Simulation of Transport in Nanodevices E-Book
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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
Linear current-voltage pattern, has been and continues to be the basis for characterizing, evaluating performance, and designing integrated circuits, but is shown not to hold its supremacy as channel lengths are being scaled down. In a nanoscale circuit with reduced dimensionality in one or more of the three Cartesian directions, quantum effects transform the carrier statistics. In the high electric field, the collision free ballistic transform is predicted, while in low electric field the transport remains predominantly scattering-limited. In a micro/nano-circuit, even a low logic voltage of 1 V is above the critical voltage triggering nonohmic behavior that results in ballistic current saturation. A quantum emission may lower this ballistic velocity.
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Table of Contents
Cover
Title
Copyright
Preface
List of Symbols
List of Abbreviations
1 Introduction: Nanoelectronics, Quantum Mechanics, and Solid State Physics
1.1. Nanoelectronics
1.2. Basic notions of solid-state physics
1.3. Quantum mechanics and electronic transport
1.4. Conclusion
1.5. Bibliography
2 Electronic Transport: Electrons, Phonons and Their Coupling within the Density Functional Theory
2.1. Introduction
2.2. Electronic structure
2.3. Phonons
2.4. Electron–phonon coupling
2.5. Semiclassical transport properties
2.6. Quantum transport
2.7. Conclusion
2.8. Appendix A
2.9. Bibliography
3 Electronic Band Structure: Empirical Pseudopotentials, k ⋅ p and Tight-Binding Methods
3.1. Band structure problem
3.2. Empirical pseudopotentials method
3.3. The k ⋅ p method
3.4. The TB method
3.5. Optimization of empirical models
3.6. Bibliography
4 Relevant Semiempirical Potentials for Phonon Properties
4.1. Introduction
4.2. Generic pair potentials: the Lennard-Jones potential
4.3. Semiconductors: Stillinger–Weber and Tersoff potentials
4.4. Oxydes: Van Beest, Kramer and van Santen potential
4.5. Metals – isotropic many-body pair-functional potentials for metals: the modified embedded-atom method
4.6. Polymers and carbon-based compounds: adaptive intermolecular reactive bond order, adaptive intermolecular REBO and Dreiding potentials
4.7. Water: TIP3P potential
4.8. Conclusion
4.9. Bibliography
5 Introduction to Quantum Transport
5.1. Quantum transport from the point of view of wavepacket propagation
5.2. The transmission formalism for the conductance
5.3. The Green’s function method for quantum transmission
5.4. Conclusion
5.5. Matlab/Octave codes
5.6. Bibliography
6 Non-Equilibrium Green’s Function Formalism
6.1. Second quantization and time evolution pictures
6.2. General definition of the Green’s functions, their physical meaning and their perturbation expansion
6.3. Stationary Green’s functions and fluctuation-dissipation theorem
6.4. Dyson’s equation and self-energy: general formulation
6.5. Some examples
6.6. The ballistic regime
6.7. The electron–phonon interaction
6.8. Bibliography
7 Electron Devices Simulation with Bohmian Trajectories
7.1. Introduction: why Bohmian mechanics?
7.2. Theoretical framework: Bohmian mechanics
7.3. The BITLLES simulator: time-resolved electron transport
7.4. Computation of the electrical current and its moments with BITLLES
7.5. Conclusion
7.6. Acknowledgments
7.7. Appendix A: Practical algorithm to compute Bohmian trajectories
7.8. Appendix B: Ramo–Shockley–Pellegrini theorems
7.9. Appendix C: Bohmian mechanics with operators
7.10. Appendix D: Relation between the Wigner distribution function and the Bohmian trajectories
7.11. Bibliography
8 The Monte Carlo Method for Wigner and Boltzmann Transport Equations
8.1. The WTE
8.2. The semiclassical limit: BTE
8.3. Scattering in Boltzmann and Wigner equations
8.4. The MC method for solving the BTE
8.5. Extension of the MC method for solving the WBTE
8.6. Bibliography
List of Authors
Index
End User License Agreement
Guide
Cover
Table of Contents
Begin Reading
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G1
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Series EditorRobert Baptist
Simulation of Transport in Nanodevices
Edited by
François Triozon
Philippe Dollfus
First published 2016 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:
ISTE Ltd27-37 St George’s RoadLondon SW19 4EUUK
www.iste.co.uk
John Wiley & Sons, Inc.111 River StreetHoboken, NJ 07030USA
www.wiley.com
© ISTE Ltd 2016
The rights of François Triozon 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 Control Number: 2016950150
British Library Cataloguing-in-Publication Data
A CIP record for this book is available from the British Library
ISBN 978-1-84821-566-5
Preface
This book gives an overview of the various methods used to simulate electronic transport in nanoelectronics devices. The miniaturization of transistors to sizes below 20 nm, together with the increasing variety of materials involved in these devices, leads to new challenges for simulation. Quantum mechanical effects are enhanced at such small length scales. Moreover, simulations at the atomic scale are needed to obtain an accurate description of new materials and of their interfaces and defects. Predicting the performances of next generation devices hence requires a multi-scale simulation strategy, from the simulation of materials at the atomic scale to the simulation of electron and heat transport at the device and even circuit scale.
In this book, we focus on the simulation of electronic transport at the device level. In Chapter 1, the field of nanoelectronics is briefly presented and basic notions of solid state physics and of electronic transport are introduced. This chapter serves as an introduction to the other chapters of the book, which are much more specialized and detailed. Before examining the various simulation methods, it is first necessary to introduce some theoretical background concerning solid state physics and the simulation and modelling of materials at the atomic scale. This is the purpose of Chapters 2, 3 and 4. The other chapters then present different methods for the simulation of electronic transport, based on quantum mechanical or semi-classical formalisms.
The book is mainly intended for Master’s or PhD students beginning a research project in nanoelectronics.
François TRIOZONPhilippe DOLLFUSSeptember 2016
List of Symbols
e, q
elementary charge
h
Planck constant
ħ
reduced Planck constant (= h/2π)
k
B
Boltzmann constant
ψ, ϕ, φ, χ
wavefunction
H
Hamiltonian
V, U
potential energy
m
0
electron mass
m, m*
effective mass
E, ε
energy
k
electron wavevector
q
phonon wavevector
p
linear momentum
ω
angular frequency
v
(
k
)
group velocity
f(E)
Fermi-Dirac distribution
E
,
Ɛ
electric field
B
magnetic field
T
temperature
n
carrier density
ρ
density matrix
G, G
r
, G
a
, G
<
, G
>
Green’s functions
j(E)
spectral current density
A(E)
spectral function
Σ
self-energy
c
k
+
, c
k
, ψ(r)
+
, ψ(r)
electron creation and annihilation operators
a
q
+
, a
q
phonon creation and annihilation operators
f
b
Boltzmann distribution function
f
w
Wigner function
V
w
Wigner potential
D
s
density of states
s
i
(
k
,
k'
)
scattering rate from state
k
to state
k'
Γ
i
(
k
,
k'
)
scattering rate from state
k
to any final state
(
k
,
k'
)
overlap integral
(
k
,
k'
)
overlap factor
List of Abbreviations
CMOS
Complementary Metal-Oxide-Semiconductor
MOSFET
Metal-Oxide-Semiconductor Field Effect Transistor
SPICE
Simulation Program with Integrated Circuit Emphasis
AC
Alternating Current
DC
Direct Current
BTE
Boltzmann Transport Equation
WTE
Wigner Transport Equation
DFT
Density Functional Theory
MC
Monte Carlo
LDA
Local Density Approximation
GGA
Generalized Gradient Approximation
FDSOI
Fully-Depleted Silicon On Insulator
EPM
Empirical Pseudopotential Method
CB
Conduction Band
VB
Valence Band
TB
Tight-Binding
LCAO
Linear Combinations of Atomic Orbitals
SCBA
Self-Consistent Born Approximation
ITRS
International Technology Roadmap for Semiconductors
DM
Density Matrix
GW
Commonly used approximation for many-body effects. G is the Green’s functions and W the screened Coulomb potential
MD
Molecular Dynamics
RTA
Relaxation Time Approximation
BZ
Brillouin Zone
PES
Potential Energy Surface
DOS
Density of States
1Introduction: Nanoelectronics, Quantum Mechanics, and Solid State Physics
1.1. Nanoelectronics
1.1.1. Evolution of complementary metal–oxide–semiconductor microelectronics toward the nanometer scale
Current microprocessors are based on the complementary metal–oxide–semi-conductor (CMOS) technology, whose main building blocks are field-effect transistors (MOSFETs). A transistor is made of a semiconducting silicon “channel” connected to “source” and “drain” electrodes. The electrical current through the channel is controlled by a voltage applied to a third electrode, called the “gate” electrode, separated from the channel by a thin insulating layer. Figure 1.1 shows transmission electron microscopy images of MOSFETs. During the past decades, the microelectronics industry has constantly reduced the size of transistors in order to increase the complexity and speed of microprocessors. Current transistors have a channel length LG of the order of 20 nm, and a channel thickness below 10 nm. Such length scales are close to the typical wavelength of the electrons’ wavefunctions propagating through the channel, which enhances quantum effects. The main quantum and atomistic effects occurring in CMOS technology are summarized in Table 1.1.
Figure 1.1.Transmission electron microscopy cross-sections of MOSFETs. Left panel: longitudinal cross-section of a fully-depleted silicon-on-insulator (FDSOI) transistor [LIU 13]. The channel is made of a thin silicon film lying on an oxide layer. Right panel: transverse cross-section of an ‘Ω-gate’ transistor. The channel is made up of a SiGe nanowire with a diameter of 12 nm [NGU 14]
Table 1.1.Phenomena occurring at different transistor gate lengths LG
L
G
> 100 nm
“Classical” microelectronics
10 nm <
L
G
< 100 nm
Quantization of energy levels in the channel Gate leakage by tunneling through the thin oxide Small and uncontrolled number of impurities
L
G
< 10 nm
Wave phenomena along the transport direction Few electrons in the channel
When decreasing the gate length, the thickness of the silicon oxide (SiO2) layer separating the gate electrode from the channel must be reduced accordingly in order to keep a good electrostatic control of conduction inside the channel. However, reducing the SiO2 thickness below 2 nm leads to detrimental current leakage through the oxide. Hence, materials with higher dielectric constant, such as HfO2, have been introduced. They allow for a good electrostatic control with larger oxide thickness, hence limiting gate leakage. The transistors shown in Figure 1.1 feature such high-κ gate stacks. This is a first example of a new material introduced in nanoelectronics devices. Other examples are SiGe alloys, used in “p-type” transistors (see the right panel in Figure 1.1) to improve the “hole” mobility, and silicidation of silicon in the source and drain regions to reduce the electrical resistance between the metal contacts and the transistor.
1.1.2. Post-CMOS nanoelectronics
While the quantum effects occurring at the nanometer scale tend to limit the performance of CMOS devices, they can be exploited to develop novel types of devices. This is the purpose of post-CMOS nanoelectronics, a research field that has grown considerably during the last two decades. Figure 1.2 shows a tentative classification of these phenomena.
Figure 1.2.Tentative classification of quantum phenomena that can be exploited in nanoelectronics
Figures 1.3 shows examples of post-CMOS devices exploiting the wave nature of the electron: a resonant tunneling diode [PAU 00] and a tunnel FET whose channel is made up of a carbon nanotube [APP 04]. Figure 1.4 shows devices exploiting the granularity of the charge: a “flash” memory with silicon nanocrystals [MOL 06] and a single electron transistor [LAV 15]. Such a variety of materials, nanostructures and quantum effects involved in nanoelectronics poses significant challenges to simulation.
1.1.3. Theory and simulation
From the very beginning, the development of transistors and microelectronics has been associated with theoretical progress in solid-state physics. Even the most basic properties of solids cannot be explained without quantum mechanics. In particular, their conducting or insulating properties are related to the wave nature of electrons [BLO 29, WIL 31]. Hence, a good knowledge of solid-state physics, including the quantum theory of solids, is needed to address the theory, simulation and modeling of electronic devices.
The mechanical and electronics properties of solids can be modeled at various degrees of refinement from the atomic scale to continuous medium models (see Chapters 2–4). Electronic transport can be described by an even broader variety of formalisms (Chapters 5–8). The semiclassical theory of electronic transport, which essentially consists of describing electron “wavepackets” as point particles (see Chapters 5 and 8), has successfully accompanied CMOS technology up to gate lengths well below 100 nm. Formalisms including the relevant quantum phenomena must be used for simulating smaller CMOS transistors and nanoelectronics devices.
Figure 1.3.Devices exploiting the wave nature of the electron. Left panel: resonant tunneling diode made up of a thin layer of silicon between two SiGe barriers [PAU 00]. Right panel: tunnel FET made up of a carbon nanotube channel controlled by an Al gate and a doped Si back gate [APP 04]
To study a given type of device, we have to choose a good approximation for modeling the electronic properties of the materials, and an appropriate formalism for simulating electronic transport. This requires a good knowledge of the available formalisms and how they capture the quantum phenomena involved in the device operation. The main purpose of this book is to give an overview of some commonly used formalisms for electronic transport.
