is the most widely used approach.

This method can be used to compute numerically the transparency of arbitrary tunnel barrier.

Chapter II: Modeling of the RTD

48

This approach can also be used in some particular case such as a double rectangular barrier, to derive analytical expression of the transparency.

Another widely applied method that compute current from transmission coefficient consists in solving Schrodinger equation by the Green function method.

Finally, among the methods that allow estimating current directly, without computing explicitly T(E) Wigner function and Bohm trajectories based methods, are presented.

II.2.1. TRANSFER MATRIX METHOD

The transmission coefficient through a barrier can be defined as the ratio between the particles flux incident in a barrier and the particles flux of the particles transmitted in the other side of the barrier.

The most applied method to compute transmission coefficient is the Transfer Matrix Method (TMM).

TMM consist in determining the transmission amplitude T through barriers of an incident wave function.

The first formulation of TMM was in 1962. Its derivation and understanding is easy due to the electromagnetic analogy, and it is widely applied because of its theoretical simplicity and low computational requirement.

This method was originally introduced to study RTD by Tsu and Esaki [Tsu73]. TMM solves the time independent (steady state) Schrodinger equation without including scattering potential, but taking into account the heterojunction band offset (and electrostatics potential if Poisson equation is solved). [Ghata88][Ando87]

Let us introduce the time –independent, 1D Schrodinger equation, considering z as transport direction:

)()()()(*)(

1

2

2

zEzzVzzzmz zΨ=Ψ+Ψ

∂∂

∂∂− (2.1)

Where m(z)*is the effective mass, Ez is the energy value.

This method consists in approximating an arbitrary potential V(z) by a series of small potential step, as shown in the figure II.1.

Chapter II: Modeling of the RTD

49

Fig II.1: Potential profile used to solve Schrodinger equation by TMM.

For a series of small potential steps, the wave function can be expressed in plane-wave form for each section i :

)()()()()( )()(

)( zikizikii iz

iz eBeAz −+=Ψ (2.2)

Where )(izk is the complex wave vector:

)(*2 )()()(

iz

ii

z

VEmk

−= (2.3)

kz has a real values for energy above the gap and imaginary values for energy inside the gap.

Ai and Bi are found by matching Ψi and imΨ1 at each i boundaries:

)()( 1)1()(

++Ψ=Ψ i

ii

i zz and 1

)(

*

1)(

*

1 )1(

)1(

)(

)(

+=

+

+=

∂Ψ∂⋅=

∂Ψ∂⋅

ii zz

i

izz

i

i z

z

mz

z

m

For each section the wave function coefficients can be related by the matrix equation:

=

+

+

)1(

)1(

)(

)(

i

i

i

i

B

AM

B

A (2.4)

In order to explain how the matrix M is computed, we have to consider two possible cases, the propagation in a region at constant potential and the reflection on a potential step. The two possible situations are shown in figure II.2.

Chapter II: Modeling of the RTD

50

Fig II.2: Constant potential profile (left) and potential discontinuity (right).

In a constant potential profile, the wave function in z = L has just a phase shift of kL, where k is the wave vector.

ikL

ikL

eBD

eAC−=

=

Thus, the propagation matrix P in each section i of potential profile is:

=

− ii

z

ii

z

zk

zik

e

eP )(

)(

10

0][ (2.5)

In both cases, we have to consider the matching of Ψi and imΨ1 at the interface z=0 leading to

consider the following relationship:

A + B = C + D and )()()1(*

)1(

)(*

)(

DCm

ikBA

m

iki

iz

i

iz −=− +

+

Where )(i+α and )(i

−α are respectively:

+= +

+

+ )1(

)(

)(

)1()(

*

*1

2

1i

z

iz

i

ii

k

k

m

mα and

−= +

+

− )1(

)(

)(

)1()(

*

*1

2

1i

z

iz

i

ii

k

k

m

The reflection matrix is thus:

=

+−

−+)()(

)()(

][ii

ii

Rαααα (2.6)

Finally the global matrix M is the product of element M(i) for each section i:

),,,(),( 1)1*()*()(

++

+ Δ= ii

ii

iii VmVmRzVPM (2.7)

In the emitter and collector region, the coefficients A and B are connected by transfer matrix as:

Chapter II: Modeling of the RTD

51

=

R

R

L

L

B

AM

B

A][ (2.8)

If we consider that there is not reflection (B=0) in the collector region, the transmission probability is given by the ratio between the transmitted flux and incident flux as :

211

2

2

||

1

*

*

*

*)(

Mk

k

m

m

A

A

k

k

m

mET

L

R

R

L

L

R

L

R

R

L

z == (2.9)

This method can be improved using instead of constant potential, linear potential. In this case, Airy functions are solutions of the Schrodinger equation. [Wang01]

II.2.1.1 ANALYTICAL APPROACH BASED ON TRANSFER MATRIX METHOD In some particular case, TMM can lead to analytical solutions of transmission coefficient. Three

examples have been presented in this section and will be used in the following to derive compact model.

Symmetric rectangular barrier

Let us begin with the simple case of a squared barrier whose height is Vo, for example the conduction band offset between two materials. [Ferry05]

Fig II.3: Potential profile of a simple squared barrier.

For energies lower that V0, in x< -a and x> a, solutions are incoming and outgoing waves,

while for -a <x< a , and for ],[ aax −∈ , solutions are evanescent waves.

If the barrier is thin enough, the propagating wave in the left side can penetrate in the barrier, continue attenuated trough the barrier and excite a propagating wave beyond the barrier. We want evaluate the transmission probability trough the barrier.

Chapter II: Modeling of the RTD

52

The wave function can be written in the three regions as:

axincondition

axforFeEe

axaforDeCe

axforBeAe

xikzikz

zz

ikzikz

−=

>+

−<<+

−<+

=Ψ−

γγ)( (2.10)

Where k and γ are the wave vectors: Em

k2

2

= ; )(

202

EVm −=

γ .

Now we have 6 coefficients to evaluate: A and B are associated to the incident and reflected waves in the first region, C and D are associated to the evanescent wave in the barrier.

The boundary conditions in a and –a , ensuring the continuity of wave function and its first order derivate at each interface, lead to:

axinconditionDCeBeAeik

DCeBeAeaaikaika

aaikaika

=−=+

+=+−−

−−

][][ γγ

γγ

γ (2.11)

Solving to seek A and B in function of C and D, leads at the follows matrix equations:

+

+

=

−−+−

+−

D

C

eik

ike

ik

ik

eik

ike

ik

ik

B

A

aikaik

aikaik

)()(

)()(

22

22

γγ

γγ

γγ

γγ

(2.12)

Applying the same boundary conditions at the other interface, we obtain:

][][ aaikaika

aaikaika

DeCeFeEeik

DeCeFeEeγγ

γγ

γ −−

−−

+=+

+=+

Again, solving to for C and D function of E and F:

+

−−

+

=

−−+−

+−

F

E

eik

eik

eik

eik

D

C

aikaik

aikaik

)()(

)()(

22

22

γγ

γγ

γγ

γγ

γγ

γγ

(2.13)

From the two matrix equation, we can finally link the coefficients A and B in the left side with the coefficients E and F in the right side, leading the following matrix equations:

=

F

E

MM

MM

B

A

2221

1211 (2.14)

The matrix elements are:

Chapter II: Modeling of the RTD

53

ikaaikaik eak

kiae

ik

ikike

ik

ik

ikM 2

22)(2)(2

11 )2sinh(2

)2cosh(2222

−−=

−−

+

+= +− γ

γγγγ

γγ

γγγ γγ (2.15)

)2sinh(22222

2222

21 ak

kie

ik

ik

ike

ik

ikikM aa γ

γγ

γγγγ

γγ γγ

+−=

+−

+= − (2.16)

Note that:

*2112

*1122 MMMM ==

The transmission coefficient can be deduced from these equations as the ratio between the flux incident from the left side in the barrier and the transmitted flux in the right side, when F = 0 (no incident wave from the left).

*

2

m

kAfinc

= and

*

2

m

kEftran

=

)2(sinh2

1

1)2(sinh

2)2(cosh

1

2

222

1

222

22

112

2

ak

ka

k

ka

MA

E

f

fT

inc

tran

γγγ

γγγγ

++

=

−+====−

(2.17)

For energies higher than barrier (E > V0), the waves in the barrier region are no longer evanescent and the transmission coefficient presents an oscillatory behavior, approaching unity as E augments (due to the interference with reflected wave):

)2(sin2

1

1)(

'2

2

'

2'20

akkk

kkVET

−+

=> (2.18)

Asymmetric rectangular barrier:

Now, we consider a barrier with a generic potential qV1 applied at the right side. The propagating

wave has a different wave vector that in the previous case. [Ferry05]: )(2

121 VEm

k +=

Chapter II: Modeling of the RTD

54

Fig II.4: Potential profile of an asymmetric squared barrier.

At interface –a, the boundary condition remain the same that in the previous case.

Applying the new boundary conditions at the interface a, leads to:

+

−−

+

=

−−+−

+−

F

E

eik

eik

eik

eik

D

C

aikaik

aikaik

)(1)(1

)(1)(1

1

1

22

22

γγ

γγ

γγ

γγ

γγ

γγ

(2.19)

Then using the same approach than in the previous example the matrix example can be derived, leading to:

aikkiaikikaikik eak

kkia

k

ke

ik

ikike

ik

ik

ikM )(

2211)2(21)2(1

11111 )2sinh(

2)2cosh(1

2

1

2222+++−+

−−

+=

−−

+

+= γ

γγγγ

γγ

γγγ γγ (2.20)

In this case the incident and transmitted flux are given by: *

2

m

kAfinc

= and

*12

m

kAfinc

=

The transmission coefficient becomes:

)2(sinh)(

))((1

)(

4

1

2

2

21

2

221

22

21

1

211

12

21

akk

kk

kk

kk

Mk

k

A

E

k

k

f

fT

inc

tran

γγ

γγ

+++

+

+=⋅=⋅== (2.21)

The asymmetry in the two sides of barriers is taken into account through the wave vector k1 and k.

The double barrier

Let us consider now two symmetric barriers. This situation approximately corresponds to the potential profile of a RTD in flat band condition.[Ferry05]

Chapter II: Modeling of the RTD

55

Fig II.5: Potential profile of a symmetric double squared barrier.

Again, the same procedure than in the previous example has been used. In this case there are more coefficients: A, B, E, F, A’, B’, E’, F’. The matrix can be obtained using results of previous section. The only difference is to connect each other A’ and E, B’ and F. As they correspond at different point, but at the same potential, one can only consider a phase shift:

ikbikb FeBEeA −== ''

This definition can be rewritten in matrix formulation as:

=

'

'

0

0

B

A

e

e

F

Eikb

ikb

(2.22)

This matrix represents de transfer matrix for the well wM

Connecting all coefficients from the left side to the right side of the double barrier, we obtain:

=

⋅⋅=

'

'

'

'

F

EM

F

EMMM

B

ATRwL (2.23)

Where LM and RM are respectively the transfer matrix for the left and right tunnel barrier.

The total transmission coefficient is deduced from 11TM , given by:

ikbRL

ikbRLT eMMeMMM 2112111111 += − (2.24)

Symmetric barriers:

In the case of symmetric barrier, the wave vector k is the same in left, right side and in the well. The barriers are equal, same height V0, thickness aL=aR, and same attenuation constant γ [Ferry05].

For sake of simplicity, for the elements 11LM and 11RM , will be noted as:

Chapter II: Modeling of the RTD

56

ϑiemM −= 1111 (2.25)

Where:

)2(sinh2

)2(cosh 2

2222

11 ak

kam γ

γγγ

−+= (2.26)

and

−= − )2tanh(2

tan22

1 ak

k γγγϑ (2.27)

So, the total coefficient M11 can be noted as:

)](2[cos4)()](2[cos241 2221

211

2221

211

2221

211

421

411

211 ϑϑ +++=++++= kbMMMmkbMMMmM T

(2.28)

The first term into brackets is the determinant of the simple barrier and is equal to 1 (as the matrix has been obtained by multiplying matrix that all have a determinant equal to 1).

So the previous expression can be simplified and the transmission coefficient has the following formulation:

)(cos4)](2[cos41

11)(

22

2

2221

211

211

θϑ −+=

++==

kbRT

T

kbMmMET

sbsb

sb

T

tot (2.29)

Where sbT is the transmission coefficient of a simple barrier.

The transmission coefficient is function of energy and depends on barriers characteristics, as mass, height, through the wave vectors calculated in each regions and thickness and distance between the barriers.

Lorentzian approximation:

Around a resonant energy level En , for symmetric double squared barriers, the transmission coefficient can be approximated by a Lorentzian function [Ferry].

In the hypothesis of infinite well 2

πθ −= , we can expand the cosine in the equation (2.29) as:

so )(sin)2

(cos 22 kbkb =+ π . (2.30)

As at the resonance, for nEE → the sine squared tends to 0, we can approximate it with is argument:

20

2 ])[()(sin bkkkb −≈ (2.31)

Developing this term:

Chapter II: Modeling of the RTD

57

−=−=− 1

*2)(

*20

n

nn E

EEmEE

mkk

(2.32)

Around the resonance: ε+= nEE

δεε +=+=+= 11nn

n

n EE

E

E

E (2.33)

where nE

εδ =

Putting equation (2.33) in equation (2.32):

nn

n

EE

mEm

2

*2)11(

*2 εδ

≅−+ (2.34)

So: n

n

E

EEm 2

2

)(

2

* −

(2.35)

Replacing in the equation (2.29):

222

22

)(*2

*2)(

nsb

nsb

sb

nsb

EEmR

ET

mR

ET

ET

−+=

(2.36)

Where we can put:

2/122

*

2

mR

ET

sb

nsbn

(2.37)

Leading to the Lorentzian formulation:

22

2

)(4/

4/)(

nn

n

EEET

−+ΓΓ= (2.38)

Where Rn

Lnn Γ+Γ=Γ . In the simplest case of symmetric and equal barrier R

nLn Γ=Γ .

In conclusion, the resonant T(E) can be approximated by Lorentzian peak around nEE ≈ .

Chapter II: Modeling of the RTD

58

Asymmetric barrier:

Fig II.6: Potential profile of an asymmetric double squared barrier.

Let us study the double asymmetric barrier. This case will be then use to model RTD potential barrier under bias.

We consider the most general situation, where height, thickness of the two barriers may be different. Five wave vectors are needed in each region of double barrier: k, k1 and k2, respectively for the left region, the well and right region. γ and γ1 for the left barrier and right barrier.

The results given for the simple asymmetric barrier is still valid, and we can use them to seek general solutions[Ferry05]. We define the matrix element in polar coordinates:

ijiijij emMϑ= , for the two barriers, i the index are L11, L11 ,R11 ,R21, leading the follows series of

formulations :

)2(sinh4

1)2(cosh)1(

4

1 2

221221

11 LLL ak

kka

k

km γ

γγγ

−++= (2.39)

)2(cosh)1(4

1)2(sinh

4

1 2212

221

12 LLL ak

ka

k

kkm γγ

γγ −+

−= (2.40)

)2(sinh4

1)2(cosh1

4

11

2

11

2121

12

2

1

211 RRR a

k

kka

k

km γ

γγγ

−+

+= (2.41)

)2(cosh)1(4

1)2(sinh

4

11

22

11

2

2

11

2112

212

RRR ak

ka

k

kkm γγ

γγ −+

−= (2.42)

LLL akkakk

kk)()2tanh(

)(arctan 1

1

21

11 ++

+−−= γγγϑ (2.43)

Chapter II: Modeling of the RTD

59

LLL akkakk

kk)()2tanh(

)(arctan 1

1

21

12 −++

−+−= πγγγϑ (2.44)

RRR akkakk

kk)()2tanh(

)(arctan 121

121

2112

11 −−

+−−= γγγϑ (2.45)

RRR akkakk

kk)()2tanh(

)(arctan 211

112

2112

21 −++

−+= πγγγϑ (2.46)

Following the approach used in the case of simple asymmetric barrier, we evaluate the net transmission matrix element as:

+++++−=

2cos)(4)( 111121122

211211112

21121111

2

11RLRL

RLRLRLRLT kbmmmmmmmmMϑϑϑϑ

(2.47)

The transmission coefficient is: 2

11

2 1)(

TMk

kET = (2.48)

In this section several approaches to compute T(E) have been presented. An additional step is needed to deduce I(V) from T(E). This is examined in the next section, where the coherent tunneling leading to the Tsu –Esaki formulation, is presented in the next paragraph.

II.2.2 COHERENT TUNNELING

Particles injected from the left reservoirs have a probability T(E) to pass through the double barrier and to reach the right contact. If we assume that tunneling occurs conserving transverse momentum and that particles maintain their phase coherence, tunneling is called coherent.

In this case, the current flowing in the double barrier can be considered as the net difference between the particles transmitted from left side to the right side and those transmitted from right to the left contact.

We first assumed that the Hamiltonian can be assumed separable perpendicular components, in z direction (transport direction) and transverse direction ( in the plan x , y). If the energy of the conduction band bottom Ecl= 0 is taken as reference, the energy of a particle in the left side of the barrier is:

*2*2

2,

22,

2

m

k

m

kEEE ltlz

tz

+=+= (2.49)

Chapter II: Modeling of the RTD

60

And in the right side:

rcltlz

tz Em

k

m

kEEE ,

2,

22,

2

*2*2++=+=

(2.50)

Where Ec,r is the conduction band in the right side.

The current density incident to the barriers is given by the number of particles [Ferry05]:

),(),( 1 kzktFkzkteNn −= (2.51)

with velocity *

),(1),( ,

,

,,,, m

k

k

kkEkkv lz

lz

ltlztzlzz

=

∂∂

= (2.52)

in a infinitesimal volume d2ktdkz

Where 3)2(

2),(

π=zt kkN is the density of state and F1(kt,kz) is the distribution function.

Leading to:

tzltlzzlzltlzlti dkdkkkvkkFkkeNj ),(),(),( ,,,,1,,−= (2.53)

The current density transmitted from the left to the right side is weighted by the transmission coefficient T(kz).

tzzlzltl dkdkkTkkFm

ej )(),(

*)2(

2,,13π

−= (2.54)

And the current density transmitted from right contact to the left:

tzzrzrtr dkdkkTkkFm

ej )(),(

*)2(

2,,23π

−= (2.55)

We can pass from dkz to dEz using the parabolic dispersion relationship, differentiating equation (2.49) and we obtain:

2/,/, /* zrlzrlz dEmdkk = (2.56)

The net difference between the left and right current density have to be integrated in z direction in Ez and the transversal direction on kt ,expressing in cylindrical coordinates the plan x, y perpendicular to the transport direction ( d2kt = dkxdky = ktdktdθ), the total current is given by :

)],(),()[()2(

221

003 tztzztzT kEFkEFETddkdE

eJ −=

∞∞

θπ

(2.57)

Assuming that both reservoirs are in equilibrium condition, the distribution F1,2 are the equilibrium Fermi functions:

Chapter II: Modeling of the RTD

61

Lbrl

Ftz TkEEEtzrle

EEF/)(, ,

1

1),(

−++= (2.58)

Where EF is the Fermi level in the two contacts, being eVEE rF

lF += , kb is the Boltzmann constant and

TL is the lattice temperature.

As Fermi function is isotropic, the integral on θ give 2π. Further, the integral in kt can be converted in Et for the parabolic dispersion relationship , giving:

)],(),()[()2(

*421

0033 tztzztzT EEFEEFETdEdE

emJ −=

∞∞

ππ (2.59)

The integral of the Fermi function in energy is

+

+−−

LBzlF

LBzlF

TkEeVE

TkEE

e

e/)(

/)(

1

1ln , and the final expression of

tunnel current is:

+

+=−−

−∞

LBzlF

LBzlF

TkEeVE

TkEE

zzTe

eETdE

emJ

/)(

/)(

032

1

1ln)(

2

*

π (2.60)

This formulation is sometimes called as the Tsu-Esaki formulation of tunneling current.

This formalism was originally elaborated to understand the transport in superlattice [Tsu73], as the result can be applied at any number of the barriers and became the most widely invoked assumption for RTD I-V characteristics computation.

In this paragraph we have considered tunneling through the double barriers as an elastic process. This is not longer true in the real structure where random inhomogeneities, interaction among electrons and ionized impurities, broaden the transmission coefficient resonance, reducing PVCR in experimental measurement compared with ideal coherent model [Ferry05].

Generally, the loss of phase coherence induced by inelastic scattering acts to reduce the peak current and increase the valley current [Bowe96].

In the next section, another popular method to calculate the transmission coefficient in quantum devices which consists in solving Schrodinger equation using the Green function approach, will be discussed.

Chapter II: Modeling of the RTD

62

II.2.3 GREEN FUNCTION II.2.3.1. INTRODUCTION The Green function’s operator G(E) of the Schrodinger equation can be define as [Ferry05] :

[ ] 1)( =− EGHE (2.61)

Where E is the energy and H is the Hamiltonian operator.

G(E) is subject to same boundary condition as the wave function. A solution of equation (2.61) is :

HEEG

−= 1

)( (2.62)

Which can be defined everywhere except in the singularity E=En. [Ferry05].

The Green’s function can be defined with a limiting procedure, with a retarded (r) and advanced (a) function:

r

r

HIiEEG

Σ+−+=

+→ )(

1lim)(

0 ηη (2.63)

And

a

a

HIiEEG

Σ+−−=

+→ )(

1lim)(

0 ηη (2.64)

Where η is an infinitesimal energy amount. H is the Hamiltonian. ΣL,R are the self energies arising from the interactions between contact and the leads [Datta95] (respectively , the left and the right side).

The advanced and retarded green functions can be interpreted as waves going out from the initial region in the left side and entering in right side (and in the other sense).

The transmission coefficient from Datta [Datta95] can be written as:

( )aq

rpt GGtraceET ΓΓ=)( (2.65)

where p corresponds to the left leads and q to the right one.

Where )( ap

rp p

i Σ−Σ=Γ (the same for q)

The main advantage of Green formalism is to make possible to introduce interactions such as (electron-electron and electron-phonon) by the means of self energies.

An accurate work that applies the Green’s function to predict current flowing in nanodevices was developed by Klimeck and co-workers and has been presented in this overview.

Chapter II: Modeling of the RTD

63

II.2.3.2. NEMO (KLIMECK)

The well know simulator elaborated by Klimeck at Texas Instrument, Dallas, is the NanoElectronic MOdeling tool NEMO.

NEMO is based on the non equilibrium Green’s functions that includes proper treatment of materials band structure (allowing to simulate III- V heterostructure [Lake98b]and silicon RTD [Lake98] taking into account for non parabolicity), scattering models (due to optical phonons, acoustic phonons, and interface roughness [Lake98b]) and any combination of potential.

A great merit of this work was a complete simulation of RTD relaxing the most invoked assumption in simulation of this device: Thomas-Fermi charge screening, Tsu and Esaki approach for current calculation, and mass effective approximation. In the following paragraph the main results obtained by Klimeck in [Bowe97], are presented.

1) Thomas Fermi versus Hartree screening

Commonly in the calculation of RTD potential profile, charge in the well is considered to be zero and in the emitter and collector a Thomas Fermi screening is considered. In fig II.7 the two conduction band profiles computed with Fermi and Hartree approximation are shown.

The bottom of the well in Hartree profile is above the Fermi potential because of non zero charge in the well. In addition as quantization state in the emitter that corresponds to a lower density of state, is taking into account in Hartree profile, in the emitter the energy of the Hartree band profile is lower than that of the Thomas-Fermi one.

Fig II.8 compares I-V characteristics computed and shows that Hartree approximation predicts current flowing in double barrier structure in better agreement with experimental data than Thomas Fermi approximation.

Fig II.7: Potential profile in the case of Hartree and Thomas Fermi approximation [Bowe97].

Fig II.8: I-V characteristics comparison in the case of Hartree and Thomas Fermi approximation and experimental data [Bowe97].

Chapter II: Modeling of the RTD

64

2) Tsu-Esaki Formulation:

As explained in the paragraph II.2.2, the widely applied Tsu and Esaki current density formula, is based on the following approximations:

- Effective mass

- Conservation of transverse momentum

- Use of the energy separable variables

In [Bowe97], it is demonstrated that this approximation can lead to some unphysical effect, in fact some spikes occur in I-V characteristics. Fig.II.9 provides a schematic explanation of the approximation made on dispersion relationship. The peak current occurs when the subband of the emitter is aligned with the subband of the well. In the case of identical dispersion (assumption made by Tsu and Esaki) the subbands are always aligned for all momenta at one bias. In the case of different dispersion, the alignment occur for different value of k and bias, as generally, the loss of phase coherence induced by inelastic scattering acts to reduce the peak current and increase the valley current [Bowe97]

Fig II.9: Illustration of the alignment of two subbands possessing identical dispersion relationships (left side) and different dispersion relationships (right side) [Bowen97].

Fig II.10: Comparison between numerical integration taking into account for different transverse dispersion relationship, Esaki-Tsu approximation of transverse momentum and experimental data [Bowen97].

Figure II.10 shows that when this assumption is relaxed, in the I-V characteristics, spike vanishes. Comparing with experimental data, we can notice that Tsu- Esaki integral overestimates current peak while valley current is under predicted. This is due to the fact that in the Tsu-Esaki integral scattering is not taking into account, generally, the loss of phase coherence induced by inelastic scattering acts to reduce the peak current and increase the valley current [Bowe97].

Chapter II: Modeling of the RTD

65

3) Effect of band non parabolicity:

A common approximation in simulation of quantum devices is non-parabolocity of bands. In [Bowe97] is demonstrated that this assumption leads to a not exact computation of energy levels, with an over estimations of the distance between resonant energy levels as explained by Klimeck and coworkers in [Lake97] as we can see in the figure II.11.

Some inaccuracy can occur from this assumption. The single parabolic band model predicts lower peak current than experiments, while multiband approach is in better agreement with measurement. [Bowe97] The second peak following this approximation, occur at higher voltage than in experiments and in multiband model.

In conclusion, the three most invoked assumptions to compute current flowing in RTD have the

great advantage of reduce the computational requirements. However, in order to perform simulation that predict current as accurately as possible, multiband structure model have to be taking into account, hartree charge screening have to be used to compute electrostatic potential and numerical integration on transverse moment is required. In the next paragraph other method to compute numerically current without pass for transmission coefficient calculation, are presented.

II.2.4. WIGNER FUNCTION

The Wigner Function Method (WFM) is based on solving the Wigner Function Transport Equation (WFTE). The one-dimensional WFTE was first applied to the simulation of RTDs by Frensley [Frens87]. The equation has the following formulation in a one-dimensional space:

Fig II.11: Effect of not parabolicity of conduction band in resonant levels computation.

Fig II.12 Effect of not parabolicity of conduction band in resonant levels computation in I-V characteristics [Bowen97].

Chapter II: Modeling of the RTD

66

0),',()',(2

'1),,(

*

),,(),,(

Drift termtermDiffusion nsInteractio termalTransition

=−+∂

+

∂∂−

∂∂

tkxfkkxV

dk

x

tkxf

m

k

t

tkxf

t

tkxfw

w

C

ww

π (2.66)

The WFTE is solved for the Wigner Distribution (WD) ),,( tkxf w , which is a function of the physical space x, the wave vector space k and time t. The term V(x,k) is called the non-local potential and is given by:

−−+=0

)]2

1()

2

1()[sin(2),( yxUyxUkydykxV (2.67)

where U(x) is the electric potential.

The WTFE can be seen as the quantum equivalent of the classical Boltzmann Transport Equation (BTE); in fact, they both show four distinct terms, as indicated in the formulation (2.66).

The transitional term indicates the variation of the WD in time, and is zero if stationary conditions are considered.

The collision term characterizes all interactions affecting the system, e.g. carrier-to-carrier and carrier-to-phonon scattering.

The diffusion term is proportional to the gradient of the WD across physical space, and is equal to its counterpart in the BTE.

The drift term denotes the effect of the electric potential on the carrier distribution; in the classical formulation, it is a gradient term across the wave vector space, whereas in the quantum formulation it is an integral term.

Once the WTFE solved, the charge and current are proportional respectively to the zero- and first-order moments of the WD across the wave vector space:

),,(2

1),( tkxfdktxc w=π

(2.67)

),,(*2

),( tkxkfdkm

qtxJ w=

π (2.68)

One significant advantage of the WFM method is the simple formulation of the WFTE, and the fact that the WD can be easily related to measurable physical quantities, such as the electric potential and charge density.

Also, the WFTE easily accounts for scattering phenomena, and allows to carry out time-dependent simulations.

On the other hand, one drawback is that solving the WFTE numerically requires important computational resources, especially in terms of system memory.

This is mostly due to the integral nature of the drift term: for example, in a one-dimensional device, such as a RTD, the drift term operator is discretized as a block matrix where the number of non-zero coefficients is proportional to the square of the number of nodes in the wave vector space.

Chapter II: Modeling of the RTD

67

As a comparison, the same term is discretized as a bi-diagonal matrix in the BTE, and the number of nodes is only directly proportional to the wave vector node count [Biegel97].

II.2.5. BOHM TRAJECTORY

Here, we briefly explain the time-dependent many-particle quantum computation technique. In particular, for the two simulations of mesoscopic devices [Oriols07], it provides simple expressions to compute DC, AC and transient currents and its fluctuations and it furnishes a simple algorithm to solve the quantum many-body electron problem [Oriols07]. Based on these two important attributes, the research team headed by Dr. Oriols is developing a powerful simulator for the quantum electron transport (BITLLES). The numerical algorithm is implemented via a QMC method in order to account for uncertainties in energies, initial positions of (Bohm’s) trajectories, etc. In this section we give a very brief introduction to the “transport via trajectories” in order to point out the basic ideas characterizing BITLLES.

Solving the time-dependent Schrödinger equation for an N-body system means to solve the partial differential equation in N+1 variables:

2

211 1

1

( ,.., , )( ,.., , ) ( ,.., , ),

2 k

NN

r N Nk

r r ti U r r t r r t

t m=

∂Φ = − ∇ + Φ ∂

(2.69)

where 1( ,.., , )Nr r tΦ is the many-particle wave function and jr

is the position of the electron j. The

potential energy 1( ,.., , )NU r r t

can be written as:

11 1 1 1

1( ,.., , ) · ( , ) · ( , ) · ( , )

2j i j k

N N M N

N i j i j i ji j j N j

U r r t qV r r qV r r qV r r

≠ ≠= = = + =

= + −

(2.70)

where the term with ½ is included to avoid self-interaction [Albareda09] and q is the electron charge

(with sign) and ( , )i jV r r

is:

( , )4

i j

i j

qV r r

r rπε=

, (2.71)

However the direct numerical solution of (2.69) is inaccessible already for very few electrons. Actually, the many-body 1( ,.., , )Nr r tΦ

and 1( ,.., , )NU r r t

is defined on the tensor product of N Hilbert’s

spaces for each time instant. This means that if M grid points are used for the discrete version of the physical space, then the discrete version of 1( ,.., , )Nr r tΦ

and 1( ,.., , )NU r r t

has to be evaluated in a

space endowed of MN points, i.e. a hard exponential algorithm is required.

An original approach to overcome this problem is reported in [Oriols07]. A conversion from the hard exponential problem MN to a polynomial problem NM is proved. Here we give a sketch of proof in order to point out the principal characteristics of the algorithm and its approximations.

Let us the write the wavefunction in a polar form 1 1 1( ,.., , ) ( ,.., , ) exp[ ( ,.., , ) / ]N N Nr r t R r r t iS r r tΦ = .

Chapter II: Modeling of the RTD

68

Since the Bohmian formalism [Bohm52]the gradient

1

( , ) ( , )k kr rv x t S x t

m= ∇r r

r r r (2.72)

can be interpreted as the velocity of the Bohm trajectory

0

0[ ] [ ] ( , ) .k

t

k k rtr t r t v x dτ τ= + rr r r r

(2.73)

The theorem proved in [Oriols07] states that any trajectory [ ]kr tr

that belongs to a particular set of N

Bohm trajectories 1[ ] [ ],..., [ ]Nx t r t r t=r r r

associated with a many-body wave function ( , )x tΦ solution

of eq. (2.69) with ( , )U x t

rewritten as

),(),,(),( txUtxrUtxU kxkkk k

+= (2.74)

can be obtained from a single-particle wave function ( , )k kr tΨ . This wave function is a solution of

the following single-particle Schrödinger equation:

2

2( , )( , [ ], ) ( , [ ], ) ( , [ ], ) ( , ),

2 k

k kr k k k k k k k k k k k

r ti U r x t t G r x t t iJ r x t t r t

t m

∂Ψ = − ∇ + + + Ψ ∂

(2.75)

with 1 1 1[ ] [ ],..., [ ], [ ],..., [ ]k k k Nx t r t r t r t r t− +=r r r r r

and kG and kJ two quantum potential (for the full

definition see [Oriols07]).

In particular, if we define ( , [ ], ) ( , [ ], ) ( [ ])k

k k k k k kRU r R t t U r R t t U R t= +

, then we realize that

( , [ ], )a aU r R t t

can be greatly simplified [Albareda09]. In particular, we can define ( , [ ], )k k kU r R t t

as a

solution of the following Poisson equation:

( ) ( )1

( )· ( , [ ], ) [ ]k k

j k

N

r k r k k k k jj

r U r R t t q r r tε δ≠=

∇ ∇ = −

, (2.76)

plus the appropriate (Neumann or Dirichlet) boundary conditions on the limits of the simulation box.

The rest of the terms ( [ ], [ ])j iV r t r t

of expression (2.76) appear in ( [ ])k

kRU R t

and they are included in

the potential ( , [ ], )k k kG r R t t

. However, this term ( [ ])k

kRU R t

has no role on the single-particle wave-

function ( , )k kr tΨ because it has no dependence on kr

and it only introduces an irrelevant global

phase on ( , )k kr tΨ .

There are two main problems when one tries to apply (2.75) in practical scenarios. First, all the terms

( , [ ], )k k kU r R t t

, ( , [ ], )k k kG r R t t

and ( , [ ], )k k kJ r R t t

depend on the rest of Bohmian trajectories [ ]jr tr

with 1,... 1, 1,..j k k N= − + . Therefore, all trajectories [ ]jr tr

have to be simultaneously computed. This

implies solving a system of N single-particle Schrödinger equations (2.75) coupled by N single-particle Poisson equations (2.76). As expected, for M grid point in the real space, the algorithm grows as polynomial problem MN. The second problem is much more complicated. The new

quantum potentials ( , [ ], )k k kG r R t t

and ( , [ ], )k k kJ r R t t

depend, in fact, on 1( ,.., , )Nr r tΦ that we cannot

compute. The problem can be overcame by using approximate algorithms similar to the ones

reported in [Oriols07] in order to provide reasonable estimators of ( , [ ], )k k kG r R t t

and ( , [ ], )k k kJ r R t t

without explicitly solving eq.(2.76). This algorithms have similarities with the original work of Kohn and Sham on the Density Functional Theory (DFT): the formidable simplification on the many-particle computations comes at the price that some terms of the potential energy of the corresponding

Chapter II: Modeling of the RTD

69

single-particle Schrödinger equations are unknown [the exchange correlation functional in the DFT

and, here, the terms ( , [ ], )k k kG r R t t

and ( , [ ], )k k kJ r R t t

] and must be approximated.

The important point is that ( , [ ], )k k kU r R t t

can be computed exactly, from the solution of the time-

dependent Poisson equation (2.76), providing a full self-consistent solution, where all electron-electron (Coulomb) correlations are included. See Ref.[Albareda09] for all technical details. Then, this many-particle Bohmian formalism can be used to compute all ensemble values associated to

1( ,.., , )Nr r tΦ without knowing 1( ,.., , )Nr r tΦ

. Finally, to properly reproduce the statistics implicit in

the many-particle Schrödinger equation the procedure of selecting N initial points 0[ ]kr tr

is repeated

many (ideally infinite) times, until reasonable averages are obtained.

In particular, let us notice that the (quantum average value) of the time-dependent electron current is computed in the volume Ω 2 by directly applying a quantum version of the Ramo-Shockley theorem discussed in Ref.[Alarcon09]. It provides numerical advantages over the direct computation of the time-dependent total (particle plus displacement) current in the particular surface ,L RA [Alarcon09].

II.2.6. NUMERICAL APPROACHES FOR RTD SIMULATIONS: CONCLUSION

The transmission coefficient of a double barrier structure can be computed as the ratio between the incident and the transmitted flux through the barriers. This quantity depends on barriers characteristics as mass, height, thickness but also on distance between the barriers.

The easiest method to compute transmission coefficient is the transfer matrix method that can be implemented numerically or analytically (in some particular case, including a double asymmetrical barriers system).

However, it suffers from several approximation (effective mass, ballisticity… ) which make it less accurate than other sophisticated approaches.

TMM combined with the common approximations made in current calculation is the most applied method and is very efficient from a numerical point of view.

Among methods applied to obtain transmission coefficient, Green’s function is widely applied and allow to introduce scattering and to perform transient simulations.

Other numerical approach are based on Wigner function or Bohm trajectories simulation (that consist in monte carlo approach), where many body effects can be included.

The main characteristics of the numerical methods presented in this section have been summarized in the table 1.

Chapter II: Modeling of the RTD

70

Characteristics TMM GF WF BT

Scattering NO YES YES YES

Transient sim. NO YES YES YES

Full band NO YES YES YES

Many Body NO NO NO YES

Complexity - ++ +++ +++

Table 1: Summary of the main numerical approaches.

Numerical simulations give a rigorous picture of physical behavior of devices, but one has to deal with code complexity, very long simulations time and high requirement in terms of stored memory.

For engineering purposes, an analytical compact model is preferable, as it allows to catch the main physical mechanism come into play in transport through the device and can be useful tool for circuit design.

In the next, the compact modeling of RTD is discussed.

II.3. ANALYTICAL MODELING OF RTD

II.3.1. WHY DO WE NEED COMPACT MODELS?

A compact model is a mathematic representation of the electrical behavior of a device. In integrated circuit, where several devices are connected together, it is necessary to predict the operation of single component.

Compact model are commonly used for two purposes: electrical characterization and for circuit simulations. It exists at least three types of compact model: [Woltjer06].

1) Table based models, where data is given by characterizations or device more complex simulations

2) Empirical fit

3) Physics-based compact models

Accurate results are the main advantage of full numerical simulations, while complexity is their drawback. On the other hand a compact model is most useful for engineering purposes, as it allows a faster and simpler parameters extraction and effective for circuit simulations.

Chapter II: Modeling of the RTD

71

From [Woltjer06], most models have to follow five main requirements: a) easy parameters extraction, b) wide range of input parameters, c) taking into account for the main characteristics, d) being suitable for several analysis and circuit simulations, e) good convergence in circuit simulators.

The number of parameters has to be small but sufficient to represent the main physics of device. The input model parameters can be various and depends on the device investigated behavior aspect and on application.

Normally a compact model has a range of given validity, but by using additional equations, one is able to extend the range of validity. In addition, DC and RF model have to be consistent.

It’s important that the RF model is an extension of DC model by introducing some further parameters. Finally for circuit purposes a good convergence is needed. Thus, collaboration between compact model and circuit simulations development are suitable [Woltjer06].

In the next section, the most important previous analytical models that can be found in literature have been presented. Then, the original analytical model, goal of this work, has been detailed.

II.3.2. PREVIOUS ANALYTICAL MODEL OF RTD

Several RTDs models have been developed to simulate circuit behavior and to evaluate the performances for various applications.

In particular, physics based analytical model can be useful also to select materials and suitable device design.

In order to introduce an analytical model in a circuit simulator, continuous functions, such as polynomial, trigonometric and exponentials, are needed, to model I-V characteristics.

The models found in the literature contain many fitting parameters and often are not physics based, but only reproduce a characteristic using a behavior model. Their accuracy depends on measurement fitted and sometimes fitting is not very good in whole the I-V characteristic.

Let us to list the main examples of analytical model found in the literature, developed for circuit purposes.

II.3.1.1 MODEL FOR PSPICE OF YAN ET AL. [YAN95] Yan et al, in this work, divided the RTD I-V characteristic in three parts: the positive differential

region (PDR), from 0V to the peak voltage, negative resistance region (NDR) from peak voltage to the valley voltage and then diode like exponential behavior. This model is totally empirical, not physics based, it is based on observation of most measured I-V characteristics.

Total current is given by the sum of two contributions, one is attributed to the tunnel and the second thermionic is model as diode like current:

)()()( VIVIVI DTRTD += (2.77)

Where ID is the diode like component:

Chapter II: Modeling of the RTD

72

−= 1)( qkTN

V

sD eIVI (2.78)

where Is is the emission current.

To model IT, Gaussian function or exponential function can be chosen for PDR and NDR regions. Thus, the total RTD current is the sum of positive (ITp) and negative(ITn) region contribution:

)()()( VIVIVI TnTpT += (2.79)

Fig.13: Schematic I-V curve for both Gaussian model and Exponential model for the RTD.

With:

( ) ( )[ ]

−− −

⋅⋅=pVVM

pp

pe

V

VVV

pTp eeII1

2 2

2

σ (2.80)

( ) ( )[ ]

−−

−⋅=

VpVM

pn

p

n

eV

V

Tp

VV

pGT eVIeII1

2)(

2

2

σ (2.81) Gaussian function

( ) ( )[ ]

−−

−⋅=

VpVM

pn

p

n

eV

V

Tp

VV

pET eVIeIVI1

2)()(

σ (2.82) Exponential function

Choice of Gaussian or exponential function depends on better fitting with measurement. The multiplier factor M should to be chosen very large (<10000) to make each step like function terms ideal.

Chapter II: Modeling of the RTD

73

The other two models that will be presented in this section, present more physical basis as in both cases, the current formula arises from a Lorentzian approximation of transparency.

II.3.1.2 BROWN ET AL. MODEL [CHANG93] This group has developed a RTD’s model to be introduced in SPICE to simulate integrated circuit.

( ) ( )[ ] mm VcVccVccVcVcfI ⋅+⋅++⋅−+⋅⋅⋅= −− 6542tan32tan1 11 (2.83)

This model basically is formed by fitting parameters c1…c6. The first 4 are determinate in function of peak voltage and peak current. The last two are determinate by valley voltage and current. Parameter f is a scale factor used to take in to account for the RTD area. The exponential m and n have to be chosen to have a good fit of peak and valley current.

II.3.1.3 SCHULMAN ET AL. MODEL [SCHULMAN96] Schulman et al., derived a simple expression directly from Tsu and Esaki formulation, for current

in RTD. Beginning from Tsu and Esaki formulation of current tunnel :

−−

++⋅=

0/)(

/)(

32

*

1

1ln),(

2dE

e

eVET

kTmeJ

kTeVEE

kTEE

vF

F

π (2.84)

T is the temperature, Tv is the transmission and EF is the Fermi level.

The transparency has been approximated by a Lorentzian function (see par II.2.1.1):

22

2

22

2),(

Γ+

−−

Γ

=eV

EE

VET

r

(2.85)

Where Γ is the resonance width and Er is the resonant energy level, assumed constant.

If one considers that transparency is very sharp (small Γ) in the resonant energy levels, putting in the integral this Lorentzian approximation for transparency and it is possible computing integral in

2/eVEE r −= . In this condition, the logarithm term is constant and can be extracted out of the integral and it rests only the lorentzian term to be integrated in energy.

Thus, current flowing in the RTD has the following formulation:

Γ

−+

++Γ= −

−−

+−

2

2tan21

1ln

4

* 1/)2/(

/)2/(

32

eVE

e

ekTemJ

r

kTeVEE

kTeVEE

rF

rF ππ

(2.86)

Chapter II: Modeling of the RTD

74

This formulation has been generalized then in a simple parametric form that reproduces a peak current and negative resistance. The parameters A, B, C, n1 are given by:

324

*

πΓ= kTem

A , FEB = , REC = , 2/1 en = ,2

Γ=D (2.87)

The A, B, C, D, n1 are used as simple fitting parameters.

( )( )

⋅−+⋅

++⋅= −

⋅−−

⋅+−

D

VnC

e

eAJ

kTqVnCB

kTqVnCB

RTD1

tan21

1ln 1.

/1

/1 π (2.88)

Fig II.14: Fit of Schulman model and measurement for a III-V RTD.

Like in other model, the valley current is included in the model, using a diode like model.

II.3.3. ANALYTICAL MODELS: CONCLUSION

An analytical model is useful tool for device and circuit design. In order to be introduced in a circuit simulator, a model has to be compact enough, composed by continuous functions, and to ensure good convergence.

To guarantee an accurate modeling of devices and circuit, a large number of input parameters is required. The main analytical models of RTD found in the literature have been reviewed.

Even if they present a compact parametric structure suitable for circuit purposes, are composed by several fit parameter and not totally physics based and their accuracy on reproducing the I-V characteristics depends on measurement fit. These models are thus not predictive.

The model proposed in this thesis is physics based and allows to compute I-V characteristics in function of physical parameters and geometrical dimensions without any fitting parameter, maintaining a compact form that provide its easy introduction in circuit simulator.

Chapter II: Modeling of the RTD

75

II.4. OUR ANALYTICAL DC MODEL OF RTD

Assumptions:

Following Tsu - Esaki formulation [Tsu73], the current density flowing through the RTD structure has been assumed equal to:

−−+−+

⋅=0

32

*

/)exp[(1

]/)exp[(1ln),(

2)( dE

keVEE

kEEVET

kemVJ

f

f

θθ

πθ

(2.89)

where m* is the electron mass, θ is the temperature, T is the transmission coefficient and Ef is the Fermi level in the contact. In our work, we propose to solve this integral and to compute transparency T analytically. The following assumptions have been made:

1) Transport is supposed coherent (ballistic)

2) Band structure is modeled by effective mass approximation

3) Only electron tunnel current is considered (in the framework of the single particle approximation)

4) calculation have been made non-self consistent (the voltage drop along the structure is supposed to be linear, accounting for different dielectric constants of each material, as it will be explained in the next paragraph).

Despite the limitation of our approach, consequences of the simplifying approximations used, we believe that our model will allow a back envelop estimation of RTD performances versus materials, technologies and structures.

In addition, the approximation of coherent transport and EMA leads to optimistic prediction of performances, leading to an estimation of the best performances achievable.

In consequence if the performances achieved using RTD according to our approach are not convincing, we are sure that experiments would be even worse.

II.4.1. POTENTIAL PROFILE

In order to develop a fully analytical model, we considered a non self consistent calculation, thus, the voltage drop along the structure is supposed to be linear, accounting for different dielectric constants of each material. In this way the computed potential presents triangular barrier for bias V≠0 :

Chapter II: Modeling of the RTD

76

Fig II.15: Triangular potential energy profile along the RTD structure.

The total structure length is:

L = t1+tsi+t2

The total potential drop along the double barrier is:

V= V1+ Vm + V2

At the region boundaries, leads to the continuity of the electric displacement field:

ε1E1= ε2E2= εmEm

In term of potential drop this formulation becomes:

m

mm

t

V

t

V

t

V εεε==

222

1

11

That can be written as:

C1V1= C2V2= CmVm

And the total voltage becomes:

mm

mmm V

C

CVV

C

CV

21

++=

We can write each voltage drop as a function of the total applied voltage:

Vm= αV

So:

VCCCCCC

CCV

mmm

1212

21

++=

Following a similar approach for V1 and V2:

Chapter II: Modeling of the RTD

77

12

11

11 V

C

CV

C

CVV

m

++=

VCCCCCC

CCV

mm

m

1212

21 ++=

VCCCCCC

CCV

mm

m

1212

12 ++=

This linear potential profile does not take into account for flat band voltage. We assumed that at V=0, there is no built in voltage drop in the structure, which may not be the case if the well is charged in equilibrium condition, or if the left and right contacts have different wave function.

II.4.2. RESONANT LEVELS IN THE WELL

The energy eigenvalues can be given numerically by the solution of the transcendental equation:

Izz

IIz mkLk

mk /2

tan)/( =

(2.90)

This equation arises from applying the matching conditions for wavefunction and its derivative at the boundary (-L/2 and L/2) for the system in the figure:

>

<<−−<

=

2/

2/2/)cos(

2/

)(

LzforBe

LzLforzkA

LzforBe

zzK

z

zK

n

z

z

ϕ (2.91)

Figure II.16: Eigen values.

Chapter II: Modeling of the RTD

78

The resonant energy levels can be also computed numerically directly as eigenvalues of the Schrodinger equations.

However, as the numerical computation of eigen values of Schrodinger equation is very easy and does not take long computation time, we choose the last method to compute the energy levels.

II.4.3. TRANSMISSION COEFFICIENT CALCULATION

Around each resonant peak En, for symmetric double squared barriers, the transmission coefficient can be approximated by a Lorentzian function [Ferry05], leading to:

22

2

)(4/

4/)(

nn

n

EEET

−+ΓΓ= (2.92)

Where 2

Rn

Ln

nΓ+Γ=Γ . In the simplest case of symmetric and equal barrier R

nLn Γ=Γ

In the more general case of asymmetric double barriers (for applied voltage V ≠ 0 ), with different barriers ( R

nLn Γ≠Γ ) , transmission coefficient can be written:

( ) 22 )(16/

4/)(

nRL

RLasi

EEET

−+Γ+ΓΓΓ= (2.93)

Leading the following formulation:

Δ−+

=2

max

)(21

),(

E

EE

TVET

n

(2.94)

2max)(

4

RL

RLTΓ+ΓΓΓ= (2.95)

2RLEΓ+Γ=Δ (2.96)

2/1

2,

22,

*)),(1(

),(2

−⋅

=ΓbmVET

EVET

nRL

nnRLn

(2.97)

Where ΔE is the full width at half maximum of the resonance associated with En, b the well width and TL,R the left and the right transparency barrier are computed as explained in the previous paragraph for asymmetric simple rectangular barrier.

Chapter II: Modeling of the RTD

79

As the Lorentzian transparency approximation is valid only in the case of squared barrier, to extend these approximation when the device is biased (V ≠ 0), the real potential profile along the double structure has been approximated by asymmetrical square double barriers (as illustrated in Fig. II.17). The effective barrier heights become voltage dependent, and are given by:

2/)()( 0 VVVeff −Φ=Φ (2.98)

where V0 is the corresponding barrier voltage drop.

Figure II.17: Triangular potential energy profile along the RTD structure (solid line), and the corresponding approximated rectangular potential, used in this work (dot line).

In order to validate the approximation of triangular barriers by squared barriers, fig II.18 shows the comparison transmission coefficient computed with our model and with a rigorous calculation using Airy matrix, exact solution for barriers of any triangular shape.

As shown in fig.II.18, the two models are in good agreement and this comparison allows to validate the Lorentzian approximation of transmission coefficient and the approximation of triangular barriers in squared barriers.

Chapter II: Modeling of the RTD

80

0 0.1 0.2 0.3 0.4 0.5 0.6 0.710-7

10-6

10-5

10-4

10-3

0.01

0.1

1

HfO2/Si (ml)

tw= 3 nm, tb = 0.5 nm

Energy (eV)

Tra

nspa

renc

y

0 0.1 0.2 0.3 0.4 0.5 0.6 0.710-7

10-6

10-5

10-4

10-3

0.01

0.1

1

HfO2/Si (ml)

tw= 3 nm, tb = 0.5 nm

Energy (eV)

Tra

nspa

renc

y

Figure II.18: Transparency of the HfO2/Si RTD (only longitudinal masse is take in to account) device (Symbol= exact solution, line= improved Lorentzian model - this work).

Contrary to previous analytical models [Schul96], the dependency of both the maximum of transparency Tmax and the resonant width ΔE with energy and applied voltage V has been taken into account, derived as a function of the left and right tunnel barriers.

The difference between the evolutions of transmission coefficient with voltage in two cases is shown in fig II.19 (ΔE, Tmax, constant with voltage) and fig II.20 (ΔE, Tmax function of voltage).

Figure II.19: Transparency computed with previous model, Tmax and ΔE function of energy but constant with voltage.

Figure II.20: Transparency computed with our model, Tmax and ΔE function of energy and voltage.

Chapter II: Modeling of the RTD

81

At the resonance, the resonant energy level pass below the bottom of the conduction band that represent the reference E=0, consequently the transmission coefficient tends to zero as shown in the fig II.20. Conversely, In the Lorentzian model transparency is always constant.

Thus, it is evident that the behavior described in fig II.20 is more physics than that considered in the previous model of fig II.19.

II.4.4. CURRENT CALCULATION

In each resonant energy level En , the transmission coefficient T(E,V) results very peaked, so we

can approximate it as a delta function.

2)(2

1

112)(

Δ−+

⋅Δ⋅=−

E

EEEEE

n

n πδ (2.99)

Replacing in

Δ−+

=2

max

)(21

),(

E

EE

TVET

n

we obtain:

)(2

)( max ETEET δπ ⋅⋅Δ= (2.100)

The integral of Tsu-Esaki equation can be solved in E = En and the I-V characteristics of RTD device have been computed according the following formula:

[ ][ ] ),()(

2/)((exp1

/))((exp1ln

*4)( max3

VETEEkeVVEE

kVEE

h

kmqVI nn

nf

nf ⋅Δ⋅

−−+−+⋅= π

θθθ (2.101)

Then, the voltage dependence can be put in the value of the quasi bound state energy as VEE nn α−= 0 , being α the amount of total voltage in the well taking in to account for different

dielectric constants of each material.

Chapter II: Modeling of the RTD

82

-2,0n 0,0 2,0n 4,0n 6,0n 8,0n 10,0n 12,0n 14,0n 16,0n

-0,3

-0,2

-0,1

0,0

0,1

0,2

0,3

0,4

0,5

0,6

Po

ten

tial

en

erg

y (e

V)

nm

EpcV=0 EpcV=0.25 EL1V=0 EL1V=0.25

En

En0

-αVREF

Vm=αV

Figure II.21: Potential profile and energy levels.

The basic principle modeled is shown schematically in the fig II.22.

En is the energy level referenced to the bottom of the well. When a voltage bias V is applied, the corresponding voltage drop in the well is VVm α= and the energy level goes down of the same amount: VEE nn α−= 0 .

Current is turn on in α

Fn EEV

−= 0 , it means when the amount of potential in the well VVm α= is

enough to pull down En below EF.

Current increases linearly until En goes down the energy reference in the bottom of conduction band

in emitter, and so 00 =−= VEE nn α , peak current occur in α

0nEV = , then current suddenly (sharply)

current cut off providing the NDR.

Chapter II: Modeling of the RTD

83

Figure II.22: Qualitative explanation of resonant effect. Figure II.23: Qualitative J-V characteristic of RTD.

This result seems similar to the formula obtained by Schulman et al. [Schul96], except that in this work, the terms ΔE, Tmax and En(V) are computed according to the previously described model. Fig II.24 shows the two I-V characteristics predicted by Schulman model and by this work. We can notice that the I-V characteristic obtained with our model presents smooth shape, instead the Schulman model provides a sharp negative resistance due to the behavior of the transparency described in fig II.19.

0,0 0,2 0,40,00

1,50x1010

3,00x1010

4,50x1010

Cu

rre

nt

De

ns

ity

(A/m

2 )

Voltage (V)

Schulman this work

Figure II.24: J-V characteristic of GaAs RTD, red curve this work, dash black line Schulman model.

Moreover, the dependency of transparency with energy and voltage, explained in the previous paragraph, allows our model to match perfectly with the exact solution, where T is calculated by Airy functions and where the integral of Eq. 2.89 is solved numerically, as it is shown in fig II.25.

This point has been found to be particularly important to accurately model the I-V characteristic, as Tmax cannot be assumed constant and tends to zero (as shown in the fig II.21) at each resonance, when the resonant level goes down the reference level E = 0 (Fig. II.26).

Chapter II: Modeling of the RTD

84

0 0.2 0.4 0.6 0.8 1 1.2109

1010

1011

1012

A

B

C

Voltage V − VFB (V)

Cur

rent

De

nsity

(A

/m2) tw= 3 nm, tb = 0.5 nm

HfO2/Si (ml)

0 0.2 0.4 0.6 0.8 1 1.2109

1010

1011

1012

A

B

C

Voltage V − VFB (V)

Cur

rent

De

nsity

(A

/m2) tw= 3 nm, tb = 0.5 nm

HfO2/Si (ml)

Figure II.25: J-V characteristic of the HfO2/Si RTD device, considering only ml resonant levels (A= Lorentzian model with Tmax and ΔE constant, B = improved Lorentzian model (this work), C= exact solution).

Figure II.26: Schematic energy band diagram of the RTD structure.

The results presented in this section take into account only for one effective mass for silicon. In order to predict more accurately current, one has to deal with the multi valley band structure of silicon.

The extension of model to the multi valley semiconductors is explained in the next paragraph.

II.4.5. EXTENSION OF THE MODEL TO THE MULTI-VALLEY SEMICONDUCTOR:

The aim of this work is estimating the performances of silicon based RTD and comparing with other materials used to realize RTDs.

Silicon has an indirect bandgap, it means that the minima of conduction band is not aligned with the valence band maxima (as in fig.). Minima of conduction band is in the direction (001) in (00k0) with k0= 0.85. As the silicon has a cubic structure, 6 equivalent directions at (001) exist:

)100(),100(),010(),010(),001(),001( . Thus silicon presents 6 minima for conduction band and consequently 6 valleys. Energy variation E(k) in function of wave vector is not isotropic. And the constant energy surfaces are ellipsoidal (as shown in fig.II.28). Thus in silicon there are two mass for electrons:

- Effective longitudinal mass *lm for electrons in the revolution axe of the ellipsoid. This mass

is 0.92 m0 or 0.98 m0 (it depends on the source [Moglestue86], [Sze81].

- Effective transversal mass *tm for electron in perpendicular at the revolution axe. This mass is

in the range 0.19 m0/0.21 m0[Sze81]

Chapter II: Modeling of the RTD

85

Figure II.27: Silicon conduction band [Sze81].

Figure II.28: Constant energy surface for SiGe (8 semi ellipsoid in the plane (111), Si (6 ellipsoid in direction (100), and GaAs isotropic [Sze81].

In order to extend our investigation rigorously at silicon and other non isotropic materials (see fig.II.28), the model obtained for one single valley, has been extended for multi valley semiconductors, in order to model correctly silicon devices taking into account for the 6 valleys of the silicon conduction band.

Energy levels computation has to take into account for both longitudinal and transversal mass.

As the energy levels are computed from the eigen values of Schrodinger equation for the two mass, two series of resonant levels exist in the well, EL corresponding to the longitudinal mass and ET corresponding to the transversal mass.

As effective longitudinal mass is lower than transversal one, more longitudinal levels is available for resonance in the well.

Chapter II: Modeling of the RTD

86

0 2 4 6 8 100,0

0,4

0,8

1,2

1,6

2,0

2,4

2,8

En

erg

y (

eV)

n

longitudinal trasversal

φox

Figure II.29: Energy resonant levels corresponding at longitudinal and transversal mass in 4.5nm silicon well.

Figure II.30: Transmission coefficient corresponding at longitudinal and transversal resonant levels in silicon well.

The total current is composed of two contributions as shown in fig.II.31. Finally, the formulation of current is:

),()(2/))(exp[(1

/))(exp[(1ln

4)( max3

VETEEkTeVVEE

kTVEE

h

kTmqVI v

nvn

vvalleynpeak

resonantvnf

vnf

vd ⋅Δ⋅⋅

−−+

−+⋅⋅⋅= ππ (2.102)

where mdv is the 2D density of states mass (2mt for transversal mass and ltmm4 for longitudinal

mass), for each valley and Tv is the transmission coefficient for each valley v.

Electrons located in 4 of 6 valley move with mass mt (transverse) and 2 of the 6 valley move with a mass ml (longitudinal).

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6103

104

105

106

107

108

109

Cu

rren

t D

ensi

ty (

A/m

2 )

Voltage-Vfb(V)

Longitudinal peaks Transversal peaks Tot current

Figure II.31: Current density corresponding to the longitudinal and transversal resonant levels ( solid red line = longitudinal levels contributions, blue dashed= transversal contribution, black dash-dot=total current).

Chapter II: Modeling of the RTD

87

II.4.6. ANALYTICAL MODEL VALIDATION

Results obtained using the analytical model have been compared with numerical self-consistent ballistic simulations. These simulations have been achieved solving the Poisson and Schrodinger coupled equations, using the transfer matrix approach. The program used for these simulation is QUANTIX, code elaborated in INL laboratory by Alain Poncet [Poncet02]. In the approach used in QUANTIX, only coherent tunneling is considered and the following approximations are made:

1) Only electron tunneling has been considered 2) Mass effective approximation 3) No scattering is included (interface roughness is neglected)

Two kinds of structures have been simulated: AlAs/GaAs/AlAs and HfO2/Si/HfO2. The physic and geometrical parameters of simulated structures can be found in table I and II. Fig II.32 shows the comparison between two AlAs/GaAs structures, one with thin barrier (1.7 nm) and one with more thick barriers (4.2nm) and as we can see in the two cases, the two models are in good agreement. Note that the analytical model has not been calibrated to reproduce simulations. These results validate some of the approximations of our approach and in particular the absence of coupling with Poisson.

0,0 0,1 0,2 0,3 0,4

107

108

109

1010

b= 4.2nm; well =5nm

Cu

rren

t D

en

sity

(A

/m2)

Voltage V - VFB

(V)

numerical analytical

b= 1.7nm; well =4.5nm

GaAs/AlAs

0,0 0,2 0,4

107

108

109

1010

Cur

rent

Den

sity

(A/m

2)

Voltage V - VFB

(V)

numerical analytical

Si/Hf02

b=1nm; well= 2nm

Figure II.32: Comparison between full quantum self consistent numerical simulation (symbol), and this analytical model (solid line) for two different GaAs/AlAs RTD structures.

Figure II.33: Comparison between full quantum self consistent numerical simulation (symbol), and this analytical model (solid line) for Si/HfO2 RTD structures.

Chapter II: Modeling of the RTD

88

TABLE I

Physics Parameters GaAs (well) AlAs (barriers) m* 0.063 m0 0.123 m0 ε 12.9 ε0 10.06 ε 0 φox 0.3 eV EF In the contact : 0.15 eV

Geometrical dimensions I structure II structure

Well = 4.5 nm Barriers= 1.7nm [North98]

Well= 5 nm Barriers= 4.2 nm

TABLE II

Physics Parameters Si (well) Hf/O2 (barriers) [Coign09] m* ml = 0.92 m0

mt = 0.19 m0 0.2 m0

ε 12.9 ε0 21 ε 0 φox 1.8 eV EF In the contact : 0.10 eV

Geometrical dimensions Well = 2 nm

Barriers= 1nm The analytical model has been also compared with numerical simulations based on Bohm Trajectories. Simulation code elaborated in UAB by Dr. Xavier Oriols.

0,0 0,1 0,2 0,3 0,4

1,0x10-5

2,0x10-5

3,0x10-5

4,0x10-5

5,0x10-5

6,0x10-5

7,0x10-5

8,0x10-5

Cu

rren

t (A

)

Voltage (V)

analytical model not self-consistent self consistent

Figure II.34: Comparison between full quantum self consistent numerical simulation, not self consistent (red symbol), self consistent (blue dash line) and this analytical model (solid black line) for GaAs/AlGaAs RTD structures (well 5.7 nm, barriers 1.2 nm).

Chapter II: Modeling of the RTD

89

As we can see in the fig II.34, the analytical model slightly overestimates the peak current compared to the numerical model, maintaining correct voltage peak. The second peak is well predicted by analytical approach as well. However valley current is underestimated and the negative resistance is sharper.

II.5. DC ANALYTICAL MODEL: CONCLUSION

An original analytical physics-based model for coherent tunneling in Resonant Tunneling Diode has been developed and allow to study RTD in function of materials parameters (effective mass, barriers height, dielectric constants, Fermi levels) and geometrical dimensions (well width, barriers thickness).

Compared to previous analytical models, the dependency of both the maximum of transparency Tmax and the resonant width EΔ with energy and applied voltage V has been taken into account, derived as a function of the left and right tunnel barriers, allowing in general good agreement with numerical simulations without any fitting parameters.

The only weak point of this approach is the modeling of the negative resistance. Indeed, this parameter is affected by self consistency, scattering, and non parabolicity effect, which are not considered.

One interest of this model is the possibility to compare materials and structures at the same basis.

The other advantage is its simplicity that allows introducing it in a circuit simulator in order to investigate the possible circuit applications.

REFERENCES OF CHAPTER II

[Ando87] Y.Ando, T.Itoh “Calculation of transmission current across arbitrary potential barriers” J. Appl. Phys. 61 (4) 1987.

[Alarcon09] A. Alarcón, X. Oriols, “Computation of quantum electron transport with local current conservation using quantum trajectories,” Journal of Statistical Mechanics: Theory and experiment, P01051 (2009)

[Biegel97] Dissertation about “Quantum Electronic Device Simulation” Stanford University” 1997.

[Bohm52] D. Bohm, “A suggested interpretation of the quantum theory in terms of "hidden" variables” Physical Review, vol. 85, no. 2, pp. 166, Jan 1952.

[Bowe97] R. Chris Bowen, Gerhard Klimeck, and Roger K. Lake William R. Frensley Ted Moise “Quantitative simulation of a resonant tunneling diode” J. Appl. Phys. 81 (7), 1 April 1997 page. 3207-3212.

Chapter II: Modeling of the RTD

90

[Chang93] Charles E,. Chang, Peter M. Asbeck, Keh-Chung Wang, Elliott R. Brown, “Analysis of Heterojunction Bipolar Transistor/Resonant Tunneling Diode Logic for Low-Power and High-speed Digital Applications” IEEE Transactions on Electron Devices. Vol. 40. No. 4. April 1993, page 685-690.

[Datta95] S.Datta “Quantum transport : Atom to transistor” . Cambridge University Press 2005.

[Ferry05] D.K. Ferry, S.M. Goodnick “Transport in nanostructures” Cambridge studies in semiconductors physics and microelectronics engineering 2005.

[Frens87] W.R. Frensley “Wigner function model of a resonant tunneling semiconductor device” Phy. Rev. B vol. 36 n°3 page 1570 – 1580.

[Ghata88] Ajoy K. Ghatak, K. Thyagarajan, And M. R. Shenoy “A Novel Numerical Technique for Solving the One-Dimensional Schroedinger Equation Using MatrixApproach-Application to QuantumWell Structures” IEEE Journal Of Quantum Electronics, Vol 24, No 8, August 1988 page. 1524- 1530.

[Lake 98 b] R. Lake, G. Klimeck, D. K. Blanks, “Interface Roughness and Polar Optical Phonon Scattering in InGaAs/AlAs/InAs RTDs” Semicond. Sci. Technology, Vol 13, pg. A163 (1998)

[Lake97] R. Lake, G. Klimeck, R. C. Bowen, D. Jovanovic, “Single and multiband modeling of quantum electron transport through layered semiconductor devices” J. of Appl. Phys. 81, 7845 (1997)

[Lake98] R. Lake, B. Brar, G. D. Wilk, A. C. Seabaugh, and G. Klimeck, “Resonant tunneling in disordered materials such as Si02/Si/Si02” 1997 IEEE Int. Symp. Comp. Semicond. (Inst. Phys. Publ.) pp. 617-620, 1998.

[Moglestue86] G. Moglestue “Self consistent calculation of electron and hole inversion charges at silicon-silicon dioxide interfaces” J. App. Phys. vol. 59 p. 3175- 3183, mai 1986

[North98] A. J. North E. H. Lienfield, M.Y. Simmons, et al., “Electron reflection and interference in the GaAs/AlAs-Al Schottky collector resonant-tunneling diode” Physical review B, Vol. 57, n°3, pp. 1847-1854 (1998).

[Oriols05] X. Oriols, A. Alarcón, and E. Fernàndez-Díaz “Time-dependent quantum current for independent electrons driven under nonperiodic conditions” Physical Review B 71, 245322 (2005).

[Oriols07] X. Oriols, Quantum-Trajectory Approach to Time-Dependent Transport in Mesoscopic Systems with Electron-Electron Interactions, Phys. Rev. Lett. 98, 6, 066803-066807, (2007)

[Poncet02] http://www.insa-lyon.fr/Laboratoires/LPM/nano/act_fichiers/tech_mod.html, “Techniques d’analyse:modélisation de dispositifs”, A. Poncet, (2002).

[Schul96] J. N. Schulman, H. J. De Los Santos, “Physics-Based RTD Current-Voltage Equation” IEEE Electron Device Letters, Vol 17, n°5, pp. 220- 222 (1996).

[Sze81] S.M.Sze “Physics of semiconductors Devices”, Wiley 1981.

[Tsu73] R. Tsu, L. Esaki, “Tunneling in a finite superlattice” Appl. Physics Letters, Vol. 22, pp. 562-564 (1973).

[Wang01] J. Wang,Y. Ma, L. Tian, and Z. Li “Modified Airy function method for modeling of direct tunneling current in metal–oxide–semiconductor structures” Appl. Phys. Lett., Vol. 79, No. 12, 17 September 2001 page 1831-1833.

[Woltjer06] R. Woltjer, L. Tiemeijer and D. Klaassen “An industrial view on compact modeling” ESSDERC 2006 proceedings.

[Yan95] Z.Yan, M. J. Deen, “New RTD Large-Signal DC Model Suitable for PSPICE” IEEE transaction on computer-aided design of integrated circuits and systems, Vol. 14 N.2 1995.

Chapter III: Transient behavior of the RTD

CHAPTER III: TRANSIENT BEHAVIOR OF THE RTD

III.1. INTRODUCTION

The high operating frequency of III–V based RTD is one of the major advantages of this device. Indeed, intrinsic cut off frequency (i.e. the highest frequency at which the diode can oscillate) up to 712 GHz has been demonstrated by [Brown91]. For this reason, since the first demonstration of tunnel current by Chang [Chang74], the transient behavior of these devices has also been extensively investigated.

Several numerical and analytical approaches have been applied to predict the intrinsic frequency limitation of RTD. An overview of the most significant contributions found in the literature will be provided in the first part of this chapter.

In order to investigate the frequency domain of application of silicon RTD compared with other materials, the DC model, described in chapter II, has been extended to the AC regime. The second part of this chapter thus details the derivation of the small signal equivalent circuit consistent with the DC analytical model.

Finally, this analytical approach has been validated and interpreted by the means of state of the art numerical simulations based on Bohm trajectories.

III.2. NUMERICAL APPROACHES

Several approaches have been proposed in literature to compute the intrinsic frequency limitation of RTD.

Frensley was one of the first to calculate the RTD frequency response by solving the time dependent Wigner equation, using a non self consistent approximation ([Fren87]). In this work, it has been found that the RTD can be modeled as a conductance in parallel with a capacitance: 121 ])/(1[)//( −− += gCgCgR ω , leading to the following equivalent circuit:

Chapter III: Transient behavior of the RTD

92

Fig. III.1: RTD small signal equivalent circuit [Frensley87].

In this approach, the frequency limitation occurs when the negative resistance can no longer exceed the series resistance Rs.

In [Kluk88], the transient response of a RTD has been calculated again solving the Wigner equation but self consistently with Poisson. An oscillatory current transient has been obtained for bias switching between peak to valley of I-V characteristics as shown in the fig.III.2.

Fig. III.2: Current transient [Kluk88].

In the work, using a Fourier analysis of the transient behavior of RTD, it has been found that at low frequency (< 2 THz), this system presents an inductive behavior, and that at high frequency (> 2 THz), it also presents a capacitive contribution, attributed to the charging and discharging of the quantum well. Indeed, the real part of the conductance becomes positive for frequency above 1.5 THz.

Chapter III: Transient behavior of the RTD

93

Fig. III.3: The (a) imaginary and (b) real parts of the complex admittance of the RTD [Kluk88].

Several works ([Frensl87], [Kluk88], [Mains88], [Chen90], [Kislov91], [Frensley90]) have considered decoupled the electron current and displacement current and that the two contribution of current are individually constant across the RTD. This assumption is not strictly speaking correct as it is the total current (electron + displacement) which is constant across the RTD. For this reason, it is important to perform self consistent simulations in DC and in AC regime as well. [Liou94].

This particular point will be discussed in more details in paragraph III.5.1.

Liou et al [Liou94] compute current and charge from DC simulations, solving Poisson and Schrodinger equations self consistently using a harmonic balance technique. This work is especially interesting because it provides an interpretation of frequency behavior of RTD, discussing the validity of the different small signal equivalent circuits shown in figure III.4. From this comparison, it turns out that the RLC type (figure III.4 d) reproduces better the experimental data.

Fig. III.4: Equivalent circuits of RTD considered in [Liou94]. Note that in circuits b and c, C and G elements may depend on frequency.

Chapter III: Transient behavior of the RTD

94

In figure III.5, we can notice that the imaginary and real parts of total current (displacement + electron current), have a non trivial dependency versus frequency. At low frequency (lower than 100GHz), the conductance is a constant: it can be modeled by a frequency independent conductance that can be obtained from the differential conductance of RTD.

Fig. III.5: Real and imaginary part of the small-signal ac current versus frequency [Liou94].

At frequency lower than 500 GHz, the susceptance varies linearly with frequency and can be approximated with: )()( DVCB ωω ≅ , C being the RTD capacitance, computed following the approach

of [Hu91][Hu92].

Thus, at low frequency, the RTD can be modeled with the circuit of figure (III.4a), where Rs is the series resistance (mostly due to the contact).

At higher frequency, a more complex equivalent circuit has to be considered. As conductance and susceptance are strongly frequency dependent (especially in the negative resistance region for the capacitance), it may be tempting to introduce an artificial frequency dependency to the circuit elements C, G (leading to fig.III. 4(b)).

Fig. III.6: Real conductance versus voltage for several frequencies [Liou94].

Fig. III.7: Susceptance versus voltage for several frequencies [Liou94].

Chapter III: Transient behavior of the RTD

95

However, the reduction of both capacitance and conductance with frequency can simply be modeled with an inductance, following the small signal equivalent circuit of fig.III.4(d), in agreement with the work of Brown in [Brown89] (which is detailed in the paragraph III.3).

In others approaches, the transient behavior is expressed as a function of intrinsic physical parameters such as charge in the quantum well and tunnel life time in the well, quantities that are not directly measurable. Thus, the small signal equivalent circuit parameters are deduced in a second step from these quantities. The extraction of these parameters will be explained in detail, in paragraph III.3 where the Small Signal Equivalent Circuit (SSEC) is presented.

Among these works, let us underline the excellent work of Zhao et al [Zhao01], where the equivalent circuit parameters (conductance, capacitance, and inductance) are extracted from self consistent Wigner-Poisson simulations, as function of bias.

Numerical approaches: conclusions

A summary of some of the main works about frequency behavior of RTD has been presented in this section.

The intrinsic cut off frequency of RTD can reach several THz. This frequency was for the first time numerically evaluated by Frensley, solving the time dependent Wigner equation. The same approach has also been used in other works, accounting for self consistency with Poisson equation.

Several techniques however have been proposed to extract the intrinsic cut off frequency from simulations. Indeed, in [Kluk88], this parameter was deduced from the transient behavior of the AC current, when in [Zhao01], it was computed by an analytical derivation, using a RLC equivalent circuit deduced from simulations

Several SSEC have been proposed in the literature. The meaning of these SSEC will be discussed in the next section in more details. However, at this step, it is interesting to note that several numerical studies such as [Liou94] tend to confirm the validity of the RLC model proposed in [Brown89].

Accuracy of the model is of course the main advantage of numerical simulations, while complexity is their drawback. A compact model is more useful for engineering purposes. In the next paragraph the main analytical models proposed in literature, usually based on SSEC, are presented.

III.3. ANALYTICAL APPROACHES

Most of the analytical models describing the frequency behavior of RTD are based on Small SSEC .Several similar SSEC can be found in literature.

Chapter III: Transient behavior of the RTD

96

In older works, the intrinsic RTD cutoff frequency was computed from a RC small signal equivalent circuit with a capacitance in parallel with the differential conductance, as in Fig.III.8 [Brown88].

Fig. III.8: Small signal equivalent circuit [Brown88].

The maximum intrinsic cut off frequency is the highest frequency at which the diode can oscillate, and has been defined as the frequency at which the real part of the impedance is equal to zero. In the RC model in Fig.III.8, the cut off frequency is given by:

2/121)2(

−−= − G

R

GCf

sRC π (3.1)

In [Brown88], G, Rs and C have been assumed constant with voltage.

Several approaches for computing capacitance have been reported. The simplest way consists in considering only the geometrical capacitance with a parallel plate model ε/L [Luryi85].

Improved models also account for the dependence of capacitance on voltage, including, as suggested in [Hu92] the depletion width in the collector side.

Other models emphasize on the importance of the charge in the well [Hu91], introducing the concept of quantum capacitance in order to model the charging and discharging mechanism of the well.

Since the Brown et al. work [Brown89], in addition to conventional RC equivalent circuits, an inductance has also been introduced in series with the differential conductance. This inductance, as discussed in more details later on, represents the physical effect of the finite quasi-bound state lifetime in the well. In other words, this element models the delay of the current with respect to the voltage, arising from the effect of lifetime in the well.

Fig. III.9: Small signal equivalent circuit of [Brown89].

Chapter III: Transient behavior of the RTD

97

The intrinsic cut off frequency in this case becomes:

2/12/1

222 )12/(

/)1(11

21

1

2

1

−+−−⋅

−=

LGC

GRGR

LG

C

LCf ss

RLC π (3.2)

If L tends to zero, this equation as expected, reduces to the previous formula, valid for the RC circuit.

In [Brown89], Brown et al. have demonstrated that the single pole RC model, used previously, tends to overestimate the maximum intrinsic cut off frequency, while the RLC circuit better corresponds to the experimental data, as shown in fig III.10.

Fig. III.10: Comparison of theoretical and experimental results of oscillation power versus frequency for an AlGaAs diode. [Brown89]. Prediction based on the RC and RLC models are shown for comparison.

Other works also consider a contact inductance in series with contact resistance [Wei95], [Gerin87], as shown in fig.III.11.

Fig. III.11: Small signal equivalent circuit [Gerin87].

Chapter III: Transient behavior of the RTD

98

The inductance parallel model of figure 9 [Brown89] constitute the commonly admitted reference in term of RTD equivalent circuit [[Liu04] [Zheng04] [Vanbe92] [Zhao01]]. Capacitance in these works include both geometrical and quantum capacitance. This last point however (the existence of a quantum capacitance) remains the more controversial aspect of this question.

More recently, other interpretation in high frequency behavior of RTD has been done by Feignov [Feign01]. In his work, he introduce the concept of displacement current modeled by another conductance in parallel at RLC well know circuit.

Fig. III.12: Small signal equivalent circuit [Feign01].

In the most of previous works, the parameters of small signal equivalent circuits are extracted from simulations or experiments. The model proposed by [Liu04] however, provides analytical expressions for inductance and capacitance derived from a physics based approach. According to the authors, using the same consistent set of parameters, a good agreement between theory and experiments has been obtained. Our approach will be based on the concept proposed in [Liu04].

Small signal equivalent circuits: Conclusions A SSEC helps to interpret and model the transient and frequency behavior of RTD. The simplest SSEC combines a capacitance in parallel with a resistance. However, this model has been shown to overestimate the intrinsic cut off frequency of the RTD. Further analysis of RTD physics have provided a better understanding of the mechanisms that limit the transient behavior of this device and Brown in [Brown89] have proposed to introduce an inductance which account for the resonant lifetime of quasi bound states in the quantum well. This RLC approach allows to evaluate in good agreement with experimental data the intrinsic cut off frequency of RTD and this is the most popular RTD small signal equivalent circuit in the literature.

Following the approach of Liu [Liu04], the AC model has been deduced from the DC analytical model taking into account for the time to charge and discharge the well, leading to the small signal circuit presented in the next section.

Chapter III: Transient behavior of the RTD

99

III.4. OUR AC ANALYTICAL MODELING OF RTD Contrary to the DC regime, the emitter current flowing through the left barrier and charging the well Je is not equal to the collector current flowing through the right barrier Jc. Following the phenomenological approach proposed by Liu et. al. [Liu04], as schematized in Fig.III.13, these currents can be written as :

* * *e L e L w c R wJ Q Q and J Q= Γ −Γ = Γ (3.3)

Where ΓL,R* represent the transition frequency through the left and right barriers and Qe,w the emitter

and well charges density.

eLQ*Γ

wLQ*ΓwRQ*Γ

Figure III.13: Schematic drawing of in incoming and outcoming current charging and discharging the quantum well.

In static operating condition, these current densities are equal, and given by : * * *

c e R w L e L wJ J J Q ( Q Q )= = = Γ ⋅ = Γ ⋅ − Γ ⋅ (3.4)

Which leads to the following expression of the static current J:

* *L R

e * *L R

J QΓ ⋅ Γ

= ⋅Γ + Γ

(3.5)

This expression is exactly the same expression than the DC model presented in the previous section, for a single resonance (see chapter II):

* *e L R

max e * *L R

2QJ E T Q

h

Γ ⋅Γ= ⋅Δ ⋅ = ⋅

Γ + Γ (3.6)

With ΓL,R* = ΓL,R / h .

In consequence, the DC model previously presented can easily be extended to the AC domain, using

ΓL,R* as electron escape rates. By this simple procedure, the phenomenological approach of Liu

[Liu04], where the parameters ΓL,R* were only simple constant fitting parameters, is generalized. The

time evolution of the charge well is thus given by:

* * *we c L e R L w

dQj j Q ( ) Q

dt= − = Γ − Γ + Γ (3.7)

Assuming that Qe is proportional to the applied voltage variation v(t): ( ) ( )eQ t v t= α , we can write:

Chapter III: Transient behavior of the RTD

100

*w wL

dQ Qv

dt+ = Γ ατ

(3.8) with τ = 1 /(ΓL*+ΓR

*) (3.9)

Using Eq. (3.7), the current density jc is then given by:

* *c cL R

dj jv

dt+ = Γ Γ ατ

(3.10)

Leading to the following admittance in AC: * *L R

* * * *L R L R

A 1Y

i 1i 1/A A

Γ Γ α= =

ωω+ τ +Γ Γ α τΓ Γ α

(3.11)

(A being the diode area). The AC equivalent circuit can be easily deduced from this last equation, as composed of a differential conductance G in series with a quantum inductance L:

* *L R

* *L R

1L

GA

G A

τ = = Γ Γ α = τ Γ Γ α

(3.12)

Note that G, given by Eq. (3.12) also corresponds to dJ/dV, according to Eq. (3.6). The equivalent circuit model has also to consider the quantum capacitance arising from the well charging and discharging:

* *w L RQ v G v A= Γ τ α = Γ (3.13)

wq *

L

dQ GC A

dv= − =

Γ (3.14)

as * * *

L R RΓ +Γ ≈ Γ then qC G≈ − ⋅ τ (3.15)

Finally, including also the geometrical capacitance C0, the global capacitance is given by:

0 qC C C= + , with w b d0

w b d

L 2L LC A /( )= + +

ε ε ε (3.16)

where Lw, Lb, Ld (resp. εw, εb εd) are the well, tunnel barrier and depletion layer (when needed) thicknesses (resp. dielectric constants), which in final, leads to the following small signal equivalent circuit.

LG

C

Rs

Figure III.14: AC RTD equivalent circuit model used in this work.

As previously mentioned, the cut off frequency corresponding to the intrinsic frequency of RTD is defined as the frequency at which the real part of small signal circuit impedance is equal to zero [Brown89]. In consequence,

Chapter III: Transient behavior of the RTD

101

2 2 2int 2

s

1 1 LGf 2LG C C 4LG (C )

2 R2LCG

= − + − ⋅ + π (3.17)

The series resistance is the only parameter that is not calculated from the DC / AC model. It obviously depends on the diode technology and architecture, and plays a role in the cut off frequency optimization.

Fig.III.15: Intrinsic cut off frequency for a GaAs/AlAs RTD.

Indeed, as shown in figure III.15, fint is a decreasing function of Rs, tending to zero when Rs tends to Rn = 1/|G|, meaning that series resistance have to be carefully minimized to improve fint. When Rs <<

Rn = 1/|G|, replacing L and C by their expressions function of G and τ, equation (3.17) may be further simplified, leading to :

1/ 4

nint

s0 n

R1 1f

2 R1 C R

= ⋅ ⋅ π⋅ τ + τ

(3.18)

Equation (3.18) suggests that the intrinsic frequency fint given by Eq. (3.18) presents a maximum value when :

n max0

RC

τ= (3.19)

For this value of Rn, fint reaches a maximum given by:

max 3/ 4 1/ 4s o

1 1f

2 2 (R C )= ⋅π τ ⋅ ⋅ ⋅

(3.20)

This result suggests several way to increase fint.

First of all, τ should be minimized, by increasing the tunnel transparency of both barriers for instance. More surprising, the negative resistance has also to be optimized, as both a very large or very small Rn can be prejudicial to the intrinsic frequency. Finally, neglected in several studies, the role of the quantum inductance L has also to be considered. Indeed, Fig. III.15 shows both the intrinsic frequency calculated considering the global R, L, C equivalent circuit (fRLC) or neglecting the quantum inductance (fRC).

Chapter III: Transient behavior of the RTD

102

It turns out that, except when Rs ≈ Rn (regime where the intrinsic frequency fint tends to zero), neglecting L always tends to overestimate the intrinsic cut off frequency, as already stated in [Brown89]. This model has been used as tool to compare Silicon and GaAs RTD performances. Results will be presented in the Chapter IV.

AC analytical model: conclusion The transient behavior of a RTD has been derived, following the approach proposed by [Liu04].

This model accounts for the mechanism of charging and discharging of the quantum well through the tunnel barriers. It allows calculating the elements of the SSEC from basic principles.

Contrary to [Liu04], the characteristic times of this approach have not been taken from experiments, but have been derived from the DC model. In consequence, the same approach can be used to model both DC and AC performances, allowing thus to compare device performance versus technology and materials in both regime.

In the next section, the validity of analytical AC model has been investigated in more details, by the mean of comparison with numerical simulations.

Chapter III: Transient behavior of the RTD

103

III.5 INTERPRETATION OF THE SMALL SIGNAL ANALYTICAL MODEL BY NUMERICAL SIMULATIONS

In order to compare our analytical model with numerical simulation, we collaborate with the research team of Dr. Oriols in Universitat Autonoma de Barcelona (UAB). In particular, we analyze in more detail the validity and role of each elements of the equivalent circuit. In particular, the impact of current conservation and charge neutrality in the procedure used to calculate the maximum operating frequency of RTD has been particularly investigated. To this aim, the self consistent numerical algorithm for electron transport based on Bohm trajectory [Oriols07] has been used named Quantum Monte Carlo (QMC) algorithm.

An accurate frequency analysis is a really very difficult task because one has to take into account the Coulomb correlation among electrons to assure (i) current conservation (the total current is the sum of the conduction mentioned before as electron current and displacement currents due to time-dependent variations of the electric field) and (ii) overall charge neutrality (screening deep inside the leads assures that the total charge tends to zero).

As shown in paragraph III.2, several numerical approaches has been proposed in the literature to evaluate the frequency response, but in most of them, results are doubtful, as total current conservation and charge neutrality have not been considered.

Introducing coulomb interaction can be very difficult, especially in quantum transport, which explains why it is neglected in previous studies.

Frensley’s work [Fren87] for example provides a not self consistent calculation of RTD frequency response from the time evolution of Wigner distribution.

In previous self consistent works, someone derives information about frequency from current computation [Kluk88] but charge neutrality is not ensured and the contacts are not included in simulations.

In addition, in the most part of these works the intrinsic frequency of RTD is computed from the well known formula used by Brown et al [Brown91] (detailed in the paragraph III.3), with some differences however in the way to determine small signal equivalent circuit elements. The cut off frequency calculated in these work has not been directly deduced from current calculation, but from an equivalent circuit analysis, which also relies on assumptions [Zhao04].

In addition, in all self consistent listed simulations, Poisson equation is solved using a mean field approximation (ignoring many body effects). This approximation is no longer true in the quantum well, while correlation electron-electron is lost. Simulations performed in UAB however are based on many body Bohm trajectory approach.

To investigate the validity of our equivalent circuit approach, self consistent time dependent simulations of electron tunneling have been carried out, extracting the intrinsic frequency limitation of RTD directly from the intrinsic response time of the device at a small step voltage signal. Results

Chapter III: Transient behavior of the RTD

104

obtained using the equivalent circuit approach has been compared to Bohm trajectory simulations, including or not current conservation, charge neutrality and many body effects.

To the best of our knowledge, this is the first time that such a detailed analysis has been carried out.

In the next paragraph, the importance of current conservation and charge neutrality is explained.

III.5.1 CONDUCTION AND DISPLACEMENT CURRENT: THE CURRENT CONSERVATION In this section, we will explain the role of displacement current in the total current. Even if this

concept should be well known, we will briefly revisit the current conservation law to make clear one more time, the importance of Poisson equation resolution.

The DC current can be simply computed from the rate at which electrons cross a given surface.

We refer to this rate as the “particle” or “conduction” current ),( trJc

. The same approach is no

longer correct when non-static current is considered. What an amperemeter measures in this case, is the particle current combined with the so called “displacement” current. The latter is the effect due to the time variation of the electric field.

Maxwell introduced the concept of displacement current to demonstrate that the total current flowing through the boundary surface of an arbitrary volume is always zero, i.e. the total current is conserved. The continuity equation of the charge density is given by:

( , )( , ) 0c

r tJ r t

t

ρ∂ +∇ =∂

(3.21) Where ),( tr

ρ is the total free charge density. The first term of the equation (3.21) can be related to

the temporal variations of the electric field ),( trE

. Let the Poisson equation

( )( ) ( , ) ( , )r E r t r tε ρ∇ =

(3.22)

where the electric permittivity )(rε is assumed to be a time-independent scalar function. Substituting

(3.22) in (3.21), the continuity equation can be rewritten as

( , )( , ) ( ) 0c

E r tJ r t r

tε ∂∇ +∇ =

(3.23)

From a careful analysis of eq.(3.23), the natural definition of the total current ),( trJT

arises:

( , )( , ) ( , ) ( ) ( , ) ( , )T c c d

E r tJ r t J r t r J r t J r t

tε ∂= + = +

(3.24)

Finally, since the divergence theorem, the total current conservation reads

Chapter III: Transient behavior of the RTD

105

( ) ( )· , , 0T TJ r t dv J r t dsΩ ∂Ω∇ = =

(3.25)

where Ω is an arbitrary compact volume in 3¡ and ∂Ω the boundary of Ω.

From expression (3.25), it turns out the importance of computing the total current in high-frequency scenarios. Let us assume a volume Ω that contains inside the wire that connects one surface of the device active region to another surface of the ammeter (see fig.III.16).

VIN(t) RL RIN Wire Wire

Amperimeter Ω

Device Active Region

SDAR

SAMP

Fig.III.16: Schematic representation of the current measurement in an electron device. Device simulators compute the current in the surface, SDAR, of the device active region, while the amperemeter measures the current in the surface, SAMP [Alarcon08].

In most simulators, the current is computed on the surface of the active region, SDAR, while in most experimental setups the current is measured on another surface, SAMP, close to the ammeter. Assuming that there is neither electric field nor current flowing outwards from the other surfaces that limit the volume Ω, we realize that expression (3.25) enforces that the total current computed on SDAR is equal to that measured in SAMP. Therefore, the computation of the total current is a mandatory requirement in order to provide experimental predictions in electron transport under high-frequency conditions.

Let us emphasize again the main conclusion drawn in this subsection. The total current, from eq.(3.24), means compute the displacement current, i.e. the time-dependent variations of the electric field in the surface SDAR. Therefore, any approach trying to predict the AC current needs to take into account, somehow, the Coulomb interaction among electrons together with the transport equation. We name such solution of coupled equations a self-consistent simulation.

Chapter III: Transient behavior of the RTD

106

III.5.2 CHARGE NEUTRALITY The well-known standard textbook expression of the DC (zero temperature) conductance through

a tunneling obstacle is known as the “two-terminal” expression because it is defined as the current divided by the voltage drop sufficiently far from the obstacle. However, the original formulation of the conductance proposed by Landauer [Landauer57], [Landauer70] in 1957 was known as the “four-terminal” conductance because its experimental validation needs two additional voltages probes to measure the voltage drop close to the tunneling obstacle. The presence of resistances in the leads explains the difference between both expressions. The ultimate origin of such resistances is the requirement of “overall charge neutrality” that transforms unbalanced charges in the leads into a voltage drop there, via the Poisson equation.

To ensure overall charge neutrality in the whole device, is not a easy task because usually the simulation box is smaller than the region where there is unbalanced charge. To solve this difficulty a Boundary Condition (BC) algorithm could be employed within a QMC method [Albareda10]. The development of the BC algorithm consists in integrating the local continuity equation in a large volume Ω that include sample, lead and reservoirs.

The BC algorithm consists in a time-dependent condition involving the total charge density ( , )r tρ ,

the electric field ( , )E r t

and the scalar potential ( , )V r t

at the interface between the sample and the

leads. We report here only the introductory preliminaries of the BC algorithm to point out in which way the “overall charge neutrality” can affect the frequency behavior of mesoscopic devices. The full treatment can be found in [Albareda10].

We start by integrating the equ.(3.21) on a large volume Ω including the sample, the leads and the reservoirs:

( , ) ( , ) 0cr t dr J r t dstρ

Ω ∂Ω

∂ + ⋅ =∂

(3.26)

.The volume Ω is limited by the surface ∂Ω . We assume that the particle current ( , )cJ r t

in (3.26) is

parallel to all the sub-surfaces of ∂Ω , except to the surfaces AS (Source) and AD (Drain) connecting leads to the reservoirs. Thus, taking , ,S D S Dr A∈

, the eq.(3.26) results

( , ) ( , ) ( , ) 0S D

c S c DA Ar t dr J r t ds J r t ds

Ω

∂ + ⋅ + ⋅ =∂

(3.27)

The current density and the electric field, deep inside the reservoirs, are related by the integral form of the Ohm’s law ( )

, ,, ,( , )· , ·

S D S Dc S D S DA A

J r t ds E r t dsσ=

(3.28)

where σ is the reservoir (frequency-independent) conductivity. To be valid, the Ohm’s law (3.28) needs time scale larger than the inelastic scattering time and imposes an important limitation on the frequency-validity of the boundary conditions:

Chapter III: Transient behavior of the RTD

107

( )( , )· ( ) ( ) 0c cS Dr t dr E t E t

tρ σ

Ω

∂ + − =∂

(3.29)

where ( ) ( , )·S

cS SA

E t E r t ds=

and ( ) ( , )·D

cD DA

E t E r t ds= −

.

To further simplify 3.29, consider the integral form of the Gauss equation in the same volume Ω : ( , )· ( , ) 0r t dr D r t dsρ

Ω ∂Ω− ⋅ =

(3.30)

with ( , ) ( )· ( , )D r t r E r tε=

and ( )rε the (frequency-independent) dielectric constant. Again, we

assume the vector ( , )D r t

parallel to the surfaces, except to AS and AD hence

( )( , )· ( ) ( ) 0c cS Dr t dr E t E tρ ε

Ω− − =

(3.31)

with ( ) ( )D Sr rε ε ε= = . Combining expression (3.29) and (3.31), we obtain:

( , )· ( , )·r t dr r t drt

σρ ρεΩ Ω

∂ = −∂

(3.32)

Eq.(3.32) provides the time-evolution of the charge ( ) ( , )·Q t r t drρΩ

=

in the whole system and its

solution has the form:

0

0( ) ( ) c

t t

Q t Q t e τ−

−= (3.33)

with the dielectric relaxation time (sometimes called Maxwell relaxation time) defined as:

c

ετσ= (3.34)

As expected, the meaning of expression (3.33) is that the total charge inside the system tends to zero and charge neutrality is achieved in periods of time related to the dielectric relaxation time. From (3.29) and (3.31), using the same procedure for Q, the electric field reads

( )0

0 0( ) ( ) ( ) ( ) c

t t

c c c cS D S DE t E t E t E t e τ

−−

− = − (3.35)

The solution (3.35) imposes that the electric field deep inside both reservoirs is asymptotically identical. Finally, we know that the time-average electric field deep inside the reservoir tends to the Drude’s value , ( )drift

S DE t [Drude]. Therefore, one possible solution of (3.35), considering

, ,( ) ( )c driftS D S DE t E t→ , when t→∞ is:

( )0

, , , 0 ,( ) ( ) ( ) ( ) c

t t

c drift c driftS D S D S D S D oE t E t E t E t e τ

−−

− = − (3.36)

The parameters σ and ε are assumed constant (time-independent and frequency independent) and determine the frequency limitation of expression (3.36).

In good reservoirs (with nanoscale samples) such frequency limitation is on the order of few THz, which is high enough for most practical electronic applications. However in the case of RTDs, where the time scales involved in the transport are on order of picoseconds, this limitation sensibly affects their frequency behavior as shown in section III.5.3.3.

Chapter III: Transient behavior of the RTD

108

III.5.3 NUMERICAL COMPLETE SIMULATIONS AND SMALL SIGNAL EQUIVALENT

CIRCUIT

In the next section, the role of current conservation and charge neutrality in the intrinsic cutoff frequency has been investigated. To extract information about the different time constants associated to the several processes characterizing the electron dynamics in the RTDs, we accounted for three different conditions employed in NDR regime. 1) Not self consistent simulation (current conservation and charge neutrality are neglected), 2) Current conservation (i.e. coulomb interaction), but charge conservation is not ensured in whole device (leads not included in simulation box), 3) Both, current conservation and charge neutrality are included.

The simulated RTD is a GaAs / AlAs heterostructure with 1.2 nm of barriers and well of 5.7 nm, with a band-offset of 0.6 eV (we assume a constant effective mass m*= 0.067 m0 along the whole structure).

III.5.3.1 NON SELF CONSISTENT SIMULATION AND CHARGE NEUTRALITY NOT INCLUDED In order to be able to distinguish the effect of interactions between electrons, first, we remove

self-consistency of the Coulomb interaction in the QMC (i.e. without solving Poisson equation for potential calculation). In this case, we assume a “hand-made” linear potential profile along the active structure.

When we apply a voltage step, the current in the RTD follows this voltage step with an intrinsic delay of about 0,035 ps (see fig. III.17). In absence of coulomb interactions, this time can be assumed to be equal to the RTD dwell time τd. In the analytical model τd = ħ/Γ where Γ is the total width of the resonant level [Brown89]. The value extracted for this structure is 0,03 ps, in excellent agreement with QMC simulations.

In this case, the SSEC can be simplified by connecting in series a negative conductance G and a negative inductance L. Indeed, without self consistency, there is no displacement current, and thus no capacitance (see fig. III.18). In addition, in the absence of leads, we do not account for series resistance. The resulting cutoff frequency of this simple filter is (2π τd)-1

= 5 THz.

Chapter III: Transient behavior of the RTD

109

0,0 2,0x10-13 4,0x10-13 6,0x10-13 8,0x10-13

3,0x10-5

3,5x10-5

4,0x10-5

4,5x10-5

5,0x10-5

5,5x10-5

6,0x10-5

6,5x10-5

0,160

0,165

0,170

0,175

0,180

Cu

rren

t (A

)

time (s)

without coulomb

Vo

ltage (V

)

Figure III.17: Response of the RTD to a voltage step, without coulomb interaction.

Figure III.18: Small signal equivalent circuit of the RTD without taking into account for coulomb interaction and contact.

More detailed study of the RTD dynamics is possible by applying the Fourier transform. In order to capture the pure effects of the dynamic in RTD, we need first to introduce some formalism.

The bias voltage versus time can be written as:

)](1[)()( 21 tVtVtV θθ −+= (3.37)

Where θ(t) is the Heaviside function:

<>

=00

01)(

tif

tiftθ (3.38)

And V1 and V2 are constant potential values. The current response can be expressed as:

I(t) = Itran(t) + I1 + I2 × [1 – θ(t)] (3.39)

Where I1 and I2 are the stationary currents corresponding to steady state voltage V1 and V2 respectively, and Itran is the transient (time dependent) current. It evolves from 0 at t = 0 to I2 – I1 after the transient regime.

Chapter III: Transient behavior of the RTD

110

Figure III.19: Currents & normalized Fourier Transform of the current in the mean field approximation with no contacts.

Two small steps 0.18V → 0.16V and 0.16V → 0.18V have been applied to the device. In fig.III.19 insets 1a and 2a show than the current responds is symmetric. The cut off frequency is equal in both side and take the value fcutoff = 5.8 THz that corresponds at a time constant τ= 2π/fcutoff = 0.027ps .

These values are in perfect accord with those extracted with our analytical SSEC.

This result confirms the reasonable physical information contained in the two elements of the analytical SSEC of figure III.18. However, the self consistent solution of the Poisson Equation and the inclusion of contacts have an important impact on the RTD transient. In the next sections, the effect of both contributions has been separately investigated.

Chapter III: Transient behavior of the RTD

111

III.5.3.2 CURRENT CONSERVATION INCLUDED AND CHARGE NEUTRALITY NEGLECTED The numerical self-consistent time-dependent QMC simulations allow an accurate calculation of

electrostatic interaction among electrons. In this case, the potential is computed by solving the Poisson equation self-consistently with the transport equation. In these preliminary simulations, the “simulation box” is very small, thus leads are neglected and the achievement of overall charge neutrality is not guaranteed.

A simple way to model in the analytical model the coulomb effect is a simple linear relationship between voltage and charge introducing in the small signal equivalent circuit a capacitance in parallel to differential conductance.

The total capacitance has two components, a geometrical one C0 (equ. 3.16) and according to the work of [Liu04], a quantum capacitance that take into account for charge and discharge of the quantum well:

dV

dQC w

q = (3.40)

That can be computed according Equ. 3.14.

The time constant of this small signal equivalent circuit drawn in figure III.21 is )( 01

qCCG +⋅− . As

the information linked to the lifetime in the well is given by quantum capacitance, in order to simplify the small equivalent circuit, and to maintain a filter behavior, the inductance in this case is neglected.

In this case the cutoff frequency found according the SSEC approach is 2.96 THz and it is consistent with the characteristic time fig. III.20 found with QMC.

0,0 2,0x10-13 4,0x10-13 6,0x10-13 8,0x10-13

2,0x10-5

2,5x10-5

3,0x10-5

3,5x10-5

4,0x10-5

4,5x10-5

0,240

0,245

0,250

0,255

0,260

0,265

0,270

Cu

rre

nt

(A)

time (s)

Vo

ltage (V

)

with coulomb

Figure III.20 Response of the RTD to a voltage step, including current conservation.

Figure III.21: Small signal equivalent circuit of the RTD taking into account for coulomb interaction, but neglecting contact.

Chapter III: Transient behavior of the RTD

112

As we can see in the figure, the system has different response to the two step voltages (from 0.27V to 0.24V and from 0.24V to 0.27V). The cut off frequency slightly differs in the two cases.

Figure III.22: Currents & normalized Fourier Transform of the current with no contacts.

The two cut off frequency found in the two cases are:

fcutof f1= 5 THz

fcutof f2 = 2.9 THz

That corresponds to the constant time: τ1 = 0.03 ps and τ2 = 0.055 ps

This asymmetric behaviour in the cut-off frequency, which is not included in the SSEC, is investigated in the next section. It points out a possible limitation of the linear (symmetric) circuits in modelling such transients (or that the voltage step is too large to assume a linear behaviour).

Chapter III: Transient behavior of the RTD

113

III.5.3.3 QUALITATIVE EXPLANATION OF IMPACT OF COULOMB INTERACTION IN DYNAMIC RESPONDS

An electron tunneling into the well from the cathode raises the potential energy of the well by an amount of e/Ceq, where e is the electron charge and Ceq the structure capacitance.

When a step like in inset 1a of figure III.22 is applied to the structure in the negative differential resistance region, the band potential is forced to tilt and the energy level that is below the bottom of band conduction of the cathode, is pushed up.

This process is helped by the electrons that entered in the well and raises (for coulomb effect) the bottom of the potential in the well and pull up further the levels. The electrons can pass in the level and the system achieves the current level corresponding to 0.24 V relatively fast.

On the contrary, when the level is above the conduction band bottom and a step like in the inset 2a of figure III.22 is applied to the structure, the level is pushed down, but the electrons that occupied the well tend to raises the potential in the well and delay the process. The resulting dynamic is slower.

In the next section effects of leads and dielectric relaxation time is reported.

III.5.3.4 CURRENT CONSERVATION AND CHARGE NEUTRALITY (LEADS RESISTANCE)

In order to ensure charge neutrality in whole device, the leads have been introduced somehow in QMC consistently with the Poisson equation as explained in paragraph III.5.2. The main idea to take into account the leads without enlarging the simulation box is to develop appropriate boundary conditions for the injected charge and the Poisson equation in a small simulation box [Albareda10].

Conversely in our small signal circuit, contacts have been included by mean of a series resistance (see fig.III.24). Thus the cut off frequency of SSEC is now 1,36 THz.

Chapter III: Transient behavior of the RTD

114

0,0 4,0x10-13 8,0x10-13 1,2x10-12 1,6x10-123,0x10-5

3,5x10-5

4,0x10-5

4,5x10-5

5,0x10-5

0,27

0,28

0,29

0,30

0,31

0,32

Cur

rent

(A

)

time (s)

Voltage (V

)

Figure III.23: Response of the RTD to a voltage step, including charge neutrality.

Figure III.24: Complete small signal equivalent circuit of the RTD.

In this case, the transient response contains two important dynamics. At first, as pointed out in figure inset 1a and 2a, there is a delay of about 0.1ps in the current response.

Figure III.25: Currents & normalized Fourier Transform of the current, including current conservation and contacts.

Chapter III: Transient behavior of the RTD

115

The phenomenon can be explained interpreting this delay as the time in which the active region potential starts to be affected by the change in the bias. Two single pole-like responses have been applied. The first one starting in t = 0 (figure III.25 inset 1a and 2a red solid lines).

The respective frequencies have been estimated as depicted in the inset 1b and 2b. From the figure III.25, insets 1a and 2a, we note that the first approximation including the delay does not match very well the current response because the single pole approximation is not appropriate. But another single pole approximation excluding the lead delay (labeled as Single pole 2), matches perfectly the transient.

Second, also in this case the responds to the step voltage in the two senses is not symmetric. Thus, two cut off frequency have to be taken into account. We have 2 series of values:

==

==

⎯→⎯

==

==

⎯→⎯

=

=⎯→⎯

=

=⎯→⎯

psf

psf

binset

psf

psf

binset

THzf

THzfbinset

THzf

THzfbinset

dcutof

fcutofd

dcutof

dcutofd

fcutof

dcutof

fcutof

dcutof

125.02

215.02

2

1.02

19.02

1

27.1

74.02

6.1

84.01

2

2

2

2

2

2

1

1

1

2

1

1

πτ

πτ

πτ

πτ

Where τd1 and τd2 are the constant time associated to the model that take into account for the leads delay and the index 1 and 2 represent the situations in the inset 1a and 2a respectively.

It is very interesting to notice that for the different input step, the difference between the times (τd1-τd2= τ2-τ1= 0.025ps ) is the same of the previous case (τ2 – τ1 = 0.025ps ) that means that the mechanism of coulomb still affect the asymmetric response of the device.

Chapter III: Transient behavior of the RTD

116

III.6. CONCLUSION OF CHAPTER III

We can conclude that several limitations come into play in the frequency response of an RTD, namely the intrinsic tunneling process, the transit time across the non-tunneling regions and time constant associated to the total capacitance of the structure.

We applied two different approaches to study the frequency behavior of RTD and to investigate the effect of current conservation and charge neutrality.

On the one hand the full time-dependent simulation provides a rigorous picture of the physics that governs the frequency behavior of RTD.

From accurate analysis of numerical simulations, the dominant dynamics involved in the transient can be identified. In particular, interesting phenomena occur when coulomb interaction and the presence of lead delays is taken into account in the responds of the systems. The price to pay to use such numerical simulations is code complexity and very long (in order of days) simulations time.

On the other hand, the equivalent small signal circuit does not allow capturing all the intrinsic mechanisms that come into play in the device dynamic, but our study shows that is able to catch characteristic times with very simple algorithms and very short simulation time. Thus, it becomes a useful tool to design RTD.

REFERENCES OF CHAPTER III

[Alarcon08] A.Alarcón and X.Oriols, “Self-consistent computation of particle and displacement currents in quantum electron devices” UPON 2008.

[Albareda10] G. Albareda, H. Lopez, X. Cartoixa, J. J. Suñé, X. Oriols, “Time-dependent boundary conditions with lead-sample Coulomb correlations: Application to classical and quantum nanoscale electron device simulators”, Phys. Rev. B, in press, (2010)

[Bohm52] D. Bohm, Physical Review 85, 166 (1952). “A Suggested Interpretation of the Quantum Theory in Terms of "Hidden" Variables”

[Brown88] E. R. Brown, W. D. Goodhue and T. Sollner “Fundamental oscillation up to 200 GHz in resonant tunnelling diodes and new estimates of their maximum oscillation frequency from stationary-state tunnelling theory” J. Appl. Phys.64 (3) 1988.

[Brown89] E.R. Brown, C.D. Parker, L.G. Sollner “Effect of quasibound state lifetime on the oscillation power of resonant tunneling diodes” Appl. Physics Lett. 54 p. 2291-2293 (1989).

[Brown91] E.R.Brown, J.R. Soderstrom, C.D. Parker, L.J. Mahoney, K.M. Molvar and T.C. McGill, “Oscillation up to 712 GHz in InAs/AlSb resonant tunneling diodes”, Appl. Phys. Lett. 58, 2291, 1991.

[Chang74] L. L. Chang, L. Esaki, and R. Tsu, “Resonant tunneling in semiconductor double bamers,” Appl. Phys. Lett., p. 593, 1974.

[Chen90] L.Y. Chen and C.S. Ting, “AC conductance of a double-barrier resonant tunneling system under a dc-bias voltage,” Phys. Rev. Lett., vol. 64, no.26, pp. 3159-3162, 1990.

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[Drude] Thomas Ihn, Semiconductor Nanostructures: Quantum states and electronic transport, Oxford University Press, New York, (2010).

[Feign01] M. N. Feignov, “Displacement currents and the real part of high-frequency conductance of the resonant-tunneling diode” Appl.Phys.Lett. 78 (21), 2001.

[Frensl87] W. R. Frensley, “Quantum transport calculation of the small-signal response of a resonant tunneling diode,” Appl. Phys. Lett., vol. 51, no. (6).

[Frensley90] W. R. Frensley, “Boundary conditions for open quantum systems,’’ Revue of Modern Physics, vol. 62, p. 764, 1990.

[Gabel06] J.Gabelli, G. Fève, J.M.Berroir,B.Plaçais, A.Cavanna et al. Science vol.313 p.499-502,2006. “Violation of Kirchhoff's Laws for a Coherent RC Circuit”

[Gerin87] J. M. Gering, D. A. Crim, D. G. Morgan, P. D. Coleman, W.Kopp, and H. Morkoc, “A small-signal equivalent-circuit model for GaAs Al Ga As resonant tunneling heterostructures at microwave frequencies,” J. Appl. Phys., vol. 61, pp. 271–276, Jan. 1987.

[Hu91]Y. Hu and S. P. Stapleton, “Quantum capacitance of resonant tunneling diodes” Appl.Phys.Lett. 58(2) 1991.

[Hu92] Y. Hu and S. P. Stapleton, “Capacitance of a resonant tunneling diode,”Japan. J. Appl. Phys., vol. 31, pp. 23-25, 1992.

[Kislov91] V. Kislov and A. Kamenev, “High-frequency properties of resonant tunneling devices,” Appl. Phys. Lett. vol. 59, no. 12, pp. 1500-1502, 1991.

[Kluk88] N. C. Kluksdahl, A. M. Kriman, D. K. Ferry, and C. Ringhofer, “Transient switching behavior of the resonant-tunneling diode,” IEEEElectron Device Lett., vol. 9, pp. 457-459, Sept. 1988.

[Landauer57] R. Landauer, “Spatial Variation of Currents and Fields Due to Localized Scatterers in Metallic Conduction”, IBM J. Res. Dev., 1, 3, 223 (1957).

[Landauer70] R. Landauer, “Electrical resistance of disordered one-dimensional lattices”, Philos. Mag., 21, 172, 863 (1970).

[Liou94] W. Liou, P. Roblin, “High Frequency Simulation of Resonant Tunneling Diodes” Electronics letters Vol. 37, n°7, pp. 1200-1201 (1994).

[Liu04] Q. Liu et. al. “Unified AC Model for the Resonant Tunneling Diode” Transactions on Electron Devices Vol.51 p. 653 - 657 (2004).

[Luryi85] S.Luryi Appl.Phys.Lett. 47 , 490 . 1985. “Frequency limit of double barrier resonant tunnelling oscillators”

[Mains88] R. K. Mains and G. I. Haddad, “Wigner function modeling resonant tunneling diodes with high peak-to-valley ratios” J. Appl. Phys., vol. 64, no. 10, pp. 50415044. 1988.

[Oriols04] X. Oriols, A. Trois, and G. Blouin, “Self-consistent simulation of quantum shot noise in nanoscale electron devices” Appl. Phys. Lett. 85, 3596 (2004).

[Oriols07] X. Oriols, “Quantum-Trajectory Approach to Time-Dependent Transport in Mesoscopic Systems with Electron-Electron Interactions” Phys. Rev. Lett. 98, 066803 (2007).

[Pretre96] A.Pretre, H.Thomas, M.Buttiker, Phys.Rev.B, 54 8130 (1996). “Dynamic admittance of mesoscopic conductors: Discrete-potential model”

[Vanbe92] O. Vanbesien, V. Sadaune, D. Lippens, B. Vinter, P. Bols, and J. Nagle,“Direct evidence of the quasibound-state lifetime effect in resonant tunneling from impedance measurements,” Microwave Opt. Technol. Lett.,vol. 5, pp. 351–354, July 1992.

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[Wei95] T. Wei, S. Stapleton, and O. Berolo, “Scattering parameter measurements of resonant tunneling diodes up to 40 GHz,” IEEE Trans. Electron Devices, vol. 42, pp. 1378–1380, July 1995.

[Zhao01] P. Zhao, H. L. Cui, D. L.Woolard, K. L. Jensen, and F. A. Buot, “Equivalent circuit parameters of resonant tunneling diodes extracted from selfconsistent Wigner-Poisson simulation,” IEEE Trans. Electron Devices, vol. 48, pp. 614–626, Apr. 2001.

[Zheng04] Yun Zheng and Roger Lake, “Self-consistent transit-time model for a resonant tunnel diode” IEEE transaction on electron devices Vol.51, n°4, 2004-6, pp. 448450, 1987.

Chapter IV: Results & Discussion

CHAPTER IV: RESULTS & DISCUSSION

IV.1. INTRODUCTION As explained in the Chapter I, RTDs have been elaborated so far with III–V materials and technologies, thanks to epitaxial technologies allowing an accurate control of device dimension at the nanometric scale. Nevertheless, the difficulty to integrate these materials in a silicon process pushes to find possible solutions to realize them with silicon compatible materials. In an attempt to realize these devices in a Silicon compatible technology, SiGe / Si heterostructures have been also investigated, leading to encouraging results [Paul02]. In this work, an alternative option to integrate this innovative device on silicon process has been investigated. Indeed, with the scaling down of CMOS technology (where double gate transistors with body thicknesses less than 10 nm have been demonstrated [Vinet05]), and considering progress on the epitaxy and high-k thin oxide deposition (using in double gate realization), it is now possible to realize a stack of nanometric dielectric layers on silicon. Consequently, it may be possible to think of realizing Si based RTD. These devices could in principle be used for the same kind of applications that were previously realized on III-V materials. The aim of this final chapter is thus to estimate and compare the theoretical static and dynamic performances of Si and III V based RTD by the means of the analytical DC and AC models presented in the previous chapters. In addition, these comparisons will be performed not only at the device but also at the circuit level. Indeed, by introducing this model in a circuit simulator, several oscillator circuit architectures have been considered, and compared with more conventional silicon oscillator circuits. The aim of this work is thus to evaluate the expected performances of silicon based RTD RF oscillators, using this technology.

Chapter IV: Results & Discussion

120

IV.2. IMPACT OF PHYSICAL PROPERTIES: COMPARISON OF DIFFERENT MATERIALS

In this paragraph, the impact of material properties of the tunnel barrier and quantum well (effective masses, dielectric constant and barrier height) are investigated, focusing on the comparison of silicon RTD with III-V heterostructures. The geometrical and physical parameters of the model are summarized in the table I.

MATERIAL PROPERTIES :Barriers Well Contacts

Effective mass Effective mass Fermi Levels

Dielectric constant Dielectric constant

Barrier height

DEVICE GEOMETRICAL PARAMETERSBarriers Well

Thickness left and right barrier Width

Table I: Parameters of the analytical model.

Impact of the tunnel barrier material:

As explained in the paragraph I.5.2.5, the structure Si/SiO2 was considered as the first candidate for Silicon RTD for integration of RTD in MOSFET technologies processing. However, in the last generation of double gate transistors [Vinet05], other dielectrics, called High K materials, such as HfO2 have also been used.

In order to estimate which insulator is more suitable for RTD application, a Si/SiO2 and Si/HfO2 RTD of same dimensions have been compared in Fig.IV.1. Device features a barrier of 1.7 nm and a well of 4.5 nm (which is a typical range of dimension in III V devices [North98]).

As seen on Fig. IV.1, the HfO2 RTD features four orders of magnitude higher current. This can easily be explained by the fact that HfO2 has much better tunnel properties than [Coign09]

• the lower tunnel effective mass : mHfO2 = 0.2 versus mSiO2 = 0.5 • tunnel barrier φHfO2 = 1.8 eV versus φSiO2 = 3.1 eV

Chapter IV: Results & Discussion

121

0,0 0,3 0,6 0,9

101

102

103

104

105

106

107

108

Cu

rre

nt

De

ns

ity

(A/m

2)

Voltage (V)

HfO2

SiO2

Figure IV.1: Comparison between I-V characteristics of SiO2/Si

and HfO2/Si RTD devices (Lb=1.7 nm, Lw=4.5 nm). However, note that in practice, in CMOS technology, a pure HfO2 insulator does not exist, as it always also include a 6 Å interfacial layer of SiO2, which has not been considered here, and would decrease of course the tunnel transparency. III-3 versus Si technologies:

To compare III-V and Si based RTD performances, I-V static characteristics of a standard AlAs / GaAs / AlAs heterostructure, and the corresponding HfO2 /Si / HfO2 RTD have been benchmarked (Fig IV.2). Using the same dimensions for tunnel barriers, (1.7 nm) and well thicknesses, (4.5 nm), and the same Fermi level (taken from [North98]), it turns out that the GaAs device offers both a significantly higher PVCR and NDR (Fig.IV.2).

Frequency performances of Si and GaAs based RTD of same dimensions (Lb=1.7 nm, Lw=4.5 nm) have also been computed by the means of small signal equivalent circuit presented in the chapter 3 and are compared on Fig.IV.3. At these dimensions, the intrinsic cut off frequency appears to be typically three orders of magnitude higher in GaAs/AlAs than in Si RTD, illustrating the superiority of GaAs device (reaching dozen of THz).

Figure IV.2: Comparison between I-V characteristics of AlAs/GaAs and HfO2/Si RTD devices with same dimensions. (Lb=1.7 nm, Lw=4.5 nm).

Figure IV.3: Comparison between the intrinsic cut off frequency (defined in [14]), versus contact series resistance of AlAs/GaAs and HfO2/s-Si RTD with the same dimension (Lb=1.7 nm, Lw=4.5 nm).

Chapter IV: Results & Discussion

122

The GaAs device superiority can be explained by: 1/The higher transparency of right and left AlAs tunnel barrier compared to HfO2 due to the lower bandoffset between GaAs/AlAs than in and Si/ HfO2 (see table 2) and smaller effective mass in the AlAs barriers compared to HfO2 (0.123m0 compared to 0.2m0) 2/ the isotropic light effective masses of GaAs (0.063 instead of 0.92 and 0.19 for Si anisotropic conduction band), which significantly reduce the number and overlap of resonant levels in the GaAs well. Si / SiGe versus Si / HfO2 performances:

As explained in Chapter I, SiGe RTD have been extensively investigated in the literature, reporting performances comparable with III-V materials. Ideal SiGe structures have also been considered in our performances comparison, and results are plotted in fig IV.4 and IV.5

It turns out that SiGe RTD can achieve similar static performance of GaAs RTD. However, GaAs maximum frequency is still one order higher than SiGe.

These results can be simply explained by the higher transparency of SiGe barriers compared to HfO2 of same dimensions. However, the lowest number of quasi bound states remains a major advantage of the AlAs/GaAs device, compared to the Si / Si Ge device. Up to now, comparisons have been performed at same device dimensions. In the following, we will try to optimize Silicon based RTD dimensions, in order to match the performance of GaAs and SiGe RTD. To this aim, in the next section, the impact of device dimensions on performance is investigated more deeply.

0,0 0,2 0,4 0,6 0,8

106

107

108

109

1010

1011

Cu

rren

t D

ensi

ty (

A/m

2 )

Voltage-Vfb(V)

Si/HfO2 GaAs SiGe

Figure IV.4: Comparison between I-V characteristics of AlAs/GaAs and HfO2/Si RTD devices and SiGe with same dimensions. (Lb=1.7 nm, Lw=4.5 nm).

Figure IV.5: Comparison between the intrinsic cut off frequency (defined in [14]), versus contact series resistance of AlAs/GaAs and HfO2/s-Si RTD and SiGe with the same dimension (Lb=1.7 nm, Lw=4.5 nm).

Chapter IV: Results & Discussion

123

IV.3.1. IMPACT OF GEOMETRICAL PARAMETERS ON I-V CHARACTERISTICS

The proposed analytical model has been used to study the impact of geometrical parameters such as the well width and barrier thickness on Si-based RTD structures and then to compare this device with III-V based RTD and SiGe RTD. In particular, first the impact of tunnel barrier thickness and the well width on peak current, the peak to valley current ratio (PVCR) and negative differential resistance (NDR) has been investigated. Then, the impact of well width and barrier thickness on the intrinsic cut off frequency has been estimated. The impact of well width and barrier thickness on the I-V characteristics and the main static figures of merit such as PVCR, NDR and peak current, are been presented in this paragraph. Fig.IV 9 compares the simulated I-V characteristics of RTD for various silicon layers thickness (in the range tw = 1-5 nm). • Well width

As explained in paragraph II.4.5, in the case of silicon, two series of resonant levels exist in the well, due to the longitudinal and transversal electron masses in silicon. Increasing the well width tends to decrease resonant energy levels. In consequence, a large well contains several resonant levels very close to each other, which may reduce the peak current and NDR, as we can see from equ.2.90 in chapter II and see in the following figure IV.6 ,IV.7 and IV.8, for well of 5,3 and 2 nm.

0 2 4 6 8 100,0

0,5

1,0

1,5

2,0

2,5

En

erg

y le

vel

s (

eV

)

n

Transv levels 5nm longit levels 5nm

well 5nm

0 2 4 6 8 100,0

0,5

1,0

1,5

2,0

2,5

En

erg

y L

evel

s (

eV

)

n

Transv levels 3 nm long levels 3nm

well 3 nm

0 2 4 6 8 10

0,0

0,5

1,0

1,5

2,0

2,5

En

erg

y le

vels

(eV

)

n

transv levels 2nm long levels 2nm

well 2 nm

Fig IV.6: Transversal and longitudinal resonant energy levels for a well of 5nm.

Fig IV.7: Transversal and longitudinal resonant energy levels for a well of 3nm.

Fig IV.8: Transversal and longitudinal resonant energy levels for a well of 2nm.

In the I-V characteristics, longitudinal and transversal peaks may overlap each other reducing PVCR and NDR, as shown in fig IV. 9.

Chapter IV: Results & Discussion

124

Fig IV.9: Impact of the well thickness tw on the I-V characteristics

of a Si/HfO2 RTD device (ml and mt resonant levels included). • Barrier thickness

The transmission coefficient of simple barrier strongly depends on the barrier thickness. Very thin barriers provide high transparency. Thus, thick tunnel barriers reduce the tunnel current. However, the position of resonant levels does not depends on barrier thickness, thus the resonance peak occurs at the same voltage for each simulated barrier thickness, as shown in fig IV.10.

Fig IV.10: Impact of tunnel barrier tb thickness on the I-V characteristics

of a Si/HfO2 RTD device (ml and mt resonant levels included).

IV.3.2. IMPACT OF GEOMETRICAL PARAMETERS ON INTRINSIC CUT OFF FREQUENCY

Geometrical parameters have a strong impact also in intrinsic frequency limitation of RTD. Frequency behavior of Si based RTD with different well width (2nm, 3nm and 5nm) and barrier thickness (1nm, 1.7nm and 2nm) have been computed using the small signal equivalent circuit presented in the chapter 3. Following formulation (3.17) in chapter III, the maximum cut off frequency depends on several geometrical parameters, series resistance and on lifetime τ in the well. A reduction of well width and barrier thickness decreases τ and consequently improves the intrinsic cutoff frequency, as shown respectively in figures IV.11 and IV.12:

Chapter IV: Results & Discussion

125

Fig IV.11: Impact of the well thickness tw on the intrinsic cut off frequency of a Si/HfO2 RTD device.

Fig IV.12: Impact of the barrier thickness tw on the intrinsic cut off frequency of a Si/HfO2 RTD device.

IV.3.3. CONCLUSION ON GEOMETRICAL PARAMETERS IMPACT

To improve device performance, the silicon well thickness has to be scaled down to few nanometers in order to limit the number of resonant peaks, and consequently to maintain a good PVCR. In addition, reduction of barrier thicknesses will increase peak current; and thus also improve the cut off frequency. In consequence, it appears as very important to control the technology process, in order to have on the same chip, the same characteristic for all RTD. In fact, variability on peak current may be prejudicial in circuit application. This point will be investigated in more details in the next. By a drastic reduction of the layer thickness, it is possible to improve current peak and to obtain a correct negative resistance. The performances that can be achieved by this approach are presented in the next paragraph.

IV.4. TOWARD SILICON BASED RTD According to the previous paragraphs, it turns out that due to the silicon material properties of silicon and insulator (transversal and longitudinal effective electron mass and higher bandoffset between silicon and oxide), silicon based RTD present poor performances compared to GaAs or SiGe RTD. However, the Si based device can still be optimized in order to achieve the same level of performances than III V devices. Indeed, in order to improve current peak and to obtain a correct negative resistance, the tunnel transparency of both barriers should be increased. To this purpose, new simulations have been performed using reduced tunnel barriers thicknesses (1 nm instead of 1.7 nm). In addition, in order to reduce the number of resonant levels in the well, the well width has been also reduced to 2 nm (instead of 4.5 nm). The comparison between the GaAs/AlAs RTD and the optimized Si RTD is shown in the figure IV.13.

Chapter IV: Results & Discussion

126

Fig IV.13: Comparison between I-V characteristics of

AlAs/GaAs and HfO2/Si RTD devices of different dimensions.

This discrepancy on intrinsic cut off frequency can be reduced to one order of magnitude (Fig. IV.14 and IV.15) by further reducing Si dimensions (Lb = 1 nm, Lw=2 nm). These results can be attributed to the values of the tunnel lifetime τ (being ∼10-14s in GaAs, ∼10-10 s in the first Si-based structure and ∼10-13 s in the second one).

1015

1014

1013

1012

1011

1010

109

108

10-3 10-2 0.1 1 10 100

fRC

fRC

fRLC

fRLC

GaAs/AlAs

Si/HfO2

G-1Si/HfO2G-1

GaAs/AlAs

Fre

quen

cym

ax (

Hz)

Contact resistance (Ω)

1015

1014

1013

1012

1011

10-4 10-3 10-2 0.1 1 10

Fre

quen

cym

ax (

Hz)

Contact resistance (Ω)

G-1Si/HfO2

G-1GaAs/AlAs

fRC

fRC

fRLC

fRLC

GaAs/AlAs

Si/HfO2

Fig IV.14: Comparison between the intrinsic cut off frequency versus series resistance of AlAs/GaAs (Lb=1.7 nm, Lw=4.5 nm) computed with RC and RLC equivalent circuit.

Fig IV.15: Comparison between the intrinsic cut off frequency versus series resistance of AlAs/GaAs (Lb=1.7 nm, Lw=4.5 nm) and HfO2/Si RTD (Lb = 1 nm, Lw=2 nm).

However, these dimensions for well and barriers thicknesses are very aggressive, and barely achievable using today’s technology (in particular, below 5 nm, silicon body thicknesses achieved by smart technologies are not very well controlled). It would be interesting to maintain reasonable layers thicknesses, using other technological solutions. An option to do so would be to use biaxially strained silicon, as presented in the next section.

Chapter IV: Results & Discussion

127

IV.5. STRAINED SILICON BASED RTD The peak-to-valley current ratio, another critical figure of merit of RTD, requires good confinement properties within the quantum well. To this respect, the six valleys of the Si conduction band and mass effective values (ml = 0.9 and mt = 0.2) leads to a larger number of resonant levels within the well, compared to GaAs for instance (one single valley with m = 0.067), prejudicial for RTD application.

In order to keep reasonably thick silicon thickness (thicker than 4 nm), one possible technological solution consists in using biaxially strained silicon (s-Si) substrate instead of relaxed silicon.

A biaxially strained silicon (s-Si) substrate can be realized by growing a Silicon layer on the top of an epitaxial relaxed Si1-x Gex substrate, which induces a tensile strain on silicon layer. This strained silicon layer can then be reported on insulator to achieve a strain SOI structure. [Tezuka01], [Maleville04]

This strain causes modification of the electron conduction band: the six-fold degenerated valleys of the unstrained silicon conduction band splits into two groups of valley with different conduction bands: one 2-fold degenerated, the other 4 fold degenerated [Takagi96]. These two groups of valleys do not have the same quantification masses.

The splitting energy between the two-degenerated valleys and the four others valleys is given by :

eVxEstrain 67.0=Δ

Where x is the Ge concentration in the Si1-x Gex substrate used to induce the strain.

Fig IV.16: The splitting energy between the two and four-degenerated valleys.

This effect is usually used to improve electron mobility in a MOSFET channel. Indeed, electrons mostly occupy the lower valleys (energetically favoured), valleys which have a lower effective mass in the transport direction.

Chapter IV: Results & Discussion

128

Fig IV.17: Constant-energy ellipses for unstrained Si and strained Si [Takagi96].

For RTD application, we believe that strain may be useful to increase the splitting between quantized 2 fold sub-band (named L energy levels) and 4 fold subband (named T energy levels).

As shown on Fig. IV.18 and IV.19, this solution significantly reduces the number of resonant peaks in the I-V characteristics at same body thickness.

0 0.2 0.4 0.6 0.8 1

Voltage V − VFB (V)

107

108

109

1010

1011

Cur

rent

Den

sity

(A

/m2 ) HfO2/Si

Well = 5 nmWell 2nm

Well 3nm

Barrier 1nm

107

108

109

1010

1011

0 0.2 0.4 0.6 0.8 1

Voltage V − VFB (V)

Cur

rent

Den

sity

(A

/m2) HfO2/s-Si Barrier1nm

Well = 5 nm

Well 2nmWell 3nm

Fig IV.18: Well width impact in Si/HfO2. Fig IV.19: Well width impact in HfO2/s-Si.

Finally, the performances of HfO2/s-Si RTD are compared with conventional AlAs/GaAs heterostructure with same barrier and well dimensions. As expected, GaAs structure offers two orders of magnitude higher peak currents (Fig.IV. 20), and maximum operating frequency (Fig. IV.21)

Chapter IV: Results & Discussion

129

0.2 0.4 0.6 0.8 1105

106

107

108

109

1010

1011C

urre

nt D

ensi

ty (

A/m

2 )

Voltage V - VFB (V)

HfO2/s-Si

AlAs/GaAs

b= 1.7nm; well =4.5nm

0.2 0.4 0.6 0.8 1105

106

107

108

109

1010

1011C

urre

nt D

ensi

ty (

A/m

2 )

Voltage V - VFB (V)

HfO2/s-Si

AlAs/GaAs

b= 1.7nm; well =4.5nm

0.01 0.1 1 10 100

barriers = 1.7nm

well = 4.5nm

Contact resistance (Ω)

HfO2/s-Si

AlAs/GaAs

10-3108

109

1010

1011

1012

1013

1014

1015

Intr

insi

c C

ut O

ff F

requ

ency

f (

Hz)

0.01 0.1 1 10 100

barriers = 1.7nm

well = 4.5nm

Contact resistance (Ω)

HfO2/s-Si

AlAs/GaAs

10-3108

109

1010

1011

1012

1013

1014

1015

Intr

insi

c C

ut O

ff F

requ

ency

f (

Hz)

Fig IV.20: Comparison between I-V characteristics of AlAs/GaAs and HfO2/s-Si RTD devices with same dimension (Lb=1.7 nm, Lw=4.5 nm).

Fig IV.21: Comparison between the intrinsic cut off frequency (defined in [14]), versus contact series resistance of AlAs/GaAs and HfO2/s-Si RTD with the same dimension (Lb=1.7 nm, Lw=4.5 nm).

However, using the strain and High K boosters, the performance of s-Si RTD of reasonably thin dimensions (barrier 1.7 nm and well 4.5 nm thick) may be sufficient for circuit application, as investigated in the next section.

Chapter IV: Results & Discussion

130

Structures Geometrical parameters Physical parameters

Si/SiGe [Berash07]

well 4nm m* ml=0.92m0 mt=0.19m0

Mox mΔ2=0.92m0 mΔ4=0.19mo

barrier 3.5nm Φox Δ2= 0.036eV Δ4= 0.09eV

SiO2/Si/SiO2 [Ishi01]

well 2nm m* ml=0.92m0 mt=0.19m0

Mox 0.5m0

barrier 1nm Φox 3.1eV

HfO2/Sistrain/HfO2 well 2nm m* ml=0.92m0 mt=0.19m0

Mox 0.2m0

Φox 1.8eV

barrier 1nm

strain 0.210eV

AlAs/GaAs/AlAs [North98]

well 4.5nm m* 0.063m0

barrier 1.7nm Mox 0.123m0

Φox 0.3eV

HfO2/Si/HfO2 well 4.5nm/2nm m* ml=0.92m0 mt=0.19m0

barrier 1.7nm/1nm Mox 0.2m0

Φox 1.8eV Table 2: Material parameters used in simulations (see annex for details about effective mass approximation for in oxide

barriers)

Chapter IV: Results & Discussion

131

IV.6. SILICON RTD: CONCLUSION

An original analytical physically-based model for coherent tunneling in Resonant Tunneling Diodes has been developed, allowing an efficient estimation of device performances (current peaks, and maximum operating frequency) versus material properties and device dimensions.

Comparing Hf02/Si/HfO2 RTD with III-V and SiGe heterostructure, it has been found that GaAs and SiGe have comparable performances, and are materials more suitable for RTDs application than double gate like Si structure, even in the assumption of full coherent tunneling.

This is mostly due to the heavy effective masses of Silicon compared to GaAs, and the low transparency of SiO2 and HfO2 tunnel barriers.

The use of strained silicon substrate can be an option to reduce the too large number of resonant peaks in silicon wells, and slightly improve the peak-to-valley current ratio (but obviously, it can not suppress the lack of efficiency of SiO2 and HfO2 tunnel barriers).

It is of course possible to compensate these drawbacks of Silicon compatible materials, by using ultra thin silicon well and HfO2 barriers, but the feasibility, reliability and variability of such ultra thin device are questionable.

The figure IV.22 summarizes the I-V characteristics of the structures studied in this section:

0,0 0,2 0,4 0,6 0,8

105

106

107

108

109

1010

1011

Cur

rent

den

sity

(A

/m2 )

Voltage-Vfb (V)

Si/HfO2 GaAs SiGe Si strain Si scaled

Fig IV.22: GaAs/AlAs, Si/SiGe, Si/HfO2, Si-s/HfO2, Si scaled/HfO2.

The frequency performances of GaAs and Si RTD have been also investigated. The intrinsic frequency of the optimized Si/HfO2 architecture may reach the THz range, but still remains below the GaAs/AlAs performance. We have thus demonstrated that double gate like silicon RTDs are not as efficient as GaAs devices. However, it may still be possible to use them in oscillator circuits, and achieve performances that may remain competitive with conventional oscillators. This perspective is investigated in the next section.

Chapter IV: Results & Discussion

132

IV.7. RF OSCILLATOR BASED ON STRAINED-SILICON RTD DEVICES In this section, the feasibility of a RF oscillator at 20 GHz (for UWB radar application for instance) based on silicon RTD is investigated, and compared with existing solutions. A strained silicon double gate RTD has been considered in this study, using reasonably thin dimensions (barrier 1.7 nm and well 4.5 nm thick). Several kinds of circuit architectures have been considered.

IV.7.1 BASIC DIFFERENTIAL OSCILLATOR

The standard differential oscillator circuit consists in a complex architecture, where the negative resistance, to compensate oscillator losses and to maintain oscillation, is obtained by means of a feedback loop (e.g. using a cross-coupled transistor pair [Bao04]) (see Figure IV.23).

Figure IV.23: Cross-coupled CMOS oscillator.

This circuit can be considerably simplified using NDR devices such as RTD (see Fig.IV. 23), leading in principle to a reduction of the oscillator circuit size (using a lower number of devices), a reduction of the power consumption and higher frequency performances.

Fig.IV.24 presents the basic core oscillator architecture based on a differential structure. With a bias point located in the NDR region of the RTD I-V characteristic (Fig. IV.25), an instability leading to an oscillation frequency determined by the passive elements: the inductance resL , sR and capacitor

resC can be achieved.

Chapter IV: Results & Discussion

133

0 0.2 0.4 0.6

bias point

0.15

0.1

0.05

0

Voltage (V)

Cur

rent

(m

A)

HfO2/s-Si

0 0.2 0.4 0.6

bias point

0.15

0.1

0.05

0

Voltage (V)

Cur

rent

(m

A)

HfO2/s-Si

Fig. IV.24: Differential oscillator architecture. Fig. IV.25: I-V characteristics of the RTD used in this

work, showing the DC bias point.

These values can be easily calculated, transforming the oscillator circuit in Fig. IV.24 in a parallel cell in Fig IV.26, using the following equations:

2 2p p

s 2 2p p

L RR

R (L )

ω=

+ ω and

2p p

res 2 2p p

L RL

R (L )=

+ ω

The oscillation conditions are given by : n pR R= and osc p p1/ L Cω =

where Rn, Rp, Lp, Cp represent respectively the negative resistance, the equivalent parallel resistance, the equivalent parallel inductance, and the equivalent parallel capacitance.

Figure IV.26: Parallel small signal equivalent oscillator.

The analytical model detailed in the section II has been implemented in Verilog-A and introduced in Cadence environment to simulate the oscillator in Fig. IV.24.

The simulated structure is a RTD with HfO2 barrier (1.7 nm) (without SiO2 interfacial layer) and strained silicon well (4.5 nm) with a surface of 1.6 μm2.

The current peak is 0.14 mA, obtained for a voltage of 0.5 V, the negative resistance has been found equal to ~122 Ω and the peak-to-valley ratio is ~17.

We choose an oscillation operating frequency of 20 GHz, fixing the value of Cres at 0.5 pF and Lres= 128 pH, (assuming a 1 kΩ load terminal).

In this work, polysilicon contacts have been considered, and the corresponding contact resistance is around 2 Ω, according to [ITRS07].

Chapter IV: Results & Discussion

134

As it turns out from the AC simulation of this structure (Fig. IV.6), the RTD intrinsic cut off frequency is much larger than the oscillator operating frequency, so the intrinsic passive elements of the RTD device do not limit the operating frequency at 20 GHz in this case.

By circuit simulation, we found that at 20 GHz oscillation frequency, the RF output power is -44 dBm (Fig. IV.27) and the static power consumption lower than 700 μW.

0 20 40 60 80 100-120

-100

-80

-60

-40

-20

0

Frequency (GHz)0 20 40 60 80 100

-120

-100

-80

-60

-40

-20

0

RF

Out

put P

ower

(dB

m)

Frequency (GHz)

Fig. IV.27: RF output power.

Reference RF output

power (dBm)

Static power consumption

(mW) s-Si RTD

based (this work)

-44 0.7

[Bao04] -17 130 [Zhan04] -37 22

Table 3: Differential Oscillator Performances.

Table 3 shows a comparison between conventional differential oscillator performances and the RTD based differential oscillator considered in this work.

As expected, It turns out that RTD differential oscillator offers a drastic power consumption reduction (divided at least by 30).

However, the RF output power remains relatively low -44 dBm, which might be prejudicial, depending on the application.

This low output power is related to the low negative resistance of the s-Si RTD. A parametric study on the VCO performances has shown that a higher negative resistance leads to increase the output power [Medjahdi07].

To overcome this limit, another architecture using RTD, the “Muramatsu” oscillator [Muramatsu05] has been considered in the next section.

Chapter IV: Results & Discussion

135

IV.7.2 MURAMATSU OSCILLATOR

More complex circuit oscillators that couple active devices, such as HEMT transistor [Muramatsu05], HBT [Delossan01], and RTD, have been realized with conventional III-V based RTD.

In this circuit configuration, oscillation is obtained associating the transistor load line at the switching from peak current to the valley current in the RTD I-V characteristics. In this way, larger output voltage amplitude can be achieved, exceeding the output power limitation of conventional RTD based oscillator (such as basic differential oscillator of section IV.7.1).

Following the oscillator circuit proposed by Muramatsu et al. [Muramatsu05], the oscillator of Fig.IV.28 that couples a MOSFET (0.25 μm technological node) with strained silicon based RTD (structure detailed in the previous section) has been investigated.

A 22 GHz oscillation frequency has been obtained using a NMOS transistor (W = 30 µm, L = 0.25 µm) and an inductance of 3.5 nH.

Simulations of this circuit with strained silicon RTD have been carried out to estimate performances and to compare them with a circuit based on GaAs/AlAs RTD.

Table 4 summarizes results obtained comparing simulations of these two oscillators. A good output voltage magnitude and low static consumption (higher that in the previous architecture, due to the use of active transistor) have been obtained using strained silicon RTD.

As GaAs RTD offers a higher current density (Fig.IV.20), higher output voltage amplitude has been obtained using this technology. However, compared to conventional silicon architecture, silicon RTD based circuits can offer significant power consumption reduction, with reasonable output voltage amplitude, making this technology very attractive for ultra low power RF oscillator.

0 0.2 0.4 0.6

0.15

0.1

0.05

0

Voltage (V)

Cur

rent

(m

A)

HfO2/s-Si

0.80 0.2 0.4 0.6

0.15

0.1

0.05

0

Voltage (V)

Cur

rent

(m

A)

HfO2/s-Si

0.80 0.2 0.4 0.6

0.15

0.1

0.05

0

Voltage (V)

Cur

rent

(m

A)

HfO2/s-Si

0.8

Fig IV.28: Muramatsu oscillator. Fig. IV.29: I-V characteristics of the RTD used in this work, showing the DC bias points for the Muramatsu oscillator.

Chapter IV: Results & Discussion

136

Structure RF output

power (dBm)

Static power consumption

(mW)

III V

-9

7.8

Strained Si(this work)

-1.9 2.5

Table 4: Muramatsu Oscillator. In conclusion, even if the performances of double gate silicon RTD are lower than their III V counterparts, circuit simulations have shown that such technology still has an interest for RF oscillator application. In addition, the weak output power of Differential Oscillator RTD circuit can be compensated using hybrid architecture, coupling transistor and RTD (called the Muramatsu oscillator).

Due to their ultra thin dimension, Resonant Tunneling device and circuit are expected to be sensitive to variability. For this reason, the impact of dimension variations on Muramatsu performances has been investigated in the next section. IV.7.2.1 MURAMATSU OSCILLATOR: VARIABILITY As explained in the previous sections, very thin barriers and well layers are needed in order to achieve significant performance of silicon RTD.

Realizing these dimensions is very challenging and it is reasonable to think that oxide barriers and well thicknesses should be affected by variability.

It may be interesting to study the impact of barrier and well thickness variability on the oscillator performances, for example on the RF output power. This analysis has been carried out in the next section considering the most promising architecture, the Muramastsu oscillator.

Chapter IV: Results & Discussion

137

• Impact of the barrier thickness variability

The impact of left and right barrier thicknesses variability between 1 nm and 2 nm have been investigated.

The transmission coefficient of a simple barrier in pure tunneling regime varies exponentially with the barrier thickness: Therefore, increasing thickness reduces the transmission and consequently the current peak decreases as shown in the fig IV.30.

1,0 1,2 1,4 1,6 1,8 2,0

1,6x108

1,8x108

2,0x108

2,2x108

2,4x108

2,6x108

2,8x108

3,0x108

3,2x108

0,0

5,0x108

1,0x109

1,5x109

2,0x109

2,5x109

3,0x109

Pe

ak

cu

rre

nt

for

dr

va

ria

ble

(A

/m2

)

dl, dr (nm)

Peak cu

rrent fo

r dl variab

le (A/m

2)

Fig. IV.30: Peak current versus right dr or left dl barrier thickness in the range 1nm -2nm @0.49V.

It is interesting to note that the peak current is more affected by the thickness variation of the left than the right barrier.

For the voltage corresponding to the peak current (0.49V), the dependency on barrier thickness of right and left transparency is shown in fig IV.31:

1,0 1,2 1,4 1,6 1,8

1E-4

1E-3

0,01

Tra

nsm

issi

on

co

eff

icie

nt

dr,dl (nm)

Tr 0.49V Tl 0.49V

Fig. IV. 31: Variation of the transparency of the left barrier for an applied voltage of 0.49 V.

Chapter IV: Results & Discussion

138

This higher sensitivity of the current peak to the left carrier can be explained as follows: first, let us note that the transmission coefficient of the second barrier is higher than the transmission coefficient of the left one. Indeed, taking into account the voltage drop across the device, the effective barrier of the second barrier (φ - V/2 – E) is lower than the first one.

As the current peak is approximately proportional to:

rl

rlpeak TT

TTI

Thus, when Tr> Tl, Ipeak depends only on Tl. In consequence, Ipeak is more sensitive to the variation of the left barrier thickness than the right barrier. Of course, this result remains true only when Tr> Tl, which stops to be valid at very small left barrier.

The RF output power variation follows the current behavior, it is more affected by the left barrier variability (Fig.IV.32) than by the right one (Fig.IV.33).

1,0 1,2 1,4 1,6 1,8 2,0

-2,0

-1,8

-1,6

-1,4

RF

ou

tpu

t p

ow

er (

dB

m)

well = 4.5 nmdl = 1.5 nm

dl (nm)

1,0 1,2 1,4 1,6 1,8 2,0

-1,95

-1,94

-1,93

-1,92

-1,91

-1,90

RF

ou

tpu

t p

ow

er

(dB

m)

dr (nm)

well = 4.5 nmdr = 1.5 nm

Fig.IV.32: RF output power for dl variable between 1 nm and 2 nm.

Fig. IV.33: RF output power for dr variable between 1 nm and 2 nm.

However, this oscillator architecture appears surprisingly quite robust to the variability on RTD barriers.

In fact, for variation of 10% in the left barrier thickness, from 1 nm to 1.1 nm, the RF output power varies only by 1.74% and for a 100% variation in thickness from 1nm to 2nm the RF output power shows a variation of only 10%.

Instead, for a variation of 100% in the right barrier, the RF output power varies by only less than 1%.

• Impact of the well width variability

Increasing well width leads to peak current reduction and some variation in PVCR, due to the merging of several resonant levels.

As Muramatsu oscillator takes his operation point switching from peak current to the valley current in the RTD, RF output power decreases increasing well width as shown in fig.IV.34.

Chapter IV: Results & Discussion

139

4,2 4,3 4,4 4,5 4,6 4,7 4,8

-1,918

-1,916

-1,914

-1,912

-1,910

-1,908

-1,906

-1,904

-1,902

-1,900

RF

out

put

pow

er

(dB

m)

well (nm)

RF output power

Fig. IV.34: RF output power versus well th ickness (4 .2 nm to 4 .8 nm).

In this case for a variation of the well between 4.5nm and 4.2nm, 6.66%, the RF output power (in dBm) varies only by 0.21%.

In both case of barrier and well variability, this oscillator is not very sensitive at small variations in layer thickness, due to the presence of transistor in this architecture IV.7.3. OSCILLATOR BASED ON STRAINED-SILICON RTD DEVICES: CONCLUSION The device compact model presented in Chapter II has been programmed in Verilog A and has been implemented in a circuit simulator (Spectre in Cadence framework).

Using this model, the performances of two RF oscillator architectures have been investigated: the differential oscillator and the Muramatsu oscillator, a more complex circuit that couples transistor and RTD.

It turns out that RTD differential oscillator can operate at 20 GHz, with a significant power consumption reduction compared to conventional non RTD solution (divided at least by 30).

The drawback of the differential oscillator is the low RF output power that can be prejudicial for applications at higher consumption.

To overcome the low output power issue of RTD differential oscillators, other architecture including active MOS transistor can be also realized, such as the “Muramatsu” oscillator. In this case, power consumption increases of one order of magnitude, remaining still very competitive compared to non RTD solution.

The impact of variability on barrier and well layer has been investigated for the last circuit. From parametric simulations, it turns out that variability on barriers thickness has not a relevant effect on

Chapter IV: Results & Discussion

140

oscillator performances. Interestingly, the RF output power has been found more affected by the left barrier variation than the right one.

The robustness to the variability of the Muramatsu oscillator is an interesting result. Indeed the differential architecture, having its operation point in the negative resistance is very sensitive at the variability of this parameter as shown in [Madjadi07]. Variability in the layer thickness leading variation of negative resistance could be prejudicial for this kind of architecture.

In summary, despite the lower performance of silicon based RTD compared to III V materials, circuit simulations using hybrid technologies combining RTD and transistors, have shown that in principle such a technology can still be competitive in term of layout efficiency and output power consumption, compared to a conventional silicon oscillator.

REFERENCES OF CHAPTER IV

[Bao04] M. Bao et al. A 21.5/43-GHz Dual-Frequency Balanced Colpitts VCO in SiGe Technology. IEEE Journal of Solid-State Circuits, Vol. 39, No. 8, 2004.

[Berash07] Berashevich et al. “Resonant tunneling versus thermally activated transport through strained Si1−xGex⁄Si⁄Si1−xGex quantum wells” Phys.rev.b 2007.

[Coignus09] J. Coignus, Clerc, R. Leroux, C. Reimbold, G. Ghibaudo, G. Boulanger, F.Analytical modeling of tunneling current through SiO2–HfO2 stacks in metal oxide semiconductor structures, J. Vac. Sci. Technol. B 27(1) Jan/Feb 2009, p. 338.

[DelosSantos01] H. J. De Los Santos, K. K. Chui, D. H. Chow, H. L. Dunlap, “An Efficient HBT/RTD Oscillator for Wireless Applications” IEEE Microwave and Wireless Components Letters, Vol. 11, pp. 193-195 (2001).

[Ishikawa01] Y. Ishikawa, T. Ishihara, M. Iwasaki and M. Tabe, “Negative differential conductance due to resonant tunnelling through SiO2/single crystalline-Si double barrier structure” Electronics Letters 13th September2001 Vol. 37 No. 19.

[ITRS07] International Technology Roadmap for Semiconductors 2007 Ed., http://www.itrs.net/.

[Maleville04] C. Maleville, C. Mazuré “Smart-Cut® technology: from 300 mm ultrathin SOI production to advanced engineered substrates”, Solid-State Electronics, Volume 48, Issue 6, June 2004, Pages 1055-1063.

[Medjadi07] A. Medjahdi, F. Calmon, N. Baboux, L. Becerra, A. Poncet, « Vers des oscillateurs intégrés e technologie silicium à base de composants à résistance négative », 15èmes Journées Nationales Microondes, Toulouse, (mai 2007).

[Muramatsu05] Muramatsu, H. Okazaki, T. Waho, “A novel oscillation circuit using a resonant tunneling diode”, ISCAS 2005 proceedings, pp. 2341 – 2344.

[North98] A. J. North E. H. Lienfield, M.Y. Simmons, et al., “Electron reflection and interference in the GaAs/AlAs-Al Schottky collector resonant-tunneling diode” Physical review B, Vol. 57, n°3, pp. 1847-1854 (1998).

[Paul02] D.J. Paul , P. See, I.V. Zozoulenko , K.-F. Berggren, B. Holla¨nder , S. Mantl, N. Griffin, B.P. Coonan, G. Redmond, G.M. Crean “n-type Si/SiGe resonant tunnelling diodes” Materials Science and Engineering B89 (2002) 26–29.

Chapter IV: Results & Discussion

141

[Tazagi96] S. Takagi, J. L. Hoyt, J. J. Welser, and J. F. Gibbons “Comparative study of phonon-limited mobility of two-dimensional electrons in strained and unstrained Si metal–oxide–semiconductor field-effect transistors” J. Appl. Phys. 80 (3), 1 August 1996, pagg.1567-1576.

[Tezuka01] T. Tezuka, N. Sugiyama, S. Takagi, “Fabrication of strained Si on an ultrathin SiGe-on-insulator virtual substrate with a high-Ge fraction” Appl. Phys. Lett. 79, p. 1798 (2001).

[Vinet05] M. Vinet, T. Poiroux, J. Widiez et al. “Bonded Planar Double-Metal-Gate NMOS Transistors Down to 10 nm” IEEE Electron Device Letters, Vol. 26, n°5, pp. 317-319 (2005).

General conclusion

General conclusion

143

GENERAL CONCLUSION In this dissertation, our researches about silicon based Resonant Tunneling Diode (RTD) have

been presented.

In the first chapter of this thesis, an overview of the main realizations of RTD has been summarized. Thanks to epitaxial technologies allowing a control of semiconductor heterostructures at the nanometric scale, RTDs have been mostly elaborated with III–V materials and technologies so far. In addition, because of the difficulty to integrate these materials on silicon, in attempt to find solutions compatible with the mainstream silicon technology, SiGe based RTD have been also investigated, reporting satisfactory results (for hole, and more recently for electron also).

Several works have been focused to the realization of RTD with silicon well and different materials for the barriers. However, it is very challenging to find materials having a good mismatch with silicon and to grow them with a good interface properties and crystallinity.

Nowadays, the amazing progress of SOI wafer technology has made possible to realize Si quantum wells of dimension lower than 10 nm. Such substrates have already been used to process double gate MOSFET, leading to a heterostructure very close to a typical RTD. In this context, the objectives of this work have been to investigate the theoretical performances of such a “double gate like” silicon RTD, at the device and circuit level as well.

To this aim, an original physically-based analytical model has been elaborated. This model is presented in the second Chapter of this manuscript, after a detailed review of numerical and analytical RTD modeling. This review has shown on one hand the complexity of a perfect numerical model of RTD, and on the other hand the poverty of existing RTD compact models.

In an attempt to fulfill the gap between this two kind of approaches, our model approximations is based on the following approximations: 1) Transport is supposed coherent (ballistic), 2) Band structure is modeled using the effective mass approximation, 3) Only electron tunnel current is considered (in the framework of the single particle approximation) 4) self consistency between electrostatics and carrier transport is neglected. The current is computed following the Tsu-Esaki approximation and the transparency is approximated by an improved Lorentzian function. Our approach extends the previous compact models based on the Lorentzian function, by including the left and right transparency variations with energy and field. This point has been found particularly critical to model accurately the current around the peak.

As shown in the last part of this chapter, the model has been benchmarked with numerical simulations and the two approaches have been found in qualitative good agreement, without introducing any fitting parameters. The analytical model can reproduce with a good precision the peak current (which is one of the major figure of merit of RTD) and predict the presence of the several resonant levels,. It however tends to underestimate the valley current, and in absence of scattering and non parabolicity effects, can not accurately predict the negative resistance.

General conclusion

144

In the Chapter III, an original extension of DC model into the AC regime has been presented. Starting from the Liu’s small signal equivalent circuit model, each elements (conductance, capacitance …) have been derived using the DC model, allowing to predict the intrinsic cut off frequency of the RTD as a function of geometrical and physical parameters. In the second part of this chapter are reported the simulations carried out from the self consistent numerical model based on Bohm trajectory elaborated by the Xavier Oriol’s team at the UAB: a time-dependent Quantum Monte Carlo (QMC) based on many-particle Bohm trajectories .This powerful simulator can include Coulomb correlations self-consistently and has been used to extract the RTD intrinsic frequency limitation directly from the current response to a small step voltage signal.

By applying the two approaches, we noticed that that several limitations come into play in the frequency response of an RTD, namely the intrinsic tunneling process, the transit time across the non-tunneling regions (leads and contacts) and time constant associated to the total capacitance of the structure.

On the one hand the full time-dependent simulation provides a rigorous picture of the physics that governs the frequency behavior of RTD. These simulations allowed noticing that the Coulomb interactions tend to slow down the dynamic of the device and to interpreted interesting phenomena occurring in these conditions.

On the other hand, our study shows that the equivalent small signal circuit is able to catch characteristic times, resulting as a useful tool to design RTD.

In the DC and AC regime as well, despite the intrinsic limitations of our approach, consequences of the simplifying approximations used to make the model as compact as possible, we believe that one of the important advantages of our model is to allow a “best case” estimation of RTD performances. In consequence if the performances predicted with our model are not sufficient to achieve a given target, we are sure that experiments would be even worse. A second advantage of our approach is also to allow to benchmark device performance versus material properties and device dimensions, a critical point to investigate the feasibility of silicon based RTD. Finally, our approach can be introduced into a circuit simulator, allowing an estimation of performance at the circuit level, and not only at the device level.

The Chapter IV gathers the main results of this work. In the first part, the AC-DC analytical is applied to investigate the performances of silicon based “double gate like” RTD and to compare them with the one of III-V and SiGe based devices. It turns out that compared to III-V and SiGe RTD of the same dimensions (4.5nm for the well and 1.7nm for the barrier thicknesses), double gate like Si/HfO2 structure offers poor performances. III-V heterostructure remains the more suitable material for RTD applications. This is mainly due to the higher transparency of right and left AlAs tunnel barrier compared to HfO2 and the isotropic light effective masses of GaAs (0.063 instead of 0.92 and 0.19 for Si anisotropic conduction band), which significantly reduces the number and interference of resonant levels in the GaAs well.

In theory, a strong reduction of the layer thicknesses of the silicon “double gate like” device (2nm for well and 1nm for the barrier) may allow to reach the performances of GaAs / AlAs RTD. These dimensions are however technologically very challenging to realize.

General conclusion

145

In order to keep reasonable dimensions for well and barriers, we have proposed to use strained silicon instead of conventional silicon as quantum well material. At same width, strained silicon would present less resonant levels, and thus would improve the peak current and Peak to Valley Current Ratio.

However, despite the lower performances of “double gate like” silicon RTD compared to III V, it may be enough to achieve RF oscillator competitive with conventional silicon similar circuits. For this reason, the performances of a 20 GHz oscillator based on silicon RTD have been investigated, focusing on two different architectures.

The first circuit is simpler and is based on a differential architecture. The second one is more complex and couples one RTD and a MOSFET. From circuit simulations, it turns out that RTD differential oscillator can operate at 20 GHz, with a huge power consumption reduction compared to conventional non RTD solution (divided at least by 30). Unfortunately, this architecture presents very low RF output power, prejudicial for many applications. However, this limitation can be overcome using the second architecture that couple a transistor MOS and a RTD, such as the “Muramatsu” oscillator. In this case, the RF output power is higher and very competitive compared to non RTD solution, maintaining lower power consumption.

Due to the ultra-thin layers needed for RTD fabrication, the application based on RTD may suffer from variability. In Chapter IV, the results of variation of barriers and well thicknesses on the RF output power of Muramatsu oscillator have been investigated. Surprisingly, this architecture has not been found too much affected by variability.

In conclusion, these encouraging results may open the way for future application of Si-strain based RTD radiofrequency oscillators. If silicon is clearly not the best material to enhance resonant tunnelling effects, for several reasons explained in our work, circuits using such device may still be competitive in term of power consumption compared to conventional oscillators. In addition, the creativity of circuit design (introducing for instance a Muramatsu oscillator instead of standard differential oscillator) may help compensating the weakness of silicon as RTD material.

Annex: Discussion on the effective mass approximation

Annex: Discussion on the effective mass approximation

147

ANNEX: DISCUSSION ON THE EFFECTIVE MASS APPROXIMATION To perform the simulations presented in chapter 4, the parameters in table 2 have been used.

Tunneling simulation has been performed in the framework of the effective mass approximation in HfO2 and SiO2 oxides.

In this paragraph the validity and limitations of this approximation are discussed.

The band structure of a dielectric is real in the valence and conduction band and imaginary in the gap. This part represents the evanescent behavior of carriers in the barrier and thus the tunnel current. It can be seen in Figure A.1 that within the gap, the imaginary relation dispersion has a parabolic shape only close to the valence and conduction band.

Fig A.1: Dispersion relationship in HfO2 in cubic phase (according [Sacconi07]).

In principle, the impact of the non parabolicity of imaginary dispersion relationship can only be modeled by the mean of numerical simulation, including Full Band effects in quantum transport [Sacconi07]. However, several analytical approaches have also been developed in order to take into account the dispersion relationship as accurately as possible. It has to be noted that these analytical expressions developed to take into account the imaginary part of dispersion relationship in the gap introduce more fitting parameters.

Parabolic approximation:

The parabolic approximation is the most applied approach and has been used in this manuscript. The dispersion relationship is completely represented by two parameters, effective masses (one for electrons in conduction band and one for the holes in valence band) and the band offset between oxide and silicon.

EmEk ez 21

)(= (A.1)

Annex: Discussion on the effective mass approximation

148

While in Silicon, the parabolic approximation is applied to compute charge in conduction and valence band (real part of the dispersion relationship), in oxide, this approach allows to model the electron tunnel current for a constant energy value computed from material conduction or valence band (imaginary part).

Tunnel current

Conduction band

Energy (eV)

Real part k (u,a) Imaginary part k (u,a)

Fig A.2: Schematic dielectric dispersion relationship and parabolic approximation taking into account the conduction band and tunnel current using mass as fitting parameter.

The parabolic approximation is thus expected to present some limitations, in particular when carriers tunnel in the middle of the gap. This occurs in particular in ultra thin barriers, where direct tunneling occurs.

Relation de Franz:

To better reproduce the non parabolic dispersion relation especially in the middle of the gap, the Franz relation dispersion has been introduced. This approach takes into account the electrons and holes transport by means of a single effective mass mf, the Franz mass. The dispersion relationship considers two symmetric bands and is expressed as function of the gap energy:

−=

gfz E

EEmEk 12

1)(

(A.2)

Non symmetric relation:

In order to take into account the dissymmetric characteristic of imaginary part of dispersion relationship in the gap, two values for masses are introduced: mns,e and mns,h. These two parameters allow weighting the influence of valence and conduction band on the imaginary part of the dispersion relationship.

1

,

,, 1112

1)(

−−

−=

ghns

ens

gensz E

E

m

m

E

EEmEk

(A.3)

Annex: Discussion on the effective mass approximation

149

The parabolic dispersion relationship, Franz and non symmetric has shown for the case of SiO2 in the Figure A.3.

Fig A.3: Typical shape of the parabolic, Franz and non symmetric tunnelling dispersion relation.

In the case of SiO2, non parabolicity effects have been reported to occur at thicknesses lower than 2 nm. In this case, nor Franz neither Asymmetrical dispersion relation correctly model the impact of non parabolicity. Instead, energy dependent effective masses have been proposed on the basis of numerical and experimental results [Simonetti02].

In HfO2, the situation is less critical [Sacconi07], because HfO2 layer are typically thicker than SiO2 dielectric, and because the gap of HfO2 is smaller. In this later case, parabolic dispersion relation has been experimentally proven to be accurate enough for electron tunneling [Coignus10].

In consequences, in this work, the parabolic approximation has been applied in the case of electron tunnel current.

[Simonetti02] O. Simonetti, T. Maurel, and M. Jourdain, “Characterization of ultrathin metal–oxide–semiconductor structures using coupled current and capacitance–voltage models based on quantum calculation,” Journal of Applied Physics, vol. 92, no. 8, pp. 4449–4458, 2002. [Sacconi07] F. Sacconi, J. Jancu, M. Povolotskyi, and A. Di Carlo, “Full-Band Tunneling in High-k Oxide MOS Structures,” IEEE Transactions on Electron Devices, vol. 54, no. 12, pp. 3168 –3176, dec. 2007. [Coignus10] J. Coignus, C. Leroux, R. Clerc, R. Truche, G. Ghibaudo, G. Reimbold and F. Boulanger “HfO2-based Gate Stacks Transport Mechanisms and Parameter Extraction” Volume 54, Issue 9, September 2010, Pages 972-978.

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RESUME EN FRANÇAIS

Modélisation analytique et simulation de diode tunnel résonante sur silicium pour application oscillateur radiofréquence

1. Introduction et contexte

La diminution incessante de la taille des composants semi-conducteurs utilisés dans les systèmes électroniques permet d’envisager une rupture en associant ou remplaçant les composants conventionnels (filière CMOS) par des composants nanométriques avec un transport des charges de nature quantique ou balistique. La diode tunnel résonante (RTD) (Fig.1), présente des caractéristiques très intéressantes et peut répondre à un grand nombre d’attentes: sa fréquence de coupure intrinsèque élevée doit permettre de travailler à de très hautes fréquences (possibilité d’atteindre la valeur théorique du THz), sa caractéristique atypique présente une Résistance Différentielle Négative (NDR) et permet d’envisager une diminution très importante en nombre de composants utilisés pour une fonction électronique donnée (électronique faible consommation).

La littérature propose de nombreuses études de structures à base de matériaux III-V. Egalement, dans l’intention de réaliser des RTD compatibles silicium, des structures à base de SiGe/Si ont été largement étudiées et ont amené des résultats encourageants. Par contre, nous dénombrons peu de travaux sur des RTD en silicium avec de bonnes caractéristiques en termes de résistance négative, courant de pic et PVCR (peak-to-valley current ratio) et leurs applications analogiques.

L’objectif de ce travail de thèse est d’évaluer les performances théoriques de la RTD sur silicium avec des barrières d’oxyde de forte permittivité et de faible épaisseur (comme par exemple HfO2) et d’analyser son utilisation dans une application oscillateur radiofréquence.

Ef emitter

Er0

Er1

Barrier Barrier

Well

Fig.1: Double barrière.

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Pour atteindre cet objectif, les travaux ont concerné: la modélisation physique des dispositifs RTD, la comparaison entre RTD à base de III-V, SiGe et silicium, l’optimisation et la proposition d’une structure compatible silicium et enfin la simulation du comportement du dispositif et de son application oscillateur radiofréquence.

L’originalité de ce projet repose sur une démarche globale (voir Fig. 2), associant la simulation physique poussée de RTD, la modélisation analytique, l’implémentation du modèle dans un environnement de conception de circuits intégrés et finalement l’analyse par simulations de plusieurs circuits de type oscillateur à base de RTD.

Architecturede la RTD(matériaux,

géométrie …)

Simulation physique

Modélisation analytique

Circuit oscillateur(structure)

Cahier des charges(fréquence, puissance,

consommation, bruit de phase etc.)

Étude paramétrique : hauteurs et largeurs des barrières, du puits etc.

Fig. 2: Démarche de la thèse.

2. Etat de l’art

Dans la littérature, nous recensons de nombreux exemples de structures RTD réalisées en matériaux III-V [North98], leur succès étant lié à l’épitaxie qui permet de réaliser des couches très minces avec ces matériaux. Ces dispositifs présentent des caractéristiques intéressantes pour des applications analogiques : convertisseur analogique-numérique, diviseur et multiplicateur de fréquence, oscillateur, [DelosSantos01] et numériques : inverseur, porte logique, mémoires, fonction basée sur le logique multi-niveau [Lin94]. En revanche, il y a peu de travaux sur des RTD compatibles silicium [Kubota06], [Rommell88], [Osten07] et leurs applications. L’évolution des technologies sur silicium, en particulier la technologie DGMOS [Vinet05], pourrait rendre possible la réalisation de structures RTD sur Silicium avec des performances comparables à celles réalisées en technologie III-V. Depuis la formulation du courant tunnel résonant de Tsu-Esaki [Tsu73], plusieurs techniques de modélisation numérique (Matrices de transfert [Ando87], fonctions de Green [Datta95], équation de Wigner [Frens87]…) et analytique (model de Yan [Yan95], de Brown [Chang93] de Schulman [Schulman94] …) ont été élaborées.

Parmi ces travaux, nous avons trouvé de l’intérêt particulier pour le modèle de Schulman [Schulman94]. Ce model simple est très compact, mais il utilise des paramètres d’ajustement extraits à partir de mesures électriques. Au contraire des travaux précédents, le modèle présenté dans notre

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travail de thèse est suffisamment compact pour être introduit dans un simulateur de circuits et considère des variables calculées sur une base complètement physique sans aucun paramètre d’ajustement.

3. Modèle analytique statique et dynamique (petit signal)

Comme indiqué dans le paragraphe précédent, la modélisation analytique de RTD reste un sujet ouvert car, à ce jour, aucun modèle de RTD n’est entièrement satisfaisant (modèle comportemental sans lien avec la physique ou modèle basé sur la physique mais incluant des paramètres d’ajustement, pas de modèle analytique pour le comportement dynamique – petit signal). Ainsi, nous focalisons notre travail sur le développement d’un modèle analytique (compact) le plus complet possible, à insérer dans un environnement de conception de circuits intégrés, qui doit permettre d’établir le cahier des charges des caractéristiques électriques de la RTD pour des performances données de notre circuit oscillateur (fréquence de fonctionnement, puissance, consommation etc.), comme expliqué en figure 2. A partir de la formulation de Tsu Esaki du courant tunnel [Tsu73], nous proposons de calculer, avec une approche physique, la transparence avec une approximation Laurentienne de la transparence quantique, conduisant à une expression analytique du courant :

[ ][ ] ),()(

2/)((exp1

/))((exp1ln

*4)( max3

VETEEkeVVEE

kVEE

h

kmqVI nn

nf

nf ⋅Δ⋅

−−+−+⋅= π

θθθ

Pour obtenir cette formulation les suivantes approximations ont été faites :

1) Le transport est considéré purement balistique

2) Approximation de masse effective

3) Seulement le tunnel d’électron est considéré

Chaque paramètre de cette expression est calculé rigoureusement analytiquement et sur une base totalement physique. Ce modèle balistique (sans interaction) a été validé avec un code de calcul numérique développé à l’INL (Quantix), basé sur la résolution auto-cohérente des équations de Poisson et de Schrödinger.

Nous avons complété le modèle analytique en incluant le comportement dynamique (petit signal – AC), à partir du schéma équivalent R, L, C, G montré en figure 3 [Liu04] dont les paramètres en petit signal sont extraits directement du modèle statique.

LG

C

Rs

Fig. 3: Circuit équivalent de petit signal.

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La partie sur l’étude temporelle de la structure a été approfondi et le circuit équivalent en petit signal a été interprété et validé par des simulations temporelles basées sur les trajectoires de Bohm [Oriols05]. Ce travail a été réalisé dans le cadre d’un séjour de 9 mois à l’Universidad Autonoma de Barcelona, dans l’équipe du Dr. Xavier Oriols.

4. Résultats

4.1 Comparaison RTD en III-V et en silicium et optimisation

Nous avons utilisé notre modèle analytique complet (statique et dynamique) pour étudier les performances statiques et en fréquence des structures RTD en fonction des paramètres géométriques et physiques des différents matériaux et en comparant, par exemple, différents dispositifs à base de matériaux III –V et des solutions compatibles silicium.

Pour augmenter la densité de courant dans la diode, les barrières doivent être très fines et réalisées avec des matériaux à basse hauteur de barrière. Nous avons comparé deux structures avec des barrières de HfO2/Si et SiO2/Si, la première structure présente une plus haute densité de courant, donc l’oxyde HfO2 sera plus adapté que le SiO2.

La supériorité de l’oxyde HfO2 est liée à:

• Une masse effective plus basse: mHfO2 = 0.2 à comparer avec mSiO2 = 0.5

• Une hauteur de barrière plus faible: φHfO2 = 1.8 eV à comparer avec φSiO2 = 3.1 eV

En revanche, comparée à une structure réalisée en III-V ou en SiGe de mêmes dimensions; la structure HfO2/Si présente de moins bonnes performances, comme on peut remarquer en figure 4 et figure 5.

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0,0 0,2 0,4 0,6 0,8

106

107

108

109

1010

1011

Cu

rren

t D

ensi

ty (

A/m

2 )

Voltage-Vfb(V)

Si/HfO2 GaAs SiGe

Fig. 4: Comparaison entre les caractéristiques I-V de structures RTD AlAs/GaAs, HfO2/Si et SiGe de mêmes dimensions. (Lb=1.7 nm, Lw=4.5 nm).

Fig. 5: Comparaison entre les fréquences intrinsèques de structures AlAs/GaAs, HfO2/Si RTD devices and SiGe de mêmes dimensions. (Lb=1.7 nm, Lw=4.5 nm).

Les meilleures performances des diodes GaAs/AlAs peuvent être expliquées par les propriétés physiques supérieures de ces matériaux, à savoir :

1) un plus haut coefficient de transmission des barrières en AlAs en raison d’un plus bas offset de bande entre le GaAs et le AlAs en comparaison avec HfO2 / Si (0.56 eV contre 1.8 eV) et une masse effective des électrons plus faible dans AlAs (0.123.m0 contre 0.2.m0 dans HfO2).

2) moins de niveaux énergétiques et des niveaux plus éloignés les uns des autres dans le puits en GaAs que dans un puits en silicium car la masse effective est plus basse dans le cas du GaAs que dans le silicium (0.063.m0 pour GaAs contre 0.92.m0 et 0.19.m0 pour la bande de conduction anisotrope du silicium).

L’impact des dimensions géométriques de la structure (épaisseur des barrières et largeur du puits) a été aussi évalué:

- En réduisant l’épaisseur des barrières; la densité de courant augmente car la transparence d’une simple barrière dépend exponentiellement de la largeur de la barrière.

- En réduisant la largeur du puits; le nombre de niveaux énergétiques dans le puits est réduit, améliorant ainsi le PVCR.

Il est donc possible, avec une forte réduction des dimensions des barrières et du puits, d’améliorer les performances de la diode en silicium jusqu’à atteindre la densité de courant de la diode III-V et augmenter la fréquence intrinsèque de quelques ordre de grandeur, comme montré dans les figures 5 et 6.

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Fig. 6: Comparaison entre les caractéristiques I-V De Structures RTD AlAs/GaAs et HfO2/Si de différentes dimensions. (Lb=1.7 nm, Lw=4.5 nm Pour AlAs/GaAs and HfO2/Si Lb=1 nm, Lw=2 nm).

1015

1014

1013

1012

1011

10-4 10-3 10-2 0.1 1 10

Fre

quen

cym

ax (

Hz)

Contact resistance (Ω)

G-1Si/HfO2

G-1GaAs/AlAs

fRC

fRC

fRLC

fRLC

GaAs/AlAs

Si/HfO2

Fig. 7: Comparaison entre les fréquences intrinsèques, calcule par un model RC et RLC de structures RTD AlAs/GaAs et HfO2/Si de différentes dimensions. (Lb=1.7 nm, Lw=4.5 nm pour AlAs/GaAs And HfO2/Si Lb=1 nm, Lw=2 nm).

Malheureusement, les tailles de puits et de barrières requises, demandent une technologie très agressive. Il serait souhaitable de maintenir des épaisseurs des couches raisonnables. Pour cette raison, nous proposons d’utiliser du silicium contraint pour le puits, ceci permet d’avoir par rapport à une épaisseur de silicium conventionnel : moins de niveaux résonants dans le puits et des niveaux plus éloignés les uns des autres, permettant d’obtenir ainsi un bon PVCR et d’améliorer la résistance négative.

Dans ce cas, même si les performances de la diode HfO2/Si-strained finales restent inferieures à celles de la diode en GaAs/AlAs (comme on peut voir dans les figures 8 et 9), elles sont suffisantes pour une application oscillateur radiofréquence (par exemple à 20 GHz). Dans la prochaine section, nous résumons les principaux résultats obtenus par simulation au niveau circuit en utilisant notre modèle de RTD.

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0.2 0.4 0.6 0.8 1105

106

107

108

109

1010

1011

Cur

rent

Den

sity

(A

/m2 )

Voltage V - VFB (V)

HfO2/s-Si

AlAs/GaAs

b= 1.7nm; well =4.5nm

0.2 0.4 0.6 0.8 1105

106

107

108

109

1010

1011

Cur

rent

Den

sity

(A

/m2 )

Voltage V - VFB (V)

HfO2/s-Si

AlAs/GaAs

b= 1.7nm; well =4.5nm

Fig. 8: Comparaison entre les caractéristiques I-V de structures RTD AlAs/GaAs et de HfO2/Si de mêmes dimensions. (Lb=1.7 nm, Lw=4.5 nm).

0.01 0.1 1 10 100

barriers = 1.7nm

well = 4.5nm

Contact resistance (Ω)

HfO2/s-Si

AlAs/GaAs

10-3108

109

1010

1011

1012

1013

1014

1015

Intr

insi

c C

ut O

ff F

requ

ency

f (

Hz)

0.01 0.1 1 10 100

barriers = 1.7nm

well = 4.5nm

Contact resistance (Ω)

HfO2/s-Si

AlAs/GaAs

10-3108

109

1010

1011

1012

1013

1014

1015

Intr

insi

c C

ut O

ff F

requ

ency

f (

Hz)

Fig. 9: Comparaison entre les fréquences intrinsèques de structures RTD AlAs/GaAs et de HfO2/Si De mêmes dimensions. (Lb=1.7 nm, Lw=4.5 nm).

4.2 Vers l’oscillateur à base de RTD en silicium

Le modèle analytique présenté dans la précédente section a été implémenté en langage Verilog-A et introduit dans le simulateur circuit Spectre de l’environnement de conception de circuits intégrés Cadence.

Utiliser les RTD pour une fonction oscillateur permet de:

• Réduire le nombre de dispositifs et donc la taille du circuit (utilisation de la résistance négative),

• D’atteindre de très haute fréquence (dans le range du THz),

• Réduire la puissance dissipée (la RTD présente de bonnes performances dynamiques sous faible polarisation).

Deux types d’oscillateurs à 20 GHz ont été simulés: une architecture différentielle simple à base de RTD et une architecture plus complexe qui couple une RTD et un transistor MOS.

D’après les simulations, nous avons trouvé que la structure différentielle permet de gagner beaucoup en puissance dissipé (0.7 mW contre 130 mW pour une architecture différentielle), mais la puissance en sortie reste très faible (-44 dBm sous 1 kΩ) et ce pourrait être préjudiciable pour certaines applications.

Par contre cette limitation peut être dépassée avec l’utilisation d’une architecture plus complexe associant transistor MOSFET et RTD (structure proposée par Muramatsu et al. [Muramatsu05].

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Fig.10: Oscillateur Muramatsu.

L’oscillateur Muramatsu permet d’avoir une puissance en sortie plus grande (-1,9 dBm sous 1kΩ), mais consomme davantage que l’oscillateur à structure différentielle (~2.5 mW), tout en réduisant tout de même la puissance dissipée par rapport à des architectures standards.

Comme il a été expliqué dans le paragraphe précédent, pour réaliser des RTD avec de bonnes performances, il est nécessaire d’utiliser des couches très fines. Cela risque d’être un grand défi technologique et des variabilités dans les épaisseurs sont inévitables. L’impact de la variabilité de l’épaisseur des barrières et du puits a été également étudié. D’après les simulations paramétriques, le circuit oscillateur Muramatsu semble assez robuste à des variations de l’épaisseur des couches. En fait on trouve que pour une variation du 10% dans l’épaisseur de la barrière de gauche de 1nm à 1.1nm, la puissance en sortie présente une variation de seulement 1.74% et pour une variation du 100% de 1nm à 2nm, la puissance varie du 10%. Pour la barrière de droite, à une variation en épaisseur du 100% corresponds une variation dans la puissance en sortie, de 1%. Pour une variation du 6.66% dans la largeur du puits, de 4.2 nm à 4.8 nm, la puissance en sortie varie à peine du 0.21%.

5. Conclusion

Dans le cadre de cette thèse, nous avons développé un modèle analytique de diode tunnel résonante totalement basé sur la physique, validé en régime statique et dynamique par des simulations numériques. Ce modèle était assez « compact » pour être implémenté en Verilog-A et introduit dans un simulateur circuit pour évaluer les performances d’une fonction oscillateur RF à base de RTD en silicium.

Le modèle analytique nous a permis d’analyser l’impact des paramètres physiques et géométriques des matériaux utilisés pour la RTD. Egalement il ressort que la limitation d’une structure RTD HfO2/Si par rapport aux matériaux III-V est essentiellement liée à la présence des deux masses, transversale et longitudinale, assez élevées, entrainant la présence de plusieurs niveaux énergétiques dans le puits, et à l’offset de barrière assez élevé entre Si et HfO2, limitant ainsi la transparence des barrières. En conséquence, à géométrie identique, les structures HfO2/Si présentent une baisse de la densité de courant et un faible PVCR par comparaison à des structures conventionnelles à base de matériaux III-V.

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A partir de cette étude de l’impact des paramètres géométriques (épaisseur des barrières et largeur du puits), nous trouvons qu’en réduisant fortement les dimensions des couches, la RTD HfO2/Si peut atteindre des performances comparables du GaAs. Mais ces dimensions requièrent des technologies trop agressives. Donc une bonne solution, qui permettrait de garder des dimensions acceptables, serait d’utiliser un puits en silicium contraint. Cela permettrait de réduire le nombre de niveaux résonants dans le puits et d’améliorer le PVCR (peak-to-valley current ratio).

L’application oscillateur de ce dispositif a été évaluée avec deux architectures. La plus simple, différentielle, permet de réduire considérablement la consommation statique mais ne peut fournir que de petites puissances en sortie. Cette limitation peut être dépassée par l’utilisation d’un circuit plus complexe qui couple transistor MOSFET et RTD et qui parait être assez robuste à la variabilité des couches de la RTD.

Bibliographie

[North98] A. J. North E. H. Lienfield, M.Y. Simmons, et al., “Electron reflection and interference in the GaAs/AlAs-Al Schottky collector resonant-tunneling diode” Physical review B, Vol. 57, n°3, pp. 1847-1854 (1998).

[DelosSantos01] H. J. De Los Santos, K. K. Chui, D. H. Chow, H. L. Dunlap, “An Efficient HBT/RTD Oscillator for Wireless Applications” IEEE Microwave and Wireless Components Letters, Vol. 11, pp. 193-195 (2001).

[Lin94] H.C. Lin, “Resonant Tunneling Diodes For Multi-Valued Digital Applications”in: Proceedings IEEE Int. Symp. Multiple Valued Logic, pp. 188-195 (1994).

[Kubota06] J. Kubota, A. Hashimoto, Y. Suda “Si1-x Gex sputter epitaxy technique and its application to RTD” Thin Solid Films 508 (2006) 20 – 23

[Rommel88] Sean L. Rommel, Niu Jin, T. E. Dillon, Sandro J. Di Giacomo, Joel Banyai, Bryan hf. Card, C. D'hperio, D. J. Hancmk, N.Kirpalani. V. Emanuele. Paul R. Berger. Phillip E. Thompon Karl D. Hobart:a and Roger Lake “Development of δB/i-Si/δSb and δB/i-Si/δSb/i-Si/δB Resonant Interband Tunnel Diodes For Integrated Circuit Applications” IEDM 98.

[Osten07] H.J. Osten, D. Kuehne, E. Bugiel, A. Fissel, “Fabrication of single-crystalline insulator/Si/insulator double-barrier nanostructure using cooperative vapor–solid-phase epitaxy” Physica E 38 (2007) 6–10.

[Vinet05] M. Vinet, T. Poiroux, J. Widiez et al. “Bonded Planar Double-Metal-Gate NMOS Transistors Down to 10 nm” IEEE Electron Device Letters, Vol. 26, n°5, pp. 317-319 (2005).

[Tsu73] R. Tsu, L. Esaki, Appl. Physics Letters, Vol. 22, pp. 562-564 (1973).

[Frens87] W.R. Frensley “Wigner function model of a resonant tunneling semiconductor device” Phy.Rev. B vol. 36 n.3 pag. 1570 - 1580.

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[Datta95] S.Datta “Quantum transport : Atom to transistor” Cambridge University Press 2005.

[Ando87] Y.Ando, T.Itoh “Calculation of transmission current across arbitrary potential barriers” J. Appl. Phys. 61 (4) 1987.

[Chang93] Charles E,. Chang, , Peter M. Asbeck, , Keh-Chung Wang, Member, IEEE, and Elliott R. Brown, “Analysis of Heterojunction Bipolar Transistor/Resonant Tunneling Diode Logic for Low-Power and High-speed Digital Applications” IEEE Transactions on Electron Devices. Vol. 40. No. 4. April 1993, pagg 685-690.

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[Liu04] Q. Liu et. al. Transactions on Electron Devices Vol.51 p. 653 - 657 (2004).

[Oriols05] X. Oriols, A. Alarcón, and E. Fernàndez-Díaz “Time-dependent quantum current for independent electrons driven under nonperiodic conditions” Physical Review B 71, 245322 (2005).

[Muramatsu05] Muramatsu, H. Okazaki, T. Waho, “A novel oscillation circuit using a resonant tunneling diode”, ISCAS 2005 proceedings, pp. 2341 – 2344.

List of Publications

LIST OF PUBLICATIONS

International reviews

E. Buccafurri, R. Clerc, F. Calmon, M. Pala, A. Poncet, G. Ghibaudo. “Performances Comparison of Si and GaAs Based Resonant Tunneling Diodes” Physica Status Solidi (c), Vol. 6, n° 6, June 2009, pp. 1408-1411.

International conferences

E. Buccafurri, L. Traversa F., X. Oriols, A. Alarcon, G. Albareda, R. Clerc, F. Calmon, A. Poncet “High frequency resonant tunneling behavior: Testing an analytical small signal equivalent circuit with time dependent many-particle quantum simulations” NanoSpain 2010, 23-26 mars 2010, Malaga (Espagne).

F. L. Traversa, E. Buccafurri, X. Oriols “Quantifying many-particle Coulomb correlation through the super-poissonian noise of electron current in resonant structures” Nanospain 2010, Malaga 23-26 march 2010 (Poster)

E. Buccafurri, A. Medjahdi, F. Calmon, R. Clerc, M. Pala, A. Poncet, G. Ghibaudo “Challenges and prospects of RF oscillators using silicon resonant tunneling diodes” European Solid-State Device Research Conference (ESSDERC), Athens, 14 - 18 September 2009.

E. Buccafurri, R. Clerc, F. Calmon, M. Pala, A. Poncet, G. Ghibaudo “ntrinsic Cut Off Frequency of Si and GaAs Based Resonant Tunneling Diodes” Ultimate Integration on Silicon Conference (ULIS), March 18-20, 2009, Aachen, Germany, Proceedings, pp. 91-94.

E. Buccafurri, R. Clerc, F. Calmon, M. Pala, A. Poncet, G. Ghibaudo. “Performances Comparison of Si and GaAs Based Resonant Tunneling Diodes” International Symposium on Compound Semiconductors (ISCS), September 21 - 24, 2008, Europa-Park, Rust, Germany.

162

A. Medjahdi, E. Buccafurri, F. Calmon, R. Clerc, N. Baboux, A. Poncet “Feasability of Silicon RTD-Based oscillator” First Transalpine Conference in Nanoscience and Nanotechnologies (Transalp'nano 2008), October 27-29 2008, Lyon, France

E. Buccafurri, R. Clerc, F. Calmon, M. Pala, A. Poncet, G. Ghibaudo “Analytical Modeling of Silicon Based Resonnant Tunneling Diodes” First Transalpine Conference in Nanoscience and Nanotechnologies (Transalp'nano 2008), October 27-29 2008, Lyon, France.

French workshops

A. Medjahdi, E. Buccafurri, F. Calmon, R. Clerc, N. Baboux, A. Poncet “Discussion on the Physical Parameters of a RTD Analytical Current-Voltage Model” Colloque annuel du GDR SOC-SIP du CNRS, 4-6 juin 2008, Paris.

E. Buccafurri, A. Medjahdi, F. Calmon, R. Clerc, M. Pala, A. Poncet, G. Ghibaudo “RF Oscillators using Silicon Resonant Tunneling Diodes” Journées GDR SOC –SIP et Nanoélectronique, Bordeaux 9-11 décembre 2009.

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FOLIO ADMINISTRATIF THESE SOUTENUE DEVANT L'INSTITUT NATIONAL DES SCIENCES APPLIQUEES DE LYON

NOM : BUCCAFURRI DATE de SOUTENANCE : 5/10/2010

(avec précision du nom de jeune fille, le cas échéant)

Prénoms : Emanuela

TITRE : Analytical modeling of Silicon based Resonant Tunneling Diodes for RF oscillator application

NATURE : Doctorat Numéro d'ordre : 2010-ISAL-0076

Ecole doctorale : Electronique, Electrotechnique, Automatique (EEA)

Spécialité : Dispositifs de l’Electronique Intégrée

Cote B.I.U. - Lyon : T 50/210/19 / et bis CLASSE :

RESUME : La diminution incessante de la taille des composants semi-conducteurs permet d’envisager une rupture en associant ou remplaçant les composants conventionnels par des composants nanométriques avec un transport des charges de nature quantique ou balistique. La diode tunnel résonante (RTD) présente des caractéristiques très intéressantes et peut répondre à un grand nombre d’attentes: sa fréquence de coupure intrinsèque élevée doit permettre de travailler à de très hautes fréquences (possibilité d’atteindre la valeur théorique du THz), sa caractéristique atypique présente une Résistance Différentielle Négative (NDR) et permet d’envisager une diminution très importante en nombre de composants utilisés pour une fonction électronique donnée (électronique faible consommation). L’objectif de ce travail de thèse a été d’évaluer les performances théoriques de la RTD sur silicium avec des barrières d’oxyde de forte permittivité et de faible épaisseur (comme par exemple HfO2) et d’analyser son utilisation dans une application oscillateur radiofréquence. Pour atteindre cet objectif, les travaux ont concerné: la modélisation physique des dispositifs RTD, la validation du modèle analytique par comparaison à des simulations numériques, et enfin la simulation du comportement du dispositif et de son application: l’oscillateur radiofréquence. La partie sur l’étude temporelle de la structure a été approfondie et validée par des simulations temporelles basées sur les trajectoires de Bohm dans le cadre d’une collaboration avec l’Universidad Autonoma de Barcelona, dans l’équipe du Dr. Xavier Oriols. Par rapport aux travaux existant, notre modèle est suffisamment compact pour être introduit dans un simulateur circuit (formulation analytique) et considère des variables calculées su une base totalement physique sans aucun paramètre d’ajustement. L’originalité de ce projet repose sur une démarche globale, allant de la simulation physique du dispositif à la simulation de circuit pour une application ciblée (oscillateur), la principale retombée concerne le modèle analytique de RTD.

MOTS-CLES : diode tunnel résonante (RTD), effets quantiques, modélisation, simulation, oscillateur

Laboratoire (s) de recherche : Institut des Nanotechnologies de Lyon (INL)

Directeur de thèse: Francis Calmon

Président de jury : Pr. Olivier VANBESIEN

Composition du jury : Pr. Olivier VANBESIEN Rapporteur & Président Pr. Fabrizio PIRRI Rapporteur M. Thomas ERNST Examinateur Pr. Xavier ORIOLS Examinateur M. Raphaël CLERC Co-encadrant de thèse Pr. Alain PONCET Co-directeur de thèse M. Francis CALMON Directeur de thèse