Table of Contents
Symbols
Abbreviations
Introduction
Acknowledgements
Chapter 1. Theoretical Framework of Quantum Transport in Semiconductors and Devices
1.1. The fundamentals: a brief introduction to phonons, quasi-electrons and envelope functions
1.2. The semi-classical approach of transport
1.3. The quantum treatment of envelope functions
1.4. The two main problems of quantum transport
Chapter 2. Particle-based Monte Carlo Approach to Wigner-Boltzmann Device Simulation
2.1. The particle Monte Carlo technique to solve the BTE
2.2. Extension of the particle Monte Carlo technique to the WBTE: principles
2.3. Simple validations via two typical cases
2.4. Conclusion
Chapter 3. Application of the Wigner Monte Carlo Method to RTD, MOSFET and CNTFET
3.1. The resonant tunneling diode (RTD)
3.2. The double-gate metal-oxide-semiconductor field-effect transistor (DG-MOSFET)
3.3. The carbon nanotube field-effect transistor (CNTFET)
3.4. Conclusion
Chapter 4. Decoherence and Transition from Quantum to Semi-classical Transport
4.1. Simple illustration of the decoherence mechanism
4.2. Coherence and decoherence of Gaussian wave packets in GaAs
4.3. Coherence and decoherence in RTD: transition between semi-classical and quantum regions
4.4. Quantum coherence and decoherence in DG-MOSFET
4.5. Conclusion
Conclusion
Appendix A. Average Value of Operators in the Wigner Formalism
Appendix B. Boundaries of the Wigner Potential
Appendix C. Hartree Wave Function
Appendix D. Asymmetry Between Phonon Absorption and Emission Rates
Appendix E. Quantum Brownian Motion
Appendix F. Purity in the Wigner formalism
Appendix G. Propagation of a Free Wave Packet Subject to Quantum Brownian Motion
Appendix H. Coherence Length at Thermal Equilibrium
Bibliography
Index
First published 2010 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:
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© ISTE Ltd 2010
The rights of Damien Querlioz and Philippe Dollfus to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.
Library of Congress Cataloging-in-Publication Data
Querlioz, Damien.
The Wigner Monte Carlo method for nanoelectronic devices : a particle description of quantum transport and decoherence / Damien Querlioz, Philippe Dollfus.
p. cm.
Includes bibliographical references and index.
ISBN 978-1-84821-150-6
1. Solid state physics--Mathematics. 2. Semiconductors. 3. Transport theory. 4. Coherent states. 5. Quantum statistics. 6. Particles (Nuclear physics) 7. Nanoelectronics. 8. Wigner distribution. 9. Monte Carlo method. I. Dollfus, Philippe. II. Title.
QC176.Q48 2010
530.4’10151--dc22
2010003704
British Library Cataloguing-in-Publication Data
A CIP record for this book is available from the British Library
ISBN 978-1-84821-150-6
A | Spectral density of states |
A_{i} | Particle affinity |
A_{w} | Weyl-Wigner transform of any two space variable quantity A |
a_{C–C} | Distance between neighboring C atoms |
â_{q} | Annihilation operator (particle of mode q ) |
Creation operator (particle of mode q ) | |
C | Chiral vector |
C | Collision operator |
C | Coherence |
C_{G} | Gate capacitance |
C_{OX} | Electrostatic oxide capacitance |
C_{Q} | Quantum capacitance |
D | Deformation potential |
D_{ij} | Coupling energy between subbands |
d | Diameter |
E | Energy |
E_{F} | Fermi energy |
E_{n} | Subband energy |
e | Elementary charge; e = 1.602×10^{−19}C |
f_{b} | Boltzmann function |
f_{mb} | Maxwell-Boltzmann function |
f_{w} | Wigner function |
G^{a} | Advanced Green’s function |
g_{m} | Transconductance |
G^{r} | Retarded Green’s function |
G^{<} | Lesser Green’s function |
G^{>} | Greater Green’s function |
h | Planck constant h = 6.626 × 10^{−34} Js |
ħ | reduced Planck constant ħ = h/2π |
Ĥ | Hamiltonian operator |
I | Identity matrix |
I_{D} | Drain current |
I_{OFF} | OFF-current |
I_{ON} | ON-current |
I_{P} | Peak current |
I_{V} | Valley current |
k | Electron wave vector |
k_{b} | Boltzmann constant; k_{b} = 1.38×10^{−23}JK^{−1} |
L | Length |
L_{coh} | Coherence length |
L_{G} | Gate length |
L_{th} | Thermal coherence length |
M | Mass of ions and atoms |
m | Mass of electrons and holes |
m* | Effective mass |
m_{l} | Longitudinal effective mass |
m_{0} | Free electron mass; m_{0} =9.1×10^{−31}kg |
m_{t} | Transverse effective mass |
n | Electron density |
n | Phonon number |
N_{A} | Acceptor impurity concentration |
N_{D} | Donor impurity concentration |
N_{imp} | Impurity density |
n_{q} | Phonon number of mode q |
n_{s} | Sheet electron density |
P | Probability |
p | Occupation probability |
p | Momentum |
Momentum operator | |
PV | Cauchy principal value |
Q | Quantum evolution operator |
q | Phonon wave vector |
Q_{scr} | Screening function |
R | Position |
r | Position |
Position operator | |
R | Reflection coefficient |
R_{c} | Contact resistance |
R_{s} | Access series resistance |
S | Subthreshold slope |
s | Scattering rate density |
S_{ij} | Coupling energy between subbands |
T | Circumference (of a tube) |
T | Temperature |
t | Time |
T | Transmission coefficient |
T_{Si} | Silicon body thickness |
U | Electrostatic potential |
Û | Displacement operator |
V | Potential energy, interaction energy |
V_{C} | Crystal potential |
V_{DD} | Power supply voltage |
V_{DS} | Drain voltage |
V_{e–e} | Interacting potential between two electrons |
V_{e–n} | Interacting potential between an electron and a nucleus |
v_{g} | Group velocity |
V_{GS} | Gate voltage |
v_{inj} | Injection velocity |
V_{n–n} | Interacting potential between two nuclei |
V_{rapid} | Rapidly varying part of the potential |
v_{s} | Sound velocity |
V_{slow} | Slowly varying part of the potential |
V_{T} | Threshold voltage |
V_{w} | Wigner potential |
β | Screening factor |
γ | Out-scattering rate |
γ | Inverse of energy relaxation time |
Γ | Scattering rate |
δ | Delta function |
Gradient operator | |
ε_{0} | Vacuum permittivity ε_{0} = 8.85×10^{−12}F/m |
ε | Dielectric permittivity |
ε_{r} | Low frequency dielectric constant |
ε_{∞} | High frequency dielectric constant |
θ | Heaviside step function |
K | Dielectric constant |
λ | Wave length |
λ_{th} | Thermal wave length |
Λ | Correlation length |
μ | Fermi energy |
ξ | Wave function |
ξ_{n} | Subband envelope wave function |
ρ | Material density |
Density operator | |
σ | Space extension |
Σ | Self-energy |
τ_{c} | Device-contact coupling energy |
φ | Wave function |
Φ | Work function |
ψ | Wave function |
ω | Angular frequency |
1D | One-dimensional |
2D | Two-dimensional |
3D | Three-dimensional |
AlAs | Aluminum arsenide |
AlGaAs | Aluminum-gallium arsenide |
AlSb | Aluminum antimonide |
BTE | Boltzmann Transport Equation |
CMOS | Complementary Metal-oxide Semiconductor |
CNT | Carbon Nanotube |
CNTFET | Carbon Nanotube Field-effect Transistor |
DG | Double-Gate |
DIBL | Drain-Induced Barrier Lowering |
DM | Density Matrix |
EMC | Ensemble Monte Carlo |
EOT | Equivalent Oxide Thickness |
FET | Field-Effect Transistor |
GaAs | Gallium arsenide |
GaInP | Gallium-indium phosphide |
GaP | Gallium phosphide |
Ge | Germanium |
GEP | Gaussian Effective Potential |
GF | Green’s Function |
HEMT | High Electron Mobility Transistor |
HfO_{2} | Hafnium oxide |
HP | High Performance |
ICF | Intra-collisional Field |
InAlAs | Indium-aluminum arsenide |
InAs | Indium arsenide |
InGaAs | Indium-gallium arsenide |
InP | Indium phosphide |
LA | Longitudinal Acoustic |
LO | Longitudinal Optical |
MC | Monte Carlo |
MOS | Metal-oxide-Semiconductor |
MOSFET | Metal-oxide-Semiconductor Field-effect Transistor |
MW | Multi-wall |
NDR | Negative Differential Resistance |
NEGF | Non-Equilibrium Green’s Function |
PEP | Pearson Effective Potential |
PVR | Peak-to-valley Ratio |
QBM | Quantum Brownian Motion |
RBM | Radial Breathing Mode |
RTD | Resonant Tunneling diode |
SCE | Short-channel Effect |
Si | Silicon |
SiO_{2} | Silicon oxide |
SOI | Silicon On Insulator |
SON | Silicon On Nothing |
SW | Single-wall |
TA | Transverse Acoustic |
TO | Transverse Optical |
WBTE | Wigner-Boltzmann Transport Equation |
WF | Wigner Function |
WTE | Wigner Transport Equation |
For many years, the semi-classical Boltzmann approach to transport in semiconductors has been very successful in interpreting the physics of electron devices, in a fast evolving context. Electrons and holes were considered to be localized particles which frequently interacted with different kinds of scatterers. Beyond analytic or semi-analytic models, the first numerical tools for device simulation were based on balance equations derived for different moments of the Boltzmann Transport Equation (BTE) [HÄN 91], [SCH 98]. In its simplest formulation, the drift/diffusion approach has been, and is still, widely used [SHO 50], [SZE 81]. To overcome its failure to describe non-stationary effects, such as velocity overshoot, hydrodynamics models have been developed [STR 62], [BLØ 70], [COO 82]. Thanks to their computational efficiency, these approaches are extensively used within technology computer-aided design (TCAD), in commercial software including models for technological processing [SEL 84], [DUT 99]. For more accurate investigations based on deep physical foundations, the stochastic solution of the BTE using the particle Monte Carlo method became very popular [JAC 89]. It has been developed by many groups to study a wide variety of transport problems in many kinds of devices. In spite of disadvantages due to large computational requirements and some limitations inherent in the finite number of simulated particles, this technique of transport simulation has turned out to be robust, versatile, essentially free from numerical difficulties, and thus suitable for device simulation even in three-dimensional (3D) real space. This approach has been the subject of intense activity in order to incorporate more and more physics, to such a point that it is impossible to summarize the most significant examples here. Extensive overviews of the method may be found in [JAC 83], [JAC 89], [MOG 93], [TOM 93], [JUN 03].
For about a decade, the emergence of nanoelectronics has led to a remarkable renaissance in the device engineering and computational electronics community. During the previous decade, the day when the validity of semi-classical models would be questioned and more rigorous quantum transport formalisms would become necessary to explain the behavior of charge carriers was expected with some concern [FIS 96]. Today, however, in nano-objects emerging from bottom-up nanotechnology as well as in ultra-scaled top-down nano-transistors, a new physics including quantum features has emerged and cannot be properly captured by the conventional models of device physics. For instance, in ultra-thin body (UTB) MOSFET, such as FD-SOI-FET, Fin-FET or Double-Gate (DG) MOSFET (which are considered the most promising device architectures likely to overcome short-channel effects that dramatically affect conventional bulk-MOSFET), a silicon channel thickness as small as 5 nm will have to be considered in the near future. It yields a strong quantization of electron gas in a direction perpendicular to the gate stack, which results in significant changes in the space and energy distributions of particles and may be reflected in the device operation and characteristics. Furthermore, for gate length aggressively downscaled in the sub-10 nm range, the wave-like nature of electrons may give rise to source-drain tunneling through the channel barrier and to quantum reflections in the channel. In this context, it can be considered meaningless still to use point particles in transport description.
While in the 1990s quantum transport was a topic of interest to the community of mesoscopic physics, essentially in the low temperature ballistic situation, it is now a major field of research in the community of electrical engineering [DAT 05], with the additional difficulty of having to carefully consider scattering mechanisms in devices operating at room temperature. To include quantum effects in device simulation, several approaches have been developed simultaneously. An early idea was to incorporate quantum corrections into a semi-classical description of transport, through the concepts of density gradient [ANC 89], [ANC 90] or effective potential [FER 00a]. Initially based on a rough Gaussian description of wave packets, the latter has been improved by considering either a quantum force formulation based on the Wigner formalism [TSU 01], a direct solution of Schrödinger’s equation [WIN 03], or a Pearson distribution for the wave packets [JAU 08]. These techniques are able to mimic some first order quantum effects but cannot describe properly advanced effects, such as resonant tunneling.
A second approach consisted of transferring quantum transport models developed for mesoscopic physics in the 1990s, as the recursive technique to compute either the wave function [GIL 04], or the Keldysh/Kadanoff/Baym Green’s functions [MAH 90], [DAT 00]. It was initially expected that such approaches may rapidly be able to replace the semi-classical ones to be universally applied to any nanoelectronic device. Since pioneering work on 2D transistor simulation [JOV 00], [SVI 00] many efforts have been made to improve the numerical techniques, and to include atomistic descriptions and scattering effects with different levels of approximation [SVI 03], [VEN 03], [WAN 04], [GIL 05], [JIN 06], [LUI 06], [BES 07], [KOS 07a], [POU 07], [BUR 08]. At the cost of huge computational resources, impressive efforts have been recently reported to include full electron and phonon spectra in quantum atomistic simulations of electron transport in silicon nanowire transistors [LUI 09]. However, these models do not yet reach the same degree of maturity as semi-classical transport simulators in terms of robustness and versatility, and very little research has investigated the transistor operation in realistic situations [KHA 07].
Within the context of a revival of Bohmian mechanics as an interpretation of quantum theory [BOH 52], another approach convenient for studying the time-dependent quantum transport and noise in mesoscopic systems consists of computing many-particle Bohm trajectories using a Monte Carlo algorithm [ORI 07 and enclosed references], which appears to be a promising technique. It has been recently implemented for the simulation of resonant tunneling diodes without any mean-field approximation and for the semi-classical simulation of double-gate MOSFET [ALB 09].
The simplest way to model the statistics of a quantum system consists of using the concept of the density matrix (DM) and the associated Liouville equation. When expressing the DM in the reciprocal space, this formalism may model the electron-phonon interaction accurately, including collisional broadening and retardation and the intra-collisional field effect [BRU 89], [JAC 92], [ROS 92a]. However, it does not allow the study of real space-dependent problems. DM-based device simulation is possible using the Pauli master equation that takes into account only the diagonal elements of the DM [FIS 98], [FIS 99]. However, in spite of recent improvements [GEB 04] the modeling of terminal contacts in an open system is difficult within this formulation, which is thought to be valid only for devices smaller than the electron dephasing length [FIS 99].
Alternatively, one option is to use the Wigner function that is defined in the phase space as a Fourier transform of the density matrix. In the classical limit, this function reduces to the classical distribution function. The dynamical equation of the Wigner function, i.e. the Wigner transport equation, is very similar to the Boltzmann counterpart, except in the influence of the potential whose rapid space variations generate quantum effects. The Wigner function is a standard tool in atomic physics [LUT 97] and in quantum optics [BER 02, DEL 08]. Quite early on, it was used in electron device simulation [RAV 85], [FRE 86], [KLU 87], [BUO 90] in spite of numerical difficulties inherent in the discretization scheme and the boundary conditions [FRE 90]. More recently, a renewed interest in this formalism has arisen from improved numerical techniques [BIE 97a], [KIM 99], [REC 05], [KIM 07], [YAM 09] and in particular from the development of particle Monte Carlo techniques [BER 99a], [SHI 03], [BER 03], [NED 04], [SVE 05], [QUE 06a], [QUE 06b], [BUS 08b], [BAU 08], [NED 08]. The strong analogy between Wigner and Boltzmann formalisms makes it possible indeed to adapt the standard Monte Carlo technique to solve the Boltzmann transport equation by just considering the Wigner function as an ensemble of pseudo-particles. Under some approximations leading to the Wigner-Boltzmann formulation, scattering effects may be included easily by using the same collision operator as in the BTE [NED 04]. It gives access to time simulation of realistic devices with possible coupling of quantum and semiclassical descriptions of transport. This approach is still limited to 1D transport problems but the possibility to compute 2D Wigner functions within a Monte Carlo algorithm has recently been suggested [NED 08].
The main focus of this book is on the development of a particle Monte Carlo device simulator able to solve the Wigner transport equation and its application to the study of quantum transport problems in some typical nanodevices. This approach benefits from the background acquired over many years in semi-classical transport regarding the treatment of scattering, and as reported in many textbooks. Hence, whilst the transition between the full quantum treatment and the standard semi-classical treatment of scattering has been described carefully, the physics of scattering will not be reported in detail here, though its consequences will take a major place, in particular in the analysis of decoherence effects. The strong connection between Wigner and Boltzmann formalisms will be widely exploited to identify the quantum contribution to device characteristics, and to analyze the transition between quantum and semi-classical regimes of transport.
In Chapter 1, the basic elements of quantum transport in nano-devices and the possible formalisms usable for their modeling are introduced, with a focus on the Wigner formalism. The fundamentals of device quantum mechanics are briefly summarized, including the concepts of quasi-electrons and envelope function. The common semi-classical approach to electron transport in semiconductor devices is then presented, together with the possible quantum corrections that it may include, which leads to a discussion on electron delocalization/localization. Three formalisms of quantum transport and their connections are described: the density matrix, the Wigner function and the non-equilibrium Green’s functions. The modeling of contacts, which is known as a crucial problem of device physics, is addressed in a dedicated sub-section. Finally, we show (i) how the effect of scattering by phonons and impurities may be integrated in quantum transport models, and (ii) how it is possible to consider a Boltzmann-like treatment of scattering, leading to the Wigner-Boltzmann formalism.
In Chapter 2 the particle Monte Carlo technique to solve the Wigner-Boltzmann equation for device simulation is described in detail. We first come back to the conventional Monte Carlo method to solve the Boltzmann transport equation for both bulk and low-dimensional multi-subband transport. In the latter case, the mode-space approximation, which consists of decoupling transport and confinement directions, is presented and its validity domain is discussed. Based on the introduction of a new quantum parameter assigned to each particle, the affinity, the extension to the Wigner-Boltzmann transport equation is then described in detail in such a way that the relationship and the compatibility between semi-classical and quantum approaches appear clearly. We focus on some specific aspects of the quantum Monte Carlo algorithm related to particle injection and boundary conditions. Finally, simple validations of the method are presented in order to show that the technique is able to correctly treat typical situations of quantum ballistic transport (interaction of a Gaussian wave packet with a tunneling barrier), as well as semi-classical transport (diffusive transport in a long enough N^{+}NN^{+} structure).
In Chapter 3 the Wigner-Boltzmann Monte Carlo simulator is applied to the simulation of some typical nanodevices where quantum effects are likely to take place with a possible influence of scattering at room temperature, i.e. (i) the resonant tunneling diode (RTD) whose operation is governed by the coherent tunneling process, (ii) the ultra-small double-gate MOSFET and (iii) the carbon nanotube transistor (CNTFET). These two types of field-effect transistors may operate in quasi-ballistic regimes which makes it possible for electron wave function to behave coherently, at least partially, over the active region. The expected quantum effects influencing the I-V characteristics in these devices are the direct source-drain tunneling through the gate-controlled potential barrier and the quantum reflections on the steep potential gradient at the drain-end of the channel. For both types of transistors comparison between quantum (Wigner) and semi-classical (Boltzmann) simulations are presented to analyze these effects. The results emphasize the role of scattering which remains surprisingly important in nanodevices, in spite of significant quantum coherence effects. For instance, though the transport may be quasi-ballistic in nano-MOSFET, it is shown that scattering in the source access region strongly impacts on the overall device performance.
The occurrence of quantum decoherence in such devices of a size smaller than the electron wave length and mean free path is becoming an important subject of experimental and theoretical research [FER 04], [KNE 08], [BUS 08a], [BUS 08b]. It may also become – in establishing a link between semi-classical and quantum transport – a ground-breaking route to understanding nanodevice behavior. In Chapter 4, the basis of the theory of decoherence is briefly introduced through academic examples and is applied to nanodevices. Wigner-Boltzmann Monte Carlo simulation is used to analyze the electron decoherence induced by the coupling of electrons to the phonon bath in typical nanostructures and nanodevices operating at room temperature: free evolution of a Gaussian wave packet in GaAs (with results compared to that given by the quantum Brownian motion theory), interaction of a wave packet with single- and double-tunnel barrier, GaAs RTD and Si double-gate (DG) MOSFET. These results emphasize the scattering-induced transition between the quantum transport regime and the semi-classical transport regime. In particular, the coupling of Boltzmann and Wigner Monte Carlo simulations within the same device allows us to examine a quantum to semi-classical space transition resulting from phonon scattering that we suggest to be essential in devices. This emergence of a semi-classical regime is finally examined for the case of DG-MOSFET.
The authors would like to warmly thank several colleagues of the nanoelectronics group at University of Paris-Sud, Orsay, in particular (in alphabetical order) Valérie AUBRY-FORTUNA, Arnaud BOURNEL, Christophe CHASSAT, Sylvie GALDIN-RETAILLEAU and Jérôme SAINT-MARTIN, together with some students and post-docs who spent time in the group over these last years, i.e., Yann APERTET, Francesca CAROSELLA, Hugues CAZIN D′HONINCTHUN, Emmanuel FUCHS, Karim HUET, Do Van Nam, Marie-Anne JAUD, Fulvio MAZZAMUTO, NGUYEN Huu Nha, NGUYEN Viet Hung, Ming SHI and Audrey VALENTIN. All of them contributed to the friendly atmosphere of impassioned discussions from which many ideas emerged in the years 2005–2009.
The authors are also indebted to several people for their encouragement and support in writing this book, and/or their kind comments and suggestions in the course of our work in progress on quantum transport and Wigner Monte Carlo device simulation. In particular, we wish to thank NGUYEN Van Lien at the Institute of Physics (Hanoi, Vietnam), David K. FERRY at Arizona State University (USA), Jean-Luc PELOUARD at the Laboratory of Photonics and Nanostructures (LPN, Marcoussis, France), Stephan ROCHE at the Institute for Nanoscience and Cryogenics (CEA-INAC, Grenoble, France), Mireille MOUIS at the Institute for Microelectronics, Electromagnetism and Photonics (IMEP, Grenoble, France), Hideaki TSUCHIYA at Kobe University (Japan) and Sylvain BARRAUD at LETI (CEA-LETI, Grenoble, France). Finally, we would like to address special thanks to Mihail NEDJALKOV at Sofia University (Bulgaria) for many stimulating discussions.