Doctoral Thesis Hot Electron Transport and … Thesis Hot Electron Transport and Performance of...

Doctoral Thesis

Hot Electron Transport and

Performance of Semiconductor Devices

A Dissertation Submitted to the Department of Electronics and

Applied Physics,

Interdisciplinary Graduate School of Science and Engineering,

Tokyo Institute of Technology

for the Degree of Doctor of Engineering

Abudureheman Abudukelimu

June 2012

Supervisor: Professor Hiroshi Iwai

Co-Supervisor: Professor Kenji Natori

i

Abstract

The ultra-scaled semiconductor devices suffer from various effects of hot electrons in

the drain that affect their performance. We have investigated these effects by Monte Carlo

simulations. We have clarified the effect and obtained the following new results.

(1) Ionized impurity scattering has a weak influence on hot electron transport at low

doping concentrations. This is owing to the fact that ionized impurity scattering is an

anisotropic scattering with a high preference for forward scattering and most hot

electrons are absorbed in the drain after the ionized impurity scattering. On the other

hand, the ionized impurity scattering approaches isotropic characteristics at

sufficiently high doping concentrations. It enhances the scattering of hot electrons

into backward direction, and severely degrades the peak of the mean velocity of

electrons in the channel and of the steady-state current.

(2) The rebound of hot electrons from the drain toward channel, which is caused by

intravalley acoustic phonon scattering, generates a high backward flow of hot

electrons and seriously degrades the mean velocity of electrons in the channel and the

drain current. In contrast, the intervalley phonon scattering can suppress the

backward flow of hot electrons and increase the drain current. The heat generation is

the main reason for these results, and the higher heat generation within drain region

can be associated with lower rebound velocity of hot electrons.

(3) Larger increases in the drain current and the mean velocity of electrons in the drain

region are observed for the strained-drain diode compared to the strained-channel

diode. This is due to reduction of intervalley scattering and electrons transported with

smaller transverse effective mass in strained drain. The reduction of intervalley

scattering also results in the lower heat generation and the parasitic resistances in

strained drain.

(4) Significant population of the non-equilibrium hot phonon is generated by hot

electrons. Re-absorption of the phonon also increases as the phonon generation

becomes significant. This increases the hot electron transport with high energy and

high velocity in the backward direction, leading to increase in the backward flow of

electrons. Consequently, the mean electron velocity in forward direction and the

drain current are suppressed.

We conclude that the scattering direction, the heat generation, the hot phonon

ii

generation and the scattering rate are important factors for hot electron transport in the

drain region of semiconductor devices. We also conclude that strained drain is an efficient

way to improve electrical performance of devices with a ballistic channel.

iii

Acknowledgment

I would first like to thank my supervisor Prof. Hiroshi Iwai and co-supervisor Prof.

Kenji Natori for agreeing to supervise this project. I am also indebted to them for the

many discussions about research, academia, and life in general during Ph.D course.

Thanks also go out to Profs. Parhat Ahmet and Kuniyuki Kakushima for serving on my

orals committee, and for providing me with precise feedback during the course of this

thesis.

My graduate career would not have happened at Tokyo Tech. without the early

support of Prof. Mamtimin Geni (Xinjiang University). Prof. Mamtimin Geni also served

as role models while I was still at Xinjiang University. My Doctoral studies were made

possible by generous financial support from my supervisor Prof. Hiroshi Iwai, the Japan

Student Services Organization (JASSO), New Energy and Industrial Technology

Development Organization (NEDO), Innovative Platform for Education and Research

(IPER) and G-COE project of Tokyo Tech.

I would like to thank Profs. Fumio Koyama, Yogaku Iwamoto, and Iwai Lab. research

collaborators Profs. Kazuo Tsutsui, Takeo Hattori, Yoshinori Kataoka, Nobuyuki Sugii,

and Akira Nishiyama for three years and fruitful discussions.

I would like to thank Profs. Masahiro Watanabe and Shun-ichiro Ohmi (Tokyo

Institute of Technology), Profs. Ming Liu (Institute of microelectronics Chinese academy

of sciences), Hei wong (City University of Hong Kong), Zhenan Tang (Dalian University

of Technology), Zhenchao Dong (University of science and technology of China),

Junyong Kang (Xiamen University), Weijie Song (Ningbo Institute of material

technology and engineering), Chandan Sarkar (Jadavpur university), Wang yang

(Lanzhou Jiaotong university) and Kenji Shirai (University of Tsukuba), and also

associate Prof. Baishan Shadeke (Xinjiang University) for their reviewing manuscript and

giving valuable comments on the manuscripts at the final examination of thesis

dissertation.

Over the years, I have also enjoyed many technical discussions with other members in

Iwai lab. I have learned much from them. I am obliged to all of them, especially, Dr.

Maimaitirexiati Maimaiti and Dr. Lee-Yeong-hun. I appreciate to Mr. Dariush Hassan

Zadeh for technical discussions and proofreading the dissertation. His proofreading

improves quality of this thesis.

iv

I remember our daily lives with all the students in our laboratory with happy

memories. I would not forget these days. Finally, I would like to thank my parents,

brothers, sisters and my wife for their help with my daily life, encouragements on my

study in Tokyo Institute of Technology for three years.

v

List of Figures

Figure 2.1 First Brillouin Zone (momentum space) of a face-centered cubic lattice.

Figure 2.2 Conduction band structure of silicon.

Figure 2.3 Schematic illustrations of electronic bands of a cubic semiconductor.

Figure 2.4 Conduction band density of states (DOS) in silicon from a full band

calculation vs. the DOS computed with the non-parabolic band

approximation (original figure from ref. 2).

Figure 2.5 Phonon dispersion in silicon along the [100] direction. Where q is the wave

vector and a is the lattice constant.

Figure 2.6 Constant energy surfaces with energy E and E + dE.

Figure 2.7 Ellipsoidal constant energy surfaces with a weakly and strongly curved

dispersion along the kx, ky, and kz axis.

Figure 3.1 Polar coordinate for k’ with respect to the polar axis k.

Figure 3.2 Scattering rate for the ionized impurities when NI = 1018

cm-3.


cm-3.

Figure 3.4 Vibrations in a crystal with two atoms per unit cell with masses m1, m2.

Acoustic vibration: the two atoms on the unit cell vibrate along the same

direction; Optical vibration: the two atoms on the unit cell vibrate in

opposing motion. a is the lattice constant.

Figure 3.5 Definition of polar angle and wave vector.

Figure 3.6 Scattering rate for the elastic acoustic phonon scattering at 300 K.

Figure 3.7 Scattering rate for the inelastic acoustic phonon scattering with LA phonon

vibration branch at 300 K.

Figure 3.8 The band valleys of silicon and intervalley scattering.

Figure 3.9 Scattering rate for the g-type phonon scattering when phonon energy is 19

meV.

Figure 3.10 Scattering rate for impact ionization scattering versus carrier energy.

vi

Figure 3.11 Relation between the laboratory frame and new frame.

Figure 4.1 Selection of a scattering mechanism algorithm flowchart.

Figure 4.2 Flowchart of the scattering selection in simulation.

Figure 4.3 Flowchart of ensemble Monte Carlo simulation. Each horizontal solid line

shows the trace of each particle. The vertical broken line shows sampling

time. The symbol ∗ shows the scattering time.

Figure 4.4 Flowchart of the drift and scattering process.

Figure 4.5 Basic Monte Carlo algorithm flowchart.

Figure 4.6 Schematic diagram showing the particle reflection and the particle exit.

Figure 4.7 Cell and grid point.

Figure 5.1 Schematic of structure of silicon ballistic channel n+-i-n

+ diode.

Figure 5.2 (a) Distribution of mean velocity of electrons along X-axis at VD = 0.3 V, and

(b) I-VD characteristics of ballistic channel n+-i-n

+ diode, for cases in which

the drain region is ballistic (case A, solid line), only acoustic phonon

scattering is considered (case B, triangles line) and both of acoustic phonon

and ionized impurity scatterings are considered (case C, squares line). Here,

the doping concentration of source/drain is ND = 1018

cm-3

.

Figure 5.3 (a) Distribution of mean velocity of all electrons along X-axis at VD = 0.3 V,

ND = 1018

cm-3

and (b) I-VD characteristics of ballistic n+-i-n

+ diode, for cases

in which the drain region is ballistic (case A, solid line), intravalley acoustic

and intervalley phonon scatterings are considered (case D, squares line) and

intravalley acoustic, intervalley phonon, and ionized impurity scatterings are

considered (case E, triangles line). Here, the doping concentration of

source/drain is ND = 1018

cm-3

.

Figure 5.4 (a) Distribution of mean velocity of all electrons along X-axis at VD = 0.3 V


+ diode, for case A (solid line),

case B (squares line) and case C (triangles line) with ND = 1020

cm-3

.

Figure 5.5 (a) Distribution of mean velocity of all electrons along X-axis at VD = 0.3 V


+ diode, for case A (solid line),

case D (squares line) and case E (triangles line) with ND = 1020

cm-3

.

vii

Figure 6.1 Mean rate of heat generation along the X-axis in ballistic channel diode at VD =

0.3 V for the cases that, (dashed line) only intravalley acoustic phonon

scattering is considered (case A), and that (solid line) intervalley phonon

scatterings is considered (case B), respectively.

Figure 6.2 (a) Distribution of mean velocity of electrons along the X-axis at VD = 0.3 V

and (b) the current-voltage characteristics for case A (dashed line) and case

B (solid line).

Figure 6.3 (a) Mean velocity of electrons along X-axis at VD = 0.3 V and (b) the

current-voltage characteristics for case A (dashed line), and (solid line) case

C where intravalley/intervalley phonon scatterings are considered in the

drain region (solid line).

Figure 6.4 Mean rate of heat generation along the X-axis at VD = 0.3 V for case A (dashed

line), and case C (solid line).

Figure 7.1 (a) Distribution of the mean velocity of electrons along the X-axis, and (b) the

drain current at VD=0.3 V for the equilibrium (Eq.) and non-equilibrium

(Noneq.) phonon occupation cases.

Figure 7.2 Phonon occupation versus time for non-equilibrium conditions.

Figure 7.3 Total scattering rates for g-LO phonon scattering under (a) equilibrium

condition (Nq=0.0945), and (b) non-equilibrium condition (Nq=0.36),

respectively. ωh = 61 meV, T=300 K, VD = 0.3 V.

Figure 7.4 (a) Distribution of mean energy of electrons, and (b) mean rate of heat

generation along the X-axis in a ballistic channel diode at VD=0.3 V for

cases that: the equilibrium (solid line) and non-equilibrium phonon (dashed

line) occupation conditions are considered in the drain.

Figure 8.1 Biaxial strain is intentionally introduced in silicon by depositing silicon on a

Si1-xGex layer.

Figure 8.2 Energy valleys of bulk silicon (left) and strained silicon (right), and the energy

splitting between the valleys.

Figure 8.3 (a) the mean velocity of electrons in the drain and (b) the drain current of

ballistic diode, for cases that the drain is strained (Dra.), the channel is

strained (Cha.), and diode without strained (UnStr.) at VD = 0.3 V.

viii

Figure 8.4 (a) the mean velocity of electrons in the drain and (b) the drain current of the

ballistic diode, for cases that the drain is strained (Dra.), the channel is

strained (Cha.), and the unstrained diode without strained (UnStr.) at VD =

1.0 V.

Figure 8.5 (a) the heat generation rate and the parasitic resistances of drain region for

cases that the diode without strain (dashed line), the channel is strained

(dotted line), and the drain is strained (solid line) at VD =0.3 V.

ix

List of Symbols

Symbol Description

c Speed of light in vacuum

cL Elastic constant

D Dielectric displacement

Do, Dij Deformation potential constant

Ek Carrier energy

Eg Band gap

Ec Conduction band minimum

Ev Valance band maxima

F Electric field

kB Boltzmann’s constant

h Planck’s constant

h π2/h

H Hamiltonian operator

k Wave vector of carrier

mo Electron mass in vacuum

m*

Effective mass

md Density of states effective mass

mc Conductive effective mass

n Electron density

NA Density of acceptor

ND Density of donor

N(Ek) Density of states

S(k,k’) Transition rate from k to k

’

T Carrier temperature

TL Lattice temperature

υ Carrier velocity

vs Sound velocity

V(r) Potential energy

W(k) Scattering rate

x

α Factor of non-parabolocity

Γ Total scattering rate

0ε Permittivity of free space

sε Permittivity of material

θ Polar angle of scattering

φ Azimuthal angle of scattering

)( kn EΛ Sum of scattering rates

Dλ Debye length

dΞ Deformation potential of acoustic phonon

ρ Mass density of material

τ Time of free flight

)( ),( rr ψΨ Wave function

Ω Volume of crystal

pω Angular frequency of plasma

qω Angular frequency of phonon

xi

Physical Constants

Symbol Description Value and Unit

q Electronic charge 1.60×10-19

C

kB Boltzmann’s constant 1.38×10-23

J/K

0ε Vacuum permittivity 8.85×10-14

F/cm

c Speed of light in vacuum 3.0×1010

cm/s

h Planck’s constant 6.63×10-34

Jこs

h Reduced Planck constant 1.054×10-34

Jこs

m0 Electron mass in vacuum 9.1×10-31

kg

kBTL/q Thermal voltage (T=300K) 0.0259 V

A& Angstrom 1 A& = 10-8

cm

nm Nanometer 1 nm = 10-7

cm

mµ Micrometer 1 mµ = 10-4

cm

mm Millimeter 1 mm = 0.1 cm

m Meter 1 m = 100 cm

eV Electron Volt 1eV = 1.6×10-19

J

xii

Material Parameters (Silicon)

Symbol Description Value and Unit

ρ Mass density 2329.0 kg/m3

sε Permittivity 11.7 0ε F/m

vs Sound velocity 9040 m/s

ml longitudinal effective mass 0.92 m0

mt transverse effective mass 0.19 m0

α Nonparabolicity factor 0.5

dΞ Deformation potential of

acoustic phonon 9.0 eV

6.5 ev

Dij Deformation potential of

optical phonon (108 eV/cm) 0.5 (f1)

3.5 (f2)

1.5 (f3)

0.3 (g1)

1.5 (g2)

6.0 (g3)

ijωh Optical phonon energy (meV) 19 (f1)

51 (f2)

57 (f3)

10 (g1)

19 (g2)

62 (g3)

xiii

Table of Contents

Abstract .................................................................................................................. i

Acknowledgment ................................................................................................. iii

List of Figures ....................................................................................................... v

List of Symbols .................................................................................................... ix

Physical Constants .............................................................................................. xi

Material Parameters (Silicon) ........................................................................... xii

Table of Contents .............................................................................................. xiii

Chapter 1

Introduction .......................................................................................................... 1

1.1 Background ................................................................................................... 1

1.2 Overview of the Methodology ...................................................................... 2

1.3 Purpose of This Work ................................................................................... 5

1.4 Organization .................................................................................................. 5

References ........................................................................................................... 7

Chapter 2

Semiconductor Fundamentals for Semi-Classical Carrier Transport.......... 10

2.1 Band Structure ............................................................................................ 10

2.2 Carrier Dynamics ........................................................................................ 14

2.3 Phonon Dispersion ...................................................................................... 15

2.4 Density of States ......................................................................................... 16

2.4.1 Density of States in Bulk Semiconductors ........................................... 17

2.4.2 Single Valley, Anisotropic, Parabolic Band .......................................... 18

2.5 Effective Mass ............................................................................................ 20

2.5.1 Conductivity Effective Mass ................................................................ 20

2.5.2 Density of States Effective Mass .......................................................... 21

Conclusion ........................................................................................................ 21

References ......................................................................................................... 22

xiv

Chapter 3

Carrier Scattering ............................................................................................... 24

3.1 Ionized Impurity Scattering ......................................................................... 26

3.1.1 Brooks-Herring Approach ..................................................................... 27

3.1.2 Kosina’s Approach ................................................................................ 28

3.2 Phonon Scattering ....................................................................................... 30

3.2.1 Intravalley Scattering by Acoustic Phonon ........................................... 31

3.2.2 Inelastic Acoustic Phonon Scattering .................................................... 33

3.2.3 Intervalley Scattering by Optical Phonon ............................................. 34

3.3 Impact Ionization ......................................................................................... 35

3.4 Wave Vector after Scattering ....................................................................... 37

Conclusion ......................................................................................................... 40

References ......................................................................................................... 40

Chapter 4

Monte Carlo Method for Devices Simulation .................................................. 42

4.1 Procedure of Monte Carlo Method.............................................................. 44

4.2 Drift Process ................................................................................................ 46

4.3 Scattering Process ....................................................................................... 47

4.4 Velocity Calculation .................................................................................... 48

4.5 Ensemble Particle Motion ........................................................................... 50

4.6 Monte Carlo Device Simulation .................................................................. 52

4.6.1 Initial Condition .................................................................................... 53

4.6.2 Boundary Condition .............................................................................. 54

4.6.3 Charge Distribution ............................................................................... 55

4.6.4 Solution of Poisson Equation ................................................................ 57

4.6.5 Electric Field Calculation ...................................................................... 59

Conclusion ......................................................................................................... 59

References ......................................................................................................... 60

Chapter 5

Effects of Scattering Direction on Hot Electron Transport ........................... 62

xv

Introduction ....................................................................................................... 62

Simulation Method ........................................................................................... 63

Results and Discussion ..................................................................................... 64

Conclusion ........................................................................................................ 69

References ......................................................................................................... 69

Chapter 6

Effects of Heat Generation on Hot Electron Transport ................................. 72

Introduction ....................................................................................................... 72



Conclusion ........................................................................................................ 78

References ......................................................................................................... 79

Chapter 7

Effects of Hot Phonon on Hot Electron Transport ......................................... 83

Introduction ....................................................................................................... 83



Conclusion ........................................................................................................ 89

References ......................................................................................................... 90

Chapter 8

Strained Drain and Hot Electron Transport ................................................... 92

Introduction ....................................................................................................... 92



Conclusion ...................................................................................................... 100

References ....................................................................................................... 100

Chapter 9

Summary ......................................................................................................... 104

xvi

For Improved Performance and Future Works ................................................ 105

Published Papers and Presentations ............................................................... 107

Appendix ............................................................................................................ 109

1

Chapter 1

Introduction

1.1 Background

The modern information technology and the electronic product depended on

tremendous progress in the technology of semiconductor device in the past half century,

and marvelously changed our daily lives and the whole world itself. Therefore,

semiconductor device has been regarded as one of the most important inventions in the

20th

century. Although the modern Metal-Oxide-Semiconductor (MOS) transistor was

first described in 1930 [1], the first transistor using silicon was reported in 1960 [2]. Until

the methods of routinely growing reliable oxide were reported and developed in 1964, the

MOS technology was not viable [3]. After that time the MOS technology and the

semiconductor industry has quickly developed. The MOS transistor has become the most

important device for the integrated circuit such as semiconductor memories. The

semiconductor technology is actually simple: scaling down the size of transistor and

increasing the integration of transistors in a single chip. The advanced semiconductor

devices have been scaled down to nanoscale size, and the device size is further shrinking

as predicted by Moore’s law [4]. A sub-100-nm modern metal oxide semiconductor field

effect transistor (MOSFET) was first developed in the 1980s. Continued success in device

scaling is indispensible for the successive improvements in the technology of

semiconductor device. The basic MOS transistor size has shrunk from a feature size of

several microns to less than 100 nm during the last two decades. This shrinkage is a result

of the advance in low temperature annealing and lithography [5].

The short channel effects, such as threshold voltage rolloff and

drain-induced-barrier-lowering, become increasingly significant as the gate length of

semiconductor devices is reduced to the nanometer regime. The short channel effect

limits the scaling capability of planar bulk MOSFETs. Using the new device structures,

such as SOI MOSFET, FinFETs or nanowire MOSFETs obtained additional performance

improvements that overcome the device scaling difficulty [6]. The application of the SiGe

technology has opened up new possibilities for band gap engineering, and enhancement

of carrier mobility by strain previously available only in expensive III-V technologies [7].

These technologies open up many new opportunities for the downsizing of devices scale.

2

The use of these devices has advantages relative to the carrier transport, mainly due to the

reduction of surface roughness scattering and the lower vertical electric field. Moreover,

the introduction of new materials, intrinsic materials to reduce phonon scattering,

Coulomb scattering and to enhance the mobility, as the strained silicon, the intrinsic

silicon channel, can result in electronic transport approaching the ballistic regime. The

previous work proposed that ballistic electron transport could be achieved in GaAs at low

temperatures at a sample length of the order of a few hundred nanometers [8]. Yokoyama

et al. provided evidence of quasi-ballistic transport through heavily doped GaAs layers

[9]. That was followed by a direct demonstration of ballistic electron transport, and a

determination of the ballistic portion of the traversing electrons. Using improved device

structures, ballistic portions greater than 75% were subsequently obtained [10].

If the channel is ballistic, then electrons that are injected from the source flow into the

drain and become hot electrons. Hot-electron phenomena have become an important issue

for the understanding of modern ultra-scale semiconductor devices. The hot electrons are

reflected back into the source-end, causing an increase in the injection barrier in the

source. The rebound of hot electrons from the drain region back into the channel owing to

scattering significantly reduces the drain current. It has been pointed out that elastic

scattering causes the backward flow of hot electrons from the drain into the channel and

seriously degrades the peak of the mean velocity of electrons in the channel and the

steady-state current. On the contrary, inelastic scattering can suppress the backward flow

of hot electrons [11]. However, the role of the scattering direction, heat generation, hot

phonon and scattering rate in hot electron transport has not been discussed in detail. In

this work, we have investigated the performance of ballistic channel devices from various

effects of hot electron transport in the drain region. It should be noted that a hot-electron

in the drain need not be a “ballistic channel” to operate without scattering. If the electrons

have sufficiently high energies and the electric field is large, the electrons may be

elastically scattered several times in the channel, undergo slight changes in their direction,

and become hot electrons in the drain region, although after somewhat longer transit time.

Therefore, the results of this work can be extended to other hot electron devices.

1.2 Overview of the Methodology

The International Technology Roadmap for Semiconductors has predicted that

3

semiconductor devices will have more than ten different types in future [ 12 ].

Development of such kind of new devices is not possible without the extensive use of

computer simulation based on computer models for both the device characteristic and the

technological processing. The computer simulation can greatly reduce the cost for the

development of new device by replacing expensive experiments by inexpensive computer

simulations. The computer simulation is the only possibility to gain insight into the

complex processes of device fabrication and carrier transport. The degree of success of

computer simulation depends on the accuracy, reliability and efficiency of the utilized

models of computer simulation. For the above mentioned new progress of device

technology, the introduction of new device types, materials, and simulation models has

time and again pushed the limits of continuous improvement or new developments. At the

beginning analytical models were sufficient for previous device structures. However

when the gate length reaches the nanoscale regime, the short channel effects become a

problem, and the previous analytical modeling had to be replaced by more accurate

models [13].

There are two main approaches in ultra-scale semiconductor simulation. One is that

the Boltzmann transport equation (BTE), which describes carrier transport in the

semiconductor devices, is solved by the semiclassical approach [14]. The BTE is

carrier-based because it is derived by careful consideration of carrier inflow and outflow.

The carrier motion is described as a series of free flights that are decided by scattering

rates.

Since the BTE is seven-dimensional, the process of direct solution is computationally

demanding extremely. However, the transport process of carrier can be characterized by

using approximate models as an assumption based on certain conditions such as

quasi-equilibrium. The other level consists of numerical models based on balance

equations derived for different moments of the semi-classical BTE. The drift-diffusion

approach is a model that most widely used this level [15]. The drift-diffusion approach is

an important method for the devices simulation. However, the new approach, the

hydrodynamic models [ 16 ], is a development to make up the shortcomings of

drift-diffusion approach, especially its failure to reproduce hot electron effects like the

velocity overshoot or parasitic currents. These two approaches, although built on a

simplified model, are still of great utility on a large scale in that the advantage with regard

to the computational cost compensates for unphysical behavior. However, these

approaches are not without their own problems. The coarse granularity, in nature, hinders

4

the models form discovering short channel effects.

Some problems such as spurious velocity overshoot or artificially enhanced

particle-diffusion are occurring in the case of deep ultra-scale devices. These ultra-scale

device problems can be avoided by directly solving the BTE, and these approaches

constitute the third level in the modeling hierarchy. These approaches can be separated

into two different groups depending on the approach used to solve the BTE. One group

comprises stochastic approaches, and the other group consists of numerical solutions

based on the Monte Carlo (MC) method [17].

The MC method is an important numerical approach; it has been widely utilized to

solve the BTE since 1960s [18]. The MC method is a statistical numerical method that

proves to be a direct simulation of the dynamics of carriers in the semiconductor structure.

Including various scattering mechanisms and thus realistic physical phenomena in the

simulation is one of the benefits in the MC method. As a result, the MC method is able to

provide a decent prediction for non-equilibrium effects such as velocity overshoot and hot

carriers that are commonly found under the short channel conditions in modern

semiconductor devices unlike the hydrodynamic approach and the drift-diffusion

approach. The numerous enhancements are added in MC simulation by the various

research communities over the past two decades, including more comprehensive physical

models, more efficient computer algorithms, boundary conditions, new scattering

mechanisms, electrostatic self-consistency in device simulations, etc. The role of impact

ionization and high energy scattering are greatly diminished as device dimensions are

scaled into the nanometer regime and supply voltages are reduced below the material’s

band gap. For these devices, the carrier transport can be adequately simulated with

analytic band models. Hence simpler, faster analytic band MC method can be employed

as engineering design-tools for future nanoscale devices. The MC method is basically a

semiclassical device simulation method, although it is based on the semiclassical BTE

containing quantum mechanical effects like the band structure, scattering rates calculated

by Fermi’s Golden Rule, or size quantization effects [19].

As the size of the device decreases, a quantum model is necessary to explain quantum

effects such as quantum interference, size quantization, and tunneling current [20]. Such

attempts are Green function [21], path integral [22], and Wigner function [23]. The

transport simulation fully based on quantum mechanics involves unresolved issues and is

computationally infeasible. Among the quantum and the semi-classical models, there is

no single model that offers a complete solution for carrier transport in nanoscale devices

5

[14]. The quantum methods lack band structures and scattering mechanisms, while the

semi-classical particle-based approaches lack quantum coherence. As results of previous

works, the various methods have been suggested as a trade-off between the two

approaches. Among them, the quantum correction method is introduced to account for the

size quantization, and it can be included in the MC method.

In this work, the effects of hot electrons on the performance of ballistic channel

device are studied by a semi-classical Monte Carlo method self-consistently coupled with

Poisson’s equation.

1.3 Purpose of This Work

The goals of this dissertation are two-fold. The first objective is to explore the

detailed effects of scattering on the hot electron transport in the drain region of ballistic

channel device. For this goal, the semi-classical Monte Carlo approach is employed. The

band structure is simplified by an analytical non-parabolic band model. The analytic

description for phonon dispersion is used when the effect of heat generation on hot

electron transport is studied. The ionized impurity scattering, phonon scattering and

impact ionization scattering are considered in the simulation. The discussion on both the

scattering process and hot electron transport within the drain region are given.

The second objective of this dissertation is to analyze the design and scaling of the

confined geometry of transistors from the point of view of scattering and hot electron

transport in the drain. The Monte Carlo work shows that, the hot electrons are generated

in the drain region of the ballistic channel devices. Hot electron transport obviously

influences the performance of devices. Some device design guidelines are proposed, and

the analysis indicates that it is possible to optimize device performance by reducing the

backward scattering of hot electron in the drain region.

1.4 Organization

This dissertation is organized as the following nine chapters:

Chapter 2 presents a brief introduction of the semiconductor fundamentals for

semi-classical carrier transport including band structures, phonon dispersion, carrier

6

dynamics, and related topics that are indispensable for the understanding of carrier

transport in semiconductors. The investigation of hot electron transport in the drain region

in this work is based on the semi-classical treatment, in which the drift process of the

carrier is analyzed by a classical approach.

Chapter 3 presents a brief introduction of the theory of carrier scattering, and also

introduces the ionized impurity scattering, the phonon scattering, the impact ionization

scattering and the calculation of wave vectors after scattering. The rate of inelastic

electron-phonon scattering is re-derived and discussed taking into account the phonon

dispersion and the scattering deformation potential, which properly reproduce

experimental data both in the bulk and the strained silicon. Scattering processes described

in this chapter were limited to those that are rather important for the carrier transport in

submicron devices. Other scattering processes, such as the neutral impurity scattering, the

alloy scattering, and the radiative and non-radiative processes, are not included in this

work.

Chapter 4 describes details of the Monte Carlo implementation in carrier transport and

the device simulation. This includes description of the drift process and the scattering

process, calculation of the velocity, and carrying out the ensemble particle simulation of

carrier transport. It also includes setting of the initial and the boundary condition,

calculation of the charge distribution, and solution of Poisson equation for evaluation of

the electric field.

Chapter 5 presents the discussion about the effect of scattering direction of hot

electrons within the drain. The ionized impurity scattering, the intravalley acoustic

phonon scattering and the intervalley phonon scattering are considered in the simulation.

An analytical non-parabolic band model is employed. The channel region is assumed to

be intrinsic and the transport is ballistic. Electrons do not suffer any scattering throughout

the channel. In the source region, the ionized impurity scattering and the phonon

scattering are considered. The scattering environment is varied for the drain region, and

the effect of ionized impurity scattering under the low and high doping concentration is

investigated.

In Chapter 6, the Monte Carlo simulation is applied to study the detailed effect of heat

generation on hot electron transport in the drain region in a bulk silicon ballistic channel

diode. The analytical non-parabolic band is employed. Both the intravalley and the

intervalley phonon scattering are considered, and are treated as inelastic. More precisely,

both of them are considered in the source. But the scattering condition is varied in the

7

drain region. Phonon dispersions are calculated analytically. The rate of heat generation is

estimated by the sum of all phonon emission events minus all phonon absorption events

per unit time and unit volume.

In chapter 7, the influence of hot phonons on hot electron transport within the drain

region is investigated. The ensemble Monte Carlo method self-consistently coupled with

Poisson’s equation is used. This MC simulation is three-dimensional in k-space and

two-dimensional in real space. The two-dimensional model is possible if there is no

significant variation in the physical quantities of interest along the third direction. The

Boltzmann transport equation for the phonon is solved and the non-equilibrium phonon

occupation is calculated. Mean heat generation rate is also calculated. The simulation

results under the equilibrium and the non-equilibrium phonon occupation condition are

compared.

In chapter 8, we have comparatively studied the effect of the biaxial strained-Si

channel and drain, which are assumed to be grown on a relaxed Si1-xGex substrate, on

performance of the ballistic channel diode. We adopted an ensemble Monte Carlo

simulation self-consistently coupled with Poisson’s equation. The method of the MC

simulation is basically the same as used in the unstrained case. Impact ionization is

considered in the simulation because it is important in the strained-Si due to the band gap

reduction.

Chapter 9 provides an overall conclusion of this thesis. The result is analyzed and

several suggestions for the enhancement of device performance and for the future

research direction are offered. It is suggested that the hot electron transport within the

drain region is an important issue for understanding the characteristics of future novel

devices and for optimizing the device design. Also, the role of hot electrons in

performance of the device must be studied more carefully in such geometrically confined

device designs.

References

[1] J. E Lilienfield, US Patent 17, 45175 issued Jan. 28, (1930).

[2] D. Kahng, and M. M. Atalla, “Silicon-silicon dioxide field induced surface device”,

8

IRE Solid State Device Research Conference, Pittsburgh, PA (1960).

[3] E. H. Snow, A. S. Grove, B. E. Deal, and C.T. Sah, “Ion Transport Phenomena in

Insulating Films”, J. Appl. Phys. 36, 1664 (1965).

[4] G. Moore, “Progress in digital integrated electronics”, IEEE International Electron.

Dev. Meeting (IEDM) Tech. Digest, (1975) pp. 11.

[5] S. M. Sze, Ed., VLSI Technology, 2nd Ed. McGraw-Hill Book Company, New York,

1988.

[6] Jing Wang, “Device Physics and Simulation of Silicon Nanowire Transistors”,

Doctoral Thesis, (2005).

[7] C. Jungemann, B. Neinhus, S. Decker, and B. Meinerzhagen, “Hierarchical 2-D DD

and HD noise simulations of Si and SiGe devices: Part II-Results”, IEEE Trans.

Electron Devices 49, no.7, 1258 (2002)

[8 ] M. S. Shur and L. F. Eastman, “Ballistic transport in semiconductor at low

temperatures for low-power high-speed logic”, IEEE Trans. Electron Devices ED-26,

1677 (1979).

[9] N. Yokoyama, K. Imamura, T. Ohshima, N. Nishi, S. Muto, K. Kondo, and S.

Hiyamizu, “Characterization of double heterojunction GaAs/AlGaAs hot electron

transistors”, IEDM Tech. Digest, 532 (1984).

[10] M. Heiblum, N. I. Nathan, D.C. Thomas, and C. M. Knoedler: “Direct observation of

ballistic transport in GaAs”, Phys. Rev. Lett. 55, 2200 (1985).

[11] T. Kurusu, and K. Natori, “Numerical Study on Ballistic n+–i–n

+ Diode by Monte

Carlo Simulation: Influence of Energy Relaxation of Hot Electrons in Drain Region

on Ballistic Transport”,Jpn. J. Appl. Phys. 45, 1548 (2006).

[12] ITRS 2011 [http://www.itrs.net/]

[13] C. Jungemann, and B. Meinerzhagen, Hierarchical device simulation: the Monte

Carlo perspective, Springre Wien New York, 2003.

[14] Hyung-Seok Hahm, “Enhancement of Monte Carlo Simulations on 3D Nanoscale

Semiconductor Devices”, Doctoral Thesis, University of Illinois at Urbana

9

Champaign, 2008.

[15] W. V. Van Roosbroeck, “Theory of flow of electrons and holes in Germanium and

other semiconductors”, Bell System Technical Journal 29, 560 (1950).

[16] W. Choi, J. Ahn, Y. Park, H. Min, and C. Hwang, "A time dependent hydrodynamic

device simulator snu-2d with new discretization scheme and algorithm," IEEE

Transactions on Computer-Aided Design of Integrated Circuits and Systems 13, no. 7,

899 (1994).

[17] J. M. Hammarsley, and D. C. Handscomb, Monte Carlo Methods, ethuem/chapman

and Hall, London, 1964.

[18] T. Kurosawa, “Monte Carlo simulation of hot electron problems”, Journal of

Physical Society of Japan 21, 527 (1966).

[19] C. Jacoboni and L. Reggiani, “The Monte Carlo method for the solution of charge

transport in semiconductors with applications to covalent materials”, Rev. Mod. Phys.

55, no. 3, 645 (1983).

[20] A. Gehring and S. Selberherr, “Evolution of current transport models for engineering

applications”, Journal of Computational Electronics 3, no. 3-4, 149 (2004).

[21] G. D. Mahan, “Quantum transport equation for electric and magnetic fields”, Physics

Reports 145, 251 (1987).

[ 22 ] V. Pevzner, F. Sols, and K. Hess, Approaches to Quantum Transport in

Semiconductor Nanostructures, New York, NY: Plenum Press, 1991.

[23] E. Wigner, “On the quantum correction for thermodynamic equilibrium”, Physical

Review 40, no. 5, 749 (1932).

10

Chapter 2

Semiconductor Fundamentals for Semi-Classical

Carrier Transport

The fundamentals of carrier transport with semi-classical treatment of electron

dynamics in bulk and strained silicon crystals, as well as in nano-scaled devices, are

described in this chapter. Moreover, a brief summary of analytic band structures, phonon

dispersions, density of states and effective masses are drafted.

2.1 Band Structure

The relation between energy band structure (Ek) and wave vector (k) can be expressed

on the first Brillouin Zone (BZ), which is a primitive cell centered at the origin at k-space,

as shown in Figure 2.1. The band structures are usually expressed with orientations such

as ∆ , Λ or L. A number of theoretical studies have reported to calculate the Ek-k

relation, as carrier transports are determined with this relation. Most of the case, a precise

extraction of Ek-k relations near Fermi level is important, except for the case where

carriers with high energy need to be modeled [1].

Band structures of Si crystals are well calculated from various numerical solutions

and confirmed by experiments. For example, conduction band structure of a Si crystal is

shown in Figure 2.2. Eg is the forbidden energy, which is termed as energy band gap, so

[100]

[001]

XΓ

Λ

L

∆

ΣK

W

[100]

[001]

XΓ

Λ

L

∆

ΣK

W

Figure 2.1 First Brillouin Zone (momentum space) of a face-centered

cubic lattice.

11

that electronic states are located above the energy band gap.

The lowest conduction band minima (CBM) for Si crystals are at ∆ directions near

the boundary of the first BZ, close to X symmetry points as shown in Figure 2.2. For

valence band, the maxima (VBM) are located at Γ point (k=0). Semiconductors, which

show different position for CBM and VBM, are called indirect band semiconductor.

Information near CBM and VBM are regarded as important ant for carrier transport

studies, since carriers with low energy are located near the band edges. Band structures

are approximated to have spherical or ellipsoidal constant-energy surfaces for both CBM

and VBM.

Near the X symmetry points, which are located at ± 85 percent away from Γ point,

six equivalent CMB can exist along three <100> axis for Si crystals. Therefore, the

X-valley along the <100> direction is centered at (0.85, 0, 0) a/2π , where a = 0.5431 nm

is the lattice constant of silicon.

Figure 2.3 shows schematic diagrams of a band model with a cubic semiconductor.

The band structure of materials is generally approximated by a quadratic function at the

minimum point of real band structures. The band energy follows Eq. (2.1) when the band

minimum lies at Γ point in momentum space as [1]

)(22

222

*

2

*

22

xxxk kkkmm

kE ++== hh (2.1)

where m* is the effective mass, kx, ky and kz are the wave vectors at x-, y-, and z-directions,

respectively. Therefore, the inverse of the effective mass can be expressed as [1]

2

2

2*

11k

E

mk

∂∂=

h (2.2)

Eg EX

EL

ΓE ΓE

Wave vector

Eg = 1.12 eV

EX = 1.2 eV

EL = 2.0 eV

eVE

eVE

2.4

4.3

=

=

Γ

Γ

Eg EX

EL

ΓE ΓE

Wave vector

Eg = 1.12 eV

EX = 1.2 eV

EL = 2.0 eV

eVE

eVE

2.4

4.3

=

=

Γ

Γ

Figure 2.2 Conduction band structure of silicon.

12

where h is the reduced Plank’s constant. The simple relation Ek -k given in Eq. (2.2) is

widely used to simplify the simulation of carrier transport.

When bands structures are lying on different directions such as Λ and ∆ , the Ek -k

relation can be given by new form as follows Eq. (2.3) [1]:

][2 *

2

*

22

t

t

l

lk

m

k

m

kE += h (2.3)

where *lm and *

tm are the longitudinal and transverse effective masses, kl and kt are the

components of k in the longitudinal and transverse directions, respectively. When *lm

and *tm are different, ellipsoidal distribution form in the momentum space as shown in

Figure 2.1

When high electric field is applied to semiconductors, the energy of carriers will be

far from the minimum energy with parabolic band so that effective mass approximation is

not suitable [2]. However, the Ek -k relation can be used with non-parabolic band

correction, by applying non-parabolicity parameter α . Detailed analytical models of the

correction are described in ref. [3]. This non-parabolic Ek-k relationship is [1]

)1(2

)(*

22

kk EEmkk αγ +== h (2.4)

α approximately equal to the inverse of the energy gap as given by

∆E

ΓELE

0. eV

L Γ X

<111> <000> <100>

∆E

ΓELE

0. eV

L Γ X

∆E

ΓELE

0. eV

L Γ X

<111> <000> <100>

Figure 2.3 Schematic illustrations of electronic bands of a cubic semiconductor.

13

2

0

*

)1(1

m

m

Eg

−=α (2.5)

therefore, Ek in non-parabolic band can be derived by solving Eq.(2.4)

ααγ2

1)(41 −+=

kEk (2.6)

The impact ionization and other high energy transport are not expected to play a

significant role in nano-scale devices at low-voltage operation. Therefore, the

non-parabolic band approximation represents a good description for electron transports

with low-voltage-operated devices. Difference in the density of states (DOS) with

non-parabolic band correction and full band calculation of silicon crystals is shown in

Figure 2.4. Non-parabolic band correction gives fairly nice approximation up to energy of

1.5 eV. Most the case, carrier energy can be up to 1.5 eV, so that non-parabolic band

correction can be used to for band calculation of silicon. The second conduction band at

L-valley of silicon can be neglected at the low drain voltage, since it lies about 1 eV

above the bottom of X-valley [4].

0 0.5 1 1.5 2 2.5 3 3.5D

OS

[cm

-3eV

-1]

1023

1022

1021

Nonparabolic

band

Full band

Energy (eV)

0 0.5 1 1.5 2 2.5 3 3.5D

OS

[cm

-3eV

-1]

1023

1022

1021

Nonparabolic

band

Full band

Energy (eV)

Figure 2.4 Conduction band density of states (DOS) in silicon from a full

band calculation vs. the DOS computed with the non-parabolic band

approximation (original figure from ref. 2).

14

2.2 Carrier Dynamics

Carriers in a lattice can be treated as particles in free space when m* are used. This

implies when the potential energy of carriers varies slowly compared to the lattice

potential and the quantum effects such as tunneling and reflection can be ignorable, the

electron transport may be described by the classical equations of transport in

semiconductors [1].

The classical electron transport is described by equations based on total energy, H =

Ek + U, where U is the potential energy. If the Hamiltonian is properly chosen, the

electron transport in a conduction band can be described as follows [5]:

)(rEEH ck += (2.7)

here, r is the position vector and Ec(r) is the CBM expressed as follows:

)()()( reVrErEc −−= χ (2.8)

where E is the electron energy, )(rχ is the electron affinity, e is the magnitude of the

electronic charge, and )(rV is the electrostatic potential. The electron affinity )(rχ can

be eliminated from Eq.(2.8) when a compositionally uniform material is considered. The

electron transport can be constructed using the analogy of Hamilton's equations as [1]

Hdtdk ∇−=

h

1 (2.9)

Hdtdr

k∇−=h

1 (2.10)

where ∇ and k∇ are the operators with respect to k and r, respectively. Eq. (2.10)

gives simple results for the group velocity of analytic bands denoted by Eq. (2.1) and Eq.

(2.3) as,

∗=m

khυ or **l

l

t

t

m

k

m

k hh +=υ (2.11)

The group velocity for non-parabolic bands expressed in Eq. (2.6) can be calculated by

15

the following equation [1]:

)(41

1*

kmk

αγυ

+= h (2.12)

2.3 Phonon Dispersion

To study the heat generation in nanoscaled devices, both types of phonon scatterings,

including intravalley and intervalley phonon scattering, should be treated as inelastic

scattering. Carriers exchange their energy with scattering events through phonon

absorption or emission processes. Intravalley scattering is a scattering that occurs within

the same conduction band valley, where only acoustic phonon scatterings are involved.

Acoustic phonon scattering is usually treated as elastic scattering events, resulting from

energy much smaller than thermal energy, kBT. However, inelastic treatments of acoustic

phonons become important for heat generation simulation as shown ref. 6. There exist six

types of intervalley phonons, where scattering within the same axis, such as <100> to

<100> is called g-type, and scattering between different axes, such as <100> and <010>

is called f-type. Both, f- and g-type phonon scatterings can be determined from

geometrical arguments [7] and are labeled in Figure 2.5. Most typical analytic and

full-band Monte-Calro codes [8, 9] treat intravalley scattering with a single type of

acoustic phonons. The longitudinal acoustic (LA) and transverse acoustic (TA) branches

are into dispersion less mode with a single velocity and deformation potential. Due to the

matrix element, TA becomes zero for scattering within a band located at the center of the

BZ, therefore, this mode can be neglected at the center. However, to precisely simulate

the scatterings, intravalley scattering with LA and TA branches should be considered

separately. Each phonon dispersion branch, LA or TA in Figure 2.5, can be expressed

with the isotropic approximation [6]:

20 cqqsq ++= υωω (2.13)

here ωq is the phonon frequency and q is the wave vector, υs and ω0 are the fitting

parameters. In <100> crystal direction, quadratic polynomials give a better fit, so that

quadratics is entirely sufficient for this isotropic approximation. The quadratics is easy to

16

invert and extract the phonon wave vector as a function of frequency. They track the

phonon dispersion data closely near the frequencies corresponding to intervalley f- and

g-type phonons and near the BZ center for long wavelength intravalley acoustic phonons.

This phonon dispersion model can be extending to confined dimensions or other materials.

Moreover, changes in the phonon dispersion due to strains or confinements can be easily

included. The challenge in here is deciding the correct phonon dispersion to use in such

circumstances, because phonon scattering rates need to be recomputed with changes in

phonon description.

Under the equilibrium condition, the number of phonon with mode q at temperature T

is given by the Bose-Einstein distribution since phonons are Bose particles:

1)/exp(

1−

=Tk

NBq

q ωh (2.14)

2.4 Density of States

In doped semiconductor devices, the description of carriers depends on a large

number of parameters such as m*, impurity concentrations, dielectric constants, and Eg.

Ionization energy, impurity concentration, as well as some other constants and materials

parameters, is determined by semiconductor statistic interdependences of the free

majority and minority carrier concentrations. Semiconductor statistics describes the

ケ

ケクゴ

ケクシ

ケクズ

ケケクゴケクシケクズ

ケ

コケ

サケ

シケ

Reduced wave vector qa/2ɽ

Ener

gy (

eV)

Fre

q. (1

014

rad

/s)

g-type

f-type

TA

LA

TOLO

ケ

ケクゴ

ケクシ

ケクズ

ケケクゴケクシケクズ

ケ

コケ

サケ

シケ

Reduced wave vector qa/2ɽ

Ener

gy (

eV)

Fre

q. (1

014

rad

/s)

g-type

f-type

TA

LA

TOLO

Figure 2.5 Phonon dispersion in silicon along the [100] direction. Where

q is the wave vector and a is the lattice constant.

17

probabilities of carrier states either at vacant or populated states [10].

Electronic states generally includes localized impurity states as well as delocalized

conduction and valence band states. Impurity states have single states with no degeneracy

in the simplest case. However, it may need to be considered that an impurity may have a

degenerate ground state as well as excited levels. The DOS describes the states of carriers

in the bands and their dependence on energy. The free motion of carriers is confined to

two, one, or zero spatial dimensions in semiconductor devices. DOS in above spatial

dimensions must be known before applying semiconductor statistics to such systems of

reduced dimensions. DOS in bulk systems will be calculated in the following section.

2.4.1 Density of States in Bulk Semiconductors

Carriers occupy either localized impurity states or delocalized continuum states in

conduction or valence bands. The density of impurity states equals to the concentration of

impurities, when impurity forms a single, non-degenerate state. When the impurities are

sufficiently far apart, where coupling can be neglected, the energy of the impurity states is

the same for all impurities. On the other hand, the density of continuum states is more

complicated. Therefore, the density of continuum states will be calculated by considering

various band structures and quantum confinements.

Wave vector variation, dk, which is determined associated with energy variation, dE,

is integrated over the entire constant energy surface to obtain the volume of k-space,

enclosed between two constant energy surfaces with energies between E and E + dE. The

volume of k-space enclosed between the E and E + dE shown in Figure 2.6 is given by

[10]

E

E + dE

ds

E

E + dE

ds

Figure 2.6 Constant energy surfaces with energy E and E + dE.

18

∫ ∂∂= ds

kEkdEEV k )(

)( (2.15)

where the ds is an area element on the energy surface as shown in Figure 2.6 . Rewriting

Eq. (2.15) in three dimensional k-space use ),,(zyx kkk

k∂∂

∂∂

∂∂=∂

∫∇=)(

)(kE

dsdEEV

k

k (2.16)

As carriers require a volume of 34π in phase space, the DOS per unit energy can be

expressed as follows:

∫∇=)(4

1)(3 kE

dsENk

k π (2.17)

The surface element ds is always perpendicular to the vector E∂ in Eq. (2.17) and has the

dimension m-2

in k-space.

2.4.2 Single Valley, Anisotropic, Parabolic Band

Dispersion relation depends on spatial directions in an anisotropic single valley band.

Such anisotropic dispersions are found in semiconductors in which the lowest minimum

point is located at the L or X point of the BZ such as silicon crystals. In this case, the

surface of the constant energy is no longer a sphere, but an ellipsoid, as shown in Figure

2.7. The main axes of the ellipsoid have different lengths, and the three dispersion

relations are curved differently. The dispersion relation can be express as Eq. (2.18), if the

main axes of the ellipsoid align with a Cartesian coordinate system

z

z

y

y

x

xk m

km

k

mk

E222

222hhh ++= (2.18)

19

The energy vector E is given by frame Ek = (z

z

y

y

x

x

mk

m

k

mk

2,

2,

2

222hhh

). The absolute values of ds

and kE∂ can be taken for the integration, since the energy vector E is perpendicular on

the surface element. Consequently, the density of states in an anisotropic semiconductor

with parabolic dispersion relations is obtained by integration of Eq. (2.17) with the

dispersion relation of Eq. (2.18)

EmmmEN zyx32

2)(hπ

= (2.19)

The constant energy surfaces are rotational ellipsoids and two of the main axes of the

ellipsoid are identical. The short and long axes are then denoted as the transversal and the

longitudinal axes, respectively. Such a rotational ellipsoid is schematically shown in

Figure 2.7. A relatively heavy mass is associated with the longitudinal axis, and a

relatively light mass is associated with the transversal axis. If the masses are denoted as

ml and mt for the longitudinal and transversal, respectively, Eq. (2.19) can be modified as

follows

EmmEN tl2

32

2)(hπ

= (2.20)

The anisotropic masses mt, ml are used to define a density of states effective mass and

kx

ky

kz

kx

ky

kz

Figure 2.7 Ellipsoidal constant energy surface with a weakly and strongly

curved dispersion along the kx, ky, and kz axis.

20

conductivity effective mass.

2.5 Effective Mass

A simple analytic band approximation can be described by the relationship between

the k and *m as Eq. (2.1) as previously shown. The effective mass m* plays a

fundamental role in several physical problems of different nature: the electron group

velocity (Eq. (2.12)) in a crystal subject to an external force, the electron density of states

in energy space (Eq. (2.20)), the ratio of non-parabolicity parameter (Eq. (2.15)). The role

of effective mass m* in the different situations may not be the same, when the band is not

described by the simple parabolic band. For this reason different types of effective mass

m* have been defined, all of which coincide with those in simple case [11].

2.5.1 Conductivity Effective Mass

The conductivity effective mass (mc) is defined by the ratio of k to its velocity as

follows [11]:

kE

mk

c ∂∂=

h

h 1 (2.21)

For simple analytic bands, mc = *m can be obtained. For an ellipsoidal, parabolic band

1/mc is the inverse effective mass tensor. This indicates that electron momentum and

velocity is not parallel in non-spherical bands. For this case, the conductivity effective

mass can be obtained by averaging over all possible directions as follows:

)1(32)1(

311

tlc mmm+= (2.22)

This is also the conductivity effective mass at any given direction for a cubic

semiconductor, averaged over the valleys equivalent by symmetry.

21

2.5.2 Density of States Effective Mass

The effective mass plays again a role in the evaluation of the number of carriers in the

conduction band due to influence of effective mass on the electron density of state in

energy space. The electron density n is given by [11]

∫= dEENEfn )()( (2.23)

where N(E) is the density of states in energy space and f(E) is the electron distribution

function. N(E) is given by Eq. (2.20).

The electron density n with the expression Eq. (2.20) for non-degenerate statistics is

results in

TkEc

BceNn /−= (2.24)

where Ec is the energy of the bottom of the band, and

2/3

2

*

)2

(2hπ

TkmN B

c = (2.25)

For complicated bands Eq. (2.23) still holds, and n has more elaborate expressions. The

density of state effective mass md is defined in as a way that Eq. (2.24) still hold with md

in place of effective mass *m . For an ellipsoidal, parabolic band md results in

3/12)( tld mmm = (2. 26)

Conclusion

In this chapter, electronic band structures, phonon dispersions, and related topics,

which are indispensable for understanding the semi-classical carrier transport, are briefly

described. The discussion in this dissertation is based on the semi-classical treatment, and

the drift process of the carrier is analyzed classically. The semi-classical approach to

22

transport is applicable when the applied potentials vary slowly on the scale of an

electron’s wavelength.

References

[1] K. Tomizawa, Numerical simulation of submicron semiconductor devices. Artech

House, 1993.

[2] C. Jungemann, A. Emunds, and W. L. Engl, “Simulation of linear and nonlinear

electron transport in homogenous silicon inversion layers”, Solid-State Electronics

36, no. 11, 1529 (1993).


transport in semiconductors with applications to covalent materials”, Rev. Mod.

Phys. 55, no. 3, 645 (1983).

[4] E.Pop, “Self-Heating and Scaling of Thin Body Transistors”, Doctoral Thesis,

Stanford University, (2004).

[5] S. Datta, Quantum Phenomena, (Addison-Wesley, New York, 1989).

[6] E. Pop, R. W. Dutton, and K. E. Goodson, “Detailed heat generation simulations via

the Monte Carlo method”, in Simulation of Semiconductor Processes and Devices,

(Boston, MA, 2003), p. 121.

[7] D. Long, “Scattering of Conduction Electrons by Lattice Vibrations in Silicon”, Phys.

Rev. 120, no. 6, 2024 (1960).

[8] N. Sano, T. Aoki, M. Tomizawa, and A. Yoshii, “Electron transport and impact

ionization in Si”, Phys. Rev. B 41, no. 17, 12122 (1990).

[9] B. Fischer and K. R. Hofmann, “A full-band Monte Carlo model for the temperature

dependence of electron and hole transport in silicon”, Appl. Phys. Lett., 76, no. 5,

583 (2000).

[10] www.ecse.rpi.edu/~schubert/Course-ECSE-6968%20Quantum%20mechanics

23

[11] C. Jacoboni and P. Lugli, The Monte Carlo Method for Semiconductor Device

Simulation (Springer, New York, 1989).

24

Chapter 3

Carrier Scattering

Scattering of carriers in semiconductor devices is an important process for the

investigation of the device performance. Chapter 3 provides basic issues about the

calculation of scattering rates of carriers based on the first-order time-dependent

perturbation theory in quantum mechanics [1]. Carriers are not scattered at all in an ideal

periodic lattice, however, carrier scatterings occur when lattices deviate from ideal

periodicity caused by various reasons, such as impurity atoms, crystal defects, and

thermal ion vibrations known as the phonon. The carrier-carrier interaction is another

type of scattering, where many body problems should be treated by the second

quantization formalism introducing additional quasi-particles. In this chapter, the first

type of the scattering is briefly described below.

Calculation of scattering rate is usually based on Fermi's Golden Rule, which was

derived by the time-dependent perturbation theory of the first order. Fermi's golden rule

gives the transition rate from the initial state to the final state. The transition rate from the

initial state k to final state k’ is expressed as [2]

)(|'|'2)',( '

2ωδπhm

hkk EEkHkkkS −><= (3.1)

where H' denotes the perturbation potential. The δ -function indicates that the transition

probability has a non-zero value only when the argument of the δ -function is zero and

assures the conservation of energy. The sign m respectively denotes the emission (upper

sign with Ek’ = Ek - ωh ) and the absorption (lower sign with Ek’ = Ek + ωh ) of the phonon

energy ωh by an electron. The transition probability Eq. (3.1) is the basic equation in

scattering theory, which is applied to calculation of the scattering rate in semiconductors

devices. Notice that the interaction must be weak when Eq. (3.1) is applied in calculation

of scattering rate, because the derivation is based on the first-order approximation. The

matrix element is given by [2]

25

)',(

)(')(|'|'

'

'

kkIU

drrHrkHk

kk

kk

−

Ω

=

>=< ∫ ψψ (3.2)

where Ω is the volume of the crystal, kψ is the corresponding eigenfuction, and H’ is

the Hamiltonian operator. (k’ - k) expresses the conservation of momentum. I(k, k’) is the

overlap integral, and

∫Ω= drrurukkI kk )()()',( *' (3.3)

I(k, k’) = 1 for a nearly parabolic band; Uk’-k can be written as:

∫Ω⋅⋅−

− = dretrUeU rikrikkk

''' ),( (3.4)

The scattering rate of carrier can be obtained by integrating the transition probability

S(k, k’), given by Eq. (3.1) with respect to final states k

’. The equation for scattering rate,

W(k), is given by

')',()2(

)(3

dkkkSkW ∫Ω=π

(3.5)

The conservation of energy and momentum should be taken into account properly

when the integration over final states k’ in Eq. (3.5) is made. The scattering rate in Eq.

(3.5) is rewritten as Eq. (3.6), when the polar coordinate with the polar axis k, as shown in

Figure 3.1, is employed.

φθπ

ππdddkkkSkW ∫∫∫

∞Ω=

00

2

03')',(

)2()( (3.6)

Where θ is the polar angle and φ is the azimuthal angle defined as in Figure 3.1. The

transition rate is assumed to be independent of φ , since the crystal of semiconductor is

treated as a continuum in the effective mass approximation. Hence, the integration over

φ can be performed directly, and the scattering rate can be rewritten again as follows [2]:

26

θπ

πddkkkSkW ∫∫

∞Ω=00

3')',(

)2()( (3.7)

The integrations over θ and k’ are mutually related to each other because of the

energy and momentum conservation, and the double integral in Eq. (3.7) can be reduced

to a single one. This is discussed further in the following sections together with the

example of various scattering mechanisms.

3.1 Ionized Impurity Scattering

Carriers in semiconductor devices transport through heavily doped regions. The

transport of carrier is substantially disturbed by scatterings due to ionized impurities,

distributed randomly in heavily doped regions. This can be evidenced from two facts, the

impurity scattering is especially pronounced for low-energy carriers, and the carriers do

not acquire high energy levels from low electric fields in heavily doped regions [2].

The electrostatic potential is Coulombic due to a point charge in vacuum. However,

electrostatic potential is more or less screened due to an impurity charge in a lattice

depending on therein many free carriers are present. The ionized impurities scattering is

an elastic process. The most popular models for ionized impurity scatterings are Brooks

and Herring [3] and Conwell and Weisskopf approaches [4].The screening effect

treatment is the main differences between the two approaches. In this work, the

Brooks-Herring approach is employed when low doping concentration with some

Figure 3.1 Polar coordinate for k’ with respect to the polar axis k.

φ

θk’

k

φ

θk’

k

27

refinements described below. For the condition of high doping concentration, Kosina’s

approach, which is explained in this chapter, is adopted for simulation.

3.1.1 Brooks-Herring Approach

Within this approach the perturbation potential is [2]

rq

s

Der

ZeH

−=πε4

' (3.8)

where the Z denotes the number of charge units of the impurity, the sε is the dielectric

constant of the semiconductor, and the 1/qD is the screening length, the e is the magnitude

of the electronic charge. Substituting Eq. (3.8) into Eq. (3.2) and integrating over the

volume of Ω , substituting the results of matrix element into Eq. (3.1), the transition rate

of ionized impurity scattering from a single ionized impurity is [2]

222

'22

)(

)()(

2)',(

D

kk

s qq

EEZekkS

+−

Ω=

δε

πh

(3.9)

δ -function simply shows the fact that the electron energy is conserved during the

scattering since the screened Coulomb potential is time independent. The q2 = (k' - k)

2 =

2k(1 - cosθ ) due to the k' = k. Making use of q2 and multiplying Eq. (3.9) by NIΩ , which

is the number of impurities in the volume Ω , finally the form of the transition rate is as

follows

222'

2

4

])cos1(2[

)(2)',(D

kk

S

I

qk

EEZeNkkS

+−−

Ω=

θδ

επh

(3.10)

The total scattering rate can be obtained after substituting Eq. (3.10) into Eq. (3.7).

)4(

1)(2)(

2222

42

DDs

kI

qkq

ENeZNkW

+=

επ

h (3.11)

28

where N(E) is the Density of states as in Eq. (2.20). The parameter qD is defined by

Tk

neq

BsD ε

02

= (3.12)

here no is the equilibrium electron density, Ze denotes the charge on the impurity atom.

Figure 3.2 shows the scattering rates of silicon calculated by Eq. (3.11) for NI = 1018

cm-3

, where we assumed no = NI.

3.1.2 Kosina’s Approach

When doping concentration is low enough, the impurity scattering is a highly

anisotropic scattering. The low energy carriers have high scattering rate as shown in

Figure 3.2. The carriers show a strong preference for small scattering angles if carriers

have high momentum energy. Direct implementation of Brooks-Herring approach in a

Monte Carlo simulation would yield several problems. Many scattering of the low energy

carrier’s events, which has a weak influence on performance of semiconductor device,

would have to be processed consuming computational time. Many short free-flight times

would be also further degrading the efficiency of the simulation. The Brooks-Herring

approach is not more suitable for high doping concentration (usually ND >>1018

cm-3

) due

to the invalidation of distribution function. The scattering approach proposed by Kosina

[5] avoids such way by treating the ionized impurity scattering as an isotropic scattering

ケクケケクケケクケケクケ

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ゲクケゲクケゲクケゲクケ

ゲクコゲクコゲクコゲクコ

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ピヤモヵヵユンリワヨァンモヵユ

ァォヴ


ァォヴ


ァォヴ


ァォヴギ

ゲギゲ

ギゲ

ギゲオオオオ ××××1014

ケクケケクケケクケケクケ

ケクコケクコケクコケクコ

ケクサケクサケクサケクサ



ゲクケゲクケゲクケゲクケ


ケケケケケクザケクザケクザケクザゲゲゲゲゲクザゲクザゲクザゲクザココココ



ァォヴ


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ァォヴ


ァォヴギ

ゲギゲ

ギゲ

ギゲオオオオ ××××1014


cm-3

.

29

with the same momentum relaxation time. This approach is been shown to be adequate

for doping concentrations up to 1020

cm−3

.

Within this approach the scattering rate is given by

)1

)1(ln(41)()(

2 bbb

kkCkW

+−+= (3.13)

where

)()(2

)(2

02

42

k

eZNkC

s

I

υεεπh= (3.14)

22 /4 skb β= (3. 15)

)(

)(

2/1

2/1

0

22

ηη

εεβF

F

Tkne

nBss

−= (3.16)

sβ is the inverse Thomas-Fermi screening length, the Fj is the Fermi integral of order j,

and the η is the reduced Fermi energy, the )(kυ denotes the average carrier’s velocity.

Figure 3.3 shows the scattering rates of silicon calculated by Eq. (3.13) for NI = 1020

cm-3

. The result shows that the scattering of low energy carrier is significantly degraded.

ケケケケ


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テロユヤヵンヰワァテワユンヨケァォユプオテロユヤヵンヰワァテワユンヨケァォユプオテロユヤヵンヰワァテワユンヨケァォユプオテロユヤヵンヰワァテワユンヨケァォユプオピヤモヵヵユンリワヨァンモヵユ

ァォゲケ


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ァォゲケゲサ

ゲサ

ゲサ

ゲサゲグヴオ

ゲグヴオ

ゲグヴオ

ゲグヴオ


cm-3

.

30

3.2 Phonon Scattering

Electrons are not scattered by ideal periodic potentials associated with the periodic

array of ions constituting the crystal when Bloch electrons are eigenstates in a perfect

crystal. However, due the lack of periodicity due to various reasons, electrons are

scattered by lattice vibrations propagating in the crystal. The deviation of the crystal

potential from pure periodicity may be expressed by the amplitude of the lattice

vibrations due to a small displacement of an ion in the crystal causes a small change in

the crystal potential. This deviation from the periodicity is generally expressed in a rather

phenomenological way as the deformation potential method because of the difficulty of

knowing the crystal potential itself. The influence of lattice vibrations on electron motions

can be expressed by a quantum process since the wave nature of the lattice vibrations can

be quantized as phonons. This process is termed the electron-phonon scattering [2].

Carriers can acquire high energy levels from high electric field applied in ultra-scaled

devices. Scatterings based on the spontaneous emission of phonons take place even

though there are only a few phonons present at low temperature. Therefore, the

electron-phonon scattering becomes important for scattering processes in such

ultra-scaled semiconductor devices. There are usually two different types of phonon

process present: acoustic phonon and optical phonon. The neighboring atoms displace in

the same direction as the acoustic mode phonons, and hence the changes in lattice spacing

Unite Cell Unite Cell Unite Cell

a a

m1 m2

Equilibrium

Acoustical Vibration

Optical Vibration

Unite Cell Unite Cell Unite Cell

a a

m1 m2

Equilibrium

Acoustical Vibration

Optical Vibration

Figure 3.4 Vibrations in a crystal with two atoms per unit cell with masses

m1, m2. Acoustic vibration: the two atoms on the unit cell vibrate along the

same direction; Optical vibration: the two atoms on the unit cell vibrate in

opposing motion. a is the lattice constant.

31

are produced by the differential displacement or the strain. The neighboring atoms

displace in opposite directions as the optical phonons, and hence the displacement

produces the change in lattice spacing directly as shown in Figure 3.4. The acoustic and

optical phonon scattering are referred to as deformation potential scattering, since they

can be expressed by the deformation potential, which relates lattice vibrations with the

changes in the band energies.

Strong interaction due to polar nature of bonds can take place in the compound

semiconductors in addition to the acoustic and optical phonon scattering. Phonon perturbs

the dipole moment between atoms due to the displacement of the lattice. Polar scattering

due to acoustic and optical phonons are termed as piezoelectric and polar optical

scattering, respectively. Polar scattering is a dominant scattering mechanism for carriers

in compound semiconductor devices. However, the polar scattering can be negligible in

uncompounded semiconductor devices.

3.2.1 Intravalley Scattering by Acoustic Phonon

In the process of intravalley scattering, the initial and final states of a carrier are

assumes within the same valley. Intravalley scattering usually involves only acoustic

phonons [6].

Combining absorption and emission process together, the transition rate due to the

acoustic phonons is given by [2]

)()'()21

21()',( '

22

qkkqq

d EEqkknq

kkS ωδδρωπ

hmmm −−+Ω

Ξ= (3.17)

The two functions can be combining into a single one as follows

)cos

2()()'(

*

2

*

22

' qqkkm

kq

m

qEEqkk ωθδωδδ hm

hhhmm ±=−− (3.18)

where 'θ is the polar angle between k and q as shown in Figure 3.5, there is relation

between 'θ and q as shown in Eq. (3.19).

32

)(21'cos

qk

Ek

q

k

qωθh

m += (3.19)

Eq. (3.19) provides the angle between two successive momentum states k and q based

on the original energy Ek, and phonons energy qωh . qω is the angular frequency of

oscillation. The acoustic phonon energy is much smaller than kBT at room temperature. If

thus phonon energy qωh is assumed to be zero, the acoustic phonon scattering can be

considered as elastic process. Based on this approximation, Eq. (3.17) can be rewritten as

follows:

)'cos2

()',(22

θδπ ±Ω

Ξ≈kq

qEk

CTkq

kkSkL

LBd

h (3.20)

Total scattering rates of acoustic phonon can be obtained after substituting Eq. (3.20)

Figure 3.5 Definition of polar angle and wave vector.

'k 'k

k kq q

θ θ

'θ'θAbsorption Emission

32

2/3*

4

)2()(

hπk

k

EmEN =

ケケケケケクコケクコケクコケクコケクサケクサケクサケクサケクシケクシケクシケクシケクスケクスケクスケクスゲゲゲゲ


ピヤモヵヵ

ユンリワヨァンモヵユ

ァォヴ

ピヤモヵヵ


ァォヴ

ピヤモヵヵ


ァォヴ

ピヤモヵヵ


ァォヴギゲ

ギゲ

ギゲ

ギゲオオオオ 1000

100

10

1

1011

32

2/3*

4

)2()(

hπk

k

EmEN =



ピヤモヵヵ


ァォヴ

ピヤモヵヵ


ァォヴ

ピヤモヵヵ


ァォヴ

ピヤモヵヵ


ァォヴギゲ

ギゲ

ギゲ

ギゲオオオオ 1000

100

10

1

1011

Figure 3.6 Scattering rate for the elastic acoustic phonon scattering at 300 K.

33

into Eq. (3.7).

)(2

)(2

kL

B ENC

TkkW

h

Ξ= π (3.21)

where N(Ek) is the Density of states, CL is the elastic constant of the material, the

proportionality constant Ξ is termed the deformation potential.

Figure 3.6 is a plot of the acoustic phonon scattering rate versus electron energy for

electron in the X-valley of Silicon at 300 K.

3.2.2 Inelastic Acoustic Phonon Scattering

In this work, the scattering rate of inelastic acoustic phonon is calculated separately

with phonon vibration branch LA and TA as shown in Figure 2.5. The total scattering rate

versus the carrier energy Ek is as follows

dqqINk

mkW qq

qs

d 3

2

2

)21

21(1

4)( m

h+Ξ= ∫ωπρ

(3.22)

where ρ is the mass density of the material, Ξ is the deformation potential include

LAΞ and TAΞ , and md is the density of state effective mass. The upper and lower signs

refer to the process of phonon absorption and emission, respectively. The Np is the

Figure 3.7 Scattering rate for the inelastic acoustic phonon scattering

with LA phonon vibration branch at 300 K.

Energy (eV)


1013

1012

1011Scattering rate (1/s)

Emission

Absorption

Energy (eV)


1013

1012

1011Scattering rate (1/s)

Emission

Absorption

34

Bose-Einstein distribution.

The overlap integral of wave function can be express by the rigid ion approximation

[7]

)]cos()[sin()(

33 sss

s

q qRqRqRqR

I −= (3.23)

where Rs is the radius of the spherical Wigner-Seitz cell, and Rs = a[3/(16π)]1/3

= 0.2122

nm for silicon.

3.2.3 Intervalley Scattering by Optical Phonon

Carriers can be scattered between different band valleys both by acoustical and optical

phonons. In the same way as intravalley scattering mechanism, intervalley scattering can

be treated as a deformation potential interaction by phonons energy. Intervalley scattering

process has two different types, the f-type and g-type. If the initial and final orientations

of scattering are different, this process is referred to as f-type, otherwise scattering process

is the g-type as illusion in Figure 3.8. The transition rate of this scattering mechanism is

given by:

)()21

21()',( '

2

jiijkkqij

jij EEENZD

kkS ∆+−+Ω

= ωδρωπ

hmm (3.24)

Figure 3.8 The band valleys of silicon and intervalley scattering.

[100]

[010]

[001]

f

g

35

Repeating the calculation in a manner similar to that for the intravalley scattering,

substituting Eq. (3.24) into Eq. (3.7), the scattering rate can be obtain the following form

[2]

)()21

21()(

2

ijkqij

ijij ENNZD

kW ωρωπ

hm ±+= (3.25)

where Z j is the number of possible equivalent final valleys of the same type. For f-type

scattering Z = 4 and for g-type scattering Z = 1 for bulk silicon. Dij is the coupling

constant, ijωh is the corresponding phonon energy. jiE∆ is the energy difference

between the valley j and i.

Figure 3.9 shows the scattering rates versus electron energy for electrons in bulk

silicon due to the intervalley phonons between two equivalent valleys (g-type).

3.3 Impact Ionization

Impact ionization is the process that the carriers with enough kinetic energy knock

bound carriers out of its bound state to create other carriers and lose their kinetic energy,

which requires a large electric field. Impact ionization is a typical non-equilibrium


テワユンヨケァォユプオテワユンヨケァォユプオテワユンヨケァォユプオテワユンヨケァォユプオピヤモヵヵユンリワヨァンモヵユァォヴ

ピヤモヵヵユンリワヨァンモヵユァォヴ


ピヤモヵヵユンリワヨァンモヵユァォヴギ

ゲギゲ

ギゲ

ギゲオオオオ

テヮリヴヴリヰワテヮリヴヴリヰワテヮリヴヴリヰワテヮリヴヴリヰワヂャヴヰンヱヵリヰワヂャヴヰンヱヵリヰワヂャヴヰンヱヵリヰワヂャヴヰンヱヵリヰワ

1000

100

10

1

1011


テワユンヨケァォユプオテワユンヨケァォユプオテワユンヨケァォユプオテワユンヨケァォユプオピヤモヵヵユンリワヨァンモヵユァォヴ



ピヤモヵヵユンリワヨァンモヵユァォヴギ

ゲギゲ

ギゲ

ギゲオオオオ

テヮリヴヴリヰワテヮリヴヴリヰワテヮリヴヴリヰワテヮリヴヴリヰワヂャヴヰンヱヵリヰワヂャヴヰンヱヵリヰワヂャヴヰンヱヵリヰワヂャヴヰンヱヵリヰワ

1000

100

10

1

1011

Emission

Absorption

Figure 3.9 Scattering rate for the g-type phonon scattering when

phonon energy is 19 meV.

36

process. It creates an electron-hole pair by colliding with an electron in the valence band

and exciting it to the conduction band.

Impact ionization is important in strained-Si simulation due to the reduction of

threshold energy. The scattering rate of impact ionization is a quadratic function of

electron kinetic energy above the ionization threshold. The scattering rate of impact

ionization is modeled using a modified threshold expression in ref. [8], which is

expressed as follow:

bthEkEPkW ])([)( −= (3.26)

where Eth is empirical threshold energy, E(k) is a carrier’s energy and P is a pre-factor

which determine the softness of the threshold. The threshold energy is 1.1 eV for

unstrained Si. The threshold energy is assumed to be proportional to the band gap, by

Eth(x) = 1.1Eg(x)/Eg(0) for the strained Si. Where fraction x defines the strain present of

the Si material, the Eg is band gap. When ionization occurs, the original electrons are

assumed to not change valley.

Figure 3.10 shows the scattering rates of impact ionization versus electron energy for

electrons in bulk silicon, where the threshold energy is Eth = 1.1 eV, the pre-factor is P =

6.25×1110

, the value of b is 2.0.

Figure 3.10 Scattering rate for impact ionization scattering versus

carrier energy.

ゲゲゲゲゲクゲゲクゲゲクゲゲクゲゲクコゲクコゲクコゲクコゲクゴゲクゴゲクゴゲクゴゲクサゲクサゲクサゲクサゲクザゲクザゲクザゲクザ

Energy (eV)

1011

107

105

109

ゲゲゲゲゲクゲゲクゲゲクゲゲクゲゲクコゲクコゲクコゲクコゲクゴゲクゴゲクゴゲクゴゲクサゲクサゲクサゲクサゲクザゲクザゲクザゲクザ

Energy (eV)

1011

107

105

109

Scattering rate (s-1)

37

3.4 Wave Vector after Scattering

The new wave vector k' of carriers is needed to determine after determined the

scattering mechanism. The magnitude of k' is determined by the new energy of carriers

after scattering. According to the laboratory frame ( LLL zyx ,, ) chosen for simulated

device, the final direction of k' is determined in terms of its components in a Cartesian

coordinate ( Lz

Ly

Lx kkk ,, ) [2].

If the scattering process is an isotropic in which the scattered carrier has the same

probability of being in any direction after scattering. The components of wave vector kx,

ky, and kz can be found by considering that the probability density p(φ ,θ )dφ dθ is

proportional to the number of available states on a sphere of radius k' , where φ and θ

are the azimuthal and polar angles of k' with respect to kz . p(φ ,θ ) equals sin(θ ), since

any φ is equally probable. Therefore, θ and φ can be determined by a couple of

uniform random numbers, r1 and r2, lying between 0 and 1 [2].

2

1

2

21cos

r

r

πφθ

=

−= (3.27)

The components of the laboratory frame (k’x, k

’y, k

’z) for the φ and θ given by Eq.

(3.27), are easily obtained as [2]

φθ cossin'' kk x =

φθ sinsin'' kk y =

θcos''

kk z =

φθ cossin'' kk x =

φθ sinsin'' kk y =

θcos''

kk z = (3.28)

These wave vector expressions of carrier after carrier scattering are only available for

the case of isotropic scattering process.

If the scattering process is an anisotropic in which the scattered carrier has the various

probability of being in scattering direction. For anisotropic scattering processes, such as

ionized impurity scattering, the final state k' is denoted by φ and θ , which are the

polar and azimuthal angles between the final state k' and initial state k. As isotropic

scattering, the final state k' is expressed in terms of the components k’x, k

’y, k

’z in the

38

chosen laboratory frame.

The transition rate is independent of the azimuthal angleφ . Therefore, the azimuthal

angle can also be determined randomly for the anisotropic scattering, thus, φ can be

written as form:

(3.29)

where r3 is a uniform random number lying between 0 and 1.

The polar angle θ for the ionized impurity scattering is given as follow:

(3.30)

When transformation the direction of the wave vector of carrier, k, in a frame

( Lz

Ly

Lx kkk ,, ), it is to work with a new frame ( r

zry

rx kkk ,, ). The kz-axis is parallel to the initial

wave vector k. The new frame ( rz

ry

rx kkk ,, ) are obtained by rotating laboratory frame

( Lz

Ly

Lx kkk ,, ) by an angle β on the vertical plane of kz-axis, and then α on the vertical

plane of kx-axis, as shown in Figure 3.1. Hence, the frame ( Lz

Ly

Lx kkk ,, ) need to be multiply

by the following matrix transform to ( rz

ry

rx kkk ,, )[2]

−

−

ααβαβαβ

βαβαβ

cossin0

cossincoscossin

sinsinsincoscos

(3.31)

where functions sine and cosine in the matrix elements of Eq. (3.31) are given as follows:

,sin

22

k

kk yx +=α ,sin22yx

x

kk

k

+=β

(3.32)

k

kz=αcos , 22

cosyx

y

kk

k

+=β

24

4

)2)(1(1

21cos

Dqkr

r

−+−=θ

32 rπφ =

39

Let θ and φ be the polar and azimuthal angles between final state k' and initial state k,

as shown in Figure 3.1. The kx, ky, and kz components in the ( rz

ry

rx kkk ,, ) frame are obtained

from [2]

)cos',sinsin',cossin'( θφθφθ kkk (3.33)

Combining Eq. (3.31), Eq. (3.32), and Eq. (3.33), the components of the wave vector

after scattering in terms of the ( Lz

Ly

Lx kkk ,, ) frame are obtained as [2]

−

−=

θφθφθ

ααβαβαββαβαβ

cos

sinsin

cossin

cossin0

cossincoscossin

sinsinsincoscos

'

'

'

'

'

'

k

k

k

k

k

k

z

y

x

(3.34)

The procedure to evaluate the components (k’x, k

’y, k

’z) for the ionized impurity

scattering is summarized as follows. The carrier has the initial energy Ek and wave vector

k and the components (kx, ky, kz) before scattering is occur. After impurity scattering is

occurred, the carrier has same energy value Ek because the ionized impurity scattering is

elastic scattering. The θcos is calculated by Eq. (3.30) and θsin is equal to 1- 2)(cosθ .

The azimuthal angle φ is calculated by Eq.(3.29), then φsin and φcos are calculated

by solving the functions sine and cosine in the matrix elements of Eq. (3.31) with Eq.

(3.32). The k' in Eq. (3.34) is equal to k. Finally, the components (k’x, k

’y, k

’z) can be

obtained by Eq. (3.34).

Lzk

Lxk

Lyk

rxk

ryk

rzk

ββ

α

α

Figure 3.11 Relation between the laboratory frame and new frame.

40

Conclusion

This chapter was devoted to illustrate the procedures and a few of the tricks

commonly employed to evaluate scattering rates for carriers in semiconductor devices.

The method employed is based on Fermi’s Golden Rule. The scattering rates are

evaluated directly from the transition rate )',( kkS . Scattering mechanisms discussed in

this chapter are limited to those that are important for carrier transport in common

semiconductors, and characteristics for bulk silicon are plotted as the scattering rate

versus energy. Various parameters necessary for calculation of the scattering rate in

silicon are listed in tables Physical Constants and Material Parameters. In practice, the

overlap integrals must be treated as an integral, and the energy bands cannot be described

as parabolic and spherical. For high energy electrons in silicon, the ellipsoidal nature of

the conduction band minima must be included. The full band numerical treatment of the

scattering mechanism is essential for the high-energy carriers in the device. Discussion on

other scattering mechanisms, such as alloy scattering, neutral impurity scattering and

radiative processes, are mostly found in refs. [9] and [10]. Many topics about the

2-Dimensional Electron Gas are found in papers cited in ref. [11].

References

[1] Schiff, L. I., Quantum Mechanics, New York, McGraw-Hili, (1968).


House, 1993.

[3] H. Brooks, “Scattering by ionized impurities in semiconductors”, Physical Review, 83,

879 (1951).

[4] E. Conwell and V. F. Weisskopf, “Theory of Impurity Scattering in Semiconductors”,

Physical Review 77, 388 (1950).

41

[5] H. Kosina and G. Kaiblinger-Grujin, “Ionized-Impurity Scattering of Majority

Electrons in Silicon”, Solid-State Electronics 42, no. 3, 331 (1998).

[6] C. Hamaguchi, Basic semiconductor physics. Springer, (2001).

[7] A. Haug, Theoretical solid state physics. Pergamon Press, 2, (1972).

[8] E. Cartier, M. V. Fischetti, E. A. Eklund, and F. R. McFeely, “Impact ionization in

silicon”, Appl. Phys. Lett. 62, no 25, 3339 (1993).

[ 9 ] Nag. B., Electron Transport in Compound Semiconductors, New York:

Springer-Verlag, 1980.

[10] K. B. Ridley, Quantum Processes in Semiconductors, London: Oxford, 1982.

[11] Ando, T., A. B. Fowler, and F. Stern, “Electronic properties of two-dimensional

systems”, Review of Modern Physics 54, no.2, 437 (1982).

42

Chapter 4

Monte Carlo Method for Devices Simulation

Computer simulation is an important approach to the device characterization and the

design of device structures. Using appropriate computer programs to analyze the device

characteristics associated with various parameters is often a more economic, efficient and

convenient method than the corresponding experimental study. Computer simulation has

been used in a wide range of application for conventional device simulations [1, 2, 3]. In

recent years, owing to the development of high-speed and ultra-size devices, conventional

simulation methods can not satisfy the demand in this area because of the limitation of

transport model. On the other hand, the Monte Carlo (MC) method can achieve more

applications in the new field.

The Monte Carlo method is a numerical statistical method that uses probability to

solve physical and mathematical problems. This approach is well suited for simulation of

physical phenomena associated with the stochastic processes. In fact, MC method was

applied to some problems in the neutron transport and the statistical physics before it was

applied to the carrier transport in semiconductor devices [4, 5]. MC simulations on

transport properties are based on the description of particle motion. The progress in

plasma simulations is helpful for the semiconductor device simulation [6].

MC methods began to be used in the semiconductor carrier transport when high

electric-field effect needs to be calculated. Most of the earlier methods assume that the

transport of electron follows the Maxwellian distribution or the Maxwellian distribution

of displacement. Under a large electric field with complex energy bands, the Maxwellian

distribution is not satisfied. One needs to turn back to semi-classical transport models of

the distribution function based on Boltzmann equation, because the semi-classical

transport problem is still a problem for solving the distribution function of the system.

Therefore, the MC approach has been adopted as a numerical method to solve the

Boltzmann equation.

The MC simulation of carrier transport has had a great progress over the past two

decades. Researchers have implemented transport mechanisms in the device simulation,

including new scattering processes, boundary conditions, electrostatic self-consistency,

more comprehensive models, more efficient simulation algorithms, etc. The MC approach

43

is regarded as the most important approach for the simulation of ultra-scale devices under

the various electric field conditions. A more accurate simulation is the introduction of full

energy electronic bands extracted by empirical pseudopotential calculations [7].

The first multi-valley MC simulation with the parabolic band, a single longitudinal

acoustic (LA) phonon, and six fixed-energy intervalley phonons was introduced in ref. [8].

Ref. [9] considered the non-parabolic band and slightly altered phonon deformation

potentials. Novel deformation potentials, which more closely match the available data on

electron diffusion in silicon, are introduced in ref. [10] a few years later. This phonon

model was the widely referenced review of the MC simulation, and it became the set of

phonon energies and deformation potentials most often employed in the literature over the

past two decades. Scattering with intervalley phonons are introduced by other workers

[11]. The full band MC simulation of silicon, computed from empirical pseudopotentials,

is firstly introduced in by Tand et al., [12]. They used the simple phonon model of LA

phonons, six fixed intervalley phonons as ref. [8], and the deformation potentials of ref.

[10]. The impact ionization in a full band MC simulation with the multi-valley

deformation potentials of ref. [13] was introduced by Sano et al. [10]. Realistic MC

device simulations using self-consistent full band were first performed by Fischetti et al.

[14]. They also make the distinction between longitudinal and transverse acoustic (TA)

phonon scattering, using a simple analytic dispersion for both LA and TA. Ref. [14]

pointed out the definition of energy valleys in the full band simulation and used two

phonon potentials, i.e. the fixed-energy optical phonon and the LA phonon including

dispersion. The most sophisticated MC simulation for carrier transport in silicon was

performed by ref. [15] and ref. [16]. They employed the full phonon dispersion obtained

from an adiabatic bond-charge model and the full band computed from empirical

pseudopotentials. The electron-phonon scattering rates were calculated as a function of

wave vector and energy in consistency with the phonon dispersion and the band structure.

Most MC simulations found in practice today employ full energy bands, yet scattering

rates and energy exchange with the lattice are still computed with simplified phonon

dispersion model. Phonon energies and deformation potentials in most frequent use are

those originally introduced in ref. [10]. MC simulation is a huge computational simulation

system that deals with the random events. Especially, the free flight time occupy a large

part of the CPU time. Reduction of CPU time for this part of the simulation is a topic

issue in MC simulation. Borsari employed the step scattering method to improve the

simulation time [17]. Kato proposed to optimize the value of self-scattering, and the CPU

44

time was further reduced significantly [18].

Full band MC simulation is necessary for high-energy carrier-transport simulations,

including impact ionization and high-energy scattering, when semiconductor devices

operate with higher external biases. As the device dimension is scaled into nanometer

regime, and the operation voltages are reduced below the band gap of material, roles of

the impact ionization and other high energy scatterings are greatly diminished. Carrier

motion at lower energies can be simulated with analytic band models. Hence, simpler

analytic-band MC simulation can be employed for future nanoscale devices [19]. In

addition, the phonon dispersion-relation is commonly simplified in practical device-

simulations despite the increasingly sophisticated treatment of electronic energy bands

over the years. Electron-phonon interaction is usually computed with the elastic

intravalley acoustic phonon scattering and the inelastic optical phonon scattering with one

or a few fixed phonon energy in silicon. In this study, we employed the MC model which

uses analytical descriptions for both the electron band structure and the acoustic phonon

dispersion relationship, when the effect of heat generation is included in the simulation.

4.1 Procedure of Monte Carlo Method

General processes of the MC simulation for carriers transport and scattering in

semiconductors devices have been well described [20]. This section provides a brief

introduction for MC algorithm. Ensemble MC method used in this work preselects

several tens of thousands “super-particles” to represent the mobile carriers inside the

devices. This number is limited by computational constraints, but good simulation results

can be obtained if the number of super-particles and simulation time are larger enough.

The particles are initialized with thermal energy distributions by expression 3kBT/2×r, r is

a random number uniformly distributed between 0 and 1, and with randomly oriented

wave vector. When the simulation is started, the particles are allowed to drift for short

free flight time (τ ), which is shorter than the average time between collisions, then one

process of scattering is selected. The selection of scattering mechanism can be made in

such a way that the scattering rate compare to the total of all scattering rates (Γ )

independent of the carrier energy. The free flight time (τ ) of each particle can be

consequently determined by total scattering rate (Γ ) and a uniform random number (r1)

as [20]

45

Γ

−= )ln( 1rτ (4.1)

The total scattering rate (Γ ) is taken to be larger than the largest value of scattering

rate WT(Ek) to avoid a negative value of scattering rate within the selected energy range of

carrier. The smaller total scattering rate is desirable to minimize the number of

self-scattering. It is possible to calculate the drift process based on the equation of motion,

which is described in section 4.2, after determined the free flight time (τ ). The carrier is

to drift under the influence of the electric fields during its free flight time. A scattering

mechanism is selected in proportion to the rate of each scattering process. Another

random number r2 between 0 and 1 is selected. Then, the random number r2 is compared

with cumulative rate of each scattering which have been pre-computed at the beginning of

the simulation. The particle continues its free flight unimpeded if the random number r2 is

larger than the rate of each scattering process. If a real scattering mechanism is selected if

the random number r2 is smaller than one scattering process, the scattering process of

carrier is include the calculation of new energy and momentum. After finish the scattering

process, another new random time of free flight is started. All particles in device system

will be repeats this procedures. The Poisson equation must be solved at every time step in

the case of a realistic device simulation to self-consistently update the electric fields. The

MC simulation can also be run on the fixed fields without the solution of Poisson

equation, although previous work has shown that the results are less accurate and

predictive [21]. When the Poisson equation is solved the super-particles are treated as

charge clouds. The super-particles are treated as single carriers during their free flights.

The cloud-in-cell method is most often employed for assigning the super-particle charge

to the grid nodes before Poisson’s equation is solved [20]. The solution for Poisson’s

equation yields a much more stringent requirement on the simulation time steps to avoid

charge imbalance. If device has high doping concentration regions, where NI= 1020

cm−3

,

short simulation time step that less than 1 fs is necessary. The density of carrier must be

updated at the end of each time step at device contacts. The density of carrier is neutrality

at the contacts of source and drain by delete or injecting carrier at the grid nodes vicinity

to the contacts. Therefore, the numbers of super-particle are updated every time step until

reaching a targeted accuracy. The MC simulations are not suited for low-field carrier

transport, where the drift-diffusion method may be preferred. However, the MC method

46

represents the most physically comprehensive simulation approach for charge transport in

semiconductors.

4.2 Drift Process

When potential energy of carriers varies slowly as a function of position, drift process

of carriers in semiconductor devices can be treated semi-classically. Thereby, carriers can

be regarded as free particles with an effective mass. Based on the equations of motion for

carriers in chapter 2, the change in the wave vector during the free flight time τ is

obtained by integrating the equation of motion with respect to time; thus [20],

∫+∇−=∆

τt

tHdtk

h

1 (4.2)

where H is the total energy of an carriers with a charge e given by

H= Ek – eV(r) (4.3)

where Ek is the kinetic energy of the carriers and V(r) is the electrostatic potential. If an

electric field F is applied on a semiconductor device, Eq. (4.2) has a solution as:

Figure 4.1 Selection of a scattering mechanism algorithm

flowchart.

Random number r2

Finish

)(12 kEr Λ<

)(22 kEr Λ<

)(2 kn Er Λ<

N

N

N

Y

Y

Y States of after

scattering

States of after

scattering

States of after

scattering

Random number r2

Finish

)(12 kEr Λ<

)(22 kEr Λ<

)(2 kn Er Λ<

N

N

N

Y

Y

Y States of after

scattering

States of after

scattering

States of after

scattering

47

τh

eFk −=∆ (4.4)

4.3 Scattering Process

In the scattering process, firstly determined what scattering mechanism by which a

carrier is to be occurred, and then identify the carrier state after the scattering. The

selection of a scattering mechanism can be made by using functions )( kn EΛ defined as

[20]:

Γ=Λ∑=

n

j

kj

kn

EW

E1

)(

)( For n = 1, 2, …., N (4.5)

which are the successive summations of the scattering rates normalized with the

maximum of sum of all scattering rates Γ . Γ is identical to the parameter defined by

∑=

=Γn

j

kj EWMax0

))(( (4.6)

and n is the total number of scattering mechanisms. A scattering mechanism for carriers

with energy Ek is selected by generating a random number r2 lying between 0 and 1, and

comparing r2 to )( kn EΛ ; if the functions )( kn EΛ is satisfied the condition as follows:

)(1 kn E−Λ < r2 < )( kn EΛ n = 1, 2, …., N 0 (4.7)

n-th scattering mechanism is chosen. The Pauli's exclusion principle is not taken into

account in Eq. (4.7), because the carrier occupancy in the final states is ignored. Selection

steps for scattering are described in the flowchart shown in Figure 4.1.

If intravalley acoustic phonon, intervalley phonon, ionized impurity and impact

ionization scattering are considered in the simulation, scattering rates of selected

scattering mechanism are W1(Ek), W2(Ek), W3(Ek) and W4(Ek) with carrier energy (Ek),

respectively. Constants of scattering rates Γ can be calculated by

48

))()()()(( 4321 kkkk EWEWEWEWMax +++=Γ Ek = 0, 0.001, 0.002, …., Ek (eV) (4.8)

Then, determine which scattering mechanism will be occurring by comparing the random

number r1 to )( kn EΛ in simulation as shown in Figure 4.2.

4.4 Velocity Calculation

In simulation of semiconductor device, MC simulation is equivalent to solving the

Boltzmann transport equation. Distribution function can be calculated by the mean

velocity of carriers, and energy can be calculated when the flight time of carriers in each

volume element of k-space is accumulated. This process demands a large amount of

memory to accumulate the data in k-space. However, it is not necessary to do this, due to

the mean values of carrier velocity and carrier energy can be calculated directly by

monitoring each carrier flight and then taking an average over all flights. The

instantaneous carrier velocity is formed by [20]

kkE∇=h

1υ (4.9)

therefore, the mean velocity of carrier during flight time τ can be formed as

.

)()()()()(

)()()()(

)()()(

)()(

123442

12332

1222

112

else

EWEWEWEWErelseif

EWEWEWErelseif

EWEWErelseif

EWErif

kkkkk

kkkk

kkk

kk

Γ+++=Λ<

Γ++=Λ<

Γ+=Λ<

Γ=Λ<

Acoustic Scattering

Intervalley Scattering

Impurity Scattering

Impact Scattering

Without Scattering

Figure 4.2 Flowchart of the scattering selection in simulation.

49

kEk

∇∇=><

h

1τυ (4.10)

where kE∇ and k∇ are small increments of the carrier energy and carrier wave vector

during flight timeτ , respectively. Substituting Eq. (4.4) into Eq. (4.10), then

τυ τ eF

Ek∇−=>< (4.11)

Making use of the mean velocity of carrier during flight timeτ given by Eq. (4.11),

the mean velocity of carriers during the total simulation time T is given as

∑ ><=>< τυυ τTT1

∑ −= )(1if EE

eFT (4.12)

where Ef is the energy of carrier at the end of the flight and Ei is the energy of carrier at

the start of the flight. The summation has to be made for all free flights. Eq. (4.12) shows

that the energy increment during each free flight time. The same reasoning leads to mean

energy of carrier being given as follows

ττ∑ ><=>< E

TE T

1 (4.13)

where <E> is given to a good approximation by

2fi EE

E+

=>< τ (4.14)

To evaluate the mean energy of carrier <E>T and mean velocity of carrier T><υ , the

calculation of τυ >< and τ>< E must be added to the drift process.

50

4.5 Ensemble Particle Motion

Ensemble Monte Carlo (EMC) method is based on the simultaneous and successive

calculations of the drift and scattering process of many super-particles during a small time

increment t∆ . The EMC method is essentially dynamic, and is therefore suitable for the

analysis of carrier transport in devices. The EMC method can also be applied to stationary

problems by continuing the calculations until the system reaches a steady state. Figure 4.3

shows schematically the flowchart of an EMC calculation process. The horizontal lines

are the trajectories of super-particles versus the time coordinate. The time increases to the

right of each line. The vertical broken lines show the time when the super-particles

system is observed. The interval between two adjacent broken lines corresponds to the

small time increment t∆ between two observations points. Each symbol ∗ on the

horizontal lines shows the time that the scattering is occurs. Hence, the interval between

two adjacent symble ∗ is the free-flight time of carrier. For the observation of the whole

carrier system, the position, velocity of carrier in real space and wave vector of carrier in

momentum space must be calculated every t∆ . Even if the electric field applied in

carrier system is uniform, t∆ should be chosen small enough so that the scattering rates

can be updated with the change of the carrier energy with interval time [20].

Scattering events take place randomly during an interval time t∆ (between t and t +

t∆ ). Therefore, the numbers of times that the carrier will be scattered during interval time

t∆ cannot be predicted. To simulate such a motion of carrier during interval time t∆

1

2

.

.

.N

timet -杝t t t + 杝t

∗

∗

∗

∗∗

∗∗

∗

1

2

.

.

.N

timet -杝t t t + 杝t

∗

∗

∗

∗∗

∗∗

∗

Figure 4.3 Flowchart of ensemble Monte Carlo simulation. Each

horizontal solid line shows the trace of each particle. The vertical broken

line shows sampling time. The symble ∗ shows the scattering time.

51

(from t to t + t∆ ), let us assume that the position, velocity, energy and momentum vectors

of carrier are known at time t and free flight time τ can be determined by Eq. (4.1) with

a random number r. If τ is larger than t∆ , then the carrier only drifts during t∆ . If τ

is less than t∆ , the particle drifts first during τ and is scattered after t + τ . The new

free flight time τ has then to be determined by another random number. Following this,

the new τ is checked again whether larger than t∆ or not and continue to do so until

the end of the simulation. A flowchart of the drift and scattering processes for the EMC

calculation based on the above consideration is shown in Figure 4.4.

The change of the position vector r∆ during free flight time τ can be calculated

according to the equations of motion. When the electric field during t∆ is assumed to be

constant, the position vector can be expressed by the average carrier velocity multiplied

by the flight time as follows:

τυ τ>=<∆r (4.15)

where τυ >< is formed by Eq. (4.11). This calculation must be added to the drift

process because carrier positions are necessary to calculate the carrier concentration.

Figure 4.4 Flowchart of the drift and scattering process.

Time step

Flight time

t∆> τ

Drift ( )

t∆τ

τ

Finish

Drift ( )

Scattering

τ

New flight time

Γ−= )ln(rτ

Y

N

Time step

Flight time

t∆> τ

Drift ( )

t∆τ

τ

Finish

Drift ( )

Scattering

τ

New flight time

Γ−= )ln(rτ

Y

N

52

4.6 Monte Carlo Device Simulation

MC method for device simulation has similar procedure with the EMC simulation.

Only a few elements have to be added for MC device simulation. Carriers spread in a

boundary less bulk semiconductor; however, transports of carrier are restricted by the

boundary condition. Therefore, to set up suitable boundary conditions, it is necessary for

carriers to reach the surface of the device [20]. Carriers should either be “exit” or “enter”

the ohmic contact area of the device or might “reflected” at the insulator surface of the

device during the simulation. Self-consistently potential and electric field calculation with

the distribution of carriers through the solution of the Poisson equation with appropriate

boundary conditions is other thing to be taken into account in device simulation.

Boundary conditions applied to the carrier motion and the Poisson equation must be

consistent each other. Therefore, the calculation of carrier motion with suitable boundary

conditions, the self-consistent Poisson calculation with charge distribution, and the

treatment of carrier associated with the delete, exit or entrance of carrier through the

surface of the device are necessary arrangements in the Monte Carlo device simulations.

Figure 4.5 shows a typical flowchart of the Monte Carlo device simulation. The

geometry of the device, the material composition, layer structure, the apply voltage, the

doping profile, and the contact regions are specifies in step “physical system” according

to the data given by the user.

Figure 4.5 Basic Monte Carlo algorithm flowchart.

Physical System

Initial Conditions

Particle Motion

(Drift and Scattering)

Charges Distribution

Potential and Electric Field

t < Total Time

Stop

Y

N

Physical System

Initial Conditions

Particle Motion

(Drift and Scattering)

Charges Distribution

Potential and Electric Field

t < Total Time

Stop

Y

N

53

The initial carrier distribution in real space and k-space, and the initial potential

profile in the device are specifies in the subroutine “initial condition”. The process of MC

device simulation are performed by iterative calculations made by the subroutines

“particle motion”, “charge distribution”, and “potential and electric field ”, as shown in

Figure 4.5. The subroutines “particle motion” is related to the carrier motion during t∆ .

The subroutines “charge distribution” and “potential and electric field” are related to the

calculation of charge distribution and potential those are carried out every t∆ after the

calculation of carrier motion as shown in Figure 4.4. The subroutine “particle motion”

includes drift and scattering process. The “particle motion” is almost identical to the one

shown in Figure 4.4, except that the boundary conditions for carrier motion is employed.

The number of carrier is always varying during the simulation because carriers are exit or

enter in ohmic contact region of device. The profile of carrier density calculated from

particle distribution in subroutine “charge distribution”. The profile of the carrier density

obtained is transferred to the subroutine “potential and electric field” for the potential and

electric field calculation. The role of each subroutine is described in following

subsections.

4.6.1 Initial Condition

At the start of the simulation, the initial condition of device as the particle

distribution in real space and k-space, and the potential profile are specifies in the

subroutine “initial condition”.

The carriers are usually distributed in spatial accordance with the density profile of

the corresponding doping concentration. The particles may be distributed spatially

according to the density profile of carriers for save computing time, which is computed in

advance by a device simulation based on the drift-diffusion method.

The initial distribution of carrier’s energy is determined by random numbers based

on the assumption that the energy of carrier is nearly at the thermal equilibrium at the

start of the simulation. Thus, the energy of each carrier Ek is calculate by [20]

)ln(5.1 rTkE Bk ⋅−= (4.16)

54

where kB is the Boltzmann constant, T is the lattice temperature (assumed to equal the

carrier temperature), r is random number uniformly distributed between 0 and 1. The

wave vector k of carrier can be determined by the Ek-k relation based on the energy value

given by Eq. (4.16). Due to the energy of carriers is located at minimum point of real

band structure, the simple spherical and parabolic band can be assumed, the k vector is

determined by the relation

h

kEmk

*2= (4.17)

and the components of the k vector can be determined by Eq.(3.28).

For non-parabolic bands, the energy of carrier is correct use the Ek-k relation given by

Eq. (4.2). Then, Eq. (4.17) has new form as follows:

h

)1(2 *kk EEm

kα+

= (4.18)

For ellipsoidal, parabolic bands, the components of k are obtained by a similar procedure

using Eq.(3.28) and Eq. (4.17).

4.6.2 Boundary Condition

Figure 4.6 schematically shows the boundary conditions of device and enter, exit or

reflection of carrier at the surface or at the contacts of the device. The carrier i colliding

with the bottom of the device is to be reflected from bottom surface, and the particle j

penetrating into the drain contacts is to be deleted because the contact is an ohmic contact.

If the reflection is not made, the carrier i will has the position (x, y) after colliding with

the bottom surface of device as shown in the Figure 4.6. If the reflection is made the

carrier i will has new position (x', z'). The new position and the wave vector (kx, ky, kz) of

the particle i after the reflection can be determined by the following calculation [20]

zzyyxx kkkkkk

yyyyxx

−===

−+=='''

max

,

)(' ,' (4.19)

55

If only the reflected and deleted carrier are considered in simulation, the total number

of particles just decreases with time, since carrier are just deleted and not injected. The

maintenance of the carrier neutrality in the vicinity of the source and drain ohmic contacts

are important for the MC device simulation. The carriers are always in the thermal

equilibrium in the vicinity of the source and drain ohmic contacts, even when the current

is flowing. No power is dissipated in the contact area caused by the voltage drop therein

is negligible. In the simulation process, these contacts must be realized and the numbers

of carrier are keeping the constant in the cells that constitute the source and drain contacts.

Therefore, carriers in cells which are in the vicinity of contacts must be deleted or

injected depending on whether the number of carrier therein is larger or smaller than the

corresponding doping density of carrier. The injected carriers are distributed in k-space

with a hemi-Maxwellian distribution at the lattice temperature.

4.6.3 Charge Distribution

The density profile of carrier is directly related to the particle distribution in the

device. The calculation of the density profile of carrier is simple and is based on counting

the number of particles for each grid point. The simplest “nearest-grid-point method” is

usually employed in the MC device simulation, in which the density profile of carrier at a

grid point (i, j) is calculated from the total number of particles in the cell surrounding the

grid point as shown in Figure 4.7. Since the particles are regarded as super-particles in the

calculation, the carrier density is obtained as [20]

i j(x’, y’)

(x, y)

(x, y)Source Drain

i j(x’, y’)

(x, y)

(x, y)Source Drain

Figure 4.6 Schematic diagram showing the particle reflection and the

particle exit.

56

yx

NjiNjin pp

∆∆×= ),(),( (4.20)

where Npp is the number of carriers per super-particle, N(i, j) is the number of particle in

the cell (i, j), and yx∆∆ is the square of the cell for the case of the two-dimensional

device.

There is inevitably statistical noise in the distribution of carrier profile since the

number of particles employed is rather limited. The statistical noise may lead to

numerical instability in some cases. Such statistical noise can be avoided using the

cloud-in-cell method. In the cloud-in-cell method, the carrier associated with a particle is

regarded as a cloud of carrier spread spatially. We report a brief description of the

“cloud-in-cell” method in follow.

The finite difference mesh is considered with the nodes located at (xi, yj). The constant

spatial step in the x-direction and y-direction are denote by x∆ and y∆ , respectively.

Then, if (x, y) the point coordinates in which one wants to compute the density of carrier,

with xi < x < xi+1 and yi < y < yi+1, the density of carrier is compute in the following way

))(()1,1(

)1,1()1,1(

))(()1,(

)1,()1,(

))((),1(

),1(),1(

))((),(

),(),(

2

112

12

112

ji

ji

ji

ji

yyxxjiA

jiNjin

yyxxjiA

jiNjin

yyxxjiA

jiNjin

yyxxjiA

jiNjin

−−++++=++

−−++=+

−−++=+

−−=

++

+

++

(4.21)

where A(i, j) = yx∆∆ . The cloud-in-cell method do exist that avoid the problems of

self-forces but they are necessary when deal with heterostructures and the spatial step is

not regular. The cloud-in-cell method reduces the amplitude of fluctuation in the density

profile of carrier during the simulation because of the spreading of the carrier cloud.

However, when the cloud-in-cell method is applied to a device with abrupt heterojunction,

the carrier density obtained by the “cloud-in-cell” scheme may be over-estimated on one

side of the junction and under-estimated on the other side.

57

4.6.4 Solution of Poisson Equation

MC device simulation requires potential profiles extracted from self-consistent

solution of the Poisson equation. Analytical solution for the Poisson equation is hardly

accepted in a realistic device structure with appropriate boundary conditions, due to the

potential profile has to be determined for a large number of charged particles. Finite

difference scheme of the Poisson equation with one- or two-dimensional form is an

effective method among various numerical methods.

Poisson equation can be solved by the following form

)],.(),,(),(),([)],,()([ tjiptjinjiNjiNqtjix AD +−−−=∇⋅∇ φε . (4.22)

There are two sources of charge in Eq. (4.22): mobile charge and fixed charge. Mobile

charges are electrons and holes, whose densities are represented by n and p. Fixed charges

are ionized donor and acceptor atoms whose densities are represented by ND and NA,

respectively. ε is the permittivity of material. The subscripts (i, j) denote the (i, j)-th

grid on the x-y plane. The discretization of the Poisson will give an algebraic system to

solve if the two-dimensional regular finite-difference grid is applied. But this method is

quite complicated to solve, because the boundary conditions are difficult to implement in

a generic simulation. Furthermore, this algebraic system is consuming from the grid view

of computer memory.

In this section, non-stationary Poisson equation will be introduces. This equation is

easy to solve and can be implemented in a general numerical context with robust and

j-1

i - 1 i i + 1

∗j

j+1

Cell Grid Point

j-1

i - 1 i i + 1

∗j

j+1

Cell Grid Point

Figure 4.7 Cell and grid point.

58

numerical schemes. The form of the non-stationary Poisson equation is shown in the

following

)],,(),,(),(),([)],,(),([1 tjiptjinjiNjiNqtjijitk AD

S

+−−−=∇⋅∇+∂∂ φεφ

, (4.23)

where the variables (n, p, ND, NA , ε ) have the same meaning as above. kS is a constant

giving the right dimensions of the potential termt∂

∂φ. The solution of this equation is

similar with those from classical Poisson equations described in the precedent paragraph.

Furthermore, both solutions have the same initial potential conditions and the same

boundary conditions. Therefore, once Eq. (4.23) is numerically solved, classical Poisson

equation will be easily solved by simply getting the solution of the non-stationary Poisson

equation for big final time.

In the context of finite difference method, same numerical scheme is obtained

applying finite-difference method of derivatives to the non-stationary Poisson equation.

In the finite difference method, the value of the potential on the grid points can be

discretized on an equally spaced mesh as

2

1,,1,

2

,1,,12 22),,(

yxtji

nji

nji

nji

nji

nji

njin

∆+−

+∆

+−=∇ −+−+ φφφφφφ

φ (4.24)

where the x∆ and y∆ are the spatial mesh size. The nji,φ is the potential computed at

time tn = ti + n t∆ , in the point (i, j). Applying these expressions to the non-stationary

Poisson equation, one gets the following numerical form [22]

]),(),([)22

(( ,,2

1,,1,

2

,1,,1,,

1,

nji

njiAD

nji

nji

nji

nji

nji

nji

jinji

nji pnjiNjiNq

yxt +−−−

∆+−

+∆

+−−∆+= −+−++ φφφφφφεφφ

(4.25)

The presented scheme is valid only in the case of homogeneous case, but it is easy to

expand it to the heterogeneous structures. Due to the fact that the initial conditions and

the boundary conditions are included in the presented scheme, it is easy to implement this

solution in simulation of semiconductor devices.

59

4.6.5 Electric Field Calculation

It is easy to get the solution of the electric field of device system by the solution of the

static Poisson equation or the non-stationary Poisson equation. The generic definition of

the electric field is as follows [20]

),(),( yxyxE φ−∇= (4.26)

So, in the context of finite-difference method, the solution of electric field in the two

dimensional cells of the grid as follows:

yjiE

xjiE

jijiy

jijix

∆−

−=

∆−

−=

−+

−+

2),(

2),(

1,1,

,1,1

φφ

φφ

(4.27)

These simple expressions are used in simulation. Although the expressions are simple, but

the result values of electric field are accurate and robust.

Conclusion

Details of the application of the Monte Carlo methods to device simulation have been

described in this chapter. Calculated results on carrier transport or performance of devices

may not precisely agree with experimental results due to uncertainty in the knowledge of

material parameters or scattering mechanism. However, the error in the MC simulation of

semiconductor devices is acceptable for many cases. The present approach is more than

an order of magnitude faster than the full-band device simulation, and is accessible on

modern computers. Monte Carlo simulations with non-parabolic bands can be applied to

the engineering of low-voltage nanoscale devices and materials, where detailed

knowledge of carrier transport including the electron-phonon interaction is required.

60

References

[1] B. T. Browne, J. J. H. Miller, Numerical Analysis of Semiconductor Devices and

Integrated Circuits, Boole Press, Dublin 1981.

[ 2 ] M. Kurata, Numerical Analysis for Semiconductor Devices, Lexington Press,

Lexington, Mass 1982.

[3] M. S. Mock, Analysis of Mathematical Models of Semiconductor Devices, Boole Press,

Dublin 1983.

[4] Yu. A. Shreider, The Monte Carlo Method, Pergamon, Oxford, 1956

[5] K. Binder, Application of the Monte Carlo Method in Statistical Physics, Springer,

Berlin, 1984

[6] R. W. Hockney and J. W. Easwood, Computer Simulation Using Particles, Mc

Graw-Hill, New York 1981.

[7] J. Y. Tang and K. Hess, “Impact ionization of electrons in silicon (steady state)”,

J.Appl. Phys. 54, no. 9, 5139 (1983).

[8] C. Canali, C. Jacoboni, F. Nava, G. Ottaviani, and A. Alberigi-Quaranta, “Electron

drift velocity in silicon”, Phys. Rev. B 12, no. 4, 2265 (1975).



55, no. 3, 645 (1983).

[10] R. Brunetti, C. Jacoboni, F. Nava, and L. Reggiani, “Diffusion coefficient of

electrons in silicon,” J. Appl. Phys. 52, no. 11, 6713 (1981).

[11] T. Yamada, J.-R. Zhou, H. Miyata, and D. K. Ferry, “In-plane transport properties of

Si/Si1−xGex structure and its FET performance by computer simulation”, IEEE

Trans. Electron Devices 41, 1513 (1994).

[12] J. Y. Tang and K. Hess, “Impact ionization of electrons in silicon (steady state)”, J.

Appl. Phys. 54, no. 9, 5139 (1983).

[13] C. Canali, C. Jacoboni, F. Nava, G. Ottaviani, and A. Alberigi-Quaranta, “Electron

61

drift velocity in silicon”, Phys. Rev. B 12, no. 4, 2265 (1975).

[14] M. V. Fischetti and S. E. Laux, “Monte Carlo analysis of electron transport in small

semiconductor devices including band-structure and space-charge effects”, Phys. Rev.

B 38, no. 14, 9721 (1988).

[15] P. D. Yoder and K. Hess, “First-principles Monte Carlo simulation of transport in Si”,

Semicond. Sci. Technol. 9, 852 (1994).

[16] T. Kunikiyo, M. Takenaka, Y. Kamakura, M. Yamaji, H. Mizuno, M. Morifuji, K.

Taniguchi, and C. Hamaguchi, “A Monte Carlo simulation of anisotropic electron

transport in silicon including full band structure and anisotropic impact-ionization

model”, J. Appl. Phys. 75, no. 1, 297 (1994).

[17] V. Borsari and C. Jacoboni, “Monte Carlo Calculations on Electron Transport in

CdTe”, Phys. Status Solidi B 54, 649 (1972).

[18] K. Kato, “Hot-carrier simulation for MOSFETs using a high-speed Monte Carlo

method”, IEEE Trans. Electr. Dev., ED 35, 1344 (1988).

[19] E. Pop, R. W. Dutton, and K. E. Goodson, “Analytic band Monte Carlo model for

electron transport in Si including acoustic and optical phonon dispersion”, J. Appl.

Phys. 96, 4998 (2004).


House, 1993.

[21] C. Jungemann and B. Meinerzhagen, “On the applicability of nonself-consistent

Monte Carlo device simulations”, IEEE Trans. Electron Devices, 49, no. 6, 1072

(2002).

[22] https://nanohub.org/resources/archimedes/supportingdocs

62

Chapter 5

Effects of Scattering Direction on Hot Electron

Transport

Introduction

Recently, advanced semiconductor devices have been scaled down to nanoscale size,

and the device size is further shrinking [1]. If the channel length is further shortened to

less than or comparable to the mean free path of carriers, frequency of scattering events in

the device is diminished, so that near ballistic transport is expected even at room

temperature [2]. In the conventional metal oxide semiconductor filed-effect transistor

(MOSFET), the influence of scattering in the drain region on carrier transport is

negligible because scattering is dominant in the channel. Carriers release their energy in

the channel and “cold” carriers flow into the drain [3]. If the channel is ballistic, carriers

flow in the channel without losing energy and become hot electrons in the drain. The hot

electrons are reflected back into the source-end, causing an increase in the injection

barrier at the source edge [4]. The rebound of hot electrons from the drain back into the

channel caused by scattering significantly reduces the drain current [5]. Kurusu and

Natori studied the influence of elastic/inelastic scattering in the drain region on the hot

electron transport. They pointed out that elastic scattering causes the backward flow of

hot electrons from the drain into the channel, and seriously degrades the peak of the mean

velocity of carriers in the channel and also the steady-state current. On the contrary,

inelastic scattering can suppress the backward flow of hot electrons [6]. However, they

have not discussed the role of the scattering- direction in hot electron transport.

In this work, the effect of the scattering-direction of hot electrons in the drain of

ballistic n+-i-n

+ diodes is studied by a semi-classical Monte Carlo method. At low doping

concentrations, the ionized impurity scattering has a weak influence on hot electron

transport, although it is an elastic scattering. At sufficiently high doping concentrations on

the other hand, the ionized impurity scattering enhances the backward flow of hot

electrons, and severely degrades the peak of mean carrier-velocity in the channel and also

the steady-state current. We argue that the scattering direction of hot electrons is the main

63

reason behind these results.

Simulation Method

The silicon n+-i-n

+ diode along [100] direction, as shown in Figure 5.1, is used in this

work. The lengths of the source, channel and drain are 100, 40 and 100 nm, respectively.

The diode width is 40 nm. The source and the drain are assumed to have ideal Ohmic

contacts [7]. The lattice temperature is assumed to be T = 300 K. The analytical

non-parabolic band model for the band structure of silicon is employed [8]. We

considered the intravalley acoustic, intervalley phonon and ionized impurity scatterings in

our simulation. We employed the parameters shown in ref. 9 for intervalley phonon

scattering and the parameters shown in ref. 10 for intravalley acoustic phonon scattering.

The electron concentration profile is calculated by the cloud-in-cell method and the

potential profile is calculated by the finite difference method scheme of the Poisson

equation [8]. The steady-state current is computed using the Ramo-Shockley formula [11,

12].

The channel region is assumed to be intrinsic and ballistic. Electrons do not suffer any

scattering throughout the channel. The different cases of scattering are studied in the drain

region.

Case A: The drain region is ballistic. Electrons do not suffer any scattering throughout the

drain.

Case B: Only intravalley acoustic phonon scattering is considered in the drain region.

Case C: Intravalley acoustic phonon and ionized impurity scatterings are considered in

Figure 5.1 Schematic of structure of silicon ballistic channel n+-i-n

+

diode.

+ +

n+

100nm 40nm 100nm

n+i

Source Channel Drain

(Ballistic)40

nm n+

100nm 40nm 100nm

n+i

Source Channel Drain

(Ballistic)40

nm

64

the drain region.

Case D: Intravalley acoustic and intervalley phonon scatterings are considered in the

drain region.

Case E: Intravalley acoustic, intervalley phonon, and ionized impurity scatterings are

considered in the drain region.

Intravalley acoustic, intervalley phonon, and ionized impurity scatterings are

considered in the source region for all cases. Intravalley acoustic phonon scattering can be

considered as an elastic scattering owing to the fact that the acoustic phonon energy is

much lower than kBT at room temperature, where kB is the Boltzmann constant and T is

the lattice temperature. The intervalley phonon energy is comparable to the average

thermal energy of carriers at room temperature and the intervalley phonon scattering is

therefore regarded as inelastic. Ionized impurity scattering is treated as an elastic process.

The doping concentrations of the source/drain are set to be ND = 1018

and 1020

cm-3

,

respectively. The scattering rate of ionized impurity scattering is computed by the

Brooks-Herring approach [8] when ND = 1018

cm-3

and using Kosina’s model [13, 14]

when ND = 1020

cm-3

. Kosina’s model has been shown to be adequate for doping

concentrations up to 1020

cm−3

.

Results and Discussion

In all cases, the channel is completely ballistic. Electrons injected from the source

flow into the drain and then become hot electrons, because electrons do not suffer any

scattering and do not lose their energy in the channel region.

Figure 5.2 (a) shows the distribution of the mean velocity of all electrons along the

X-axis for bias voltage of 0.3 V, and Figure 5.2 (b) shows the current-voltage

characteristics of a ballistic n+-i-n

+ diode for cases A, B, and C with ND = 10

18 cm

-3.

For all cases, the mean velocity of electrons is decreased in the drain region, because

there are many “cold” electrons with low velocity in the drain region.

When the drain region is ballistic (case A), all hot electrons are absorbed in the drain

region and are not transported in the backward direction since scattering does not occur.

Therefore, case A has the largest peak of the mean velocity of electrons in the channel as

well as the largest steady-state current.

When only the intravalley acoustic phonon scattering is considered in the drain region

65

(case B), results are in accordance with the theory of elastic/inelastic scattering as

described by Kurusu and Natori [6]. The intravalley acoustic phonon scattering can be

considered as elastic scattering and moreover the scattering motion is random. Therefore,

some hot electrons can rebound from the drain back into the channel; some of them even

have sufficient energy to return to the source region with high energy and velocity. For

these reasons, intravalley acoustic phonon scattering can sufficiently increase the

backward flow of hot electrons and decrease the peak of the mean velocity of electrons in

the channel as well as the steady-state current.

Figure 5.2 (a) Distribution of mean velocity of electrons along X-axis at VD

= 0.3 V, and (b) I-VD characteristics of ballistic channel n+-i-n

+ diode, for

cases in which the drain region is ballistic (case A, solid line), only acoustic

phonon scattering is considered (case B, triangles line) and both of acoustic

phonon and ionized impurity scatterings are considered (case C, squares

line). Here, the doping concentration of source/drain is ND = 1018

cm-3

.

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プユロヰヤリヵケァォゲケジジジジァヤ

ヮグヴオ

ァヤヮグヴオ

ァヤヮグヴオ

ァヤヮグヴオッモロクッモロクッモロクッモロク

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シシシシ

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ツヶンンユワヵァォヂ

グヤヮオ


グヤヮオ


グヤヮオ


グヤヮオッモロクッモロクッモロクッモロク

ヂヤヰクァナヮヱクヂヤヰクァナヮヱクヂヤヰクァナヮヱクヂヤヰクァナヮヱクヂヤヰクヂヤヰクヂヤヰクヂヤヰク

(a)

(b)

66

When the intravalley acoustic phonon and ionized impurity scatterings are considered

in the drain region (case C), the results, however, cannot be explained on only the basis of

the theory of elastic scattering as for case B, although ionized impurity scattering is

elastic. Indeed, the peak of the mean velocity of electrons in the channel is slightly larger

than that in case B. In addition, when the drain voltage is high, the steady-state current of

case C is also obviously higher than that in case B.

Figure 5.3 (a) Distribution of mean velocity of all electrons along X-axis at VD

= 0.3 V, ND = 1018

cm-3


+ diode,

for cases in which the drain region is ballistic (case A, solid line), intravalley

acoustic and intervalley phonon scatterings are considered (case D, squares

line) and intravalley acoustic, intervalley phonon, and ionized impurity

scatterings are considered (case E, triangles line). Here, the doping

concentration of source/drain is ND = 1018

cm-3

.

ケケケケ

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グヤヮオ


グヤヮオ


グヤヮオ

ッモロクッモロクッモロクッモロクパラヰクパラヰクパラヰクパラヰクパラヰクァナヮヱクパラヰクァナヮヱクパラヰクァナヮヱクパラヰクァナヮヱク

ケケケケ




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プユロヰヤリヵケァォゲケジジジジヤヮグヴ

ヤヮグヴ

ヤヮグヴ

ヤヮグヴッモロクッモロクッモロクッモロク

パラヰクパラヰクパラヰクパラヰク

パラヰクァナヮヱクパラヰクァナヮヱクパラヰクァナヮヱクパラヰクァナヮヱク(a)

(b)

67

We argue that the scattering direction plays a significant role in the results of case C.

Since more than 90% of scattering events are ionized impurity scattering that is a strongly

anisotropic process with a high probability for forward-scattering events, ionized impurity

scattering occurs more frequently than intravalley acoustic phonon scattering, the

occurrence of which is suppressed. Eventually, the rebound of hot electrons in the

backward direction is suppressed and most hot electrons are absorbed in the drain.

Therefore, case C has a slightly larger peak of the mean velocity of electrons in the

channel and a higher steady-state current at a high drain voltage relative to case B.

Figure 5.3 (a) shows the mean velocity for bias voltage of 0.3 V, and Figure 5.3 (b)

shows the current-voltage characteristics for cases A, D, and E with ND = 1018

cm-3

.

Figure 5.4 (a) Distribution of mean velocity of all electrons along X-axis at

VD = 0.3 V and (b) I-VD characteristics of ballistic n+-i-n

+ diode, for case A

(solid line), case B (squares line) and case C (triangles line) with ND = 1020

cm-3

.

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ツヶンンユワヵァォヂグヤヮオ



ツヶンンユワヵァォヂグヤヮオッモロクッモロクッモロクッモロク

ヂヤヰクヂヤヰクヂヤヰクヂヤヰク

ヂヤヰクァナヮヱクヂヤヰクァナヮヱクヂヤヰクァナヮヱクヂヤヰクァナヮヱク

ケケケケ



ゲクスゲクスゲクスゲクス






プユロヰヤリヵケァォゲケジジジジァヤヮグヴオ

ァヤヮグヴオ

ァヤヮグヴオ

ァヤヮグヴオッモロクッモロクッモロクッモロク

ヂヤヰクヂヤヰクヂヤヰクヂヤヰクヂヤヰクァナヮヱクヂヤヰクァナヮヱクヂヤヰクァナヮヱクヂヤヰクァナヮヱク(a)

(b)

68

When both intravalley acoustic and intervalley phonon scatterings are considered in

the drain region (case D), the results are also in accordance with the theory of

elastic/inelastic scattering. If hot electrons are scattered by inelastic phonon scattering,

most of them will release their energy and will be unable to return to the channel or the

source. Eventually, they are absorbed by the drain. This will decrease the backward flow

of hot electrons. Therefore, case D has a larger peak of the mean velocity of electrons in

the channel and higher steady-state current relative to case B.

When intravalley acoustic, intervalley phonon, and ionized impurity scatterings are all

considered in the drain region (case E), ionized impurity scattering suppresses the

occurrences of both intravalley acoustic and intervalley phonon scatterings, which is

similar to what happened in case C. Therefore in case E, the final effect of ionized

Figure 5.5 (a) Distribution of mean velocity of all electrons along X-axis

at VD = 0.3 V and (b) I-VD characteristics of ballistic n+-i-n

+ diode, for

case A (solid line), case D (squares line) and case E (triangles line) with

ND = 1020

cm-3

.

ケケケケ



ゲクスゲクスゲクスゲクス


ケケケケスケスケスケスケゲシケゲシケゲシケゲシケコサケコサケコサケコサケ

ベギモクリヴァォワヮオベギモクリヴァォワヮオベギモクリヴァォワヮオベギモクリヴァォワヮオ




プユロヰヤリヵケァォゲケジジジジヤヮグヮオ

ヤヮグヮオ

ヤヮグヮオ

ヤヮグヮオッモロクッモロクッモロクッモロク

パラヰクパラヰクパラヰクパラヰクパラヰクァナヮヱクパラヰクァナヮヱクパラヰクァナヮヱクパラヰクァナヮヱク

ケケケケ

ササササ

スススス

ゲコゲコゲコゲコ

ゲシゲシゲシゲシ

ケケケケケクゴケクゴケクゴケクゴケクシケクシケクシケクシケクズケクズケクズケクズプヰロヵモヨユァォプオプヰロヵモヨユァォプオプヰロヵモヨユァォプオプヰロヵモヨユァォプオ




ツヶンンユワヵァォヂグヤヮオッモロクッモロクッモロクッモロク

パラヰクパラヰクパラヰクパラヰク

パラヰクァナヮヱクパラヰクァナヮヱクパラヰクァナヮヱクパラヰクァナヮヱク

(a)

(b)

69

impurity scattering leads to a minor change in the peak of the mean velocity of electrons

in the channel and the steady-state current.

It must be mentioned that ionized impurity scattering approaches the isotropic state

and obviously increases the rebound of hot electrons in the backward direction at

sufficiently high doping concentrations. These effects will severely degrade the peak of

the mean velocity of electrons in the channel and the steady-state current. Figure 5.4 (a),

(b), Figure 5.5 (a), and (b) show that both the peak of the mean velocity of electrons in

the channel and the steady-state current are substantially reduced when the ionized

impurity scattering is considered in the drain region when the doping concentration is

ND=1020

cm-3

.

Conclusion

Intravalley acoustic phonon scattering within drain severely degrades peak of the

mean velocity of electrons in the channel and the steady-state current of the device,

because the scattering substantially increases the backward flow of hot electrons. The

reason is that the intravalley acoustic phonon scattering is not only an elastic scattering,

but is an isotropic scattering and has a random scattering direction. At low doping

concentrations, the ionized impurity scattering has a weak influence on hot electron

transport because of its anisotropic characteristics with a high probability for

forward-scattering events. On the other hand, the scattering assumes an isotropic

character at sufficiently high doping concentrations, and increases the scattering of hot

electrons in the backward direction, severely degrading the peak of the mean velocity of

electrons in the channel and also the steady-state current. It is concluded that the

scattering direction is an important factor for the hot electron transport within drain of

semiconductor devices. Both the peak of the mean velocity of electrons in the channel

and the steady-state current are decreased if the rebound of hot electrons in backward

direction is increased in the drain region.

References

70

[1] ITRS 2011 [http://www.itrs.net/].

[2] K. Natori, “Scaling Limit of the MOS Transistor-A Ballistic MOSFE”, IEICE Trans.

Electron. E84-C, 1029 (2001).

[3] P. Palestri, D. Esseni, S. Eminente, C. Fiegna, E. Sangiorgi, and L.Slemi, “A

Monte-Carlo study of the role of scattering in deca-nanometer MOSFETs”, IEDM

Tech. Dig. 2004, p. 605.

[4] M. Fischetti, and S. Laux, “Monte Carlo study of sub-band-gap impact ionization in

small silicon field-effect transistors”, IEDM Tech. Dig. 1995, p. 305.

[5] A. Svizhenko, and M. P. Anantram, “Role of scattering in nanotransistors”, IEEE



+ Diode by Monte



[7] T. Gonzalez, and D. Pardo, “Physical models of ohmic contact for Monte Carlo device

simulation”, Solid-State Electron. 39, 555 (1996).

[8] K. Tomizawa, Numerical simulation of submicron semiconductor devices. (Artech

House, 1993).



55, no. 3, 645 (1983).

[10] G. Donnarumma, J. Wozny, and Z. Lisik, “Monte Carlo simulation of bulk

semiconductors for accurate calculation of drift velocity as a parameter for

drift-diffusion, hydrodynamic models”, Mater. Sci. Eng. B 165, 47 (2009).

[11] W. Shockley, “Currents to Conductors Induced by a Moving Point Charge”, J. Appl.

Phys. 9, 635 (1938).

[12] S. Ramo, “Currents Induced by Electron Motion”, Proc. IRE 27, 584 (1939).

[13] H. Kosina, “A method to reduce small-angle scattering in Monte Carlo device

analysis”, Transaction on Electron Devices 46, 1196 (1999).

71

[14] H. Kosina, “Efficient Evaluation of Ionized-Impurity Scattering in Monte Carlo

Transport Calculations” Physica Status Solidi A 163, 475 (1997).

72

Chapter 6

Effects of Heat Generation on Hot Electron Transport

Introduction

The phonon emission is the dominant route of energy relaxation of carriers in

semiconductor devices [1]. The presence of phonon generation changes the energy of

carriers and strongly affects all kinds of carrier transport [ 2 , 3 ]. The advanced

semiconductor devices have been scaled down to nanoscale size, and the device size is

further shrinking. The channel length of MOSFETs is forecasted to reach sub-20 nm

region in a couple of years according to the prediction of International Technology

Roadmap for Semiconductors (ITRS) [4]. As the channel length of devices is scaled

toward the scattering length of electrons, hot electrons, which are substantially away from

the thermal equilibrium with lattice, will be produced in the drain region, since electrons

less suffer or do not suffer scattering in the channel region. The rebound of hot electrons

from the drain region back into the channel due to scattering significantly reduces the

drain current [5]. The hot electrons transfer some of their energy to lattice after phonon

scattering. Consequently, the energy relaxation affects the hot electron transport and the

behavior of the devices [6]. The heat (phonon) generation has recently been addressed in

several ways, mainly from a system design point of view [7] and from the reliable

operation of the integrated circuits [8]. However, the role of heat generation in hot

electron transport has not been discussed in detail. We argue that the presence of heat

generation in the drain extension region is inevitable, and the heat generation has crucial

effects on the hot electron transport and the characteristics of nanoscale devices.

In this section, we expand the results reported in refs. 5 and 6 and discuss the effects

of heat generation on the hot electron transport and the behavior of nanoscale devices.

The heat generation is caused by the inelastic phonon scattering inside the drain region.

Intravalley acoustic phonon scattering enhances the backward flow of hot electrons and

severely degrades the peak of mean velocity of electrons in the channel and the

magnitude of drain current. In contrast, intervalley phonon scattering could suppress the

backward flow of hot electrons and increase the drain current. We argue that the heat

generation of hot electrons is the main reason behind these results.

73

Simulation Method

The heat generation is often simulated with the drift-diffusion [9, 10, 11] or

hydrodynamic approach [ 12 ]. But neither of these approaches gives information

regarding the frequencies of phonons emitted. The Monte Carlo (MC) method is useful

for simulating carrier transport in such scenarios, including heat generation at various

electric fields. Therefore, all simulations are performed by the MC method

self-consistently coupled with 2D Poisson’s equation in this work. More details about the

MC method have been described in previous works [13, 14]. The bulk silicon n+-i-n

+

diode along [100] direction as shown in Figure 5.1 is employed in this work. The doping

concentrations of the source/drain are set to be ND = 1018

cm-3

. Electrostatic potential is

updated every 2 fs using Poisson’s equation. The lattice temperature is 300 K. We

considered the intravalley acoustic and intervalley phonon scatterings in our simulation.

The energy bands are modeled with the analytical non-parabolic band with respect to six

equivalent X-valleys of bulk silicon [14]. The previous works ref. 13 and 15 shows that

the non-parabolic band provides a reasonable approximation to the density of states

(DOS) for the conduction bands of bulk silicon⅕which determines the scattering rate of

intravalley acoustic phonon scattering, below electron energy 2 eV. The non-parabolic

band is a good approximation of electron transport when the voltage of the devices is near

or below the band gap [16, 17]. In the present work, the maximum energy of electrons is

limited to be below approximately 1 eV. If an electron momentum is exceeds the first

Brillouin Zone, its momentum appropriately rescaled and return to the first BZ. Since

electrons with energy larger than the band gap and affected by the L valley will be rare at

the low drain voltage and non-parabolic band approximation is suitable for low-energy

studies, the second conduction band (the L-valley) of silicon is neglected during the

simulation. Thereby, the non-equivalent intervalley scattering is also neglected. The

deformation potential of g-type LO phonon scattering is approximately 40% lower than

the value reported by ref. 18 and that of f-type LA/LO phonon scattering is stronger than

that of f-type TO phonon scattering according to the ref. 17. Therefore, we employed the

parameters of deformation potentials shown in ref. 19 for intervalley phonon scattering.

The previous studies have estimated shear deformation potential in the range of 7.3 to

10.5 eV and dilation deformation potentials has been previously cited both as -11.7 eV

74

and near 1.1 eV [20,21], but only the latter yields the correct mobility for both for

electrons and holes when the dilation deformation potential is 1.1 eV and shear

deformation potential is 6.8 eV. With these values the isotropically averaged intravalley

deformation potential are DLA = 6.39 eV and DTA = 3.01 eV [19], and these values are

employed in present work for intravalley acoustic phonon scattering.

The intravalley acoustic and intervalley phonon scatterings are considered in the

source region during all simulation processes. For the drain region, the different cases of

scattering are studied.

Case A: only the intravalley acoustic phonon scattering is considered in the drain region.

Case B: only the intervalley phonon scattering is considered in the drain region.

Case C: the intravalley acoustic and intervalley phonon scatterings are considered in the

drain region.

Quantum effects, such as two-dimensional effect and quantum confinement effect,

are usually present in inversion layers of channel and they are important at low field

(ohmic) regime [22]. In present work, the transport of electron is not complicated by these

quantum effects, which allows for the heat generation mentioned above to be better

isolated and understood.

The impact ionization caused by high voltage and high energy transport are not

expected to play a significant role in these low voltage nanoscale devices, and

consequently they can be neglected in the present work. The impact ionization caused by

kink effects in the channel is also neglected in this work due to the reasons that the

generated holes are main causes a reduction in the threshold voltage and a strong increase

in the drain current as described in ref.23, and the generated electrons by kink effects

have week influence on the heat generation. In the present work, the maximum energy of

electrons is limited to be below approximately 1 eV. Therefore, the effect of Delta

symmetry line on deformation potential is negligible. The intravalley optical phonon

scattering can also be neglected because it occurs only in the conduction band valleys

along the <111> direction [24] the intravalley optical deformation potential constant is

zero for the X conduction bands because symmetry restrictions forbid [25]. In Si,

intravalley optical scattering contributes only at high electron energies. The carrier-carrier

scattering, ionized impurity scattering and the roughness scattering are ignored to reduce

the complexity.

Since we are interested in the influence of heat generation on hot electron transport,

we treat all phonon scatterings as inelastic. Hence, the electrons exchange their energy

75

during the phonon scatterings. The intravalley acoustic phonon scattering is usually

considered as elastic at room temperature because the phonon energy is much smaller

than kBT, where kB is the Boltzmann constant and T is the lattice temperature. But treating

the intravalley acoustic phonons as inelastic is also important for heat generation

calculations, as shown in ref. [26]. The phonon energy involved in these phonon

scatterings transition can be determined from geometrical arguments in ref. 27 and are

labeled in Figure 2.5. The phonon frequency of the intravalley acoustic phonon can be

determined from phonon dispersion branch shown in Figure 2.5 with the analytic

approximation (Section 2.3)

20 cqqsq ++= υωω , (6.1)

where the fitting parameter υs = 9.01×105 cm/s, ω0 = 0 rad/s and c = -2.0×10

-3 cm

2/s for

the longitudinal acoustic phonon; υs = 5.23×105 cm/s, ω0 = 0 rad/s and c = -2.26×10

-3

cm2/s for the transverse acoustic phonon [19]. The phonon dispersion model is otherwise

assumed isotropic since a previous work has shown that the anisotropic effects of the

phonon dispersion are rather small [15, 17] The phonon dispersion is used when

computing the scattering rate and the final electrons state after the phonon scattering,

taking energy and momentum conservation into account. For the intervalley phonon

scattering, the numerical values of phonon energies of three g-types, which are LA, TA

and LO at 0.3 of the distance to the edge of the BZ, and of three f-types, which are TA,

LA/LO and TO at the edge of the BZ, shown in Figure 2.5, are similar to the ones shown

in ref. 19. They are also used at computing the scattering rate and the final electrons state.

During each simulation, if all phonons absorbed and emitted are tallied, the total heat

generation rate can be obtained from the sum of all phonon emission events minus all

phonon absorption events per unit time and unit volume as shown in Eq. (6.2) [26].

∑ −= )( ...sup

'''absems

simVtNNQ ωω hh (6.2)

where N is the total number of mobile charges in the device, Nsup is the number of

super-particles used in the simulation, so (N/Nsup) is the scaling ratio of the real Q′′′ and

the simulatedQ′′′ , V is the volume element at each grid node, tsim is the total simulation

76

time, and .emsωh is the electron emitted phonon energy and .absωh is the electron

absorbed phonon energy.


The intravalley/intervalley phonon scatterings are an isotropic scattering, and

scattering motion is random, they have therefore the same probability of scattering in any

direction. The heat generation in our simulation occurred almost in the drain region as

shown in Figure 6.1 and Figure 6.4. This could be explained by the fact that there are

many hot electrons in the drain region, and hot electrons release their energy to the lattice

over several inelastic scattering length paths. The inelastic scattering length is usually

about 5-10 nm.

When only the intravalley acoustic phonon scattering is considered in the drain region

(case A), the emission/absorption rate of phonon energy at room tempurature is low. The

heat generation rate is therefore low, as shown in Figure 6.1, and most rebounded hot

electrons from the drain region still have high energy and high velocity. Consequently, the

backward flow of electrons is increased. Therefore, the peak of the mean velocity of

Figure 6.1 Mean rate of heat generation along the X-axis in ballistic

channel diode at VD = 0.3 V for the cases that, (dashed line) only

intravalley acoustic phonon scattering is considered (case A), and that

(solid line) intervalley phonon scatterings is considered (case B),

respectively.

-1

1

3

5

7

0 50 100 150 200X-axis (nm)

Hea

t gen

. ra

te (eV

/cm

3/s

)

Inter.Intra.

1028

-1

1

3

5

7

0 50 100 150 200X-axis (nm)

Hea

t gen

. ra

te (eV

/cm

3/s

)

Inter.Intra.

1028

77

electrons in the channel and the drain current are relatively low as shown in Figure 6.2 (a)

and Figure 6.2 (b).

In contrast, when only the intervalley phonon scattering is considered in the drain

region (case B), the emitssion/absorption rate of phonon energy is higher than the

intravalley acoustic phonon scattering case, and the heat generation rate is higher than

case A, too. Here, most rebounded hot electrons from the drain region have lower energy,

and they will transport with lower velocity, which will suppresses the backward flow of

electrons. Therefore, the peak of the mean velocity of electrons in the channel and the

drain current are clearly higher than case A as shown in Figure 6.2 (a) and (b).

When both intravalley acoustic and intervalley phonon scatterings are considered in

the drain region (case C), the peak of the mean velocity in the channel and the drain

Figure 6.2 (a) Distribution of mean velocity of electrons along the

X-axis at VD = 0.3 V and (b) the current-voltage characteristics for case

A (dashed line) and case B (solid line).

ケケケケ





Velocity (107 cm/s)

Inter.Intra.

(a)

ケケケケ





Velocity (107 cm/s)

Inter.Intra.Inter.Intra.

(a)

ケケケケ

ココココ

ササササ

ケケケケケクコケクコケクコケクコケクサケクサケクサケクサケクシケクシケクシケクシケクスケクスケクスケクス

Inter.Intra.

Current (A/cm) (b)

ケケケケ

ココココ

ササササ


Inter.Intra.Inter.Intra.

Current (A/cm) (b)

X-axis (nm)

Voltage (V)

78

current are higher than those in case A as shown in Figure 6.3 (a) and Figure 6.3 (b),

because intervalley phonon scattering can increase the heat generation in the drain region

as shown in Figure 6.4, which is similar to what happened in case B.

Conclusion

Intravalley acoustic phonon scattering severely degrades the peak of the mean

Figure 6.3 (a) Mean velocity of electrons along X-axis at VD = 0.3 V and

(b) the current-voltage characteristics for case A (dashed line), and (solid

line) case C where intravalley/intervalley phonon scatterings are

considered in the drain region (solid line).

ケケケケ





Velocity (107 cm/s)

(a)

Inter. Intra.Intra.

ケケケケ





Velocity (107 cm/s)

(a)

Inter. Intra.Intra.Inter. Intra.Intra.

ケケケケ

ココココ

ササササ


Current (A/cm)

Inter. Intra.Intra.

(b)

ケケケケ

ココココ

ササササ


Current (A/cm)

Inter. Intra.Intra.Inter. Intra.Intra.

(b)

X-axis (nm)

Voltage (V)

79

velocity of electrons in the channel and the magnitude of drain current because intravalley

acoustic phonon scattering has relatively lower heat generation, and most rebounded hot

electrons from the drain region can transport with high velocity. In contrast, when they

are undergoing intervalley phonon scatterings, they transport with low velocity because

the intervalley phonon scattering has a relatively higher heat generation. Therefore, we

conclude that heat generation is an important factor for hot electron transport in the drain

of semiconductor devices. The peak of the mean velocity of electrons in the channel and

the drain current can be increased if there is a high heat generation within drain region.

The heat generation rate acts as an index that represents the influence of inelastic phonon

scattering on electron transport. Obviously, a high heat generation rate can be associated

with, on average, high mean velocity in the channel and high drain current.

References

[1] J. Shah: Ultrafast Spectroscopy of Semiconductors and Semiconductor Nanostructures,

Springer, Berlin, 1998.

Figure 6.4 Mean rate of heat generation along the X-axis at VD = 0.3 V for

case A (dashed line), and case C (solid line).

-1

1

3

5

0 50 100 150 200X-axis (nm)

Hea

t g

en.

rate

(eV

/cm

3/s

)

Inter. Intra.

Intra.

1028

-1

1

3

5

0 50 100 150 200X-axis (nm)

Hea

t g

en.

rate

(eV

/cm

3/s

)

Inter. Intra.

Intra.

1028

80

[2] J. A. Kash, Proc. SPIE 942, 138 (1988).

[3] W. Potz, and P. Kocevar, “Electronic power transfer in pulsed laser excitation of polar

semiconductors”, Phys. Rev. B. 28, 7040 (1983).

[ 4 ] International Technology Roadmap for Semiconductors (ITRS), 2011 update,

http://www.itrs.net




+ Diode by Monte



[7] R. Mahajan, R. Nair, V. Wakharkar, J. Swan, J. Tang, and G. Vandentop, “Emerging

Directions for Packaging Technologies”, Intel Technology J. 6, 62 (2002).

[8] S. Borkar, “Low power design challenges for the decade”, Design Automation

Conference, Feb. 2001, p. 293.

[ 9 ] G. K. Wachutka, “Rigorous thermodynamic treatment of heat generation and

conduction in semiconductor device modeling”, IEEE Trans. Electron Devices 9,

1141 (1990).

[10] U. Lindefelt, “Heat generation in semiconductor devices”, J. Appl. Phys. 75, 942

(1994).

[11] P. G. Sverdrup, Y. S. Ju, and K. E. Goodson, “Sub-Continuum Simulations of Heat

Conduction in Silicon-on-Insulator Transistors”, J. Heat Transfer 123, 130 (2001).

[ 12 ] J. Lai and A. Majumdar, “Concurrent thermal and electrical modeling of

sub-micrometer silicon devices”, J. Appl. Phys. 79, no. 9, 7353 (1996).

[13] M. Lundstrom, Fundamentals of Carrier Transport (Cambridge Univ. Press, 2000).

[14] C. Jacoboni and P. Lugli, The Monte Carlo Method for Semiconductor Device

Simulation (Springer, New York, 1989).

[15] T. Kunikiyo, M. Takenaka, Y. Kamakura, M. Yamaji, H. Mizuno, M. Morifuji, K.

81

Taniguchi, and C. Hamaguchi, “A Monte Carlo simulation of anisotropic electron

transport in silicon including full band structure and anisotropic impact-ionization

model”, J. Appl. Phys. 75, no. 1, 297 (1994).

[16] M. V. Fischetti and S. E. Laux, “Monte Carlo analysis of electron transport in small

semiconductor devices including band-structure and space-charge effects”, Phys.

Rev. B 38, no. 14, 9721 (1988).

[17] H. Mizuno, K. Taniguchi, and C. Hamaguchi, “Electron-transport simulation in

silicon including anisotropic phonon scattering rate”, Phys. Rev. B 48, 1512 (1993).

[18] R. Brunetti, C. Jacoboni, F. Nava, and L. Reggiani, “Diffusion coefficient of

electrons in silicon”, J. Appl. Phys. 52, no. 11, 6713 (1981).



Phys. 96, 4998 (2004).

[20] M. V. Fischetti, and S. E. Laux, “Band structure, deformation potentials, and carrier

mobility in strained Si, Ge, and SiGe alloys”, J. Appl. Phys. 80, 2234 (1996).

[21] M. V. Fischetti, and S. E. Laux, “Monte Carlo study of electron transport in silicon

inversion layers”, Phys. Rev. B 48, 2244 (1993).

[ 22 ] M. V. Fischetti, and S. E. Laux, “Monte Carlo simulation of transport in

technologically significant semiconductors of the diamond and zinc-blende

structures. II. Submicrometer MOSFET's”, lEEE Transaction on Electron Devices

38, no. 3, 650 (1991).

[23] L. Wang, T. A. Fjeldly, B. Iniguez, H. C. Slade, and M. Shur, “Self-heating and kink

effects in a-Si:H thin film transistors”, IEEE Transaction on Electron Devices 47, no.

2, 387 (2000).

[24] W. A. Harrison, “Scattering of Electrons by Lattice Vibrations in Nonpolar Crystals”,

Phys. Rev. 104, no.5, 1281 (1956).

[25] K. B. Ridley, Quantum Processes in Semiconductors, London: Oxford, 1982.

[26] E. Pop, R. W. Dutton, and K. E. Goodson, “Detailed heat generation simulations via

82

the Monte Carlo method”, in Simulation of Semiconductor Processes and Devices,

(Boston, MA, 2003), p. 121.

[27] G. Dolling, Symposium on Inelastic Scattering of Neutrons in Solids and Liquids,

Sept. 1963, p. 37.

83

Chapter 7

Effects of Hot Phonon on Hot Electron Transport

Introduction

Modern semiconductor devices already operate at lengths comparable to the

electron’s mean free path at room temperature [1, 2], and future novel devices will

continue to advance further in this regime. At such length scales, ballistic transport of

carriers will be dominant within the device channel. Carrier scattering is suppressed and

generation of hot electrons will be significant in the drain region. Scatterings of these hot

electrons significantly influence the drain current [3]. It is well known that the localized

phonon emission from hot electrons near drain is the source of heat generation in the

device. The dimension of this heat source is expected to be of the order of a few

nanometers. When these hot electrons collide with the semiconductor lattice, hot phonons

are produced, causing an extremely high rate of heat generation. Hot phonons can occur

in an extended, essentially homogeneous, region in bulk semiconductor or in a

two-dimensional heterolayer [4]. Hot phonon effects, both in bulk semiconductors and

microstructures, have been investigated by several authors [5, 6]. Hot phonon emission

can effectively disturb the phonon distribution in thermal equilibrium. The hot and

non-equilibrium phonon problems have been the subject of considerable interest over last

few years. Understanding the problem is essential for solving the self-heating and

reliability issues in nanoscale and thin-film transistors. Hot phonons (mainly longitudinal

optical (LO) phonons) dramatically affect the electron transport in small-scale

semiconductor structures, e.g. the high-field electron transport in semiconductor quantum

wells, wires and resonant hetero-structure tunneling diodes[7, 8]. Most of the previous

researches in this area are basically concentrated on influence of hot phonons on the

electron energy relaxation after initial electro- or photo-excitation.

In this work, the influence of hot phonons on hot electron transport within the drain

region is investigated. The ensemble Monte Carlo (MC) method self-consistently coupled

with Poisson’s equation is used. This MC simulation is three-dimensional in k-space and

two-dimensional in real space. The two-dimensional model is possible if physical

quantities have no significant variation along the third direction. This MC method has

84

been described in greater detail in previous works [9, 10]. The phonon Boltzmann

transport equation (BTE) is solved and the mean heat generation rate is calculated. The

simulation compares the results of two cases, 1: The non-equilibrium that is actually

present around the drain edge is neglected, and 2: The non-equilibrium

phonon-occupation conditions are correctly considered at drain edge. Our result shows

that the mean electron energy within the drain is larger when the non-equilibrium

conditions are correctly considered (case 2). Heat generation, mean electron velocity and

drain current are all estimated to have lower values if the non-equilibrium conditions are

considered. We argue that the increase of the hot phonon population and the hot-phonon

re-absorption rate are main reasons for these results.

Simulation Method

A bulk silicon n+-i-n

+ diode along the [100] direction, as shown in Figure 5.1, is used

for simulation purpose. The lengths of the source, channel, and drain are 100, 40, and 100

nm, respectively and diode width is 40 nm. Doping concentrations of the source and drain

are set to ND=5×1020

cm-3

. The channel is assumed to be both intrinsic and ballistic at

lattice temperature of 300 K. The electrostatic potential is updated every 2 fs using

Poisson’s equation. . The band structure is modeled using the non-parabolic band

approximation, including the six conduction X-valleys of silicon [ 11 ]. The

non-parabolicity factor α is set to 0.5 eV-1

. The low device voltage ranges means that

impact ionization and other high-energy scatterings can be neglected. The intravalley

acoustic and intervalley phonon scattering are considered within the source and drain

region [11, 12]. Since in this work we intend to explore the details of the hot phonon

generation, all phonon scattering events are assumed inelastic, and the electrons exchange

energy with the lattice are determined by the phonon dispersion and scattering selection

rules. For simplicity, the carrier-carrier scattering, ionized impurity scattering and

roughness scattering are not taken into account. Scatterings are treated in the standard

way using Fermi’s Golden Rule. The respective forms of the scattering rates for the

intravalley and intervalley phonon scattering are given by Eq. (7.1) and Eq. (7.2),

respectively [9]. (Section 3.2)

85

)()21

21)(()(

2

ijkqij

ijij ENTNZD

kW ωρωπ

hm ±+= (7.1)

dqqITNk

mkW qq

qs

d 3

2

2

)21

21)((1

4)( m

h+Ξ= ∫ωπρ

(7.2)

where ρ is the mass density of the material, Ξ is the deformation potential including

Ξ LA and Ξ TA, md is the density-of-states effective mass, Iq is the overlap integral of the

wave function, Zij is the number of possible equivalent final valleys of the same type, Dij

is the coupling constant, ijωh is the corresponding phonon energy, andω is the phonon

frequency. Here for f-type scattering, Z=4, and for g-type scattering Z=1 for bulk silicon.

The values for the effective scattering deformation potentials are extracted from ref. [13].

The upper and lower signs correspond to the phonon absorption and phonon emission

processes, respectively. Nq is the average phonon occupation number given by the

Bose-Einstein distribution. Under equilibrium conditions, Nq(T) = [exp(h ωq/kBT)−1]−1

.

Under non-equilibrium conditions, Nq must be determined by solving the BTE.

Intravalley acoustic phonon scatterings were treated individually and the phonon

frequency dependence on the wave vector were taken into account. Each branch of the

acoustic phonon dispersion was modeled with the analytical approximation [13] (Section

2.3). For intervalley phonon scattering, the numerical values of the phonon energies for

the three g-types and for the three f-types are similar to those shown in ref. [13]. The

phonon dispersion is also used when computing the final electron state after considering

both momentum and energy conservation. The mean heat generation rate can be obtained

from Eq. (6.2)

The non-equilibrium phonon occupation number dominates the transport near the

hotspots within the drain region and is determined by solving the phonon BTE in the

relaxation-time approximation given by:

ph

qq

phph

q TNN

t

N

τ)(−

−=∂∂

−

(7.3)

where phτ is the phonon lifetime, which is assumed to be equal for all values of q, Nq is

the non-equilibrium phonon occupation, and Nq(T) is the equilibrium phonon occupation

at the lattice temperature T. Several theoretical approaches for the solution of Eq. (7.3)

86

have been presented in the literature. Details of the various methods can be found in ref.

[14]. For all optical modes, a lifetime in the order of 10 ps can be assumed [17]. The

lifetime of zone-center optical modes in silicon at room temperature is almost on the same

order as well. This time is long enough to assume semi-equilibrium, and hence the right

side of Eq. (7.3) can be equaled to zero. The phonons do not simply decay and disappear

from the system; they are also created via emission from hot electrons. By introducing

this factor and then rewriting Eq. (7.3), the phonon occupation number can be expressed

as [15]

Figure 7.1 (a) Distribution of the mean velocity of electrons along the

X-axis, and (b) the drain current at VD=0.3 V for the equilibrium (Eq.) and

non-equilibrium (Noneq.) phonon occupation cases.

1.0

1.1

1.2

1.3

1.4

Eq. Noneq.

Current (103 A/cm)

0.0

0.5

1.0

1.5

2.0

0 60 120 180 240

X-axis (nm)

Velocity (107 cm/s)

Eq.

Noneq.

(a)

(b)

1.0

1.1

1.2

1.3

1.4

Eq. Noneq.

Current (103 A/cm)

0.0

0.5

1.0

1.5

2.0

0 60 120 180 240

X-axis (nm)

Velocity (107 cm/s)

Eq.

Noneq.

(a)

(b)

87

)(6..

3max

2

sup

''' TNtqN

nQN qsim

ph

q +=τπ (7.4)

where '''Q is the sum of the generated phonon energy at the last time step tsim, n is the

doping concentration, and qmax is the maximum wave vector of the carriers.

To clarify the role of hot phonons inside the drain region, the simulation is performed

under different conditions in the source, channel and drain regions. The channel is

assumed to be completely ballistic. The intravalley acoustic, three g-type, and three f-type

intervalley phonon scatterings are considered in the source region for all simulations. For

the drain region, only the intervalley g-LO phonon scattering is considered under the

equilibrium and non-equilibrium phonon occupation conditions, respectively.


Figure 7.1 (a) shows the distribution of the mean electron velocity along the X-axis

and Figure 7.1 (b) shows the drain current under both equilibrium and non-equilibrium

phonon occupation conditions. When the non-equilibrium phonon occupation is

considered, the simulation results for the mean electron velocity and drain current are

lower than equilibrium condition. We propose that the hot phonon population and the

change in phonon occupation rates are the main reasons for these results. As the number

of hot electrons in the drain region increases, hot phonon will also increase because the

emission rate is larger than the absorption rate of the intervalley g-LO phonons. Thus, the

0

0.1

0.2

0.3

0.4

0 1 2 3 4 5

Time (ps)

Nq

Figure 7.2 Phonon occupation versus time for non-equilibrium conditions.

88

phonon equilibrium condition is disturbed and the non-equilibrium phonon occupation

increases, as shown in Figure 7.2. In this case, the absorption rate of the g-LO phonon is

increased and the values are larger than those under the equilibrium condition (Figure 7.3).

Therefore, the mean electron energy within the drain region under the non-equilibrium

condition is larger than equilibrium condition, as shown in Figure 7.4 (a). It should be

noted that the emission rate of the g-LO phonon is almost unchanged. Figure 7.4 (b)

shows that the heat generation within the drain region is smaller for non-equilibrium

phonon effect case. Intervalley g-LO phonon scattering is isotropic scattering, and has the

same scattering probability in any direction. Considering the results above, it can be

Figure 7.3 Total scattering rates for g-LO phonon scattering under (a)

equilibrium condition (Nq=0.0945), and (b) non-equilibrium condition

(Nq=0.36), respectively. ωh = 61 meV, T=300 K, VD=0.3 V.

0.1

1

10

100

0 0.4 0.8 1.2

Energy (eV)

Scat. rate (1012 1/s)

Emi.Abs.

0.1

1

10

100

0 0.4 0.8 1.2

Energy (eV)


Emi.Abs.

(a)

(b)

0.1

1

10

100

0 0.4 0.8 1.2

Energy (eV)


Emi.Abs.

0.1

1

10

100

0 0.4 0.8 1.2

Energy (eV)


Emi.Abs.

(a)

(b)

89

concluded that for non-equilibrium case when hot phonon generation is dominated within

the drain region, the hot electrons have higher possibility of scattering in the source

direction with high energy. This will increase the flow of electrons in reverse direction

and degrade the mean electron velocity and drain current.

Conclusion

In this work, we investigated the influence of hot phonon distribution on the hot

-0.1

0.3

0.7

1.1

1.5

0 60 120 180 240

X-axis (nm)

Eq.

Noneq.

Heat gen. rate (1012W/cm3)

0

0.05

0.1

0.15

0 60 120 180 240

X-axis (nm)

Energy (eV)

Eq.Noneq.

(a)

(b)

-0.1

0.3

0.7

1.1

1.5

0 60 120 180 240

X-axis (nm)

Eq.

Noneq.


-0.1

0.3

0.7

1.1

1.5

0 60 120 180 240

X-axis (nm)

Eq.

Noneq.


0

0.05

0.1

0.15

0 60 120 180 240

X-axis (nm)

Energy (eV)

Eq.Noneq.

(a)

(b)

Figure 7.4 (a) Distribution of mean energy of electrons, and (b) mean rate of

heat generation along the X-axis in a ballistic channel diode at VD=0.3 V for

cases that: the equilibrium (solid line) and non-equilibrium phonon (dashed

line) occupation conditions are considered in the drain.

90

electron transport within the drain region of a ballistic channel diode, by implementing a

Monte Carlo method equipped with the analytical electronic band and the phonon

dispersion. The mean electron velocity and the drain current are degraded as the

non-equilibrium phonon occupation at the drain edge increases. These results are

explained by the non-equilibrium g-LO phonon distribution caused by the high

re-absorption rate of phonons and the low heat generation within the drain region. The

surplus hot electrons are rebounded from the drain region and are transported toward

source with high energy and velocity. We conclude that the hot phonon effect should not

be neglected in the study of hot electron transport within the drain region if the hot

phonon generation is significantly increased.

References

[1] M. Lundstrom: Fundamentals of carrier transport, (Cambridge University Press,

2000).

[2] Y. S. Ju, and K. E. Goodson, “Phonon scattering in silicon films with thickness of

order 100 nm”, Appl. Phys. Lett. 74, no. 20, 3005 (1999).



[4] W. Potz, and P. Kocevar, “Electronic power transfer in pulsed laser excitation of polar

semiconductors”, Phys. Rev. B. 28, 7040 (1983).

[5] D. Y. Oberli, G. Bohm, and G. Weimann, “Role of interface optical phonons in

cooling hot carriers in GaAs-AlAs quantum wells”, Phys. Rev. B 47, 7630 (1993).

[6] N. S. Mansour, Y. M. Sirenko, K. W. Kim, M. A. Littlejohn, J. Wang, and J. P.

Leburton, “Carrier capture in cylindrical quantum wires”, Appl. Phys. Lett. 67, 3480

(1995).

[ 7 ] S. Koch and T. Mizutani, “InGaAs resonant tunneling transistors using a

coupled-quantum-well base with strained AlAs tunnel barriers”, IEEE Trans.

91

Electron Devices 41, 1498 (1994).

[ 8 ] K. Kurishima, H. Nakajima, T. Kobayashi, Y. Matsuoka, and T. Ishibashi,

“Fabrication and characterization of high-performance InP/InGaAs

double-heterojunction bipolar transistors”, IEEE Trans. Electron Devices 41, 1319

(1994).

[9] K. Tomizawa: Numerical Simulation of Submicron Semiconductor Devices (Artec

House, London, 1993).

[10] M. Lundstrom: Fundamentals of Carrier Transport (Cambridge Univ. Press, 2000).

[11] C. Jacoboni, and L. Reggiani, “The Monte Carlo method for the solution of charge


55, 645 (1983).

[12] C. Hamaguchi, Basic semiconductor physics. (Springer, 2001).



Phys. 96, 4998 (2004).

[ 14 ] W. Cai, C. M. Marchetti, and M. Lax, “Nonequilibrium phonon effect on

time-dependent relaxation of hot electrons in semiconductor heterojunctions”, Phys.

Rev. B 35, 1369 (1987).

[ 15 ] Ole Christian Norum, “Monte Carlo Simulation of Semiconductors–Program

Structure and Physical Phenomena”, Master’s Thesis, Norwegian University of

Science and Technology, (2009).

92

Chapter 8

Strained Drain and Hot Electron Transport

Introduction

As the size of conventional semiconductor devices are down-scaled into the

nanometer regime, maintaining the performance enhancement of these nanoscale devices

becomes extremely difficult. Innovative materials and device structures have been

proposed including multi-gate field-effect-transistors (FETs), high-dielectric-constant

(high-k) gate dielectrics, strained silicon FETs, etc, to extend the device scaling limit and

improve device performance. IBM group fabricated Si metal-oxide-semiconductor FET

with a 20-nm channel length using a novel step/edge technique in 1990 [1].

A double

gate MOSFET is one of the most promising new device structures. It has been studied

extensively in the last couple of years, which allows down-scaling to dimensions below

10 nm [2, 3]. Ref. 4 and ref. 5 introduce the application of strained-Si and SixGe1-x in

sub-20nm n-MOS and p-MOS FETs. Strained-Si has been recognized as an important

technology to improve electrical performance of the device and to achieve the goal of the

International Technology Roadmap for Semiconductors (ITRS) [ 6 ]. Strained-Si

Si1-xGex Si1-xGex

Si Strained-Si

Si1-xGex Si1-xGex

Si Strained-Si

Figure 8.1 Biaxial strain is intentionally introduced in silicon by depositing

silicon on a Si1-xGex layer.

93

technology has been studied both experimentally [7], and theoretically [8]. The use of

epitaxial SiGe can induce compressive strain for p-channel devices and improve the drain

current [9]. Ref. [10] also reported using SiGe for the source and drain extensions, and

further improving the p-MOS performance by 35%. The uniaxial strain increases the

mobility of holes by about 50% for the 45 nm gate length transistors at high fields [11].

Strained-Si improves carrier transport by altering the scattering rate and band structure of

the material. The strained-Si layer in the devices is usually obtained by growing Si on a

relaxed Si1-xGex substrate as shown in Figure 8.1, where mole fraction x defines the strain

present in the Si material. Figure 8.2 depicts the strain mechanism on Si band structure,

where four of the six conduction band valleys are lifted upwards and two of them

downward. About 96% of electrons will be kept in the two lower valleys when x reaches

0.15 [12]. If the channel length is reduced to less than or comparable to a carrier’s mean

free path, near ballistic transport is expected at room temperature [13].In a ballistic

channel, electrons do not encounter any scattering before flowing into the drain, and they

become hot electrons. Scatterings of these hot electrons significantly influence the drain

current [14]. A previous work points out that the elastic scattering causes backward

scattering of hot electrons from the drain, and substantially degrades the peak of electron

X4

X2

Si Strained-Si

Si

X6

Strained-Si

X4

X2

∆E

X4

X2

X4

X2

X4

X2

Si Strained-Si

Si

X6

Strained-Si

X4

X2

∆ESi

X6

Strained-Si

X4

X2

∆E∆E

Figure 8.2 Energy valleys of bulk silicon (left) and strained silicon (right), and

the energy splitting between the valleys.

94

mean velocity in the channel and also the steady-state current [15].

In this work, we comparatively study the effect of biaxial strain in the Si channel and

drain on performance of ballistic channel diode. We have adopted an ensemble Monte

Carlo (MC) simulation self-consistently coupled with Poisson’s equation. The MC

method is basically the same as used in the unstrained bulk Silicon. It is

three-dimensional in k-space, and two-dimensional in real-space. More details about the

MC method are described in previous works [16, 17]. The simulated result shows that the

strained drain can suppress the intervalley scattering of electrons and increase the drain

current as well as the mean velocity of electrons in drain. This is due to the fact that

energy levels of the strained-Si valley are split and the most electrons occupy the lowest

X2 valleys with a smaller transverse effective mass. We conclude that using the strained

drain is an efficient method for improving electrical performance of ballistic channel

devices.

Simulation Method

The silicon n+-i-n

+ diode along [100] direction as shown in Figure 5.1 is used in this

work. The length of the source, channel, and drain are 100, 20, and 100 nm, respectively

and diode width is 40 nm. Doping concentrations of the source/drain are set to ND = 1018

cm-3

. The super-particle number is 105. Total simulation time is 1.0 ps. The channel is

assumed to be intrinsic and ballistic. Electrons do not undergo any scattering within

channel; however, ionized impurity scattering, intravalley acoustic phonon scattering,

intervalley phonon scattering and impact ionization are considered within the source and

drain region. Intravalley acoustic phonon scattering can be treated as elastic scattering.

The energy bands are modeled by the analytical non-parabolic band with respect to six

equivalent X-valleys of bulk silicon [18]. The second conduction band (the L-valley) of

silicon is safely neglected in simulations. The non-equivalent intervalley scattering is

neglected as well. The longitudinal/transverse effective masses of the electrons, the

various phonon coupling constants and phonon energies are assumed to be the same as in

unstrained Si in scattering rate calculations [19]. We used the parameters shown in ref.

[20] for intervalley phonon scattering and the parameters shown in ref. [18] for

intravalley acoustic phonon scattering. The non-parabolic parameter is inversely

proportional to the band gap Eg(x). Under the biaxial strain, Eg(x) = 1.11-0.4x. The

95

splitting energy between the lowered and the raised valleys is empirically represented by

∆E = 0.67x.21 Impact ionization is important in the strained-Si because of the band gap

reduction. The scattering rate of impact ionization is a quadratic function of electron

kinetic energy above the ionization threshold energy. The scattering rate )(kWimpact of

impact ionization is modeled using a modified threshold expression [22] (Section 3.3).

2])([)( thimpact EkEPkW −= (8.1)

where Eth is empirical threshold energy, E(k) is carrier’s energy and P is a prefactor

which determine the softness of the threshold. The threshold energy is 1.1 eV for

unstrained Si. The threshold energy is assumed to be proportional to the band gap, by

Eth(x) = 1.18Eg(x)/Eg(0) for the strained-Si. When ionization occurs, the original electrons

are assumed to not change valley. The respective forms of the scattering rates for the

ionized impurity, intravalley acoustic and intervalley phonon scattering are given by [16]

(Sections 3.1 and 3.2 ):

)4(1)(2

)(2222

42

ionized

DDs

kI

qkq

ENeZNkW

+=

επ

h (8.2)

)(2

)(2

intra. kL

B ENC

TkkW

h

Ξ= π (8.3)

)()21

21)(()(

2

inter. EENTNZD

kW ijkqij

ijij ∆−±+= ωρωπ

hm (8.4)

Eq. (8.2) is for ionized impurity scattering, Eq. (8.3) is for intravalley acoustic phonon

scattering, and Eq. (8.4) is for intervalley phonon scattering. The density of states N(Ek) is

given by

32

2/3

4

)2()(

hπkd

k

EmEN = (8.5)

where, md is density of states effective mass and is defined as 3/12)( tld mmm = , here ml and

96

mt are the longitudinal and transverse effective mass, respectively, for a parabolic,

ellipsoidal band. For both strained-Si and unstrained-Si, md = 0.321m0 (ml = 0.916m0 and

mt = 0.190 m0 at room temperature), here m0 is the free electron mass.

For nanoscaled devices it is necessary to include quantum effects in the simulation.

An effective potential approach for correcting the quantum effects has the advantage of

easy numerical implementation and almost guaranteed convergence [23]. In this work, we

use this effective potential approach for the modeling of these quantum effects. The

intravalley optical phonon scattering which occurs in the L valley of silicon is neglected.

[24]. The carrier-carrier scattering and roughness scatterings are also ignored to minimize

complexity and allow for the effects of strained-Si to be better isolated and understood.

Figure 8.3 (a) the mean velocity of electrons in the drain and (b) the drain

current of ballistic diode, for cases that the drain is strained (Dra.), the

channel is strained (Cha.), and diode without strained (UnStr.) at VD = 0.3 V.

1.0

2.0

3.0

4.0

UnStr. Cha. Dra.

Vel

oci

ty (

106 m

/s)

1.5

2.0

2.5

UnStr. Cha. Dra.

Curr

ent (A

/cm

)

(a)

(b)

1.0

2.0

3.0

4.0

UnStr. Cha. Dra.

Vel

oci

ty (

106 m

/s)

1.5

2.0

2.5

UnStr. Cha. Dra.

Curr

ent (A

/cm

)

(a)

(b)

97


A strained silicon crystal with x = 0.6 is assumed in this work. With the assumed

strain, the valley splitting in the silicon energy band is ∆E = 0.402 eV, the band gap is

Eg(0.6) = 0.842 eV, so that the threshold voltage is Eth(0.6) = 0.887 eV. The main effects

on performance of the device are from the electrons in the X2 valleys, as most electrons

are kept in the lowest X2 valleys [12, 25]. The X4 valleys have only a weak influence on

Figure 8.4 (a) the mean velocity of electrons in the drain and (b) the

drain current of the ballistic diode, for cases that the drain is strained

(Dra.), the channel is strained (Cha.), and the unstrained diode

(UnStr.) at VD = 1.0 V.

4

6

8

10

UnStr. Cha. Dra.

Vel

oci

ty (

106

cm

/s)

2

4

6

UnStr. Cha. Dra.

Cu

rren

t (A

/cm

)

(a)

(b)

4

6

8

10

UnStr. Cha. Dra.

Vel

oci

ty (

106

cm

/s)

2

4

6

UnStr. Cha. Dra.

Cu

rren

t (A

/cm

)

(a)

(b)

98

the performance of device.

In Figure 8.3, three diode structures are compared at the drain bias voltage of 0.3 V.

The rectangle on the left represents the diode without strain (case A), the middle one

represents the diode with a strained channel (case B), and the rectangle on the right

represents the diode with the strained drain (case C). Figure 8.3(a) and 8.3(b) respectively

show the mean electron velocity and the drain current for these three cases. Case C has a

larger mean electron velocity in the drain region, and a larger drain current than other

cases A and B. Ionized impurity scattering has a weak influence on the electron transport,

Figure 8.5 (a) the heat generation rate and the parasitic resistances of

drain region for cases that the diode without strain (dashed line), the

channel is strained (dotted line), and the drain is strained (solid line) at VD

=0.3 V.

-1

1

3

5

7

120 160 200Drain Region (nm)

Hea

t g

en.

rate

(eV

/cm

3/s

)

Unstr.Cha.Dra.

0

10

20

30

120 160 200

Drain Region (nm)

Res

ista

nt

(108

Ω/c

m) Unstr.Cha.Dra.

(a)

(b)

-1

1

3

5

7


Hea

t g

en.

rate

(eV

/cm

3/s

)

Unstr.Cha.Dra.

0

10

20

30

120 160 200

Drain Region (nm)

Res

ista

nt

(108

Ω/c

m) Unstr.Cha.Dra.

(a)

(b)

1028

Res

ista

nce

s(1

08ţ

/cm

)

-1

1

3

5

7


Hea

t g

en.

rate

(eV

/cm

3/s

)

Unstr.Cha.Dra.

0

10

20

30

120 160 200

Drain Region (nm)

Res

ista

nt

(108

Ω/c

m) Unstr.Cha.Dra.

(a)

(b)

-1

1

3

5

7


Hea

t g

en.

rate

(eV

/cm

3/s

)

Unstr.Cha.Dra.

0

10

20

30

120 160 200

Drain Region (nm)

Res

ista

nt

(108

Ω/c

m) Unstr.Cha.Dra.

(a)

(b)

1028

Res

ista

nce

s(1

08ţ

/cm

)

99

since the impurity scattering is an anisotropic scattering and has a large

forward-scattering probability for selected doping concentrations [26 ]. The impact

ionization has also a week influence on electron transport, because the selected drain

voltage is smaller than the threshold voltage of the ionization in the strained and

unstrained silicon. We argue that the small transverse effective mass of electrons in the

lowest X2 band valleys and the suppression of intervalley scattering are the main causes

of these results. Suppression of intervalley scattering is due to the fact that the energy of

valleys splitting (∆E = 0.402 eV) is larger than the energy of most electrons (including

hot electrons). For these reasons, the energy relaxation and the scattering of electrons are

effectively suppressed in case C. The electron velocity is inversely proportional to the

effective mass ( *k/mh=υ ) in semi-classical transport condition. Therefore, the electron

accelerates easier in strained-Si.

When drain bias is 1 V, the energy of most hot electrons within drain is larger than the

threshold voltage of impact ionization and the splitting energy between the X2 and X4

band valleys of the strained-Si. Hot electrons readily cause impact ionization and

intervalley scattering. If hot electrons experience the impact ionization, they will release

their energy and become low energy electrons within drain. Although hot electrons in

drain region have enough energy to cause intervalley scattering, larger increases in drain

current and mean velocity of electrons are observed for the strained drain as shown in

Figure 8.4 (a) and Figure 8.4 (b), since the most electrons still remain in the X2 valleys

and have smaller transverse effective mass even at drain bias 1 V [12].

Parasitic drain-resistance strongly influences performance of the device when the

channel length is reduced. The drain resistance does not decrease in proportion to the

channel length [27], but can be estimated using the heat generation [28]. The heat

generation rate can be obtained from Eq. (6.2).

The drain resistances can be estimated using

2''' / DD IQR = (8. 6)

where DI is drain current.

Case C has a lower heat generation in the drain region compared to other two cases at

VD = 0.3 V as shown in Figure 8.5 (a). And it has the lowest drain-resistance (Figure 8.5

(b)) since the intervalley phonon scattering is sufficiently suppressed.

100

Conclusion

A nanoscale silicon diode with the ballistic channel, and with the strain separately

applied to the channel or to the drain, was studied by the self-consistent Monte Carlo

simulation. The large performance improvement in the mean electron velocity, and in the

mean electric current in the drain, was achieved by applying strain to the drain region.

The improvements are due to the smaller transverse effective mass of electrons in the

lowest X2 valleys, and to the suppressed electron intervalley scattering in drain region.

The strained-Si in drain region can improve performance of the device, and it is a viable

candidate for the future novel device structure.

References

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[2] Z. Ren, R. Venugopal, S. Data, and M. Lundstrom, “Examination of design and

manufacturing issues in a 10 nm double gate MOSFET using nonequilibrium Green's

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[3] D. Hisamoto, “FD/DG-SOI MOSFET-a viable approach to overcoming the device

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[4] T. Tezuka, N. Sugiyama, T. Mizuno, S. Takagi, “Ultrathin body SiGe-on-insulator

pMOSFETs with high-mobility SiGe surface channels”, IEEE TED 50, no.5, 1328

(2003).

[5] K. Rim, K. Chan, L. Shi, D. Boyd, J. Ott, N. Klymko, F. Cardone, L. Tai, S. Koester,

M. Cobb, D. Canaperi, B. To, E. Duch, I. Babich, R. Carruthers, P. Saunders, G.

Walker, Y. Zhang, M. Steen, M. Ieong, “Fabrication and mobility characteristics of

ultra-thin strained Si directly on insulator (SSDOI) MOSFETs”, IEDM, 3.1.1, (2003).

101

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[7] J. L. Hoyt, H. M. Nayfeh, S. Eguchi, I. Aberg, G. Xia, T. Drake, E. A. Fitzgerald, and

D. A. Antoniadis, “Strained silicon MOSFET technology”, IEDM Tech. Dig. p 23

(2002).

[8] M. V. Fischetti, and S. E. Laux, “Band structure, deformation potentials, and carrier

mobility in strained Si, Ge, and SiGe alloys”, J. Appl. Phys. 80, 2234 (1996).

[9] T. Ghani, M. Armstrong, C. Auth, M. Bost, P. Charvat, G. Glass, T. Hoffmann, K.

Johnson, C. Kenyon, J. Klaus, B. McIntyre, K. Mistry, A. Murthy, J. Sandford, M.

Silberstein, S. Sivakumar, P. Smith, K. Zawadzki, S. Thompson, and M. Bohr, “A

90nm high volume manufacturing logic technology featuring novel 45nm gate length

strained silicon CMOS transistors”, IEDM Tech. Dig. p 11.6.1 (2003).

[10] P. R. Chidambaram, B. A. Smith, L. H. Hall, H. Bu, S. Chakravarthi, Y. Kim, A. V.

Samoilov, A. T. Kim, P. J. Jones, R. B. Irwin, M. J. Kim, A. L. P. Rotondaro, C. F.

Machala, and D. T. Grider, “35 drive current improvement from recessed-SiGe drain

extensions on 37 nm gate length PMOS”, IEEE VLSI Tech. Symp. Dig. p 48 (2004).

[11] R. Oberhuber, G. Zandler, and P. Vogl, “Subband structure and mobility of

two-dimensional holes in strained Si/SiGe MOSFET’s”, Phys. Rev. B 58, 9941

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[12] V. Aubry-Fortuna, P. Dollfus, S. Galdin-Retailleau, “Electron effective mobility in

strained Si/Si1-xGex MOS devices using Monte Carlo simulation” Solid State

Electronics 49 (8), 1320 (2005).

[13] K. Natori, “Scaling Limit of the MOS Transistor-A Ballistic MOSFET”, IEICE Trans.

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on Ballistic Transport”, Jpn. J. Appl. Phys. 45, 1548 (2006).

102

[16] K. Tomizawa, Numerical Simulation of Submicron Semiconductor Devices (Artec

House, London, 1993).

[17] M. Lundstrom, Fundamentals of Carrier Transport (Cambridge Univ. Press, 2000).

[18] C. Jacoboni, and L. Reggiani, “The Monte Carlo method for the solution of charge


55, 645 (1983).

[19] Tadashi Maegawa, Tsuneki Yamauchi, Takeshi Hara, “Strain Effects on Electronic

Bandstructures in Nanoscaled Silicon: From Bulk to Nanowire”, IEEE Transactions

on Electron Devices 56, no. 4, 553 (2009).



Phys. 96, 4998 (2004).

[21] G. Abstreiter, H. Brugger, T. Wolf, H. Jorke, and H. J. Herog, “Strain-Induced

Two-Dimensional Electron Gas in Selectively Doped Si/SixGe1-x Superlattices Phys.

Rev. Len. 54, 2441 (1985).

[22] E.Cartier, M. V. Fischetti, E. A. Eklund, and F. R. McFeely, “Impact ionization in

silicon”, Appl. Phys. Lett. 62, no. 25, 3339 (1993).

[23] Yiming Li, Ting-wei Tang, and Xinlin Wang, “Modeling of quantum effects for

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Nanotechnology 1, no 4, 238 (2002).

[24] B. K. Ridley, Quantum Processes in Semiconductors (London: Oxford, 1982).

[25] T. Yamada, Z. Jing-Rong, H. Miyata, and K. F. David, “In-Plane Transport Properties

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[26] Abudureheman Abudukelimu, Wufuer Yasenjiang, Kuniyuki Kakushima, Parhat

Ahmet, Mamtimin Geni, Kenji Natori, and Hiroshi Iwai, “Effects of Scattering

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+ Diode”, Jpn. J. Appl. Phys.

50, 104301 (2011).

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104

Chapter 9

Summary

In Chapters 2, 3 and 4, the semiconductor fundamentals, the mechanism of carrier

scattering, and the Monte Carlo method and other related topics for semi-classical carrier

transport in the device are briefly introduced, respectively.

In Chapter 5, the effect of the scattering direction of hot electrons on the

performance of ballistic channel diode is investigated and discussed. At low doping

concentrations, the ionized impurity scattering shows an anisotropic characteristics and it

has a weak influence on hot electron transport. At sufficiently high doping concentrations

on the other hand, the ionized impurity scattering shows an isotropic characteristics, and

it enhances the backward flow of hot electrons. It severely degrades the peak of the mean

velocity of electrons in the channel, and decreases the steady-state current.

In Chapter 6, the detailed effect of heat generation on hot electron transport in the

drain region is investigated. Heat generation rates due to the inelastic intravalley acoustic

phonon scattering and to the intervalley phonon scattering are computed. It was shown

that these two phonon scatterings have different influences on hot electron transport. The

intravalley acoustic phonon scattering enhances the backward flow of hot electrons, and

severely degrades the peak of mean velocity of electrons in the channel and the drain

current, as a result of low heat generation within drain region. In contrast, the intervalley

phonon scattering has a high heat generation rate within the drain, and it could suppress

the backward flow of hot electrons increasing the drain current.

In chapter 7, the influence of hot phonons on hot electron transport within the drain

region is investigated. The simulated results under the equilibrium and the

non-equilibrium phonon occupation condition are compared. According to the result,

when the non-equilibrium condition is considered in the simulation, the mean electron

energy within the drain is larger than that under the equilibrium condition. However, the

heat generation, the mean electron velocity and the drain current are all lower than

corresponding values under the equilibrium condition. The increased hot phonon

population and the increased hot phonon re-absorption rates are main reasons for these

results.

In chapter 8, the effect of biaxial strain within the Si channel and drain on

performance of a ballistic channel diode is comparatively studied. It was shown that the

105

strained drain can suppress rebound of hot electron and increase the drain current and

mean velocity of electrons in the drain. The effect is more remarkable compared to the

strained ballistic channel, because the strained drain can obviously suppress the scattering.

The effect is due to the fact that energy valleys of the strained-Si are split into two groups

and electrons in the lowest 2-fold valleys have a small effective mass.

For Improved Performance and Future Works

The carrier scattering in the drain plays an important role in hot electron transport and

strongly affects the performance in the ballistic channel device. There are several methods

to reduce the backward flow of hot electrons from drain, and to achieve improvement of

the device performance. They are summarized here based on conclusions in this

dissertation.

(1) Reducing the operating voltage to reduce the high-energy scattering;

(2) Controlling the doping concentration to reduce the ionized impurity scattering;

(3) Reducing phonon scattering by using a material which has a smaller effective mass

and/or smaller deformation potentials;

(4) Using the strained material to reduce the scattering;

(5) Using materials which have a larger phonon-energy, to increase heat generation in the

hotspot and reduce backward flow of hot electrons.

In this work, a simplistic band structure for carriers is employed. In simulation of the

device, a bulk regime is assumed and the low-dimensional structure is not considered.

The scattering mechanism is limited to those that are important for rather low energy

carrier transport. In order to achieve a more accurate simulation of the device, there are

several points that need to be considered in future works.

(1) If the device scale is further reduced, two-dimensional effects related to both the

electrostatics and the quantum confinement are likely to influence the hot electron

transport significantly. This is because the final density-of-states in scattering will be

significantly different, and therefore the scattering rate itself will be different. In the

case of bulk devices this effect may be a secondary consideration, since the quantum

confinement is not so remarkable in the drain region. In UTB SOIs or DGFETs,

however, the difference may be substantial.

106

(2) Phonon scatterings are sensitive to selection of the deformation potential. The

arguments about deformation potential are important, and one must be fully careful

in selection of the deformation potential in future works.

(3) The intravalley optical phonon scattering is suppressed by the selection rule along the

Delta symmetry line in Si. But if the energy of the hot electron is far away from the

Delta minima, the non-polar optical phonon scattering has to be considered in the

simulation.

(4) The realistic band structure of Si in the wide range is quite different from any

analytical band approximation. When the energy of hot electron is far away from the

minimum point in the band structure, one should be careful to employ a better band

structure model, e.g. the full-band model, in the simulation.

(5) Because of thermalization of the energy distribution through the carrier-carrier

scattering, hot electrons may easily be affected by the high energy valley carriers,

even when a relatively small bias is applied.

(6) Although relatively small biases are applied, the impact ionization cannot be ignored

because the carrier-carrier scattering can produce high energy electrons exceeding the

applied bias near the drain-end of the channel.

107

Published Papers and Presentations

Published

Abudureheman Abudukelimu, Wufuer Yasenjiang, Kuniyuki Kakushima, Parhat Ahmet,

Mamtimin Geni, Kenji Natori, and Hiroshi Iwai, “Effects of Scattering Direction of Hot

Electrons in the Drain of Ballistic n+–i–n+ Diode”, Japanese Journal of Applied Physics,

50 (2011) 104301.

Abudureheman Abudukelimu, Wufuer Yasenjiang, Kuniyuki Kakushima, Parhat Ahmet,

Mamtimin Geni, Kenji Natori, and Hiroshi Iwai, “Influence of strained drain on

performance of ballistic channel diode”, Semicond. Sci. Technol. 27 (2012) 055001.

.

International Conference

Abudureheman Abudukelimu, Kuniyuki Kakushima, Parhat Ahmet, Mamtimin Geni,

Kazuo Tsutsui, Akira Nishiyama, Nobuyuki Sugii, Kenji Natori, Takeo Hattori and

Hiroshi Iwai, “Performance of Silicon Ballistic Nanowire MOSFET with Diverse

Orientations and Diameters”, China Semiconductor Technology International Conference

(CSTIC), Mar. 18 - 19, 2010.


Kazuo Tsutsui, Akira Nishiyama, Nobuyuki Sugii, Kenji Natori, Takeo Hattori and

Hiroshi Iwai, “The effect of isotropic and anisotropic scattering in drain region of ballistic

channel diode”, International Conference on Solid-State and Integrated Circuit

Technology (ICSICT), Nov. 1- 4, 2010.

Domestic Conference

Abudureheman Abudukelimu, Kuniyuki Kakushima, Parhat Ahmet, Kazuo Tsutsui,

Akira Nishiyama, Nobuyuki Sugii, Kenji Natori, Takeo Hattori and Hiroshi Iwai,

“Current-Voltage Characteristics of Ballistic Nanowire FETs by Numerical Simulation”,

70th

Japan Society of Applied Physics (JSAP), Japan, Sep. 8~11, 2009.


Kenji Natori, and Hiroshi Iwai, “Improvement of the performance of Ballistic channel

108

Device by Strained Drain”, 59th

Japan Society of Applied Physics (JSAP), Japan, Mar.

15~18, 2012.

109

Appendix

Monte Carlo Program for Simulation of the Electrons Transport within 2D Diode

Coded By Abudukelimu

Supervisor: Hiroshi Iwai (1)

Co-Supervisor: Kenji Natori (1)

Adviser: Mamtimin Geni (2)

(1) Tokyo Institute of Technology

(2) Xinjiang University (China)

Important bibliography: Numerical Simulation of Submicron Semiconductor Devices,

Kazutaka Tomizawa, Artech House (English book).

This program was written in C and compiled on the Cygwin operating system with the

freely available C Compiler (GCC). To run program typing in the shell command line:

# gcc main.c

# ./a.exe diode.b

++++++++++++++++++++++ constants.h +++++++++++++++++++++++++++++

#ifndef _CONSTANTS_H_

#define _CONSTANTS_H_

// maximum number of particles

const long int pmax=1000000;

//number of divide energy

const int iemax=1000;

// Potential error

const double err=1.e-8;

// electron energy step (eV)

const double de=0.002;

// initial value for random number generator

double iseed=38467;

// Pi number

110

const double pi=3.14159;

// Boltzmann constant (J/K)

const double kb=1.38066e-23;

// Electron charge (C)

const double q=1.60219e-19;

// Reduced Planck constant (J*s)

const double h=1.05459e-34;

// Vacuum Permittivity (F/m)

const double ep0=8.85419e-12;

// Free Electron Mass (Kg)

const double am0=9.10953e-31;

// Band gap energy of Silicon

const double egSi=1.12;

// Relative Permittivity of Silicon

double epSi=11.7;

// Silicon Schottky contact density (1/m^3)

const double SchoDenSi=3.9e11;

// Silicon intrinsic density

const double n0Si=1.e16;

// Silicon sound velocity (m/s)

const double VSi=9040.;

// Silicon Density (Kg/m^3)

const double rouSi=2330.;

// Silicon acoustic deformation potential (J)

const double Sida=6.55*1.60219e-19;

// const double Sida=9.0*1.60219e-19;

// effect mass for Silicon electron

// const double emlSi=0.916;

// const double emtSi=0.196;

const double emSi= 0.32;

// non-parabolicity coefficient for Silicon

const double afSi=0.5;

// Silicon optical coupling constants (eV/m)

111

const double Sido[6]=0.05e11,0.08e11,1.1e11,0.03e11,0.20e11,0.20e11;

// Silicon optical phonon scattering energy (eV)

const double Sihwo[6]=0.01206,0.01853,0.06204,0.01896,0.04739,0.05903;

// Silicon optical phonon Z-factor

const double zf[6]=1.,1.,1,4.,4.,4.;

#endif //_CONSTANTS_H_

++++++++++++++++++++++ define.h ++++++++++++++++++++++++++++++

#ifndef _DEFINE_H_

#define _DEFINE_H_

// #define SILICON 1 // material is silicon

// #define ELEC 1 // particle is electron

#define POT 0 // potential

#define DEN 1 // density

#define VELX 2 // X-axis velocity

#define VELY 3 // Y-axis veloctiy

#define ENER 4 // Energy

#define EFX 5 // X-component of the electric Field

#define EFY 6 // Y-component of the electric Field

#define DOWN 0 // Bottom edge

#define RIGHT 2 // Right edge

#define UP 1 // top edge

#define LEFT 3 // left edge

#define INSU 0 // insulator

#define OHMI 1 //ohmic contact

// #define min(x,y) x>=y ? y:x

// #define max(x,y) x>=y ? x:y

112

#endif //_DEFINE_H_

++++++++++++++++++++++ definition.h ++++++++++++++++++++++++++++

#ifndef _DEFINITION_H_

#define _DEFINITION_H_

int nx,ny,iv,step;

int in,bn, ballistic,savestep,start,np1;

long int inum, sct[5][10];

int npt[200][4];

double te,ft,dt,t0=0., lx, ly,dx,dy, cimp,epp;

double bktq,qh,x,y,z,kx,ky,kz,ts,temp,qd2,bkt,amd,amc;

double isx[7],isy[7],iex[7],iey[7];

double bsx[7],bsy[7],bex[7],bey[7];

double mv[7],smh,hhml,hhmt,hhm,hm[5],gm,gme[15],tm[5];

double ***value,***bcs,***sio2,**swk;

double **n0d,**p,**psi;

#endif //_DEFINITION_H_

++++++++++++++++++++++++++++ extern.h ++++++++++++++++++++++

#ifndef _EXTERN_H_

#define _EXTERN_H_

extern int nx,ny,iv,step;

extern int in,bn, ballistic, savestep,start,np1;

extern long int inum, sct[5][10];

extern int npt[200][4];

extern double te,ft,dt,t0, lx, ly,dx,dy, cimp,epp;

113

extern double bktq,qh,x,y,z,kx,ky,kz,ts,temp,qd2,bkt,amd,amc;

extern double isx[7],isy[7],iex[7],iey[7];

extern double bsx[7],bsy[7],bex[7],bey[7];

extern double mv[7],smh,hhml,hhmt,hhm,hm[5],gm,gme[15],tm[5];

extern double ***value,***bcs,***sio2,**swk;

extern double **n0d,**p,**psi;

#endif // END _EXTERN_H_

+++++++++++++++++++++++++ functions.h ++++++++++++++++++++++++++

#ifndef _FUCTIONS_H_

#define _FUCTIONS_H_

void input(char*);

void array(void);

void *GetArray(int, int);

void **GetArray2( int, int, int );

void ***GetArray3( int , int , int, int );

void FreeArray( void * );

void FreeArray2( int, void ** );

void FreeArray3( int row_num, int col_num, void *** );

void initial(void);

void scatrate(void);

void poisBC(void);

void poisson(void);

void charge(void);

void emc(void);

void drift(double );

double check(void);

void scat(void);

void meanvalue(void);

void output(int );

114

void creat(int ,double ,int,int );

double rnd(void);

#endif //_FUCTIONS_H_

+++++++++++++++++++++++++ main.c +++++++++++++++++++++++++

#include<stdio.h>

#include<stdlib.h>

#include<string.h>

#include<math.h>

#include<time.h>

#include<memory.h>

#include"define.h"

#include"constants.h"

#include"definition.h"

#include"extern.h"

#include"functions.h"

time_t times;

struct tm *now;

int main(int argc,char* argv[])

// memset(sct,0,sizeof(sct));

// Read the input file

input(argv[1]);

// ==============================

times=time(NULL);

now=localtime(&times);

scatrate();

115

initial();

printf("¥nSimulation Started at %s¥n",asctime(now));

for(step=1;step<=(int)(ft/dt)+1;step++)

poisson();

charge();

emc();

// renew();

if(step>=start) meanvalue();

if(savestep!=0 && step%savestep==0 && step>start)

output(step);

printf("Output number %d has been saved¥n",step);

times=time(NULL);

now=localtime(&times);

printf("Computation Finished at %s¥n",asctime(now));

return(0);

#include "extrafuc.c"

#include "inputfile.c"

#include "alloc.c"

#include "scatrate.c"

#include "initial.c"

#include "poisson.c"

#include "output.c"

#include "ecm.c"

#include "drift.c"

#include "scattering.c"

#include "meanvalue.c"

116

+++++++++++++++++++++++++ input.c +++++++++++++++++++++++++

// read the date file (diode.b)

void input(char *infile)

FILE *fp;

char s[150];

int i,j;

in=1,bn=1, ballistic=0, savestep=0,start=0,np1=1;

fp=fopen(infile,"r");

if(fp==NULL)

printf("Can't opening the input file %s¥n",infile);

exit(EXIT_FAILURE);

do

// read the current row

fscanf(fp,"%s",s);

// # is a comment

if(strcmp(s,"#")==0)

fgets(s,100,fp);

else if(strcmp(s,"TEMPERATURE")==0)

fscanf(fp,"%lg",&te);

printf("Absolute temperature is %g (K)¥n",te);

else if(strcmp(s,"FINALTIME")==0)

fscanf(fp,"%lg",&ft);

printf("FINAL TIME = %g (s)¥n",ft);

else if(strcmp(s,"TIMESTEP")==0)

fscanf(fp,"%lg",&dt);

printf("TIME step = %g (s)¥n",dt);

else if(strcmp(s,"XLENGTH")==0)

fscanf(fp,"%lg",&lx);

printf("LENGTH of X axis = %g (m)¥n",lx);

else if(strcmp(s,"YLENGTH")==0)

fscanf(fp,"%lg",&ly);

117

printf("LENGTH of Y axis = %g (m)¥n",ly);

else if(strcmp(s,"XMESH")==0)

fscanf(fp,"%d",&nx);

dx=lx/nx;

printf("XMESH = %d (m) ",nx);

printf("dx = %g (m) ¥n",dx);

else if(strcmp(s,"YMESH")==0)

fscanf(fp,"%d",&ny);

dy=ly/ny;

printf("YMESH = %d (m) ",ny);

printf("dy = %g (m) ¥n",dy);

//++++++++++++++++++++++++++++++

array();

else if(strcmp(s,"Dopping")==0)

double x1,y1,x2,y2,temp;

fscanf(fp,"%lg %lg %lg %lg %lg",&x1,&y1,&x2,&y2,&temp);

isx[in]=x1; isy[in]=y1;

iex[in]=x2; iey[in]=y2;

for(i=1;i<=nx+1;i++)

for(j=1;j<=ny+1;j++)

if((i-0.5)*dx>=x1 && (i-1.5)*dx<=x2

&&(j-0.5)*dy>=y1 && (j-1.5)*dy<=y2)

value[DEN][i][j]=n0d[i][j]=temp;

printf("Dopping DENSITY %lg %lg %lg %lg %lg (m-3)¥n",

isx[in],isy[in],iex[in],iey[in],temp); in++;

else if(strcmp(s,"CIMP")==0)

fscanf(fp,"%lg",&cimp);

printf("Impurity Concentration = %lg (m-3)¥n",cimp);

else if(strcmp(s,"BOUNDARY")==0)

int i,j,k,k1,k2;

double l,volt;

char c1[30], c2[30];

118

fscanf(fp,"%s %s %lg",c1,c2,&volt);

if(strcmp(c1,"DOWN")==0) i=DOWN; k1=nx;l=lx;

else if(strcmp(c1,"RIGHT")==0) i=RIGHT;k1=ny;l=ly;

else if(strcmp(c1,"UP")==0) i=UP; k1=nx;l=lx;

else if(strcmp(c1,"LEFT")==0) i=LEFT; k1=ny;l=ly;

if(strcmp(c2,"INSULATOR")==0) k=INSU;

else if(strcmp(c2,"OHMIC")==0) k=OHMI;

for(j=1;j<=k1;j++)

bcs[i][j][0]=k;

if(k==OHMI) bcs[i][j][OHMI]=volt;

if(k==INSU)printf("bcs %s 0. %lg %s ¥n",c1,l,c2);

else if(k==OHMI) printf("bcs %s 0. %lg %s %lg (v) ¥n",c1,l,c2,volt);

/*++++++++++++++++++++++++++++++++++

else if(strcmp(s,"BALLISTIC")==0)

ballistic=1;

fscanf(fp,"%lg %lg %lg %lg",&bsx[bn],&bsy[bn],&bex[bn],&bey[bn]);

printf("ballistic REGION %lg %lg %lg %lg %d ¥n",bsx[bn],bsy[bn],

bex[bn],bey[bn],bn);

bn++;

++++++++++++++++++++++++++++++++++++ */

else if(strcmp(s,"SAVESTEP")==0)

fscanf(fp,"%d",&savestep);

printf("Save AT EACH savestep %d ¥n",savestep);

else if(strcmp(s,"PARTICLENUMBER")==0)

fscanf(fp,"%d",&np1);

printf("Particle number in each cell %d ¥n",np1);

else if(strcmp(s,"STARTSTEP")==0)

fscanf(fp,"%lg",&start);

119

printf("START step = %d ¥n",start);

while(!feof(fp));

fclose(fp);

printf("Input file is readed¥n");

+++++++++++++++++++++++++ alloc.c ++++++++++++++++++++++++++

void *GetArray( int nitems, int nsize )

void *ptr;

ptr = (void *)malloc( nitems * nsize );

if( ( nitems != 0 ) && ( ptr == NULL ) )

fprintf( stderr, "Failed to allocate memory¥n" );

exit( 1 );

return ptr;

void **GetArray2( int row_num, int col_num, int nsize )

int i;

void **ptr;

ptr = (void **)malloc( row_num * sizeof(void *) );

for( i = 0; i < row_num; ++i )

ptr[i] = (void *)malloc( col_num * nsize );

if( ( row_num != 0 ) && ( ptr == NULL ) )


exit( 1 );

return ptr;

120

void ***GetArray3( int row, int col, int z, int nsize )

int i,j;

void ***ptr;

ptr = (void ***)malloc( row * sizeof(void **));

for( i = 0; i < row; ++i )

ptr[i] = (void **)malloc( col * sizeof(void *) );

for( i = 0; i < row; ++i )

for( j = 0; j < col; ++j )

ptr[i][j] = (void *)malloc( z * nsize);

if( ( row != 0 ) && ( ptr == NULL ) )


exit( 1 );

return ptr;

void FreeArray( void *ptr )

free ( ptr );

void FreeArray2( int nitems, void **ptr )

int i;

for( i = 0; i < nitems; ++i )

FreeArray( ptr[i] );

FreeArray( ptr );

121

void FreeArray3( int row_num, int col_num, void ***ptr )

int i,j;

for( i = 0; i < row_num; ++i )

for(j=0;j< col_num; ++j)

FreeArray( ptr[i][j] );

FreeArray( ptr[i] );

FreeArray( ptr );

void array(void)

value = (double ***)GetArray3(8,nx+4,ny+4,sizeof(double));

bcs = (double ***)GetArray3(5,nx+4,nx+4,sizeof(double));

swk = (double **)GetArray2(15,iemax,sizeof(double));

n0d = (double **)GetArray2(nx+4,ny+4,sizeof(double));

p = (double **)GetArray2(pmax,12,sizeof(double));

psi = (double **)GetArray2(nx+4,ny+4,sizeof(double));

+++++++++++++++++++++++++ initial.c +++++++++++++++++++++++++

/* ++++++ Initial condition for electrons +++++++++++++++ */

void initial(void)

long int n=0;

int i,j,npi,m;

122

double ei,ak,cb,sb,fai,sf,r3,

epp=cimp*dx*dy/np1;

// printf("epp = %g ",epp);



npi=(int)(n0d[i][j]*dx*dy/epp+0.5);

if((i==1) || (i==nx+1)) npi/=2;

if((j==1) || (j==ny+1)) npi/=2;

if(npi<=0) continue;

for(m=1;m<=npi;m++)

n++;

if(n>pmax)

printf("Number of electron exceeds %d¥n",n);

exit(EXIT_FAILURE);

iv=1;

ei=-bktq*log(rnd())*1.5;

ak=smh*sqrt(ei*(1.+afSi*ei));

cb=1.-2.*rnd();

sb=sqrt(1.-cb*cb);

fai=2.*pi*rnd();

sf=sin(fai);

p[n][0]=iv;

p[n][1]=ak*cb*sf;

p[n][2]=ak*sb*sf;

p[n][3]=ak*cos(fai);

p[n][4]=-log(rnd())/gm;

p[n][5]=dx*(rnd()+(double)(i)-1.5);

p[n][6]=dy*(rnd()+(double)(j)-1.5);

if(i==1) p[n][5]=dx*0.5*rnd();

if(j==1) p[n][6]=dy*0.5*rnd();

if(i==nx+1) p[n][5]=lx-dx*0.5*rnd();

if(j==ny+1) p[n][6]=ly-dy*0.5*rnd();

123

/************************************

r3=3.*rnd();

if (r3 < 1.)

p[n][0]=1;

p[n][1]=tm[1]*p[n][1];

p[n][2]=tm[2]*p[n][2];

p[n][3]=tm[3]*p[n][3];

else if(r3 < 2.)

p[n][0]=2;

p[n][1]=tm[3]*p[n][1];

p[n][2]=tm[1]*p[n][2];

p[n][3]=tm[2]*p[n][3];

else

p[n][0]=3;

p[n][1]=tm[2]*p[n][1];

p[n][2]=tm[3]*p[n][2];

p[n][3]=tm[1]*p[n][3];

*********************************************************/

inum = n; for (i=inum+1; i<=pmax;i++) p[n][0]=9;

//for (i=1; i<=inum;i++) printf("%g ",p[i][6]); exit(0);



value[VELX][i][j]=0.;

value[VELY][i][j]=0.;

value[ENER][i][j]=0.;

printf("Number of particles = %d¥n", n);

124

+++++++++++++++++++++++++ ecm.c +++++++++++++++++++++++++

/* +++++++++++ Creating new electron +++++++++++++++*/

inline void creat(int i,double time,int edge,int n)

double ei,ak,cb,sb,fai,sf;

ei=-bktq*log(rnd())*1.5;

ak=smh*sqrt(ei*(1.+afSi*ei));

cb=1.-2.*rnd();

sb=sqrt(1.-cb*cb);

fai=2.*pi*rnd();

sf=sin(fai);

iv=1;

kx=ak*cos(fai);

ky=ak*cb*sf;

kz=ak*sb*sf;

ts=time-log(rnd())/gm;

x=dx*(rnd()+(double)(i)-1.5);

y=dy*(rnd()+(double)(i)-1.5);

if(edge==DOWN)

y=dy*0.5*rnd();

if(i==1) x=dx*0.5*rnd();

else if(i=nx) x=lx-dx*0.5*rnd();

else if( edge==RIGHT)

x=lx-dx*0.5*rnd(); kx=-kx;

if(i==1) y=dy*0.5*rnd();

else if(i==ny) y=ly-dy*0.5*rnd();

else if( edge==UP)

y=ly-dy*0.5*rnd(); ky=-ky;

if(i==1) x=dx*0.5*rnd();

else if(i=nx) x=lx-dx*0.5*rnd();

else if( edge==LEFT)

x=dx*0.5*rnd();

125

if(i==1) y=dy*0.5*rnd();

else if(i==ny) y=ly-dy*0.5*rnd();

p[n][0]=iv;

p[n][1]=kx;

p[n][2]=ky;

p[n][3]=kz;

p[n][4]=ts;

p[n][5]=x;

p[n][6]=y;

/* +++++++++++++++++++++++++++++++++++++++++ */

/* +++++++ Electron motion during dt +++++++++++++++ */

void emc(void)

long int n=1;

int i,ni,j,bi;

double tdt,ti,tau;

int sn1=0;

memset(&npt,0,sizeof(npt));

tdt=t0+dt;

do

iv=(int)(p[n][0]);

kx=p[n][1];

ky=p[n][2];

kz=p[n][3];

ts=p[n][4];

x=p[n][5];

y=p[n][6];

ti=t0;

sn1=0;

126

while(ts<tdt)

tau=(ts-ti);

drift(tau);

scat();

ti=ts;

ts=ti-log(rnd())/gm;

sn1=1;

tau=tdt-ti;

//if(sn1==0 ) sct[0][1]=sct[0][1]+1;

drift(tau);

if(iv==9)

for(i=0;i<=6;i++) p[n][i]=p[inum][i];

p[inum][0]=9;

inum--;

else if(iv==1)

p[n][0]=iv;

p[n][1]=kx;

p[n][2]=ky;

p[n][3]=kz;

p[n][4]=ts;

p[n][5]=x;

p[n][6]=y;

n++;

while(n<inum);

//create electron at ohmic contacts of the edge

for(i=1;i<=ny+1;i++)

if(bcs[RIGHT][i][0]==OHMI)

ni=(np1/2)-npt[i][RIGHT];

127

if(ni>0)

for(j=1;j<=ni;j++)

n=inum+j;

creat(i,t0,1,n);

inum += ni;

//create electron at ohmic contacts of the edge

if(bcs[LEFT][i][0]==OHMI)

ni=(np1/2)-npt[i][LEFT];

if(ni>0)

for(j=1;j<=ni;j++)

n=inum+j;

creat(i,t0,3,n);

inum += ni;

printf("Actual number of electron = %d¥n",inum);

if(inum>pmax)

printf("too big actual number of electron¥n");

exit(EXIT_FAILURE);

+++++++++++++++++++++++++ drift.c +++++++++++++++++++++++++++++

// check whether electrons are out of the device

inline double check()

128

int i,j;

i=(int)(x/dx)+1.5;

j=(int)(y/dy)+1.5;

if(i>=nx) i=nx;

if(j>=ny) j=ny;

// Boundary conditions for the Electrons

// left edge

// ++++++++++++++++++++++++++++++++++++++++

// --- ohmic contact---

if(x<=0.)

iv=9;

if(npt[j][LEFT]<np1/2) npt[j][LEFT]++;iv=1;

return;

// right edge

// --- ohmic contact---

else if(x>=lx)

iv=9;

if(npt[j][RIGHT]<np1/2) npt[j][RIGHT]++;iv=1;

return;

// bottom edge

// ---Insulator---

else if(y<=0. )

y=-y;

ky=-ky;

return;

//upper edge

129

// ---Insulator---

else if(y>=ly)

y=ly-(y-ly);

ky=-ky;

return;

/* ++++++++++++ Calculation of drift process +++++++++ */

void drift(double tau)

int i,j;

double dkx,dky,hmt,ksquared,gk,thesquareroot;

double vx,vy,skx,sky,skz;

if(iv==9) return;

i=(int)(x/dx)+1;

j=(int)(y/dy)+1;

if(i>=nx) i=nx;

if(j>=ny) j=ny;

//+++++++++++++++++++++++++++++++++++++++++++++++

dkx=-qh*value[EFX][i][j]*tau;

dky=-qh*value[EFY][i][j]*tau;

//++++++++++++++++++++++++++++++++++++++++++++++++

// Electron drift process

/************************************************************

skx=kx*kx;

sky=ky*ky;

sky=kz*kz;

cp=hm[iv]*tau;

130

if(iv==1) gk=hhml*skx+hhmt*(sky+skz);

else if (iv==2) gk=hhml*sky+hhmt*(skx+skz);

else if (iv==3) gk=hhml*skz+hhmt*(sky+skx);

************************************************/

hmt=hm[0]*tau;

ksquared=kx*kx+ky*ky+kz*kz;

gk=hhm*ksquared;

thesquareroot=sqrt(1.+4.*afSi*gk);

// Second order Runge-Kutta method

x+=hmt*(kx+0.5*dkx)/thesquareroot;

y+=hmt*(ky+0.5*dky)/thesquareroot;

kx+=dkx;

ky+=dky;

//++++++++++++++++++++++++++++++++++

check();

+++++++++++++++++++++++++ scatrate.c +++++++++++++++++++++++++

/* ++++++++++++ Calculation of scattering rate +++++++++ */

void scatrate(void)

double wo,no,dos,aco,aml,amt,dog,ei,ef,oe[7],oa[7];

double sei,sef;

double eps,bimp,qd, ak,qq,wk;

int ie,i;

//int z2=4, iem=1,n0;

bktq=kb*te/q;

131

qh=q/h;

eps=epSi*ep0;

// aml=emlSi*am0;

// amt=emtSi*am0;

// Anisotropy of conduction band

amd=pow(aml,1./3.)*pow(amt,2./3.);

amc=3./(1./aml+2/amt);

/****************************************************

tm[1]=sqrt(aml/amd);

tm[2]=sqrt(amt/amd);

tm[3]=sqrt(amt/amd);

hhml=h/aml/q*h/2.;

hhmt=h/amt/q*h/2.;

hm[1]=h/aml;

hm[2]=h/amt;

hm[3]=h/amt;

*********************************************/

smh=sqrt(2.*emSi*am0*q)/h;

hhm =h/(emSi*am0)/q*h/2.;

hm[0]=h/(emSi*am0);

aco=2.*pi*(Sida/q)*Sida*(bktq/h)*(q/(rouSi*VSi*VSi));

//Parameters for impurity scatterings

dos=pow((sqrt(2.*emSi*am0*q)/h),3.)/(4.*pi*pi);

qd=sqrt(q*cimp/bktq/eps);

qd2=qd*qd;

bimp=2.*pi*cimp*q*q/h*q/eps/eps;

/* +++++ Non-parabolicity of conduction band +++++++ */

132

for(i=1;i<=6;i++)

wo=Sihwo[i-1]*q/h;

no=1./(exp(Sihwo[i-1]/bktq) - 1.);

dog=Sido[i-1]*q;

oe[i]=zf[i-1]*pi*dog/wo*dog/rouSi/q*(no+1.);

oa[i]=oe[i]*no/(1.+no);

for(ie=1; ie<=iemax; ++ie) swk[0][ie]=0.;

for(ie=1; ie<=iemax; ++ie)

ei=de*(double)ie;

sei=sqrt(ei);

// intervalley scattering (inelastic)

for(i=1;i<=6;i++)

//Emission

ef=ei-Sihwo[i-1];

swk[i*2-1][ie]=swk[i*2-2][ie];

if(ef>0.)

sef=sqrt(ef*(1.+afSi*ef));

swk[i*2-1][ie]=swk[i*2-2][ie]+oe[i]*sef*dos*(1.+2.*afSi*ef);

//Absorbtion

ef=ei+Sihwo[i-1];

swk[i*2][ie]=swk[i*2-1][ie];

if(ef>0.)


swk[i*2][ie]=swk[i*2-1][ie]+oa[i]*sef*dos*(1.+2.*afSi*ef);

// ********************************************************

// intravalley Acoustic phonon (elastic)

ef=ei;


wk=aco*sef*dos*(1.+2.*afSi*ef);

133

swk[13][ie]=swk[12][ie]+wk;

// if(wk>1.e14) wk=1.e14;

// Ionized impurity scattering

ef=ei;


ak=smh*sef;

qq=qd2*(4.*ak*ak+qd2);

wk=bimp/qq*sef*dos*(1.+2.*afSi*ef);

if(wk>1.e14) wk=1.e14;

swk[14][ie]=swk[13][ie]+wk;

// Evaluation of gamma

gm=swk[14][1];

for(ie=1;ie<=iemax;++ie)

if(swk[14][ie]>gm) gm=swk[14][ie];

printf("GAMMA = %g ¥n",gm);

for(ie=1;ie<=iemax;ie++)

for(i=1;i<=14;i++) swk[i][ie]/=gm;

//+++++++++++++++++++++++++++++++++++++++++++++

+++++++++++++++++++++++++ scattering.c +++++++++++++++++++++++++

/* +++++++++++ Calculation of scattering process ++++++++ */

void scat(void)

int j=0;

int bi=0;

int impu=0;

int i,ie=0;

double ksquared,thesquareroot,superparticle_energy;

double r1,finalenergy=0.,finalk,cosinus,sinus,fai;

double f,cb,cf,sf,skk,a11,a12,a13,a21,a22,a23,a31,a32,a33,x1,x2,x3,sb,r2;

134

double ki,kf,cs,sn,kx1,ky1,kz1;

int si=0;

int acn=0;

if(iv==9) return;

/* +++++++++++++++++++++++++++++++++++++++++++

if(ballistic ==1 )

for (bi=1;bi<bn;bi++)

if( x>=bsx[bi] && x<=bex[bi] && y>=bsy[bi] && y <=bey[bi])

return;

++++++++++++++++++++++++++++++++++++++++++++++ */

/*++++++++++++++++++++++++++++++++++++

if(x<iex[1]) si=1;

else if (x<iex[2]) si=2;

else if (x<iex[3]) si=3;

+++++++++++++++++++++++++++++++++++++*/

// Selection of scattering process

// sct[0][0]=sct[0][0]+1;

/*****************************************

skx=kx*kx;

sky=ky*ky;

sky=kz*kz;




*****************************************/

ksquared=kx*kx+ky*ky+kz*kz;

kx1=kx;ky1=ky;kz1=kz;

ki=sqrt(ksquared);

135

thesquareroot=sqrt(1.+4.*afSi*hhm*ksquared);

superparticle_energy=(thesquareroot-1.)/(2.*afSi);

if(superparticle_energy<=0.) return;

ie=((int)(superparticle_energy/de))+1;

if(ie>iemax) ie=iemax;

r1 = rnd();

// intervalley optical phonons

for(i=1;i<=6;i++)

// Emission of an optical phonon

if(r1<=swk[i*2-1][ie])

finalenergy=superparticle_energy-Sihwo[i-1];

//sct[si][1]=sct[si][1]+1;

if(finalenergy<=0. ) return;

// determination of the final states

finalk = smh*sqrt(finalenergy*(1.+afSi*finalenergy));

cosinus = 1.-2.*rnd();

sinus = sqrt(1.-cosinus*cosinus);

fai = 2.*pi*rnd();

kx = finalk*cosinus;

ky = finalk*sinus*cos(fai);

kz = finalk*sinus*sin(fai);

return;

// Absorbation of an optical phonon

else if(r1<=swk[i*2][ie])

finalenergy=superparticle_energy+Sihwo[i-1];

//sct[si][2]=sct[si][2]+1;

if(finalenergy<=0. ) return;




136


fai = 2.*pi*rnd();




return;

/*******************************************************************/

// Acoustic phonon

if(r1<=swk[13][ie])

finalenergy=superparticle_energy;

// sct[si][3]=sct[si][3]+1;

if(finalenergy<=0.) return;





fai = 2.*pi*rnd();




return;

/*********************************************************/

// Impurity scattering

if(r1<=swk[14][ie])

sct[si][4]=sct[si][4]+1;

finalenergy=superparticle_energy;

137

r2=rnd();

cb=1.-r2/(0.5+(1.-r2)*ksquared/qd2);

kf=smh*sqrt(finalenergy*(1.+afSi*finalenergy));


sb=sqrt(1.-cb*cb);

fai=2.*pi*rnd();

cf=cos(fai);

sf=sin(fai);

skk=sqrt(kx*kx+ky*ky);

a11=ky/skk;

a12=kx*kz/skk/ki;

a13=kx/ki;

a21=-kx/skk;

a22=ky*kz/skk/ki;

a23=ky/ki;

a32=-skk/ki;

a33=kz/ki;

x1=kf*sb*cf;

x2=kf*sb*sf;

x3=kf*cb;

kx=a11*x1+a12*x2+a13*x3;

ky=a21*x1+a22*x2+a23*x3;

kz=a32*x2+a33*x3;

/************************************************

if (iv==1)

kx=tm[1]*kx;

ky=tm[2]*ky;

kz=tm[3]*kz;

else if (iv==2)

kx=tm[3]*kx;

ky=tm[1]*ky;

kz=tm[2]*kz;

else if (iv==3)

138

kx=tm[2]*kx;

ky=tm[3]*ky;

kz=tm[1]*kz;

************************************************/

return;

/************************************************

if (iv==1)

kx=tm[1]*kx;

ky=tm[2]*ky;

kz=tm[3]*kz;

else if (iv==2)

kx=tm[3]*kx;

ky=tm[1]*ky;

kz=tm[2]*kz;

else if (iv==3)

kx=tm[2]*kx;

ky=tm[3]*ky;

kz=tm[1]*kz;

************************************************/

return;

+++++++++++++++++++++++++ poisson.c +++++++++++++++++++++++++++

/* +++++++++++++++ Charge distribution ++++++++++++++ */

void charge(void)

139

double x0,y0,x1,y1,xb,yb;

int i,j,n;

epp=cimp*dx*dy/np1;



value[DEN][i][j]=0.;

/* ++++++++++ Cloud in cell scheme ++++++++++ */

for(n=1;n<=inum;n++)

x0=p[n][5]/dx;

y0=p[n][6]/dy;

i=(int)(x0+1.);

j=(int)(y0+1.);

if(i<1 || i > nx) continue;

if(j<1 || j > ny) continue;

x1=1.-(x0-(double)(i-1));

y1=1.-(y0-(double)(j-1));

value[DEN][i][j]+=x1*y1;

value[DEN][i+1][j]+=(1.-x1)*y1;

value[DEN][i][j+1]+=x1*(1.-y1);

value[DEN][i+1][j+1]+=(1.-x1)*(1.-y1);



value[DEN][i][j]*=epp/(dx*dy);

/*++++++++++++++++++++++++++++++++++++++++++++

if(j==1) values[DEN][i][j]=BCs[DOWN][j][2];

if(j==ny+1) values[DEN][i][j]=BCs[UP][j][ny];

if(i==1) values[DEN][i][j]=BCs[LEFT][j][2];

if(i==nx+1) values[DEN][i][j]=BCs[RIGHT][j][nx];

140

++++++++++++++++++++++++++++++++++++++++++++*/

if(i==1 || i==nx+1) value[DEN][i][j]*=2.;

if(j==1 || j==ny+1) value[DEN][i][j]*=2.;

value[DEN][nx+1][ny+1]=value[DEN][nx][ny+1];

value[DEN][1][ny+1]=value[DEN][1][ny];

/*+++++++++++++++++++++++++++++++++++++++++++++++

values[DEN][nx+1][ny+1]=values[DEN][nx+1][ny];

values[DEN][nx][ny+1] =values[DEN][nx][ny];

values[DEN][1][ny+1] =values[DEN][1][ny];

values[DEN][2][ny+1] =values[DEN][2][ny];

values[DEN][1][1] =values[DEN][1][2];

++++++++++++++++++++++++++++++++++++++++*/

/* +++++++++++++++++++++++++++++++++++++++++++

DOWN // Bottom edge

RIGHT // Right edge

UP // top edge

LEFT // left edge

INSU // insulator

SCHO // schotiky contact

OHMI //ohmic contact

++++++++++++++++++++++++++++++++++++++++++ */

void poisBC(void)

int i,j;

// Down edge


if(bcs[DOWN][i][0]==INSU)

141

//insulator

value[POT][i][0]=value[POT][i][3];

value[POT][i][1]=value[POT][i][2];

else if(bcs[DOWN][i][0]==OHMI)

// ohmic contact

value[POT][i][0]=bcs[DOWN][i][1];

value[POT][i][1]=bcs[DOWN][i][1];

// Right edge


if(bcs[RIGHT][j][0]==INSU )

//insulator

value[POT][nx+1][j]=value[POT][nx-1][j];

value[POT][nx+2][j]=value[POT][nx][j];

else if(bcs[RIGHT][j][0]==OHMI)

// ohmic contact

value[POT][nx+1][j]=bcs[RIGHT][j][1];

value[POT][nx+2][j]=bcs[RIGHT][j][1];

// exit(0);

// Top edge


if(bcs[UP][i][0]==INSU )

//insulator

value[POT][i][ny+1]=value[POT][i][ny];

value[POT][i][ny+2]=value[POT][i][ny-1];

else if(bcs[UP][i][0]==OHMI)

// ohmic contact

value[POT][i][ny+1]=bcs[UP][i][1];

value[POT][i][ny+2]=bcs[UP][i][1];

142

// Left edge


if(bcs[LEFT][j][0]==INSU)

//insulator

value[POT][0][j]=value[POT][2][j];

value[POT][1][j]=value[POT][3][j];

else if(bcs[LEFT][j][0]==OHMI)

// ohmic contact

value[POT][0][j]=bcs[LEFT][j][1];

value[POT][1][j]=bcs[LEFT][j][1];

/*+++++++++++++++++++++++++++++++++++++++++++++++++++++++*/

/* +++++++ Potential calculation ++++++++++++++++ */

void poisson(void)

int i,j,k,is,js,ks;

double fac,ka,del,rho,error1,error2,terr1,terr2;

error2=1.;

poisBC();

// ===============================

fac=1.;

ka=epSi*ep0/q;

del=0.9*0.5/ka/(1./(dx*dx)+1./(dy*dy));

while(error2>err)

poisBC();


for(j=0;j<=ny+2;j++) psi[i][j]=value[POT][i][j];

143

for(i=2;i<=nx;i++)

for(j=2;j<=ny;j++)

rho=(value[DEN][i][j]-n0d[i][j]);

value[POT][i][j]=psi[i][j]-del*rho+del*ka*

((psi[i+1][j]-2.0*psi[i][j]+psi[i-1][j])/(dx*dx)

+(psi[i][j+1]-2.0*psi[i][j]+psi[i][j-1])/(dy*dy));

terr1=fabs(value[POT][2][2]-psi[2][2]);

terr2=fabs(value[POT][i][j]-psi[i][j]);

if (terr2>terr1) terr1=terr2;

error2=terr1;

// EFX // X-component of the electric Field

// EFY // Y-component of the electric Field

poisBC();


for(j=0;j<=ny+1;j++) psi[i][j]=value[POT][i][j];

for(i=2;i<=nx;i++)


value[EFX][i][j]=-1.*(value[POT][i+1][j]-value[POT][i-1][j])/(2*dx);


value[EFX][1][j]=value[EFX][2][j];

value[EFX][nx+1][j]=value[EFX][nx][j];


for(j=2;j<=ny;j++)

value[EFY][i][j]=-1.*(value[POT][i][j+1]-value[POT][i][j-1])/(2*dy);

144


value[EFY][i][1]=value[EFY][i][2];

value[EFY][i][ny+1]=value[EFY][i][ny];

+++++++++++++++++++++ meanvalue.c ++++++++++++++++++++++++++++

/*++++++++++++++Computation of the mean values +++++++*/

void meanvalue()

register int i,j,n;

int cont[nx+2][ny+2];

double x,y,xvel,yvel,sk,gk,ei,skx,sky,skz,sq;

double vex[nx+2][ny+2], vey[nx+2][ny+2], en[nx+2][ny+2];



cont[i][j]=0;

vex[i][j]=0.;

vey[i][j]=0.;

en[i][j]=0.;

for(n=1;n<=inum;n++)

x=p[n][5]/dx;

y=p[n][6]/dy;

skx=p[n][1]*p[n][1];

sky=p[n][2]*p[n][2];

skz=p[n][3]*p[n][3];

/*++++++++++++++++++++++++++++++++++++

iv=p[n][0];


145



sq= sqrt(1.+4.*afSi*gk);

xvel=p[n][1]*hm[iv]/sq;

yvel=p[n][2]*hm[iv]/sq;

+++++++++++++++++++++++++++++++++++++*/

sk=skx+sky+skz;

sq=sqrt(1.+4.*afSi*hhm*sk);

ei=(sq-1.)/(2.*afSi);

xvel=p[n][1]*hm[0]/sq;

yvel=p[n][2]*hm[0]/sq;

i=(int)(x+1.5);

j=(int)(y+1.5);

if(i<1) i=1;

else if(i>nx+1) i=nx+1;

if(j<1) j=1;

else if(j>ny+1) j=ny+1;

cont[i][j]++;

en[i][j]+=ei;

vex[i][j]+=xvel;

vey[i][j]+=yvel;

p[n][7]=xvel;

p[n][8]=yvel;

// vex[1][j]=vex[2][j];

// vex[nx+1][j]=vex[nx][j];

// Mean Value of the variables



vex[i][j]/=cont[i][j];

146

vey[i][j]/=cont[i][j];

en[i][j]/=cont[i][j];



value[VELX][i][j]+=vex[i][j];

value[VELY][i][j]+=vey[i][j];

value[ENER][i][j]+=en[i][j];

++++++++++++++++++++ extrafuc.c +++++++++++++++++++++++++++++++++

inline double rnd(void)

double mi=1048576., in=1027.;

iseed = fmod(in*iseed, mi);

return (iseed/mi);

+++++++++++++++++++++++++ output.c ++++++++++++++++++++++++

//double unit=1.e+6; //um

double unit=1.e+9; //nm

void output(int num)

double mean[15];

double dvel,fvel[110],cur;

int cstart, ci,cj,nvel=100, mn=0,i,j;

char s[50];

memset(&mean,0.,sizeof(mean));

147

memset(&fvel,0.,sizeof(fvel));

cstart=(int)(ft/dt)+1-start;

FILE *va;

FILE *pi;

FILE *mp;

FILE *sn;

if(num<10)

sprintf(s,"value00%d.xy",num);

va=fopen(s,"w");

sprintf(s,"partical00%d.xy",num);

pi=fopen(s,"w");

sprintf(s,"mvalue00%d.xy",num);

mp=fopen(s,"w");

// sprintf(s,"scattering00%d.xy",num);

// sn=fopen(s,"w");

else if( num<100)

sprintf(s,"value0%d.xy",num);

va=fopen(s,"w");

sprintf(s,"partical0%d.xy",num);

pi=fopen(s,"w");

sprintf(s,"mvalue0%d.xy",num);

mp=fopen(s,"w");

// sprintf(s,"scattering0%d.xy",num);

// sn=fopen(s,"w");

else

sprintf(s,"value%d.xy",num);

va=fopen(s,"w");

sprintf(s,"partical%d.xy",num);

pi=fopen(s,"w");

148

sprintf(s,"mvalue%d.xy",num);

mp=fopen(s,"w");

// sprintf(s,"scattering%d.xy",num);

// sn=fopen(s,"w");

for(j=2;j<=ny;j++)

for(i=2;i<=nx;i++)

fprintf(va,"%lg %lg %lg %lg %lg %lg %lg %lg %lg¥n",unit*(i-1.)*dx,unit*(j-1.)*dy,

value[POT][i][j],value[DEN][i][j],value[VELX][i][j]/cstart,

value[VELY][i][j]/cstart,value[ENER][i][j]/cstart,value[EFX][i][j],

value[EFY][i][j]);

fprintf(va,"¥n");

fprintf(mp,"X(nm) Den(m-3) Xve(m/s) Yve(m/s) Eev(eV) Pot(eV) EFx(eV/m)

EFy(eV/m) Cur(A/m)");

for(i=2;i<=nx;i++)

mn=0;

memset(&mean,0,sizeof(mean));

for(j=1;j<=ny;j++)

mean[1]+=value[DEN][i][j];

mean[2]+=value[VELX][i][j]/cstart;

mean[3]+=value[VELY][i][j]/cstart;

mean[4]+=value[ENER][i][j]/cstart;

mean[5]+=psi[i][j];

mean[6]+=value[EFX][i][j];

mean[7]+=value[EFY][i][j];

mean[8]+=value[DEN][i][j]*value[VELX][i][j];

mn++;

//

mean[1]=mean[2]/mn;

mean[2]=mean[3]/mn;

149

mean[3]=mean[4]/mn;

mean[4]=mean[5]/mn;

mean[5]=mean[6]/mn;

mean[6]=mean[7]/mn;

mean[7]=mean[8]/mn;

mean[8]*=q*dy/cstart;

fprintf(mp,"%.6g %.6g %.6g %.6g %.6g %.6g %.6g %.6g %.6g¥n",

unit*(i-1.)*dx,mean[1],mean[2],mean[3],mean[4],mean[5],

mean[6],mean[7],mean[8]);

fclose(va);

fclose(mp);

// partical all information

/*++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+++++++++++++++++++++++++++++++++++++++++++++

fprintf(pi,"i IV KX KY KZ Time X Y VelocityX

VelocityY Energy inum= %d¥n",inum);

for(i=1;i<inum;i++)

if(isnan(p[i][1])||isnan(p[i][2])||isnan(p[i][3])||isnan(p[i][4])||

isnan(p[i][5])||isnan(p[i][6])||isnan(p[i][7])||isnan(p[i][8])||

isnan(p[i][9]));

fprintf(pi,"%d %d %g %g %g %g %g %g %g %g %g¥n",i,(int)p[i][0], p[i][1], p[i][2],

p[i][3], p[i][4], p[i][5]*unit, p[i][6]*unit,p[i][7],p[i][8],p[i][9]);

fclose(pi);

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++*/

Doctoral Thesis Hot Electron Transport and … Thesis Hot Electron Transport and Performance of...

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