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Empirical pseudopotential method for modern electron simulation

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Empirical pseudopotential method for modern electron simulation
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Strickland, Adam Lee ( author )
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The topic of this thesis is to develop a framework for finding critical values that can be used in simulating modern semiconductor device structures. Many current simulation tools use effective mass approximations to model the propagation of electrons through a device. Because modern devices take advantage of different material, geometries and sizes, a band structure calculation becomes necessary for representing effects such a strained materials and hot carriers. While highly accurate, calculation-intense methods exist, a band structure may need to be recalculated on-the-fly creating the need for a fast, simple algorithm that can be tailored for the situation at hand. Fortunately, there is a very well established method for calculating band structure based on physical properties of the material. Historically, these properties were measured, but modern techniques involve using algorithms to converge to the proper values. This allows the simulation of devices with interesting properties while maintaining relatively simple models.
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Thesis (M.S.)--University of Colorado Denver. Electrical engineering
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Includes bibliographic references.
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Department of Electrical Engineering
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by Adam Lee Strickland.

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Full Text
EMPIRICAL PSEUDOPOTENTIAL METHOD FOR MODERN
ELECTRON SIMULATION
by
ADAM LEE STRICKLAND
B.S., University of Colorado Denver, 2007
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillment
of the requirements for the degree of
Master of Science
Electrical Engineering
2014


11
This thesis for the Master of Science degree by
Adam Lee Strickland
has been approved for the Electrical Engineering Program by
Hamid Z. Fardi, Chair
Miloje Radenkovic
Jae-Do Park
April 30, 2014


Ill
Strickland, Adam Lee (M.S. Electrical Engineering)
Empirical Pseudopotential Method for Modern Electron Simulation
Thesis directed by Professor Hamid Z. Fardi
ABSTRACT
The topic of this thesis is to develop a framework for finding critical values that can be used in
simulating modern semiconductor device structures. Many current simulation tools use effective
mass approximations to model the propagation of electrons through a device. Because modern
devices take advantage of different materials, geometries and sizes, a band structure calculation
becomes necessary for representing effects such as strained materials and hot carriers. While highly
accurate, calculation-intense methods exist, a band structure may need to be recalculated on-the-fly
creating the need for a fast, simple algorithm that can be tailored for the situation at hand.
Fortunately, there is a very well established method for calculating band structure based
on physical properties of the material. Historically, these properties were measured, but modern
techniques involve using algorithms to converge to the proper values. This allows the simulation of
devices with interesting properties while maintaining relatively simple models.
The form and content of this abstract are approved. I recommend its publication.
Approved: Hamid Z. Fardi


IV
CONTENTS
Chapter
1 Introduction........................................................................... 1
1.1 Device Simulation................................................................. 2
1.2 A Breif History of Pseudopotentials and Band Structure Calculation................ 3
2 Empirical Pseudopotential Method....................................................... 5
2.1 EPM Proof......................................................................... 5
2.2 Method for Band Structure Calculation............................................ 6
2.3 Band Structure of Common Semiconductors........................................... 8
2.4 Finding Pseudopotentials.......................................................... 8
3 Applications.......................................................................... 17
3.1 Alloy Materials.................................................................. 18
3.2 Genetic Algorithms............................................................... 19
3.3 Future Work...................................................................... 19
Bibliography 22
Appendix
A Acronyms......................................................................... 26
B Code Listing and Examples........................................................ 27
B.l EPM Band Structure in C++.................................................... 27
B.2 Gnuplot, Example............................................................. 32


V
TABLES
Table
2.1 Form factors of Si, Ge, Sn, GaP and GaAs at 300K from [11]....................... 8
3.1 Form factors of SiGe Alloy for different concentrations at 300K.................. 18


VI
FIGURES
Figure
1.1 A graphical representation of pseudopotential simplification.......................... 4
2.1 Si band structure with form factors from [11]......................................... 9
2.2 Ge band structure with form factors from [11]......................................... 9
2.3 Sn band structure with form factors from [11]........................................ 10
2.4 GaP band structure with form factors from [11]....................................... 10
2.5 GaAs band structure with form factors from [11]...................................... 11
2.6 AlSb band structure with form factors from [11]...................................... 11
2.7 InP band structure with form factors from [11]....................................... 12
2.8 GaSb band structure with form factors from [11]...................................... 12
2.9 InAs band structure with form factors from [11]...................................... 13
2.10 InSb band structure with form factors from [11]..................................... 13
2.11 ZnS band structure with form factors from [11]...................................... 14
2.12 ZnSe band structure with form factors from [11]..................................... 14
2.13 ZeTe band structure with form factors from [11]..................................... 15
2.14 CdTe band structure with form factors from [11]..................................... 15
3.1 The Band Structure for Different Concentrations of Silicon-Germanium Alloy . 21


1
1. Introduction
Knowing the band structure of a material is important for relating its optical and electronic
properties. As new and exotic materials are studied, there is a greater need for energy band
calculations. Special physical properties can be studied without building devices. Band structure
calculations can be used to engineer custom band structures to tailor optical or electrical devices.
Though calculation of band structure is valuable in its own right, in this paper the focus will
be on its importance with regard to semiconductor simulation. In order to understand, design and
model new devices it is imperative to know the limitations and applications of different methods.
For calculating band structure there are two basic approaches: ab initio and empirical methods.
Ab initio, or from the beginning methods involve calculation of band structure by use of first-
principles without using measured data. Empirical methods take advantage of experimental data
to give more accurate band structure representation. Generally, ab initio methods are calculation
intensive but give better insight on how the structure arrives. Both methods have their place in
calculating and engineering transport properties in semiconductor devices.
Examples of ab initio methods are the Hartree-Fock[44] and Density Functional Theory
(DFT)[39]. Hartree-Fock method is Based on Linear Combination of Atomic Orbitals (LCAO)
and uses the muffin-tin atom orbital as the method. Several other methods based on LCAO can
be employed. The DFT method projects the system of interacting particles into a system of non-
interacting particles. Ab initio methods are useful for looking into properties of new materials,
such as the currently popular graphene material. QuantumWise[l] is an example of a program that
uses ab initio methods for simulations. It will be shown that empirical methods can also be used
for exciting materials without being as bulky as first principle based methods.
Some common empirical methods are k-p and EPM. k-p takes advantage of effective mass
theory in its calculations. EPM is the topic of this paper and its applications and limits will be


2
discussed further in the following sections.
1.1 Device Simulation
Semiconductor device simulation can be grouped into three basic categories: Classical, Semi-
classical and Quantum. The manufacturing process for semiconductor devices requires that logic
and electrical performances be simulated long before they go into production. Most simulators
use classical techniques with parameters belted on to account for special situations. The classical
approach is still used because it enables complex devices to be simulated quickly. Semi-classical
approaches have evolved as a way of engineering, studying and even revealing parameters for
classical simulations in new materials. In all of these methods the Poisson equation is used to
describe the electrostatic part of the system,
eSip = q(n p C) (1.1)
where the space charge density, p = q(n p C) consists of the charge of an electron q multiplied
by the difference of the number of electron (n), hole (p) carriers and the concentration of charges C.
Each method takes moments of the Boltzmann equation and simplifies them to self-consistently
solve the system. The Drift-Diffusion model is the simplest and it uses current continuity to solve
both static devices and slow changes in current. The model focuses on two mechanisms, namely
drift of charge carriers due to an external electric field and diffusion caused by a gradient in charge
carrier concentration.
The major simplification of Drift-Diffusion is that it assumes that lattice temperature and
charge carrier temperature is equal. However, this is almost never the case. The Hydrodynamic
model is somewhat more complex but accounts for the constant change in temperature inherent in
all devices.
Semi-classical simulations involve calculating the density of states of the system (and therefore
the band structure) and applying it the moments of the Boltzmann Transport Equation. For
whichever assumptions and simplifications that are made the equations will be solve self consistently


3
with the Poisson equation.
Finally, The Wigner function approach removes all ballistic, classical properties of the system
by replacing the Poisson equation with the Wigner-Boltzmann equation.
1.2 A Breif History of Pseudopotentials and Band Structure Calculation
Since core electrons in the inner shells of atoms are strongly bound, they do not play a signif-
icant part in the electronic and optical properties. The Philips Kleinman cancellation theorem[37]
creates a smooth wave function to represent the core energies and potentials by taking advantage
of crystal symmetry, and allows for simple calculation of wave energies. The previous method pro-
posed by Herring in 1940 (and still of importance when core electrons must be taken into account)
involved orthogalization of each plane wave creating a linear combination of core orbitals[28]. This
method was successful in improving accuracy for specific applications, but was admittedly labo-
rious. Spherical symmetry of the core was lost due to signalizing a single plane wave to the core
functions, and this limited its use to metallic structures. In 1953, Slater effectively solved the sym-
metry issue by using Bloch waves to represent the periodicity of a crystal. The final step to what
we call EPM was to smooth the contribution of the core atoms by using an effective potential. The
secondary benefit of doing this is that molecules can also be modeled in the system. [37] discusses
the possibility of modifying the repulsive potential to reproduce the free-atom term values. This
ironically hints at the powerful adaptability of the model.


4
Figure 1.1: A graphical representation of pseudopotential simplification


5
2. Empirical Pseudopotential Method
2.1 EPM Proof
Herrings Orthogonalized Plane Wave (OPW) method is the starting point for EPM. By
assuming s or p atomic symmetry 4/fc is constructed to be orthogonal to the core states,
= (pk y ' (2-i)
t
where ak^ are the orthogal coefficients and 4>k,t are the core wave functions. We seek a solution
that gives pk as a smooth function. Using the Schrodinger equation,
Hmk = Emk (2.2)
and substituting (2.1) into (2.2) we get,
Hpk + dktt(E Ef)(j)k>t = Epk (2-3)
t
Introducing
W = y ' Et)(f)ktt/pk (2.4)
t
where Vr represents the short-range repulsive potential; we have the desired form to find the
smooth function,
(H + Vr)pk = Epk (2-5)
By adding the Hamiltonian operator,
2
H=^- + Vc (2.6)
2 m
where Vc represents the core potential. Inserting (2.6) into (2.5), we get the eigenvalue problem,
2
(w---h Vc + Vr)pk = Eipk
(2.7)


6
It can be seen from (2.7) that the core and the short-range potentials can be added together to equal
a pseudopotential Vp = Vc V.n which is known as the Phillips-Kleinman cancellation theorem.
The eigenvalue energy of (2.7) is the actual energy of the crystal wave and tpk can be thought
of as the pseudo-wave function.
The actual contributions of each orbital can be added up through LCAO to create Vp, but his-
torically EPM used potentials measured empirically. Modern methods use other known properties
to converge to useful pseudopotentials for different applications or materials.
2.2 Method for Band Structure Calculation
Solving the Schrodinger equation is all that is needed to find the band structure of a particular
material. Unfortunately, this is a nontrivial exercise. We will focus on discreet points along the
Brillouin Zone (BZ). Starting with the Time Independent Schrodinger Equation (TISE),
[-^v2 + U(r)]*(k, r) = E(k)Â¥(k, r) (2.8)
Both the wave function and the potential must be descritized. Since the wave function is periodic,
it can be written in Bloch form and as a series,
^(k, r) = eikruk(r) = eikr A/leiKhr (2.9)
h
where k is the crystal wave vector, r is the position and -Uk(r) is a periodic function with the same
periodicity as the crystal. Similarly the potential is periodic,
U(r) = ^ UmeiKmr (2.10)
m
By substituting (2.9) and (2.10) into (2.8) we arrive at a nearly discrete function,
^ ^|k + Kh|2Ahei(k+Kh)r + ££. Ahei(k+Kh+Km)r = E(k) Ahei(k+Kh)r (2.11)
h m h h
To take advantage of orthogonality (2.11) is multiplied by e^k+Kh)r and integrated to give,
2 rri
T> + Kh\2 Ah5h,i + ££ VmAh5m,i-h
h m h
E(k)Y,Ah6h'i
h
(2.12)


7
where Sh,i are Kronecker delta functions. In other words,
5h,i
'
1 if h = l
0 if h ^ l
(2.13)
This can be simplified further since h = l only happens once for the first term of the left-hand side,
h times for the second term and once for the right-hand side,
\k + Kh\2Ah + Y,VmAh = E(k)Ah (2.14)
h
(2.14) is an eigenvalue problem. All that is needed to find E{k) is to construct the Hamiltonian
and solve for the eigen values,
Hi
1,3
-|k + Kh| Sij-\-V
2m
(2.15)
where,
V = Vs cos (Km t) + iVA sin (Km r)
(2.16)
and,
d 1 C ,
T = 0Wl) (2'17)
(2.16) accounts for two atoms per lattice point where the maximum offset is r. Vs are form factors
and are measured experimentally. The actual code starts by creating two tables used during the
actual calculation process. The first table contains all of the k values. The second table is the
precalculated bottom half of the Hamiltonian matrix. The eigen solver library[24] only needs the
lower half if the matrix is Hermitian. Since these values do not change for different k, they can be
precalculated. Creating these tables reduces the execution time and simplifies the creation of the
Hamiltonian matrix in the main execution loop.
Once these tables are created, the main execution loop is started. The diagonals for a given
k are calculated and added to the the template H matrix. After the Hamiltonian is built, the eigen
library finds the eigenvalues for a given k. It should be noted the the number of points used for
k determines the resolution for the Brillouin Zone, while the number of K points determines the
accuracy for the Energies of Eigen Values.


8
Table 2.1: Form factors of Si, Ge, Sn, GaP and GaAs at 300K from [11]
a(A) Vss Vus v4A Vn A
Si 5.43 -0.21 0.04 0.08 0 0 0
Ge 5.56 -0.23 0.01 0.06 0 0 0
Sn 6.49 -0.20 0 0.04 0 0 0
GaP 5.44 -0.22 0.03 0.07 0.12 0.07 0.02
Get As 5.64 -0.23 0.01 0.06 0.07 0.05 0.01
AlSb 6.13 -0.21 0.02 0.06 0.06 0.04 0.02
InP 5.86 -0.23 0.01 0.06 0.07 0.05 0.01
GaSb 6.12 -0.22 0.00 0.05 0.06 0.05 0.01
InAs 6.04 -0.22 0.00 0.05 0.08 0.05 0.03
IsSb 6.04 -0.20 0.00 0.04 0.06 0.05 0.01
ZnS 5.41 -0.22 0.03 0.07 0.24 0.14 0.04
ZnSe 5.65 -0.23 0.01 0.06 0.18 0.12 0.03
ZnTe 6.07 -0.22 0.00 0.05 0.13 0.10 0.01
CdTe 6.41 -0.20 0.00 0.04 0.15 0.09 0.04
2.3 Band Structure of Common Semiconductors
A calculation replicating the work of [11] was made to verify the validity of the code. Figures
2.1 through 2.14 show the results of program.
The average run time for the program was approximately 1 minute and 10 seconds. This
includes calculating the eigen energies for 70, 124 x 124 Hamiltonian matrices or about 10 seconds
per lattice point. For materials such as Si or Ge where VA values are 0 the calculation reduces to
about 1 second per lattice point. For both cases 98% of these times were spent in the eigen library
calculating energies.
2.4 Finding Pseudopotentials
Ab initio techniques to find band structures can be used, but this can be both time and
memory intensive. It is sometimes desirable to calculate band structure inside of a simulation
for certain applications. Examples of this include studying hetero-structures with different doping
gradients [13], Strained Latices [26] or studying the effects of surface roughness [22] in a device.
EPM is a powerful method for calculating band structure and studying the effects, and there


relative energy [eV] relative energy [eV]
Si Bandstructure
Figure 2.1: Si band structure with form factors from [11]
Ge Bandstructure
Figure 2.2: Ge band structure with form factors from [11]


relative energy [eV] relative energy [eV]
10
Sn Bandstructure
Figure 2.3: Sn band structure with form factors from [11]
GaP Bandstructure
Figure 2.4: GaP band structure with form factors from [11]


relative energy [eV] relative energy [eV]
11
GaAs Bandstructure
Figure 2.5: GaAs band structure with form factors from [11]
AlSb Bandstructure
Figure 2.6: AlSb band structure with form factors from [11]


relative energy [eV] relative energy [eV]
12
InP Bandstructure
Figure 2.7: InP band structure with form factors from [11]
GaSb Bandstructure
Figure 2.8: GaSb band structure with form factors from [11]


relative energy [eV] relative energy [eV]
13
InAs Bandstructure
Figure 2.9: InAs band structure with form factors from [11]
InSb Bandstructure
Figure 2.10: InSb band structure with form factors from [11]


relative energy [eV] relative energy [eV]
14
ZnS Bandstructure
1 1 1 c
1 /
i
L G X U,K G
wave vector, k
Figure 2.11: ZnS band structure with form factors from [11]
ZnSe Bandstructure
\ 7 7 1 1
y

1 1 1
L
G
X U,K
wave vector, k
G
Figure 2.12: ZnSe band structure with form factors from [11]


relative energy [eV] relative energy [eV]
15
InP Bandstructure
Figure 2.13: ZeTe band structure with form factors from [11]
CdTe Bandstructure
Figure 2.14: CdTe band structure with form factors from [11]


16
are several modern approaches to finding form factors for different applications. First-principles
calculations such as LCAO can be used to find pseudopotentials. Curve fitting and Genetic al-
gorithms can be employed to find fine grain values of band structure where data have already be
acquired [34],


17
3. Applications
Utilizing exciting new materials such as graphene and device structures such as Silicon On
Insulator (SOI) necessitates being being able predict their operation as actual devices. Ab initio
approaches give insight into engineering effective geometries, but they are limited to structures in
the range of thousands of atoms[20]. For any kind of large scale integration these methods fall
short. EPM offers a solution for devices > 105 atoms. There are many techniques for finding
pseudopotentials that will work for different systems. Most of these methods stretch the definition
of an empirical method. Ab initio calculations can be used to find pseudopotentials. Another
procedure that will be shown below is to make a simple hybrid representation of the atomic structure
in the system. Finally, stochastic methods for finding pseudopotentials have become increasingly
popular. Indeed this is no longer empirical, but shows the versatility of EPM for modeling modem
materials. Remarkably, pseudopotentials for armchair nanoribbons of graphene have been used to
calculate band structure[31] and simulate the properties of devices.
There are situations where it becomes desirable to do band calculations inside device simu-
lations where the band structure can change as a result of the operation of the device. This is a
good application for modeling using first principles to discover pseudopotentials that could work
for using EPM in the actual device.
Once band structure is known (either by ab initio or EPM) it can be mapped into effective
mass and/or mobility to lessen the computational burden in actual device simulation. Work has
been carried out to calculate mobility in strained materials by finding the dilation deformation po-
tentials from unstrained materials, and then applying the strained band structure[23] to recalculate
the mobility. Another method is to use Full Band Monte Carlo (FBMC) to calculate the mobility
for specific common geometries[45].


18
3.1 Alloy Materials
First principles techniques are able to make calculations based on specific atom locations.
This is of great use in studying boundary deformations. For some alloy materials there is a
known concentration, and there may be a known strain. For example SiGe with highly con-
trolled concentrations can be epitaxially grown through the use of ultrahigh-vacuum chemical va-
por deposition[33]. There may even be a gradient concentration in the material. Because these
materials are quasi-homogeneous virtual crystal approximation (VCA) allows the properties of two
atoms to be used to make a hybrid atom[26]. For this method the strained lattice and the form
factors are a function of the concentration percentage. The strained lattice can be represented as:
a!(x) = ai(l x) + a,2X (3.1)
Similarly the form factors can be calculated as:
V'{x) = Vi(l-x) + ViX (3.2)
Table 3.1: Form factors of SiGe Alloy for different concentrations at 300K
X a(A) kgs Vus %3A V4A VllA
0.00 5.4300 -0.210 0.0400 0.080 0 0 0
0.25 5.4625 -0.215 0.0325 0.075 0 0 0
0.50 5.4950 -0.220 0.0250 0.070 0 0 0
0.75 5.5275 -0.225 0.0175 0.065 0 0 0
1.00 5.5600 -0.230 0.0100 0.060 0 0 0
Table 3.1 shows the parameters of SiGe alloy at differing concentrations, and figure 3.1 shows
progression of how the band structure changes. The method for calculating the band structure is
the same as above. It can be seen that the top band in figure 3.1(a) starts to pull down in 3.1(b),
and by 3.1(c) at a concentration of x = 50% that same band is below the normal conduction band of
Silicon. At x = 75% the alloy becomes Germanium-like as the direct band gap begins to takeover.


19
3.2 Genetic Algorithms
Genetic algorithms have become highly utilized in many aspects of semiconductor devices.
They have been used to find device parameters[19], in band structure engineering[30], in direct
band gap discovery[14], in reconstruction of Scanning Tunneling Microscope (STM) imagery[10]
and even in optimization of manufacturing devices[50]. They have been used to because of their
ability to hone into special systems by evolving from similar parent states. A detailed overview of
genetic algorithms is given in [12]. Here is the abbreviated description of a genetic algorithm:
(1) Start. An initial population is made. This can be either a random population or a
population that already has desired traits.
(2) Breeding. Breeding takes place in three steps:
(a) Selection. Parents are selected at random and weighted by their fitness and Copies.
(b) Crossover. Children exchange chromosomes.
(c) Mutation. Each chromosome has the chance to mutate by some probability.
(3) Repeat. Breed until a desired trait is optimized to the selected quantity or by a predeter-
mined number of generations.
There are many variations on this system. One of its main advantages is that many generations
can be done in parallel.
Recently, genetic algorithms applied to EPM has been used to calculate the band structure
for 4H-SiC [34],
3.3 Future Work
Future work will be focused on applying the band structures to actual device simulation. A
framework for this purpose needs to be developed. Spin-orbit, coupling is necessary for effectively
simulating holes in many materials such a germanium and is an easy addition to the algorithm.


20
A density of states calculation is also necessary. Using a genetic algorithm find different devices
properties and simulate operation is the end goal.


21
Si Bandstructure
SiGe25 Bandstructure
(a) x = 0%
(b) x = 25%
(c) x = 50%
(d) x = 75%
Ge Bandstructure
(e) x = 100%
Figure 3.1: The Band Structure for Different Concentrations of Silicon-Germanium Alloy


22
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[25] Salvador Gonzalez. Empirical pseudopotential method for the band structure calculation of
strauned-slilicon germanium materials. Masters thesis, Arizona State University, 2001.
[26] Salvador Gonzalez, Dragica Vasileska, and AlexanderA. Demkov. Empirical pseudopotential
method for the band structure calculation of strained-silicon germanium materials. Journal of
Computational Electronics, 1(1-2):179-183, 2002.
[27] Suyog Gupta, Blanka Magyari-KApe, Yoshio Nishi, and Krishna C. Saraswat. Achieving
direct band gap in germanium through integration of sn alloying and external strain. Journal
of Applied Physics, 113(7), 2013.
[28] Conyers Herring. A new method for calculating wave functions in crystals. Phys. Rev., 57:1169-
1177, Jun 1940.
[29] Jiseok Kim and Massimo V. Fischetti. Empirical pseudopotential calculations of the band
structure and ballistic conductance of strained [001], [110], and [111] silicon nanowires. Journal
of Applied Physics, 110(3):-, 2011.


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25
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conductivity of silicon by nanostructure patterning. Applied Physics A, 110(1):93 98, 2013.


26
A. Acronyms
BZ Brillouin Zone..................................................................6
DFT Density Functional Theory.......................................................1
EPM Empirical Pseudopotential Method
FBMC Full Band Monte Carlo.........................................................17
LCAO Linear Combination of Atomic Orbitals...........................................1
OPW Orthogonalized Plane Wave.......................................................5
SOI Silicon On Insulator...........................................................17
STM Scanning Tunneling Microscope..................................................19
TISE Time Independent Schrodinger Equation...........................................6
VCA virtual crystal approximation..................................................18


27
B. Code Listing and Examples
It is necessary to have the Eigen Library[24] to compile and run the simulator. To compile
(using the gnu compiler) and run a simulation type (from a UNIX shell):
l g++ -o epm epm.cpp
./epm < Si.epm > Si.dat
The Si.epm file is a parameter file used to deliver the lattice constant, form factors and the lattice
traversal info to the program. It is a very roughly parsed into the simulator.
resources/Si.epm
LC 5.431e-10 ; lattice constant
V 3 -0.21 0 ; Vs3 Va3
3 V 8 0.04 0
V 11 0.08 0
L 20 G 20 X 10 U 0 K 20 G ; crystal points to traverse
LC is a keyword to tell the program that the lattice constant follows after some amount of white
space. V tells it that a form factor follows. If L, G, X, U, K or G starts a line, the parser the
will build the k vector and will calculate the number of points specified inside the Brillouin zone.
If 0 is used, the program will still calculate one point from the reciprocal lattice.
B.l EPM Band Structure in C-|\-
resources/epm.cpp
#include
#include
#include
#include "Eigen/Dense


5
10
15
20
25
30
35
40
28
using namespace std;
using namespace Eigen;
// constants
#define mO
#define q
#define hbar
#define hbareV
#define pi
#define nN 124
double dot(double *A, double *B) {
return A[0]*B[0] + A[1]*B[1] + A[2]*B[2];
}
int main(int argc char *argv[]) {
double T [3] = { 0.125, 0.125, 0.125 >;
int Nn = 124;
double k[Nn] [3] ;
double K[nN*2+l] [3] ;
double H[nN][nN];
complex V;
double i,j,1 [3];
int dd,d,e,f,g,h;
double *p;
typedef Matrix Matrixl
typedef Matrix, 124, 124
Matrixl24cd HH;
char 1ine [256] ;
char *tok;
double lc ; // lattice constant
double C; // constant used for building H
int ppc = 0; // to keep track of the number of pseudopotentials
// 20 pseudopotentials max
// structure factor
// k vectors
// lookup table for K vectors
// Hamaltonian
// psudopotential
// generic counters
24d ;
> Matrixl24cd;
9.109e-31 // free electron mass
1.602 e-19 // carge of an electron
1.054e-34 // Plank *s constant
6.581e-16
3.1415926535898
int
pp[20] ;


45
50
55
60
65
70
75
29
double Vs [20] ;
double Va [20] ;
// get parameters: lattice constant pseudopotentials and lattice traversal
Nn = 0;
while ( c in get 1 ine (1 ine 256) ) {
tok = strtok(line "u") ;
if(!tok) break;
// get lattice constant
if(!strcmp(tokLC ")) {
lc = at of ( st rt ok (NULL u")) ;
C = hbar*hbar*2*pi*pi/(lc*lc*m0*q);
>
// get pseudopotential values
if (! strcmp (tok "V") && ppc<20) {_
pp[ppc] = at of(strtok(NULL u")) ;
Vs[ppc] = atof(strtok(NULLu")) ;
Va[ppc++] = atof(strtok(NULL,"u"));
>
// get and generate k matrix
if(!strcmp(tok "L")) { k[0] [0] = .5; k [0] [1] = .5; k [0] [2] = 5 ; Nn = 1; >
if(!strcmp(tok,"G")) { k[0] [0] = 0; k [0] [1] = 0; k [0] [2] = 0; Nn = 1; >
if(!strcmp(tok,"X")) { k [0] [0] = 1; k [0] [1] = 0; k[0] [2] = 0; Nn = 1; >
if(!strcmp(tok,"W")) { k [0] [0] = 1; k [0] [1] = .5; k[0] [2] = 0; Nn = 1; >
if(!strcmp(tok,"K")) { k [0] [0] = 75; k [0] [1] = 75 ; k[0] [2] = 0; Nn = 1; >
if(!strcmp(tok,"U")) { k [0] [0] = 1; k [0] [1] = 25 ; k [0] [2] = 25 ; Nn = 1; >
if(Nn) {
Nn=0;
tok = strtok(NULL,"u");
while(isdigit(*tok)) {
h = atoi(tok);
tok = strtok(NULLu") ;
if(h==0) h=l;
if(!strcmp(tok,"L")) { 1[0]= (.5-k [Nn] [0])/h; 1[1]= (.5-k [Nn] [1])/h; 1[2]= (.5-
k[Nn][2])/h; >


80
85
90
95
100
105
110
if(!strcmp(tok, " G" )) { 1 [0] = (0-k [Nn] [0] ) /h; 1 [1] = (0-k[Nn][l])/h;
k[Nn][2])/h; >
if(!strcmp(tok, " X" )) { 1 [0] = (1 -k [Nn] [0])/h; 1 [1] = (0-k[Nn][l])/h;
k[Nn][2])/h; >
if(!strcmp(tok," W" )) { 1 [0] = (1 -k[Nn] [0])/h; 1 [1] = (.5-k[Nn][l])/h;
k[Nn][2])/h; >
if(!strcmp(tok, " K" )) { 1 [0] = ( .75-k[Nn][0])/h; 1[1]=( . 75-k[Nn] [l])/h;
k[Nn][2])/h; >
if(!strcmp(tok, " U" )) { 1 [0] = (1-k[Nn][0])/h; 1[1]=( .25-k[Nn][l])/h;
k[Nn][2])/h; >
if (-le-15 if (-le-15 if (-le-15 while(h-- kk (Nn<124)) {
Nn++;
k[Nn][0] = k [Nn-1] [0] + 1 [0] ;
k [Nn] [1] = k[Nn-l] [1] + 1 [1] ;
k [Nn] [2] = k [Nn-1] [2] + 1 [2] ;
if(-le-15 if (-le-15 if(-le-15 >
tok = strtok(NULL,"u");
}
Nn++;
}
>
// verify pseudopotentials
//f=ppc; iuhil e (f--) cout << pp [f] << " << Vs [f] << " << Va[f] << endl;
// verify k vector generation
// for(h=0; h // cout << endl;
// build a look up table for building K vector
// caution/ this section must be modified if nN is modified
// needs to be updated for variable nN
f = -5;


115
120
125
130
135
140
145
31
g = 0;
h=l;
p=&K [0] [0] ;
while(l) {
*p++ = -f+g+h;
*p++ = f-g+h;
*p++ = f+g-h;
if(f==5 && g==0 && h==-l) break;
if(++h>2) {
h= -2;
if(++g>2) {
g=-2;
f ++ ;
>
>
>
// verify K vector generation
//print/ ("\n"); for(h=0; h<24-9; h++) printf("%i %g %g %g\n", h-124, K [h] [0] K[h] [1] K [h
] [2]);
// generate Hamiltonian with out diagonals
for(f=0; f // calculate pseudopotential
h = f+124-g;
// search pseudopotential table
V = 0;
d = ppc;
e = (int)dot(&K[h] [0] ,&K[h] [0]) ;
while(d--) if(e == pp[d]) {
// V = Vs [d] 13 6059;
V.real(Vs[d]*13.6059);
V.imag(Va[d]*13.6059);
break;
}
// V = V* cos (2* pi dot (&K [h] [0] ,&T [0] ) ) ;
V.real(real(V)*cos(2*pi*dot(&K[h][0],&T[0])));


150
155
160
165
170
32
V.imag(imag(V)*sin(2*pi*dot(&K[h][0],&T[0])));
HH(f,g) = V;
>
// create diagonals for each k space and find energies !
for(f=0; f // calculate diagonals
for(g=0; g h = g + 62;
1 [0] = k[f ] [0] + K [h] [0] ;
1 [1] = k[f] [1] + K [h] [1] ;
1 [2] = k[f ] [2] + K [h] [2] ;
// H[g][g] = C* dot (&l [0] ,&l [0] ) ;
HH(g,g) = C*dot(&1 [0] ,&1 [0]) ;
}
SelfAdjointEigenSolver eigensolver(HH);
cout << eigensolver.eigenvalues().transpose() << endl;
>
// varify full Hamiltonian
//for(f=0; f biock(f,g,1,1) << " << endl;
B.2 Gnuplot Example
[38] was used to generate the plots in this paper. Which can be done by:
gnuplot Si.gp
resources/Si.gp
set key off
set title "SiuBandstructure\n"
set xlabel "waveuvector,uk"
4 set ylabel "relativeuenergyu[eV]


33
unset x2tics
set grid x
set yr [-10:10]
set xtics ("L" 0 "G' 20, "X" 40 , "U K" 51
9 plot "Si.dat" us ing ($1-10.54) with lines
replot "Si.dat" us ing ($2-10.54) with 1 ines
replot "Si.dat" us ing ($3-10.54) with 1 ines
replot "Si.dat" us ing ($4-10.54) with 1 ines
replot "Si.dat" us ing ($5-10.54) with 1 ines
14 replot "Si.dat" us ing ($6-10.54) with 1 ines
replot "Si.dat" us ing ($7-10.54) with 1 ines
replot "Si.dat" us ing ($8-10.54) with 1 ines
pause -1
set term postscript eps enhanced
19 set output "Si.eps"
"Si.dat" us ing ($1-10.54) with lines , "Si.dat" using ($2-10. 54) with lines , "Si.dat"
using ($3-10 . 54) with lines , "Si.dat" using ($4-10.54) with 1 ines , "Si.dat" using ($5
-10.54) with lines "Si.dat " using ($6 -10.54) with lines, "Si . dat " using ($7 -10.54) with
lines "Si . dat" using ($8- 10.54) with 1 ines


Full Text

PAGE 1

EMPIRICALPSEUDOPOTENTIALMETHODFORMODERN ELECTRONSIMULATION by ADAMLEESTRICKLAND B.S.,UniversityofColoradoDenver,2007 Athesissubmittedtothe FacultyoftheGraduateSchoolofthe UniversityofColoradoinpartialfulllment oftherequirementsforthedegreeof MasterofScience ElectricalEngineering 2014

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ii ThisthesisfortheMasterofSciencedegreeby AdamLeeStrickland hasbeenapprovedfortheElectricalEngineeringProgramby HamidZ.Fardi,Chair MilojeRadenkovic Jae-DoPark April30,2014

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iii Strickland,AdamLeeM.S.ElectricalEngineering EmpiricalPseudopotentialMethodforModernElectronSimulation ThesisdirectedbyProfessorHamidZ.Fardi ABSTRACT Thetopicofthisthesisistodevelopaframeworkforndingcriticalvaluesthatcanbeusedin simulatingmodernsemiconductordevicestructures.Manycurrentsimulationtoolsuseeective massapproximationstomodelthepropagationofelectronsthroughadevice.Becausemodern devicestakeadvantageofdierentmaterials,geometriesandsizes,abandstructurecalculation becomesnecessaryforrepresentingeectssuchasstrainedmaterialsandhotcarriers.Whilehighly accurate,calculation-intensemethodsexist,abandstructuremayneedtoberecalculatedon-the-y creatingtheneedforafast,simplealgorithmthatcanbetailoredforthesituationathand. Fortunately,thereisaverywellestablishedmethodforcalculatingbandstructurebased onphysicalpropertiesofthematerial.Historically,thesepropertiesweremeasured,butmodern techniquesinvolveusingalgorithmstoconvergetothepropervalues.Thisallowsthesimulationof deviceswithinterestingpropertieswhilemaintainingrelativelysimplemodels. Theformandcontentofthisabstractareapproved.Irecommenditspublication. Approved:HamidZ.Fardi

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iv CONTENTS Chapter 1Introduction...............................1 1.1DeviceSimulation...........................2 1.2ABreifHistoryofPseudopotentialsandBandStructureCalculation......3 2EmpiricalPseudopotentialMethod......................5 2.1EPMProof.............................5 2.2MethodforBandStructureCalculation..................6 2.3BandStructureofCommonSemiconductors................8 2.4FindingPseudopotentials........................8 3Applications...............................17 3.1AlloyMaterials............................18 3.2GeneticAlgorithms..........................19 3.3FutureWork.............................19 Bibliography 22 Appendix AAcronyms................................26 BCodeListingandExamples.........................27 B.1EPMBandStructureinC++......................27 B.2GnuplotExample...........................32

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v TABLES Table 2.1FormfactorsofSi,Ge,Sn,GaPandGaAsat300Kfrom[11].........8 3.1FormfactorsofSiGeAlloyfordierentconcentrationsat300K.........18

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vi FIGURES Figure 1.1Agraphicalrepresentationofpseudopotentialsimplication..........4 2.1Sibandstructurewithformfactorsfrom[11]................9 2.2Gebandstructurewithformfactorsfrom[11]...............9 2.3Snbandstructurewithformfactorsfrom[11]...............10 2.4GaPbandstructurewithformfactorsfrom[11]...............10 2.5GaAsbandstructurewithformfactorsfrom[11]..............11 2.6AlSbbandstructurewithformfactorsfrom[11]...............11 2.7InPbandstructurewithformfactorsfrom[11]...............12 2.8GaSbbandstructurewithformfactorsfrom[11]..............12 2.9InAsbandstructurewithformfactorsfrom[11]...............13 2.10InSbbandstructurewithformfactorsfrom[11]...............13 2.11ZnSbandstructurewithformfactorsfrom[11]...............14 2.12ZnSebandstructurewithformfactorsfrom[11]...............14 2.13ZeTebandstructurewithformfactorsfrom[11]...............15 2.14CdTebandstructurewithformfactorsfrom[11]..............15 3.1TheBandStructureforDierentConcentrationsofSilicon-GermaniumAlloy..21

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1 1.Introduction Knowingthebandstructureofamaterialisimportantforrelatingitsopticalandelectronic properties.Asnewandexoticmaterialsarestudied,thereisagreaterneedforenergyband calculations.Specialphysicalpropertiescanbestudiedwithoutbuildingdevices.Bandstructure calculationscanbeusedtoengineercustombandstructurestotailoropticalorelectricaldevices. Thoughcalculationofbandstructureisvaluableinitsownright,inthispaperthefocuswill beonitsimportancewithregardtosemiconductorsimulation.Inordertounderstand,designand modelnewdevicesitisimperativetoknowthelimitationsandapplicationsofdierentmethods. Forcalculatingbandstructuretherearetwobasicapproaches:abinitioandempiricalmethods. Abinitio,orfromthebeginning"methodsinvolvecalculationofbandstructurebyuseofrstprincipleswithoutusingmeasureddata.Empiricalmethodstakeadvantageofexperimentaldata togivemoreaccuratebandstructurerepresentation.Generally,abinitiomethodsarecalculation intensivebutgivebetterinsightonhowthestructurearrives.Bothmethodshavetheirplacein calculatingandengineeringtransportpropertiesinsemiconductordevices. ExamplesofabinitiomethodsaretheHartree-Fock[44]andDensityFunctionalTheory DFT[39].Hartree-FockmethodisBasedonLinearCombinationofAtomicOrbitalsLCAO andusesthemun-tinatomorbitalasthemethod.SeveralothermethodsbasedonLCAOcan beemployed.TheDFTmethodprojectsthesystemofinteractingparticlesintoasystemofnoninteractingparticles.Abinitiomethodsareusefulforlookingintopropertiesofnewmaterials, suchasthecurrentlypopulargraphenematerial.QuantumWise[1]isanexampleofaprogramthat usesabinitiomethodsforsimulations.Itwillbeshownthatempiricalmethodscanalsobeused forexcitingmaterialswithoutbeingasbulkyasrstprinciplebasedmethods. Somecommonempiricalmethodsarek pandEPM.k ptakesadvantageofeectivemass theoryinitscalculations.EPMisthetopicofthispaperanditsapplicationsandlimitswillbe

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2 discussedfurtherinthefollowingsections. 1.1DeviceSimulation Semiconductordevicesimulationcanbegroupedintothreebasiccategories:Classical,SemiclassicalandQuantum.Themanufacturingprocessforsemiconductordevicesrequiresthatlogic andelectricalperformancesbesimulatedlongbeforetheygointoproduction.Mostsimulators useclassicaltechniqueswithparametersbeltedontoaccountforspecialsituations.Theclassical approachisstillusedbecauseitenablescomplexdevicestobesimulatedquickly.Semi-classical approacheshaveevolvedasawayofengineering,studyingandevenrevealingparametersfor classicalsimulationsinnewmaterials.InallofthesemethodsthePoissonequationisusedto describetheelectrostaticpartofthesystem, = q n )]TJ/F23 10.9091 Tf 10.909 0 Td [(p )]TJ/F23 10.9091 Tf 10.909 0 Td [(C .1 wherethespacechargedensity, = q n )]TJ/F23 10.9091 Tf 10.732 0 Td [(p )]TJ/F23 10.9091 Tf 10.732 0 Td [(C consistsofthechargeofanelectron q multiplied bythedierenceofthenumberofelectron n ,hole p carriersandtheconcentrationofcharges C Eachmethodtakesmoments"oftheBoltzmannequationandsimpliesthemtoself-consistently solvethesystem.TheDrift-Diusionmodelisthesimplestanditusescurrentcontinuitytosolve bothstaticdevicesandslowchangesincurrent.Themodelfocusesontwomechanisms,namely driftofchargecarriersduetoanexternalelectriceldanddiusioncausedbyagradientincharge carrierconcentration. ThemajorsimplicationofDrift-Diusionisthatitassumesthatlatticetemperatureand chargecarriertemperatureisequal.However,thisisalmostneverthecase.TheHydrodynamic modelissomewhatmorecomplexbutaccountsfortheconstantchangeintemperatureinherentin alldevices. Semi-classicalsimulationsinvolvecalculatingthedensityofstatesofthesystemandtherefore thebandstructureandapplyingitthemomentsoftheBoltzmannTransportEquation.For whicheverassumptionsandsimplicationsthataremadetheequationswillbesolveselfconsistently

PAGE 9

3 withthePoissonequation. Finally,TheWignerfunctionapproachremovesallballistic,classicalpropertiesofthesystem byreplacingthePoissonequationwiththeWigner-Boltzmannequation. 1.2ABreifHistoryofPseudopotentialsandBandStructureCalculation Sincecoreelectronsintheinnershellsofatomsarestronglybound,theydonotplayasignificantpartintheelectronicandopticalproperties.ThePhilipsKleinmancancellationtheorem[37] createsasmoothwavefunctiontorepresentthecoreenergiesandpotentialsbytakingadvantage ofcrystalsymmetry,andallowsforsimplecalculationofwaveenergies.ThepreviousmethodproposedbyHerringin1940andstillofimportancewhencoreelectronsmustbetakenintoaccount involvedorthogalizationofeachplanewavecreatingalinearcombinationofcoreorbitals[28].This methodwassuccessfulinimprovingaccuracyforspecicapplications,butwasadmittedlylaborious.Sphericalsymmetryofthecorewaslostduetosignalizingasingleplanewavetothecore functions,andthislimiteditsusetometallicstructures.In1953,SlatereectivelysolvedthesymmetryissuebyusingBlochwavestorepresenttheperiodicityofacrystal.Thenalsteptowhat wecallEPMwastosmooththecontributionofthecoreatomsbyusinganeectivepotential.The secondarybenetofdoingthisisthatmoleculescanalsobemodeledinthesystem.[37]discusses thepossibilityofmodifyingtherepulsivepotentialtoreproducethefree-atomtermvalues.This ironicallyhintsatthepowerfuladaptabilityofthemodel.

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4 Figure1.1:Agraphicalrepresentationofpseudopotentialsimplication

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5 2.EmpiricalPseudopotentialMethod 2.1EPMProof Herring'sOrthogonalizedPlaneWaveOPWmethodisthestartingpointforEPM.By assuming s or p atomicsymmetry k isconstructedtobeorthogonaltothecorestates, k = k + X t a k;t k;t .1 where a k;t aretheorthogalcoecientsand k;t arethecorewavefunctions.Weseekasolution thatgives k asasmoothfunction.UsingtheSchrodingerequation, H k = E k .2 andsubstituting.1into.2weget, H' k + X t a k;t E )]TJ/F23 10.9091 Tf 10.909 0 Td [(E t k;t = E' k .3 Introducing V r = X t a k;t E )]TJ/F23 10.9091 Tf 10.91 0 Td [(E t k;t =' k .4 where V r representstheshort-rangerepulsivepotential;wehavethedesiredformtondthe smooth"function, H + V r k = E' k .5 ByaddingtheHamiltonianoperator, H = p 2 2 m + V c .6 where V c representsthecorepotential.Inserting.6into.5,wegettheeigenvalueproblem, p 2 2 m + V c + V r k = E' k .7

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6 Itcanbeseenfrom.7thatthecoreandtheshort-rangepotentialscanbeaddedtogethertoequal apseudopotential V p = V c + V r ,whichisknownasthePhillips-Kleinmancancellationtheorem. Theeigenvalueenergyof.7istheactualenergyofthecrystalwave k .and k canbethought ofasthepseudo-wavefunction. TheactualcontributionsofeachorbitalcanbeaddedupthroughLCAOtocreate V p ,buthistoricallyEPMusedpotentialsmeasuredempirically.Modernmethodsuseotherknownproperties toconvergetousefulpseudopotentialsfordierentapplicationsormaterials. 2.2MethodforBandStructureCalculation SolvingtheSchrodingerequationisallthatisneededtondthebandstructureofaparticular material.Unfortunately,thisisanontrivialexercise.Wewillfocusondiscreetpointsalongthe BrillouinZoneBZ.StartingwiththeTimeIndependentSchrodingerEquationTISE, [ )]TJ/F35 10.9091 Tf 11.884 7.38 Td [(~ 2 2 m r 2 + V r ] k ; r = E k k ; r .8 Boththewavefunctionandthepotentialmustbedescritized.Sincethewavefunctionisperiodic, itcanbewritteninBlochformandasaseries, k ; r = e i kr u k r = e i kr X h A h e i K h r .9 where k isthecrystalwavevector, r isthepositionand u k r isaperiodicfunctionwiththesame periodicityasthecrystal.Similarlythepotentialisperiodic, V r = X m V m e i K m r .10 Bysubstituting.9and.10into.8wearriveatanearlydiscretefunction, ~ 2 2 m X h j k + K h j 2 A h e i k + K h r + X m X h V m A h e i k + K h + K m r = E k X h A h e i k + K h r .11 Totakeadvantageoforthogonality.11ismultipliedby e i k + K h r andintegratedtogive, ~ 2 2 m X h j k + K h j 2 A h h;l + X m X h V m A h m;l )]TJ/F24 7.9701 Tf 6.586 0 Td [(h = E k X h A h h;l .12

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7 where h;l areKroneckerdeltafunctions.Inotherwords, h;l = 8 > > > < > > > : 1if h = l 0if h 6 = l .13 Thiscanbesimpliedfurthersince h = l onlyhappensonceforthersttermoftheleft-handside, h timesforthesecondtermandoncefortheright-handside, ~ 2 2 m j k + K h j 2 A h + X h V m A h = E k A h .14 .14isaneigenvalueproblem.Allthatisneededtond E k istoconstructtheHamiltonian andsolvefortheeigenvalues, H i;j = ~ 2 2 m j k + K h j 2 i;j + V .15 where, V = V S cos K m + iV A sin K m .16 and, = a 1 8 ; 1 8 ; 1 8 .17 .16accountsfortwoatomsperlatticepointwherethemaximumosetis V S areformfactors andaremeasuredexperimentally.Theactualcodestartsbycreatingtwotablesusedduringthe actualcalculationprocess.Thersttablecontainsallofthekvalues.Thesecondtableisthe precalculatedbottomhalfoftheHamiltonianmatrix.Theeigensolverlibrary[24]onlyneedsthe lowerhalfifthematrixisHermitian.Sincethesevaluesdonotchangefordierentk,theycanbe precalculated.Creatingthesetablesreducestheexecutiontimeandsimpliesthecreationofthe Hamiltonianmatrixinthemainexecutionloop. Oncethesetablesarecreated,themainexecutionloopisstarted.Thediagonalsforagiven karecalculatedandaddedtothethetemplateHmatrix.AftertheHamiltonianisbuilt,theeigen libraryndstheeigenvaluesforagivenk.Itshouldbenotedthethenumberofpointsusedfor kdeterminestheresolutionfortheBrillouinZone,whilethenumberofKpointsdeterminesthe accuracyfortheEnergiesofEigenValues.

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8 Table2.1:FormfactorsofSi,Ge,Sn,GaPandGaAsat300Kfrom[11] a A V 3 S V 8 S V 11 S V 3 A V 4 A V 11 A Si 5.43 -0.21 0.04 0.08 0 0 0 Ge 5.56 -0.23 0.01 0.06 0 0 0 Sn 6.49 -0.20 0 0.04 0 0 0 GaP 5.44 -0.22 0.03 0.07 0.12 0.07 0.02 GaAs 5.64 -0.23 0.01 0.06 0.07 0.05 0.01 AlSb 6.13 -0.21 0.02 0.06 0.06 0.04 0.02 InP 5.86 -0.23 0.01 0.06 0.07 0.05 0.01 GaSb 6.12 -0.22 0.00 0.05 0.06 0.05 0.01 InAs 6.04 -0.22 0.00 0.05 0.08 0.05 0.03 IsSb 6.04 -0.20 0.00 0.04 0.06 0.05 0.01 ZnS 5.41 -0.22 0.03 0.07 0.24 0.14 0.04 ZnSe 5.65 -0.23 0.01 0.06 0.18 0.12 0.03 ZnTe 6.07 -0.22 0.00 0.05 0.13 0.10 0.01 CdTe 6.41 -0.20 0.00 0.04 0.15 0.09 0.04 2.3BandStructureofCommonSemiconductors Acalculationreplicatingtheworkof[11]wasmadetoverifythevalidityofthecode.Figures 2.1through2.14showtheresultsofprogram. Theaverageruntimefortheprogramwasapproximately1minuteand10seconds.This includescalculatingtheeigenenergiesfor70,124 124Hamiltonianmatricesorabout10seconds perlatticepoint.FormaterialssuchasSiorGewhere V A valuesare0thecalculationreducesto about1secondperlatticepoint.Forbothcases98%ofthesetimeswerespentintheeigenlibrary calculatingenergies. 2.4FindingPseudopotentials Abinitiotechniquestondbandstructurescanbeused,butthiscanbebothtimeand memoryintensive.Itissometimesdesirabletocalculatebandstructureinsideofasimulation forcertainapplications.Examplesofthisincludestudyinghetero-structureswithdierentdoping gradients[13],StrainedLatices[26]orstudyingtheeectsofsurfaceroughness[22]inadevice. EPMisapowerfulmethodforcalculatingbandstructureandstudyingtheeects,andthere

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9 Figure2.1:Sibandstructurewithformfactorsfrom[11] Figure2.2:Gebandstructurewithformfactorsfrom[11]

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10 Figure2.3:Snbandstructurewithformfactorsfrom[11] Figure2.4:GaPbandstructurewithformfactorsfrom[11]

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11 Figure2.5:GaAsbandstructurewithformfactorsfrom[11] Figure2.6:AlSbbandstructurewithformfactorsfrom[11]

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12 Figure2.7:InPbandstructurewithformfactorsfrom[11] Figure2.8:GaSbbandstructurewithformfactorsfrom[11]

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13 Figure2.9:InAsbandstructurewithformfactorsfrom[11] Figure2.10:InSbbandstructurewithformfactorsfrom[11]

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14 Figure2.11:ZnSbandstructurewithformfactorsfrom[11] Figure2.12:ZnSebandstructurewithformfactorsfrom[11]

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15 Figure2.13:ZeTebandstructurewithformfactorsfrom[11] Figure2.14:CdTebandstructurewithformfactorsfrom[11]

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16 areseveralmodernapproachestondingformfactorsfordierentapplications.First-principles calculationssuchasLCAOcanbeusedtondpseudopotentials.CurvettingandGeneticalgorithmscanbeemployedtondnegrainvaluesofbandstructurewheredatahavealreadybe acquired[34].

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17 3.Applications UtilizingexcitingnewmaterialssuchasgrapheneanddevicestructuressuchasSiliconOn InsulatorSOInecessitatesbeingbeingablepredicttheiroperationasactualdevices.Abinitio approachesgiveinsightintoengineeringeectivegeometries,buttheyarelimitedtostructuresin therangeofthousandsofatoms[20].Foranykindoflargescaleintegrationthesemethodsfall short.EPMoersasolutionfordevices & 10 5 atoms.Therearemanytechniquesfornding pseudopotentialsthatwillworkfordierentsystems.Mostofthesemethodsstretchthedenition ofanempirical"method.Abinitiocalculationscanbeusedtondpseudopotentials.Another procedurethatwillbeshownbelowistomakeasimplehybridrepresentationoftheatomicstructure inthesystem.Finally,stochasticmethodsforndingpseudopotentialshavebecomeincreasingly popular.Indeedthisisnolongerempirical,butshowstheversatilityofEPMformodelingmodern materials.Remarkably,pseudopotentialsforarmchairnanoribbonsofgraphenehavebeenusedto calculatebandstructure[31]andsimulatethepropertiesofdevices. Therearesituationswhereitbecomesdesirabletodobandcalculationsinsidedevicesimulationswherethebandstructurecanchangeasaresultoftheoperationofthedevice.Thisisa goodapplicationformodelingusingrstprinciplestodiscoverpseudopotentialsthatcouldwork forusingEPMintheactualdevice. OncebandstructureisknowneitherbyabinitioorEPMitcanbemappedintoeective massand/ormobilitytolessenthecomputationalburdeninactualdevicesimulation.Workhas beencarriedouttocalculatemobilityinstrainedmaterialsbyndingthedilationdeformationpotentialsfromunstrainedmaterials,andthenapplyingthestrainedbandstructure[23]torecalculate themobility.AnothermethodistouseFullBandMonteCarloFBMCtocalculatethemobility forspeciccommongeometries[45].

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18 3.1AlloyMaterials Firstprinciplestechniquesareabletomakecalculationsbasedonspecicatomlocations. Thisisofgreatuseinstudyingboundarydeformations.Forsomealloymaterialsthereisa knownconcentration,andtheremaybeaknownstrain.ForexampleSiGewithhighlycontrolledconcentrationscanbeepitaxiallygrownthroughtheuseofultrahigh-vacuumchemicalvapordeposition[33].Theremayevenbeagradientconcentrationinthematerial.Becausethese materialsarequasi-homogeneousvirtualcrystalapproximationVCAallowsthepropertiesoftwo atomstobeusedtomakeahybridatom[26].Forthismethodthestrainedlatticeandtheform factorsareafunctionoftheconcentrationpercentage.Thestrainedlatticecanberepresentedas: a 0 x = a 1 )]TJ/F23 10.9091 Tf 10.909 0 Td [(x + a 2 x .1 Similarlytheformfactorscanbecalculatedas: V 0 x = V 1 )]TJ/F23 10.9091 Tf 10.909 0 Td [(x + V 2 x .2 Table3.1:FormfactorsofSiGeAlloyfordierentconcentrationsat300K x a A V 3 S V 8 S V 11 S V 3 A V 4 A V 11 A 0.00 5.4300 -0.210 0.0400 0.080 0 0 0 0.25 5.4625 -0.215 0.0325 0.075 0 0 0 0.50 5.4950 -0.220 0.0250 0.070 0 0 0 0.75 5.5275 -0.225 0.0175 0.065 0 0 0 1.00 5.5600 -0.230 0.0100 0.060 0 0 0 Table3.1showstheparametersofSiGealloyatdieringconcentrations,andgure3.1shows progressionofhowthebandstructurechanges.Themethodforcalculatingthebandstructureis thesameasabove.Itcanbeseenthatthetopbandingure3.1astartstopulldownin3.1b, andby3.1cataconcentrationof x =50%thatsamebandisbelowthenormalconductionbandof Silicon.At x =75%thealloybecomesGermanium-likeasthedirectbandgapbeginstotakeover.

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19 3.2GeneticAlgorithms Geneticalgorithmshavebecomehighlyutilizedinmanyaspectsofsemiconductordevices. Theyhavebeenusedtonddeviceparameters[19],inbandstructureengineering[30],indirect bandgapdiscovery[14],inreconstructionofScanningTunnelingMicroscopeSTMimagery[10] andeveninoptimizationofmanufacturingdevices[50].Theyhavebeenusedtobecauseoftheir abilitytohoneintospecialsystemsbyevolvingfromsimilarparentstates.Adetailedoverviewof geneticalgorithmsisgivenin[12].Hereistheabbreviateddescriptionofageneticalgorithm: Start. Aninitialpopulationismade.Thiscanbeeitherarandompopulationora populationthatalreadyhasdesiredtraits. Breeding. Breedingtakesplaceinthreesteps: a Selection. ParentsareselectedatrandomandweightedbytheirtnessandCopies. b Crossover. Childrenexchangechromosomes. c Mutation. Eachchromosomehasthechancetomutatebysomeprobability. Repeat. Breeduntiladesiredtraitisoptimizedtotheselectedquantityorbyapredeterminednumberofgenerations. Therearemanyvariationsonthissystem.Oneofitsmainadvantagesisthatmanygenerations canbedoneinparallel. Recently,geneticalgorithmsappliedtoEPMhasbeenusedtocalculatethebandstructure for4H-SiC[34]. 3.3FutureWork Futureworkwillbefocusedonapplyingthebandstructurestoactualdevicesimulation.A frameworkforthispurposeneedstobedeveloped.Spin-orbitcouplingisnecessaryforeectively simulatingholesinmanymaterialssuchagermaniumandisaneasyadditiontothealgorithm.

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20 Adensityofstatescalculationisalsonecessary.Usingageneticalgorithmnddierentdevices' propertiesandsimulateoperationistheendgoal.

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21 a x =0% b x =25% c x =50% d x =75% e x =100% Figure3.1:TheBandStructureforDierentConcentrationsofSilicon-GermaniumAlloy

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22 BIBLIOGRAPHY [1]Quantumwiseatomistixtoolkitatk. [2]WandaAndreoniandR.Car.Similarityofga,al,asalloysandultrathinheterostructures: Electronicpropertiesfromtheempiricalpseudopotentialmethod. Phys. Rev. B,21:3334{3344, Apr1980. [3]GabrielBester.Electronicexcitationsinnanostructures:anempiricalpseudopotentialbased approach. Journal of Physics: Condensed Matter,21:023202,2009. [4]S.BloomandT.K.Bergstresser.Bandstructureof^ -sn,insbandcdteincludingspin-orbit eects. Solid State Communications,6:465{467,1968. [5]PBowlan,EMartinez-Moreno,KReimann,MWoerner,andTElsaesser.Terahertzradiative couplinganddampinginmultilayergraphene. New Journal of Physics,16:013027,2014. [6]MadsBrandbyge,Jose-LuisMozos,PabloOrdejon,JeremyTaylor,andKurtStokbro.Densityfunctionalmethodfornonequilibriumelectrontransport. Phys. Rev. B,65:165401,Mar2002. [7]J.Chelikowsky,D.J.Chadi,andMarvinL.Cohen.Calculatedvalence-banddensitiesof statesandphotoemissionspectraofdiamondandzinc-blendesemiconductors. Phys. Rev. B, 8:2786{2794,Sep1973. [8]JamesR.ChelikowskyandMarvinL.Cohen.Electronicstructureofsilicon. Phys. Rev. B, 10:5095{5107,Dec1974. [9]JamesR.ChelikowskyandMarvinL.Cohen.Nonlocalpseudopotentialcalculationsforthe electronicstructureofelevendiamondandzinc-blendesemiconductors. Phys. Rev. B,14:556{ 582,Jul1976. [10]F.C.Chuang,C.V.Ciobanu,V.B.Shenoy,C.Z.Wang,andK.M.Ho.Findingthereconstructionsofsemiconductorsurfacesviaageneticalgorithm. Surface Science,573:L375{L381, 2004. [11]MarvinL.CohenandT.K.Bergstresser.Bandstructuresandpseudopotentialformfactorsfor fourteensemiconductorsofthediamondandzinc-blendestructures. Phys. Rev.,141:789{796, Jan1966. [12]E.A.B.Cole.Overviewofdevicemodelling.In Mathematical and Numerical Modelling of Heterostructure Semiconductor Devices: From Theory to Programming,pages3{20.Springer London,2009. [13]RichardA.Coles. Theory of the Electronic States of Semiconductor Heterostructures.PhD thesis,DurhamUniversity,March1999. [14]Mayeuld'Avezac,Jun-WeiLuo,ThomasChanier,andAlexZunger.Genetic-algorithmdiscoveryofadirect-gapandopticallyallowedsuperstructurefromindirect-gapsiandgesemiconductors. Phys. Rev. Lett.,108:027401,Jan2012.

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23 [15]A.DeandCraigE.Pryor.Predictedbandstructuresofiii-vsemiconductorsinthewurtzite phase. Phys. Rev. B,81:155210,Apr2010. [16]StephenM.GoodnickDragicaVasileska,editor. Nano-Electronic Devices: Semiclassical and Quantum Transport Modeling.SpringerScience+BusinessMedia,2011. [17]N.D.Drummond,V.Zolyomi,andV.I.Fal'ko.Electricallytunablebandgapinsilicene. Phys. Rev. B,85:075423,Feb2012. [18]AnielloEsposito. Band Structure Eects and Quantum Transport.PhDthesis,ETHZURICH, 2010. [19]FangFeng-boandWuTao.Geneticalgorithmandsemiconductordevicemodelparameter extraction.In Genetic and Evolutionary Computing, 2009. WGEC '09. 3rd International Conference on,pages97{100,Oct2009. [20]MassimoVFischetti,JiseokKim,SudarshanNarayanan,Zhun-YongOng,CatherineSachs, DavidKFerry,andShelaJAboud.Pseudopotential-basedstudiesofelectrontransportin grapheneandgraphenenanoribbons. Journal of Physics: Condensed Matter,25:473202, 2013. [21]MassimoV.FischettiandStevenE.Laux.Montecarloanalysisofelectrontransportin smallsemiconductordevicesincludingband-structureandspace-chargeeects. Phys. Rev. B, 38:9721{9745,Nov1988. [22]MassimoV.FischettiandSudarshanNarayanan.Anempiricalpseudopotentialapproachto surfaceandline-edgeroughnessscatteringinnanostructures:Applicationtosithinlmsand nanowiresandtographenenanoribbons. Journal of Applied Physics,110:{,2011. [23]M.V.FischettiaandS.E.Laux. Band structure, deformation potentials, and carrier mobility in strained Si, Ge, and SiGe alloys,volume80.J.Appl.Phys.,1996. [24]etalGelGuennebaud,BenoitJacob.Eigenv3,2010. [25]SalvadorGonzalez.Empiricalpseudopotentialmethodforthebandstructurecalculationof strauned-slilicongermaniummaterials.Master'sthesis,ArizonaStateUniversity,2001. [26]SalvadorGonzalez,DragicaVasileska,andAlexanderA.Demkov.Empiricalpseudopotential methodforthebandstructurecalculationofstrained-silicongermaniummaterials. Journal of Computational Electronics,1-2:179{183,2002. [27]SuyogGupta,BlankaMagyari-K ~ Ape,YoshioNishi,andKrishnaC.Saraswat.Achieving directbandgapingermaniumthroughintegrationofsnalloyingandexternalstrain. Journal of Applied Physics,1137,2013. [28]ConyersHerring.Anewmethodforcalculatingwavefunctionsincrystals. Phys. Rev.,57:1169{ 1177,Jun1940. [29]JiseokKimandMassimoV.Fischetti.Empiricalpseudopotentialcalculationsoftheband structureandballisticconductanceofstrained[001],[110],and[111]siliconnanowires. Journal of Applied Physics,1103:{,2011.

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24 [30]KwiseonKim,PeterA.Graf,andWesleyB.Jones.Ageneticalgorithmbasedinverseband structuremethodforsemiconductoralloys. Journal of Computational Physics,208:735{ 760,2005. [31]YoshiyukiKurokawa,ShintaroNomura,TadashiTakemori,andYoshinobuAoyagi.Largescalecalculationofopticaldielectricfunctionsofdiamondnanocrystallites. Phys. Rev. B, 61:12616{12619,May2000. [32]RitaMagri.Pseudopotentialsforbandstructurecalculations.TMCSIII,January2012. [33]BernardS.Meyerson,KevinJ.Uram,andFrancoiseK.LeGoues.Cooperativegrowthphenomenainsilicon/germaniumlowtemperatureepitaxy. Applied Physics Letters,53:2555{2557, 1988. [34]G.Ng,D.Vasileska,andD.K.Schroder.Empiricalpseudopotentialbandstructureparameters of4h-sicusingageneticalgorithmttingroutine. Superlattices and Microstructures,49:109 {115,2011. [35]M.AliOmar. Elementry Solid State Physis.Addison-WesleyPublishingCompany,Inck,1993. [36]R.H.Parmenter.Symmetrypropertiesoftheenergybandsofthezincblendestructure. Phys. Rev.,100:573{579,Oct1955. [37]JamesC.PhillipsandLeonardKleinman.Newmethodforcalculatingwavefunctionsin crystalsandmolecules. Phys. Rev.,116:287{294,Oct1959. [38]JeRacine.gnuplot4.0:Aportableinteractiveplottingutility. Journal of Applied Econometrics,21:133{141,January/February2006. [39]I.N.RemediakisandEfthimiosKaxiras.Band-structurecalculationsforsemiconductorswithin generalized-density-functionaltheory. Phys. Rev. B,59:5536{5543,Feb1999. [40]MartinM.RiegerandP.Vogl.Electronic-bandparametersinstrainedsubstrates. Phys. Rev. B,48:14276{14287,Nov1993. [41]MarkSilver. Application of the Pseudopotential Method to the Theory of Serniconductors. PhDthesis,UniversityofSurrey,1991. [42]J.C.Slater.Anaugmentedplanewavemethodfortheperiodicpotentialproblem. Phys. Rev., 92:603{608,Nov1953. [43]D.L.SmithandC.Mailhiot.Theoryofsemiconductorsuperlatticeelectronicstructure. Rev. Mod. Phys.,62:173{234,Jan1990. [44]A.Svane.Hartree-fockband-structurecalculationswiththelinearmun-tin-orbitalmethod: Applicationtoc,si,ge,and-sn. Phys. Rev. B,35:5496{5502,Apr1987. [45]EnzoUngersboeck,SiddharthaDhar,GerhardKarlowatz,ViktorSverdlov,HansKosina,and SiegfriedSelberherr. The Eect of General Strain on the Band Structure and Electron Mobility of Silicon,volume54.IEEETransactionsonElectronDevices,Sep2007. [46]DavidVanderbilt.Theoryofpseudopotentials.Technicalreport,July2006.

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25 [47]DragicaVasileska.Empiricalpseudopotentialmethod. [48]DragicaVasileska.Tutorialforsemi-empiricalband-structurecalculation. [49]P.Vogl,HaroldP.Hjalmarson,andJohnD.Dow.Asemi-empiricaltight-bindingtheoryofthe electronicstructureofsemiconductors^a. Journal of Physics and Chemistry of Solids,44:365 {378,1983. [50]Cheng-ShuoWangandRehaUzsoy.Ageneticalgorithmtominimizemaximumlatenessona batchprocessingmachine. Computers and Operations Research,29:1621{1640,2002. [51]Y.W.Wen,H.J.Liu,L.Pan,X.J.Tan,H.Y.Lv,J.Shi,andX.F.Tang.Reducingthethermal conductivityofsiliconbynanostructurepatterning. Applied Physics A,110:93{98,2013.

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26 A.Acronyms BZ BrillouinZone..........................................................................6 DFT DensityFunctionalTheory..............................................................1 EPM EmpiricalPseudopotentialMethod FBMC FullBandMonteCarlo................................................................17 LCAO LinearCombinationofAtomicOrbitals.................................................1 OPW OrthogonalizedPlaneWave.............................................................5 SOI SiliconOnInsulator...................................................................17 STM ScanningTunnelingMicroscope........................................................19 TISE TimeIndependentSchrodingerEquation................................................6 VCA virtualcrystalapproximation..........................................................18

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27 B.CodeListingandExamples ItisnecessarytohavetheEigenLibrary[24]tocompileandrunthesimulator.Tocompile usingthegnucompilerandrunasimulationtypefromaUNIXshell: 1 g++-oepmepm.cpp ./epmSi.dat TheSi.epmleisaparameterleusedtodeliverthelatticeconstant,formfactorsandthelattice traversalinfototheprogram.Itisaveryroughlyparsedintothesimulator. resources/Si.epm LC5.431e-10;latticeconstant V3-0.210;Vs3Va3 3 V80.040 V110.080 L20G20X10U0K20G;crystalpointstotraverse LC"isakeywordtotelltheprogramthatthelatticeconstantfollowsaftersomeamountofwhite space.V"tellsitthataformfactorfollows.IfL,G,X,U,KorGstartsaline,theparserthe willbuildthe k vectorandwillcalculatethenumberofpointsspeciedinsidetheBrillouinzone. If0isused,theprogramwillstillcalculateonepointfromthereciprocallattice. B.1EPMBandStructureinC++ resources/epm.cpp #include #include #include #include"Eigen/Dense"

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28 5 usingnamespacestd; usingnamespaceEigen; //constants 10 #definem09.109e-31 //freeelectronmass #defineq1.602e-19 //cargeofanelectron #definehbar1.054e-34 //Plank'sconstant #definehbareV6.581e-16 #definepi3.1415926535898 15 #definenN124 doubledotdouble*A,double*B{ returnA[0]*B[0]+A[1]*B[1]+A[2]*B[2]; 20 } intmainintargc,char*argv[]{ doubleT[3]={0.125,0.125,0.125}; //structurefactor 25 intNn=124; doublek[Nn][3]; //kvectors doubleK[nN*2+1][3]; //lookuptableforKvectors doubleH[nN][nN]; //Hamaltonian complexV; //psudopotential 30 doublei,j,l[3]; //genericcounters intdd,d,e,f,g,h; double*p; typedefMatrixMatrix124d; typedefMatrix,124,124>Matrix124cd; 35 Matrix124cdHH; charline[256]; char*tok; doublelc; //latticeconstant 40 doubleC; //constantusedforbuildingH intppc=0; //tokeeptrackofthenumberofpseudopotentials intpp[20]; //20pseudopotentialsmax

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29 doubleVs[20]; doubleVa[20]; 45 //getparameters:latticeconstant,pseudopotentialsandlatticetraversal Nn=0; whilecin.getlineline,256{ tok=strtokline,""; 50 if!tokbreak; //getlatticeconstant if!strcmptok,"LC"{ lc=atofstrtokNULL,""; 55 C=hbar*hbar*2*pi*pi/lc*lc*m0*q; } //getpseudopotentialvalues if!strcmptok,"V"&&ppc<20{ 60 pp[ppc]=atofstrtokNULL,""; Vs[ppc]=atofstrtokNULL,""; Va[ppc++]=atofstrtokNULL,""; } 65 //getandgeneratekmatrix if!strcmptok,"L"{k[0][0]=.5;k[0][1]=.5;k[0][2]=.5;Nn=1;} if!strcmptok,"G"{k[0][0]=0;k[0][1]=0;k[0][2]=0;Nn=1;} if!strcmptok,"X"{k[0][0]=1;k[0][1]=0;k[0][2]=0;Nn=1;} if!strcmptok,"W"{k[0][0]=1;k[0][1]=.5;k[0][2]=0;Nn=1;} 70 if!strcmptok,"K"{k[0][0]=.75;k[0][1]=.75;k[0][2]=0;Nn=1;} if!strcmptok,"U"{k[0][0]=1;k[0][1]=.25;k[0][2]=.25;Nn=1;} ifNn{ Nn=0; tok=strtokNULL,""; 75 whileisdigit*tok{ h=atoitok; tok=strtokNULL,""; ifh==0h=1; if!strcmptok,"L"{l[0]=.5-k[Nn][0]/h;l[1]=.5-k[Nn][1]/h;l[2]=.5k[Nn][2]/h;}

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30 80 if!strcmptok,"G"{l[0]=0-k[Nn][0]/h;l[1]=0-k[Nn][1]/h;l[2]=0k[Nn][2]/h;} if!strcmptok,"X"{l[0]=1-k[Nn][0]/h;l[1]=0-k[Nn][1]/h;l[2]=0k[Nn][2]/h;} if!strcmptok,"W"{l[0]=1-k[Nn][0]/h;l[1]=.5-k[Nn][1]/h;l[2]=0k[Nn][2]/h;} if!strcmptok,"K"{l[0]=.75-k[Nn][0]/h;l[1]=.75-k[Nn][1]/h;l[2]=0k[Nn][2]/h;} if!strcmptok,"U"{l[0]=1-k[Nn][0]/h;l[1]=.25-k[Nn][1]/h;l[2]=.25k[Nn][2]/h;} 85 if-1e-15
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31 g=0; h=1; 115 p=&K[0][0]; while1{ *p++=-f+g+h; *p++=f-g+h; *p++=f+g-h; 120 iff==5&&g==0&&h==-1break; if++h>2{ h=-2; if++g>2{ 125 g=-2; f++; } } } 130 //verifyKvectorgeneration //printf"n";forh=0;h<249;h++printf"%i%g%g%gn",h-124,K[h][0],K[h][1],K[h ][2]; //generateHamiltonianwithoutdiagonals forf=0;f
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32 150 V.imagimagV*sin2*pi*dot&K[h][0],&T[0]; HHf,g=V; } //creatediagonalsforeachkspaceandfindenergies! 155 forf=0;feigensolverHH; cout<
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33 unsetx2tics setgridx setyr[-10:10] setxtics"L"0,"G"20,"X"40,"U,K"51,"G"71 9 plot"Si.dat"using$1-10.54withlines replot"Si.dat"using$2-10.54withlines replot"Si.dat"using$3-10.54withlines replot"Si.dat"using$4-10.54withlines replot"Si.dat"using$5-10.54withlines 14 replot"Si.dat"using$6-10.54withlines replot"Si.dat"using$7-10.54withlines replot"Si.dat"using$8-10.54withlines pause-1 settermpostscriptepsenhanced 19 setoutput"Si.eps" plot"Si.dat"using$1-10.54withlines,"Si.dat"using$2-10.54withlines,"Si.dat" using$3-10.54withlines,"Si.dat"using$4-10.54withlines,"Si.dat"using$5 -10.54withlines,"Si.dat"using$6-10.54withlines,"Si.dat"using$7-10.54with lines,"Si.dat"using$8-10.54withlines