ABSTRACT
Name: Wesley J. Fabella Department: Physics
Title: Ab-Initio Supercell Calculation of an Isolated Neutral Silicon Vacancy for the
Investigation of the Properties Relating to Deep Centers
Major: Physics Degree: Master of Science
Approved by: Date:
___________________________ ________________________Thesis Director
NORTHERN ILLINOIS UNIVERSITY
ABSTRACT
The imperfections of materials, commonly called defects or impurities,
become more important as the size of electronic devices continues to decrease. The
vacancy constitutes a defect in which its properties affect the local atomic
environment; the perturbations about the vacancy result in electronic states that lie
within the bandgap of semiconductors. In the present study a single vacancy was
introduced into the bulk of crystal silicon. The local structural perturbations and
electronic properties produced by the isolated neutral silicon vacancy is the basis for
this thesis. The charge associated with the vacancy is neutral and the vacancy image
interaction is negligible. This is an ab-initio supercell calculation of the properties
associated with the vacancy. In the process of calculation, the forces due to local
perturbations are relaxed, or minimized. The results of this study show that the
relaxation leads to calculated distortion in the initial point group symmetry of the
vacancy, reduction in volume of the vacancy site including nearest neighbor atomic
variations, total energy, defect energy, and local density of states. In addition, it is
shown that the density of states becomes delocalized farther from the vacancy site;
the vacancy-induced states reduce to nearly bulk-like properties, indicating a
dependence on the distance, or range, of the states from the vacancy site.
NORTHERN ILLINOIS UNIVERSITY
AB-INITIO SUPERCELL CALCULATION OF AN ISOLATED NEUTRAL
SILICON VACANCY FOR INVESTIGATION OF THE PROPERTIES
RELATING TO DEEP CENTERS
A THESIS SUBMITTED TO THE GRADUATE SCHOOL
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE
MASTER OF SCIENCE
DEPARTMENT OF PHYSICS
BY
WESLEY JAMES FABELLA
DEKALB, ILLINOIS
AUGUST 2005
Certification: In accordance with departmental and Graduate School
policies, this thesis is accepted in partial fulfillment of
degree requirements.
________________________________________
Thesis Director
________________________________________
Date
ACKNOWLEDGEMENTS
A special thanks to all who have helped me complete the present thesis. First
of all, I would like to thank my graduate research advisor, Prof. Yasuo Ito, and the
graduate committee members, Prof. Dennis Brown and Prof. Michel van Veenendaal
in preparation of this manuscript. Yasuo has assisted me greatly in the
commencement of my research as a graduate student. He has given me the freewill
in order to become skilled at independent research. Michel aided me in the
clarification of my results and insight into the theoretical means of a calculation. I
also thank Yoshie Murooka for the informative conversations on various topics in
experimental physics.
This thesis was supported in part by the Laboratory of Nano-Science,
Engineering, & Technology (LnSET) through the U.S Department of Education
granted by Director Clyde Kimball. Another form of support came from my mother
and father, without whom none of this would have been possible. Also, a special
thanks to my wife and daughter for lending me the time needed in order to
accomplish this step in my academic endeavors.
DEDICATION
To all my family, mother, father, daughter Sofia, wife Michelle, and in loving
memory of my brother Maurice
TABLE OF CONTENTS
Page
LIST OF TABLES ……………………………………………………………... vii
LIST OF FIGURES ……………………………………………………………. viii
LIST OF APPENDICES ………………………………………………………. x
Chapter
I. INTRODUCTION TO THE VACANCY ………………………….. 1
Motivation …………………….……………………………….. 2
Background: Past and Present …………………………………. 6
II. WIEN2K OVERVIEW …………………………………………….. 15
Self-Consistent Field .…………………………………………... 20
Wien2k Tasks …………………………………………………... 21
III. VACANCY CALCULATION …………………………..………… 25
Vacancy Model ………………………………………………… 27
Results ………………………………………………………….. 34
Summary of Results …………………………………….……… 65
IV. DISCUSSION ……………………………………………………… 68
Comparative Works …………………………………………….. 69
V. CONCLUSION ……………………………………………………... 77
vi
Page
REFERENCES ………………………………………………………………….. 80
APPENDICES …………………………………………………………………... 83
LIST OF TABLES
Table Page
1. Amount of D2d Character ………………………………………………. 37
2. Calculated Defect Energy ……………………………………………… 48
3. Atomic Variation During the Relaxation Process ……………….……… 49
4. R Dependence of the Vacancy-Induced States in the Bandgap ………… 65
LIST OF FIGURES
Figure Page
1. Moore’s Law and the Metal-Oxide Semiconductor Field EffectTransistor (MOSFET) …….…………………………………………… 4
2. G. D Watkins’s Model of a Silicon Vacancy (LCAO-MO) …………… 8
3. Wien2k Flow of Programs …………………………………………….. 16
4. Diamond C-K Edge X-ray Emission Spectrum ……………………….. 23
5. Isolated Silicon Vacancy Model (1st NN) ……………………………. 29
6. Energy Level Diagram for Td to D2d Symmetry ………………………. 31
7. Convergence of the 1st NN Forces …………………………………….. 35
8. Vacancy Model of 1st NN Bond Lengths - D2d Character …………….. 38
9. Vacancy Model of 1st NN Bond Angles - D2d Character ……………… 40
10. Volume of the Vacancy ……………………………………………….. 43
11. The Convergence of the Formation Energy …………………………… 46
12. Convergence of Successive xNN Shells ………………………………. 50
13. Density of States for Crystal Silicon ………………………………….. 53
14. Density of States for the Isolated Neutral Silicon Vacancy …………... 55
15. A1 (Bonding) and T2 (Anti-Bonding) Vacancy-Induced States ……….. 58
16. Difference Plot of the Local Density of States - Bandgap Region ……. 60
ix
Figure Page
17. R Dependence on the Vacancy Density of States ……………………… 63
18. D2d Character in 1st NN Bond Lengths ………………………………… 74
LIST OF APPENDICES
Appendix Page
A. DENSITY FUNCTIONAL THEORY - KOHN–SHAM EQUATIONS .. 83
B. THE FULL POTENTIAL METHOD AND BASIS FUNCTIONS OFWIEN2K …………………….………………....….……………………. 86
C. HELLMAN–FEYNMAN THEOREM …………………………………. 90
CHAPTER I
INTRODUCTION TO THE VACANCY
Today a large part of condensed matter research is focused on the
imperfections of materials, commonly called defects or impurities. These become
more important as the size of electronic devices continues to decrease.1 The defects
in materials introduce local structural perturbations that induce electronic bound
states that lie within the forbidden bandgap of semiconductors.2 The vacancy
constitutes a defect in which properties affect the local atomic environment; the
perturbations about the vacancy result in electronic states within the bandgap. In the
present study, a single vacancy was introduced into the bulk of crystal silicon. The
local structural perturbations and electronic properties produced by the isolated
neutral silicon vacancy is the basis for this thesis. The charge associated with the
vacancy is neutral and the vacancy image interaction is negligible. This calculation
is based on a theoretical technique devised by the developers of Wien2k, ab-initio
software based on density functional theory (DFT) for the calculation of material
1 (Packan, 1999, p. 2079)
2 (Watkins, 1986, p. 147)
2
properties.3 The analysis of the properties associated with the vacancy pertains to
calculations done on the relaxation of the vacancy site. These include distortion in
the initial point group symmetry of the vacancy, reduced volume of the vacancy site
including nearest neighbor (NN) atomic variations, the total energy, defect energy,
and (local) density of states ([L]DOS).
Motivation
The properties of silicon (Si) are the most frequently studied because it is the
foremost material used in the semiconductor industry. In result many scientists have
led the way for silicon vacancy research because of its technological significance.4
At the demand of modern electronics, the focus is on scaling down the size of
electronic devices. There is a demand to increase switching speed for data
processing, increase memory storage capacity, and make energy storage devices
more efficient.5 The experimental and theoretical studies of the vacancy are
instrumental to this progression.6, 7
3 (Blaha et al., 2002)
4 (Pantelides, 1986, pp. 1-79)
5 (Schulz, 1999, p. 729)
6 (Freitag et al., 2002, pp. 1-4)
7 (Mueller, Alonso, & Fichtner, 2003, pp. 1-8)
3
The properties of defects are implicated in the functionality of modern
electronic devices. In the metal-oxide field effect transistor (MOSFET), a voltage
applied to the gate terminal regulates the drain-source voltage through the transistor.
The gate-oxide acts as an insulator in order to impede the gate terminal voltage when
desired. The thickness of a SiO2 gate-oxide is to be reached within a decade,
following from Moore’s law shown in Figure 1. At the cost of the potential end to
the use of silicon in electronic devices there is a demand to study electronic
properties approaching this limit, “nanotechnology.” The experimental technique
known as scanning transmission electron microscopy (STEM) measured the electron
energy loss spectrum (EELS) in order to set a limit on the thickness of a silicon
dioxide (SiO2) gate-oxide. The study done by Muller et al. found that the O K and Si
L 2,3 EELS edges produced dominant spectral differences when probing across a gate
stack containing thin SiO2.8 The study established evidence of interfacial states
beyond the gate-oxide interface. These interfacial states were found to be related to
the structural and electronic properties generated via the reduction of oxygen atoms
in SiO2 gate-oxide beyond the SiO2/Si interface. The O-O bonding substantially
decreased beyond the interface, implying evidence that the O atoms were no longer
in pairs. The EEL spectra showed that farther from the gate-oxide the interfacial
states represented the properties of bulk silicon. In order for the gate-oxide to be an
insulating layer, the absolute critical thickness of the gate-oxide was measured to be
7 Å (~ 5 atoms across), 1.2 nm when correcting for interfacial roughness. These
8 (Muller et al., 1999)
Figure 1. Moore’s Law and the Metal-Oxide Semiconductor Field Effect Transistor(MOSFET).9
9 (Intel Technology and Research, 2005)
5
6
results implicated significant structural and electronic dependencies on the properties
of defects at the interface. This work inspired the calculations done in this thesis in
order to simulate the structural and electronic properties of a single isolated defect
like the silicon vacancy. The study also demonstrated that STEM is an ideal
experimental technique for studying isolated defects because it directly probes the
local atomic dimensions and electronic properties on a subnanometer scale.
In this thesis the focus is on an intrinsic impurity, a defect that is native to the
host material, an isolated neutral silicon vacancy. The outcome of this study is also
relevant to the analysis of the properties of an extrinsic impurity. For an extrinsic
impurity one introduces a foreign atom into an atomic site or interstitial; this
effective “doping” induces local charged states.10 The local charged states
introduced in the material depend on the properties of both the host and foreign
atoms. The same fundamental theory applies to charge states. The intrinsic (deep
center) and extrinsic impurities (shallow center) both have electronic states within
the forbidden gap of the semiconductor.
Background: Past and Present
The compilation of research for studying the properties of the vacancy has spanned
over many decades.11 There are various experimental techniques that can be used to
10 (Kaxiras, 2003, p. 325)
11 (Pantelides, 1986, pp. 1-79)
7
study vacancies, including photoluminescence, electron paramagnetic resonance
(EPR), and x-ray spectroscopy.12, 13, 14 G.D. Watkins was one of the primary
investigators studying the silicon vacancy. In the mid-70’s Watkins’s research
focused on radiation damage effects in semiconducting materials. He used EPR
measurements to study the chemical and local atomic properties by detecting
electron spin resonance created by the vacancy-charged states.15 Watkins found that
the vacancy has interesting characteristics: their properties induce electronic states
that lie within the bandgap of commonly know semiconductors. This was referred to
as “deep centers.” Within his EPR studies he showed that the vacancy could exist in
several possible charged states. The familiar notation used for the vacancy (V) and
its charged states is V+, V0, and V- (the vacancy can also be multiply ionized). The
notation for the neutral charged vacancy is V0, characteristic of the initial neutral
charge of an atom. The charged states are like excited states, the final state in optical
absorption or the initial state in optical emission. In accordance, he conceived a
direct model of the silicon vacancy in its neutral and charged states. Figure 2 is
representative of the V+, V0, and V- charged states. It is the neutral charge vacancy,
V0, which is the principal concern of this thesis. Watkins used the single particle
description with a linear combination of atomic orbital – molecular orbital (LCAO-
12 (Choyke, 1971, p. 1843)
13 (Watkins, 1986, pp. 147 –155)
14 (Y. Ito & Y. Murooka, 2005, personal communications)
15 (Watkins, 1976, p. 203)
Figure 2. G. D Watkins’s Model of a Silicon Vacancy (LCAO-MO).16
16 (Watkins, 1986, p. 153)
9
10
MO) treatment. The single particle approach described the vacancy and its charged
states in a simple form. This approach developed the qualitative existence of the
localized electronic states in the bandgap and could be applied to point group
symmetry operations. To this day, the single particle approach still describes the
molecular model of the vacancy. The molecular model of the vacancy treats the
local “dangling bonds” as sp3 hybrids pointing toward the vacancy site. The four
dangling hybrids of the silicon vacancy initially form two states, A1 that is non-
degenerate and T2 states that are 3-fold degenerate under the Td point group
symmetry. Watkins found that in his EPR studies the vacancy splits the degeneracy
of the T2 depending on the structure and charged state of the vacancy; the lifting of
degeneracy is the process related to the Jahn-Teller effect.17 In the vacancy, the
energy degeneracy of the T2 state is sacrificed for orientational distortion, obeying
point group symmetry arguments. Watkins found that the orientational distortion of
the Vo leads to D2d point symmetry. The local perturbations are relaxed, splitting the
T2 state into E and B2 states. For the undistorted Vo, the localized states are referred
to as A1 (singlet, bonding) and T2 (triplet, anti-bonding) states which obey the Td
point symmetry.18 The A1 state is doubly occupied in the valence band and T2 is
purely localized, partially occupied, in the bandgap. More on the point symmetry
and electronic states of the Vo are discussed in the results section of Chapter III.
17 (Watkins, 1986, pp. 147 –155)
18 (Lannoo & Bourgoin, 1981, p. 80)
11
The investigation of defects has led to advances in experimental techniques,
similar to that done by Muller et al. The experimental electronic probing devices
like scanning tunneling microscopy (STM) and STEM have shed incredible insight
into defect research. Specifically this research has implicated applications to
electronic devices. A variation of STM using scanning impedance microscopy by
Freitag and Johnson used carbon nanotubes on a Si/SiO2 substrate to measure the
potential difference as a function of tip-gate position while scanning over the surface
about defects.19 They showed clearly resolved voltage steps at the location of
defects. Concurrently Batson and Bruley used STEM to study EEL difference
spectra between vacancy and bulk sites at the atomic scale around a defect. They
concluded that the spectral difference in pre-Si K-edge was due to vacancy-induced
states.20 Batson improved the atomic resolution of the STEM probe to study the
interface of Si/SiO2/a-Si for vacancy-induced states.21 This paved the way for Muller
and colleagues’ research that looked at the interfacial states in order to estimate the
critical thickness of a SiO2 gate-oxide.
In review of theoretical calculations of the vacancy done by Probert and
Payne, it is apparent that they share the same motivation for studying point defects.
In their words, “It is essential to have a detailed understanding of both the electronic
19 (Freitag et al., 2002)
20 (Batson & Bruley, 1989)
21 (Batson, 1993)
12
and ionic structure of the defect.”22 The theoretical counterpart used to study
material properties is density functional theory (DFT). The formulation of DFT is
found in two articles done by Hohenberg, Kohn, and Sham.23, 24 The single-particle
Kohn-Sham equations are shown in equation 2.1:
[− 2me
∇r2 +V eff (r,ρ(r))]ϕ i (r) = εiϕ i (r) (2.1)
For a short description of the derivation, refer to Appendix A. The main reason for
using DFT is that the total energy can be represented as a functional of the electron
density. Instead of the many-body wavefunction using the Hartree-Fock
approximation, 25 one uses a pseudo-wavefunction (noninteracting fermions) that
represents the electronic density about the atoms, accounting for many-body
interactions (electron correlations) separately. Wien2k uses the linearized
augmented plane wave (LAPW) method to solve equation 2.1, discussed in
Appendix B. In the DFT formalism, the Kohn-Sham equations are solved
computationally under self-consistent field (SCF) cycle, performing a variational
technique over many iterations, minimizing the total energy of the system for the
electron density. With the electron density and total energy one can calculate the
electronic properties of the system. The equation in 2.1 also includes the electron-
correlation potential for the many-body interactions. The local density
22 (Probert & Payne, 2003)
23 (Hohenberg & Kohn, 1964)
24 (Kohn & Sham, 1965)
25 (Slater, 1929)
13
approximation (LDA) is the original treatment for electron correlations in DFT,26
also known as the local spin density approximation (LSDA). The LDA was
improved by taking gradients of the local electronic density; this is known as the
generalized gradient approximation (GGA). The GGA method was devised by
Perdew, Burke, and Ernzerhof.27 Both approximations can be applied within the
Wien2k code. The most popular is the GGA method, used in the present calculation
in this thesis.
The theoretical techniques mentioned are implemented to improve simulation
of the properties of materials, in this case the calculation of an isolated neutral silicon
vacancy and its interaction with the local environment. To simulate the local
environment some theorists have used cluster calculations. In cluster calculations
one has to get rid of boundary states in order to study the localized nature of defects,
which could mean the addition of boundary dopants to absorb surface states.28
Others find that supercell calculations give more desired results. In supercell
calculations one can make the unit cell sufficiently large so that within calculation
the self-interactions of the defect/vacancy image are negligible between periodic
cells, making it more plausible to simulate interfaces and defect sites in the bulk. 29
26 (Kohn & Sham, 1965)
27 (Perdew, Burke, & Ernzerhof, 1996)
28 (Orlando et al., 1996)
29 (Puska et al., 2003)
14
This is a substantial improvement to the molecular model of the vacancy.
The qualitative description of the vacancy is now constructed within a self-consistent
formulation. The Watkins model served as a basis for the vacancy calculation. In
this thesis the Wien2k code is implemented to simulate the simplicity of this model
and illustrate the eminent properties of the silicon vacancy.
CHAPTER II
WIEN2K OVERVIEW
The Wien2k code is an exceptional tool for calculating material properties.
In order to calculate these properties successfully one must be familiar with theory
and application of the code. There are three major processes: first is the
initialization, second is the self-consistent field cycle, and third is the implementation
of the calculated data from the SCF calculation to run specific subroutines including
the calculation of DOS, electron density plots, bandstructure, and optical properties.
The interpretation of the data is an essential part not included in the three-step
process. The effective methodology of a typical Wien2k calculation can be followed
from the flow chart diagram in Figure 3.
This diagram is coordinated such that in the process of the calculation one can
simply follow the chart in order to grasp both the application of Wien2k and
theoretical framework of the code. There are many Wien2k references on DFT and
(L)APW methods, but the paper most recommended is written by Cottenier.30 The
paper contains methods for improving calculations, including adding atomic orbitals
about specific atomic sites to various methods of convergence. Typically elements
like carbon and silicon show modest differences when using optimization methods
30 (Cotteneir, 2002)
Figure 3. Wien2k Flow of Programs.31
31 (Blaha et al., 2002)
17
18
compared to the original data.32 The addition of atomic orbitals is most useful in
situations containing transition metals, not covered within the thesis. There are other
techniques for including spin-orbit coupling, LDA+U, etc. Depending on the
material of choice, one must find the best technique to apply; more information on
various techniques can be found in the textbook resources from Wien2k’s website33.
The first step in the process is the initialization of the calculation. The
general initialization process is performed using either the UNIX-based terminal or
via the w2web graphical interface provided by Wien2k.34 For the remainder of this
section Figure 3 can be referred to for the language used by Wien2k. To initialize a
case file, the crystal structure (space group, P, B, F, etc.) and geometry (lattice
parameters and basis coordinates) have to be known/determined. For documented
materials, lattice constants or structure information may be found in Wyckoff’s
series of volumes 35 or on the web.36, 37 After these values are found, the StructGen
(structure file) can be created, which is the foundation of the calculation, containing
all of the structure data included in the calculation. The order of calculations is
performed sequentially. The initialization starts with the calculation of nearest
32 Calculations were done on crystal diamond and silicon.
33 (Wien2k Textbooks, 2005)
34 (Blaha et al., 2002)
35 (Wyckoff, 1963)
36 (Kroumova et al., 2003)
37 (Center for Computational Materials Science of the United States Naval ResearchLaboratory, 2005)
19
neighbor (NN) distances, which checks for overlapping spheres and equivalency of
atoms. The equivalency of the atoms is determined by the nature of the initial space
group used in StructGen. Once equivalency is determined, a value for the
multiplicity of each atom is obtained and labeled within the StructGen file. The NN
distances will depend on muffin-tin radius (RMT) and lattice constants. Inside the
spheres with RMT lie the core and semicore states. Within these spheres, 99.9% of
the total core charge is contained; everything outside of the RMT is valence and
interstitial states. In general, as a guide, RMT is related to the atomic radii. In some
circumstances one can use the bond radius. If there are a number of different atoms
in a cell it is recommended to keep the RMTs within 20% of one another (50% leads
to divergence or errors).
The next significant part of the initialization is the series of SGROUP and
SYMMETRY operations. The SGROUP calculation determines the best space group
that will define the structure file. This could mean a reduction of atoms representing
the crystal unit cell. It was useful in the initialization of the vacancy calculation,
described in the next chapter. The SYMMETRY element determines the symmetry
operations given for a space group (mirrors, rotations, inversions, etc.). Directly
following these series of calculations is the LSTART command. The LSTART
calculates discrete energy bands (atomic eigenvalues) where the electrons (atomic
densities) lie either in core, semi-core, or valence. It will generate case.inx_st files
that will be the default input values for the following series of initializations. The
default input values for case.inx are usually adequate; there are reference materials
20
that can be found on further optimizing these parameters.38 The default input values
are used for the vacancy calculation; this way a general calculation using Wien2k
could be examined. After input files are generated, a proper kmesh (KGEN) is
inserted, the number of k-points in the first brillouin zone, in general, for metals, a
large kmesh (2000 pts.), and for semiconductors, not so large kmesh (less than
1000). It should be intuitive that for metals more k-points are needed to define more
states that are like free electrons. In semiconductors these states are localized;
therefore, not as many states are needed. Another important factor is the size of the
unit cell; for supercells one can use a smaller kmesh. This is because a supercell can
be quite large; therefore, the reciprocal coordinates are small and do not require a
large number of k-points to sample the brillouin zone. In addition, there is
significant time expense for using a large kmesh. This typically depends on the
number of orbitals cubed. The last part of the initialization is the DSTART
command, which calls info from all of the previous calculations and input files to
determine a proper starting potential to run SCF. Once DSTART is calculated, the
convergence criteria must be chosen before running the SCF cycle.
Self-Consistent Field
In a self-consistent calculation, the criterion for convergence is required. In
most cases energy convergence is the typical parameter to satisfy, but the charge
38 (Wien2k Textbooks, 2005)
21
convergence can also be used. If the energy does not fluctuate much in case.scf
analysis then SCF can be run again with charge convergence criteria. There are
other parameters that can be tested for proper convergence, which are all discussed
by Cottenier.39 If the calculation contains forces in the previous case.scf file, the
structure must be relaxed to determine the final structure/properties. The calculation
is done in conjunction with MINI, using the PORT algorithm to minimize the forces,
implementing the force convergence criteria.40 The PORT method is significant to
the minimization of the forces within the supercell; the atomic positions are adjusted
within PORT such to “relax” (minimize) the forces. The convergence of the forces
for the supercell containing the vacancy is discussed in the results section of the
vacancy calculation.
Wien2k Tasks
The subroutines are implemented following the SCF calculation once
convergence has been satisfied. Within the subroutines one can calculate DOS,
electron density, bandstructure, and much more. For the experimental DOS, Wien2k
uses the Fermi’s Golden Rule method to calculate the occupied and unoccupied
states. The theoretical DOS is calculated using a modified tetrahedron method.41
39 (Cotteneir, 2002)
40 (Blaha et al., 2002, pp. 126)
41 (Blochl, Jepsen, & Andersen, 1994)
22
The electron density is taken straight forward from the calculated charge density that
minimizes the total energy. The next chapter provides the analysis of the isolated
neutral silicon vacancy. The results section of the vacancy calculation includes
analysis of the raw data42 calculated by Wien2k. The user can go beyond the
standard methods of analyzing results while using Wien2k.
The validity of Wien2k varies for each calculation; the results obtained
should be tested against actual physical phenomena whenever possible. For
example, the diamond theoretical C-K x-ray emission spectrum was compared
directly to the experimental diamond C-K edge x-ray emission spectrum shown in
Figure 4. The experimental soft x-ray emission spectrum was obtained from the
high-resolution soft x-ray spectrometer attached to the (S)TEM (JEM2000FX)
housed in the Northern Illinois University Electron Microscopy Lab. The Wien2k
software has proven to be a powerful tool for comparing to experimental results.
42 The data pertaining to an external file via the calculation from Wien2k.
Figure 4. Diamond C-K Edge X-ray Emission Spectrum. (a) Experimental. (b)Theoretical.
24
250 275 300
18000
20000
unregistered
-20 -10 0
Energy (eV)
a.
b.
Coun
tsIn
tens
ity (A
rb. U
nits)
CHAPTER III
VACANCY CALCULATION
The calculation of the material properties of the isolated neutral silicon
vacancy are done using Wien2k, based on DFT and the corresponding LAPW and
APW basis sets found in Appendix B. The exchange correlation is treated using the
GGA method. Following the overview of the Wien2k code one is now familiar with
the flow of programs for calculation to self-consistency. Therefore, before the
results of this calculation are presented the initialization of the supercell containing
the vacancy shall be discussed.
The original silicon unit cell consists of two atoms in FCC structure. The
value of the lattice parameters used is 5.29 Å. The lattice parameters used in the
initial unit cell are within 3% of the experimentally measured 5.44 Å. This unit cell
was used to create a supercell using the “supercell” command in Wien2k. A 4x4x4
128-atom supercell was constructed with an FCC structure. After the supercell was
complete, an atom was removed to create the Si vacancy. With the help of
SGROUP, the number of atoms needed to represent the FCC 127/128-atom supercell
was modified to the final supercell model of the vacancy. The resultant supercell
was adjusted such that there was one vacancy per supercell. The supercell used for
the calculation had a BCC structure, space group #44, with 49/50-atoms. This model
26
is equivalent to the FCC 127/128-atom cell through multiplicity and symmetry.
Wien2k saves vital computational time through useful symmetry operations
implemented in the code. The kmesh used consists of 100 k-points in the first
brillouin zone. This kmesh is suitably dense to describe the properties of the
vacancy. The supercell is 4 times the size of the initial silicon unit cell, therefore
needing approximately 1/4 of the number of k-points to describe the brillouin zone.
Typically a silicon unit cell needs only 250-500 k-points to describe the DOS. The
cell is partitioned into silicon atomic spheres that include one vacancy per supercell.
The RMT = 1.11 Å for each silicon sphere; the value corresponding to the periodic
table is 1.17 Å. In this calculation a slightly smaller sphere size was chosen so that
the atomic spheres do not overlap. The PORT program must efficiently move the
spheres during the relaxation process (minimization of forces). In the construction of
the supercell the vacancy image was separated by 7.48 Å. The periodicity of the
vacancy in the supercell calculation is large enough to separate the vacancy image
with a worthy amount of bulk. This way the vacancy is isolated and the interaction
between vacancy image is negligible. Since the vacancy has such localized
properties, as mentioned previously, it will be deduced in the results section that the
interaction is in fact negligible for this distance separating the periodic vacancy.
This calculation serves as a model to bring insight into the associated
properties of the vacancy during relaxation and the effect of vacancy-induced states
in the bandgap. The vacancy geometry and characterization of the local electronic
properties will be the focus of the vacancy calculation results section. Although it
27
would be difficult to measure such geometries and electronic states using the present
experimental techniques, these characteristics can be modeled using theoretical
methods such as Wien2k. There are various dependencies that introduce
shortcomings to the calculation. These are not limited to but include many of the
initial conditions of the calculation, such as supercell size, symmetry, kmesh, and
methods of calculated forces and relaxation. The results will be presented,
accomplished by portraying some of the most fundamental properties of the vacancy.
Vacancy Model
The results presented in this thesis are a combination of both qualitative and
quantitative aspects of the isolated neutral silicon vacancy calculation. The elegance
of a self-consistent calculation can lead to varying results; the calculation strives to
present the most significant results that were obtained. A supercell model of the
isolated neutral silicon vacancy was constructed using Wien2k. The details of the
construction of this supercell were presented in the previous section. This section
serves to give one a vivid depiction of the fundamental model of the vacancy. The
quantitative results in the next section will be given such that the fundamental
properties known to the vacancy are projected. In summary, the reader should reflect
on a broad view of the results in order to understand the dynamical geometry and
electronic properties of the vacancy.
28
To realize the physical aspects of the vacancy it was helpful to develop a
model of the vacancy exemplary of Watkins’s model. This model is shown in Figure
5. Only the 1st NNs are depicted in the model but in fact there are 2nd NNs and 3rd
NNs to the vacancy site that are considered in the total calculation. Atoms 40a and
40b (19a and 19b) are equivalent; their atomic neighborhoods are the same. The
numbers are used to label the silicon atomic sites used in the Wien2k calculation.
The most distinctive effects of relaxation and DOS are predominantly in the 1st NNs
while the 2nd and 3rd NNs mirror these properties to a significantly less order of
magnitude. Therefore, the model will focus on the 1st NNs in Figure 5, then leave it
towards the end of this section to show some of the properties including the farther
2nd and 3rd NN effects due to the vacancy site.
The model of the vacancy depicts symmetry that permits one to describe the
properties of lattice about the vacancy in terms of symmetry operations. These
operations relate to the symmetry of the lattice sites, specifically about the vacancy,
in terms of various rotations, inversions, diagonals, and mirrors. The model of the
vacancy for Group IV elements is typically described by the Td point group
symmetry in its initial undistorted state. The A1 and T2 states represent the local
point symmetry about the vacancy for the Td point symmetry.
During the relaxation process, the vacancy undergoes a distortion known as
the Jahn-Teller effect. The Jahn Teller distortion results in a splitting of the T2 state
into E and B2, shown in Figure 6, lowering the symmetry from Td to D2d. The point
group symmetry expressed by Td has four sets of rotation axes of three-fold
Figure 5. Isolated Silicon Vacancy Model (1st NN).43
43 (Crystal Maker Software, 2005)
30
Figure 6. Energy Level Diagram for Td to D2d Symmetry.
32
v(A1) = 12
(ψ 10 +ψ 20 +ψ 30 +ψ 40 )
tx (T2 ) = 12
(ψ 10 +ψ 20 −ψ 30 −ψ 40 )
ty (T2 ) = 12
(−ψ 10 +ψ 20 +ψ 30 −ψ 40 )
tz (T2 ) = 12
(ψ 10 −ψ 20 +ψ 30 −ψ 40 )
Splitting of levels dueto coupling of thevacancy-crystalenvironment
Ψ; v, tx , ty , tz (4) B2 (1)
E (2)
D2dTd -->
T2; tx, ty, tz (3)
A1; v (1)
Δ
γ γ
A1 (1)
Splitting of levelsdue to orientationaldistortion
33
symmetry and the D2d character has two-fold axes at right angles to another axis.44
The amount of D2d character in the vacancy is found by distinctive evidence of two
sets of bond lengths. For example, there are a total of six equal sides in the
tetrahedral (Td) point group symmetry. In the D2d symmetry there are two sets of
bond lengths. Set I (2 sides) has equal but longer lengths than set II (4 sides) of
equivalent but shorter lengths than set I. Other evidence of D2d character would exist
in the DOS, and it will be clearly depicted that the T2 state does not split in the
following calculation, leaving only A1 and T2 states reminiscent of the initial Td
point symmetry; therefore, the D2d character calculated is only minutely involved in
the relaxation pertaining to the Wien2k-calculated results.
Similarly to Watkins’s model, the A1 and T2 states are formed from linear
combinations of s, px, py, pz atomic orbitals. The systems of equations are coupled in
order to express the eigenfunctions that form sp3 hybrids representing the vacancy
site; ψ i0 (i=1 to 4) denotes the coupled vacancy hybrids, forming A1 bonding and T2
anti-bonding states.45 The A1 state forms a singlet and is symmetric under symmetry
operations of the tetrahedral point group Td. The T2 state forms a 3-fold degenerate
triplet state under the Td point group.
44 (Kaxiras, 2003, pp. 114-115)
45 (Lannoo & Bourgoin, 1981, p. 83)
34
Results
The simple model of the vacancy (molecular model) is used in order to
project the vacancy’s physical simplicity while as a whole all calculations of the
properties of this vacancy have been done using a large supercell (previous section).
The main focus of the results will be on the relaxation leading to the final structure
of the vacancy site. Then the calculation will deduce the electronic properties from
the resulting relaxed structure. In the first chapter it was mentioned how the creation
of a vacancy affects the initially tetrahedral structure. The effect is the process
(Jahn-Teller effect) that exchanges energy degeneracy for orientational distortion
(relaxation). The degeneracy of the T2 state perturbs the vacancy site, producing
forces in the calculation. In order to correctly describe the vacancy’s properties,
these forces must be relaxed. In effect this relaxation produces an acute amount of
distortion due to the relaxation of the forces about each atom. Wien2k uses the
calculated Hellmann-Feynman forces found in Appendix C to calculate the local
force on each atom within the crystal lattice. The PORT algorithm determines the
atomic positioning of the atoms, effectively minimizing the total energy in order to
relax the forces to self-consistency. The result is a fully relaxed vacancy. The force
was plotted for the 1st NNs during this relaxation; the forces converged to 0.30 eV/Å,
shown in Figure 7. Similarly the 2nd and 3rd NNs also converged to 0.30 eV/Å.
Figure 7. Convergence of the 1st NN Forces.
36
1NN Forces
0.00
2.00
4.00
6.00
8.00
10.00
12.00
0 2 4 6 8 10Iteration
Avg
. Fo
rce o
n 1
NN
s (e
V/
Å)
37
In this calculation the distortion is larger within the 1st NNs, to a lesser order
of magnitude in the 2nd and 3rd NNs. The bond lengths and bond angles had to be
examined in order to investigate the magnitude of the distortion. During the
relaxation the distortion is labeled to be the amount of D2d character introduced into
the initial Td symmetry of the vacancy site. The calculated amount of D2d character
is shown in Figure 8 and Figure 9, included in the vacancy model. It was determined
that this minute amount of D2d character is present in both the bond length and bond
angles. Later it was determined in the DOS calculation that this magnitude of D2d
character does not significantly alter the initial Td symmetry of the vacancy site (the
difference is quoted in Table 1). From the table of calculated differences, the total
amount of D2d character accounts for about 2% of distortion of the Td point
symmetry. Therefore, the final point group symmetry is effectively Td. The minute
distortion in the purely Td point symmetry is not sufficient to split the T2 state.
Table 1
Amount of D2d Character
Bond Lengths (Å) Unrelaxed Relaxed|40a-19b| = |19a-40b| = R13 3.741 3.002|19a-19b| = |40a-40b| = R12 3.741 2.953D2d Character (Difference) 0.00 0.05Bond Angles (deg)A 109.3 111.2B 109.3 108.4D2d Character (Difference) 0 2.8
Figure 8. Vacancy Model of 1st NN Bond Lengths - D2d Character.
39
R12
R13
D2d Character
2.90
3.00
3.10
3.20
3.30
3.40
3.50
3.60
3.70
3.80
0 2 4 6 8 10Iteration
Bo
nd
Len
gth
(Å
)
R12
R13
Figure 9. Vacancy Model of 1st NN Bond Angles – D2d Character.
41
A
B
D2d Character
108.00
108.50
109.00
109.50
110.00
110.50
111.00
111.50
0 2 4 6 8 10Iteration
An
gle
s (
deg
)
AB
42
The minute amount of distortion of the initial Td point symmetry implies that
the Δ splitting in Figure 6 is about zero; this is clearly representative in the DOS that
will be shown. The factor γ includes the vacancy-crystal coupling between bonding
and anti-bonding states. There are a number of shortcomings as to why the
calculated point symmetry had a small distortion, amount of D2d character, and
effective final Td symmetry. In many supercell calculations such as the present the
initial conditions may alter the final results. These were mentioned, including
supercell size, supercell space group symmetry, and brillouin zone sampling to name
a few. Techniques for improving vacancy calculations may be in superior relaxation
methods. More of these factors are discussed in the comparative works section of
the discussion chapter.
Included within the relaxation process is the change in the amount of vacancy
volume. The volume of the vacancy before relaxation is initially just the volume due
to a silicon sphere (spheres are nearly touching). The initial volume significantly
decreases as the 1st NNs move inward during relaxation, shown in Figure 10. The
volume was calculated using equation 3.1 for tetrahedral symmetry:
Volume = 16r40b − r19a( ) ⋅ r19b − r19a( ) × r40a − r19a( ) (3.1)
After a total 50% decrease in volume, the vacancy has finally relaxed. Between the
3rd and 4th iterations there was a significant change in the volume during relaxation.
This monotonic decrease in volume is due to the relaxation of the initial forces
calculated by the Wien2k code. The remainder of the calculation produced a minute
Figure 10. Volume of the Vacancy.
44
Volume of Vacancy
3.00
3.50
4.00
4.50
5.00
5.50
6.00
6.50
7.00
7.50
8.00
0 2 4 6 8 10Iteration
Vo
lum
e (
A^
3)
45
amount of D2d character (distortion). In the discussion section following the results,
these ideas will be reiterated along with a comparison of the results to other
literature.
The quantitative results thus far have depicted the substantial relaxation
undergone in the vacancy calculation. In order to realize the amount of energy put
back into the system during relaxation, one must consider the formation energy of
the vacancy. The formation energy represents the total energy difference between
the defective and crystal supercells, corrected for the intrinsic energy of the defective
site. Equation 3.2 was used to calculate the formation energy (Ef ), which is plotted
in Figure 11:
Ef = EN −1 −N −1N
⎛⎝⎜
⎞⎠⎟EN (3.2)
This is also known as the defect energy. Various methods for calculating the defect
energy using DFT have produced a broad range of values.46 Once again, these
values are dependent on the initial kmesh, supercell size, symmetry, exchange
correlation, and relaxation methods. There is still a large part of debate on the
correct method to calculate the defect energy. The calculated values of this
calculation of the defect energy during convergence are quoted in Table 2.
46 (Puska et al., 1998)
Figure 11. The Convergence of the Formation Energy.
47
Convergence of the Formation Energy
2.00
2.20
2.40
2.60
2.80
3.00
3.20
3.40
0 1 2 3 4 5 6 7 8
Itertion
Ef
(eV
)
48
Table 2
Calculated Defect Energy
Isolated Neutral Silicon Vacancy Unrelaxed RelaxedSymmetry of Vacancy Td ~Td
Volume of Vacancy (Å3) 7.67 3.82Total Energy Ideal Silicon (Ry) -74247.271Total Energy w/ Silicon Vacancy (Ry) -73666.974 -73667.060Defect Energy (eV) 3.26 2.10Relaxation Energy (eV) 1.16
The results obtained thus far should present an intuitive description of the
properties of the vacancy structure during relaxation. Now the model will be
expanded to include the farther NNs, mentioned earlier in this section. For example,
how the 2nd and 3rd NNs fit into this model of the vacancy can be shown. The
vacancy measures an average distance before and after relaxation from each NN,
which is recorded in Table 3. It should be noted that initially the distance between
the vacancy and xNNs in the unrelaxed state are most likely underestimated. As
mentioned earlier, the lattice parameters were approximately 3% less than the
experimentally measured value. Therefore the quantitative values obtained from the
present calculation may not be exact with respect to the experimental, but the physics
of the vacancy described here is valid.
49
Table 3
Atomic Variation During the Relaxation Process
Distance from Vacancy (Å) Unrelaxed Relaxed % Relaxation
1NN 2.29 1.82 20.73
2NN 3.74 3.64 2.80
3NN 4.38 4.36 0.41
Vacancy 7.48 7.48 0.00
The Wien2k software stores the data pertaining to the positions of the atoms
into a set of consecutive structure files during the relaxation process. By analyzing
the data in these structure files, the distance between the vacancy and xNN shells
were calculated for each successive structure file during the relaxation. The
calculated convergence of the xNN shells is shown in Figure 12. The overall
variation in the 2nd and 3rd NNs is of significantly less magnitude than the 1st NNs.
The positive (outward) and negative (inward) values on the y-axis indicate a slight
breathing motion of the vacancy. Predominantly, this relaxation is inward and thus
can obviously be seen due to the significant reduction in the volume of the vacancy
site.
The most fascinating physical property of the vacancy is the electronic states
created within the forbidden bandgap. The originally degenerate four dangling
bonds of the silicon vacancy form the two states mentioned earlier, A1 and T2. These
are the bonding and anti-bonding states located about the Fermi energy. In the DFT
ground state calculation, the Fermi energy is equal to zero. It should be noted,
Figure 12. Convergence of Successive xNN Shells.
51
Convergence of succesive xNN shells
-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0 1 2 3 4 5 6 7 8
Iteration
Ato
mic
Vari
ati
on
(Å
)
1NN
2NN
3NN
52
however, that self-consistent methods within DFT are well known for the error in
estimating the absolute value of the bandgap, which is beyond the scope of the
present thesis. (Approaches for solving this problem may be found in the
generalized Kohn-Sham scheme for DFT.47) The calculated bandgap of
semiconductors is usually underestimated; in fact, this is the case in the present
calculation of the vacancy. The experimental bandgap of silicon is 1.2 eV. The
bandgap calculated here is about 0.70 eV. The underestimate of the bandgap is due
to approximations and continuity of the exchange correlation. Although this
underestimate of about 43% is significant, the bandgap for silicon is usually
underestimated by a factor of 2. The inclusion of this shortcoming does not render
the overall qualitative behavior of the vacancy calculation. Figure 13 shows total
DOS for crystal silicon followed by the LDOS of the 1st NNs.
Once the vacancy is introduced into the crystal silicon, the calculated DOS
includes the vacancy-induced states. The calculated DOS for the supercell
containing the vacancy in Figure 14 appears to be like the crystal DOS except that
the vacancy-induced states dominate the former bandgap region. Hence, the
introduction of the silicon vacancy produces electronic states within and about the
bandgap region; these are the vacancy-induced states. As a result, electronic
properties are now dependent on the properties of the vacancy.
The creation of the vacancy showed evidence of the vacancy-induced states
introduced into the DOS of the supercell containing the vacancy. These are the A1
47 (Stadele et al., 1997)
Figure 13. Density of States for Crystal Silicon. (a) Total DOS. (b) 1st NN LDOS –Bandgap Region.
54
b.
a.
Bandgap
Figure 14. Density of States for the Isolated Neutral Silicon Vacancy. (a) Relaxed.(b) Unrelaxed.
56
a.
b.
57
and T2 states about the bandgap region for the approximate Td point symmetry. The
remainder of this section is focused on the LDOS about the bandgap region in Figure
13b. In order to clearly illustrate the “deep center” states about the bandgap region,
Figure 15 magnifies the total DOS of the isolated neutral silicon vacancy in Figure
14. The A1 states bond the vacancy site; the peak exists in the occupied states just
below the Fermi energy contributing to the valence band (NN bonding). The T2
states are partially filled in the unoccupied states, just above the Fermi energy
contributing to the conduction band. During relaxation of the vacancy, the LDOS in
the region of the bandgap shifts to the left. This is the expected outcome because
during relaxation the bonding at the vacancy site strengthens, therefore increasing
the density of A1 bonding states.
The LDOS of the vacancy calculation will be further examined.48 The scale
of vacancy states contributing to the 1st, 2nd, and 3rd NN LDOS could also be
calculated, shown in Figure 16, plotted such to include only those vacancy-induced
states. This plot represents the supercell containing the vacancy LDOS subtracted by
the crystal LDOS about the bandgap region. It is important to illustrate that the
properties of the vacancy-induced states have a finite range; this range was
calculated by using the xNNs LDOS. For example, from plots 16a, 16b, and 16c,
one can calculate how the dominating peaks A1 and T2 gradually decrease as a
function of distance; i.e., the DOS in the bandgap region delocalizes farther from the
vacancy site. The delocalization of the vacancy-
48 The data pertaining to the DOS stored in an external file via the calculation fromWien2k.
Figure 15. A1 (Bonding) and T2 (Anti-Bonding) Vacancy-Induced States.
59
A1
Relaxed
--->
T2 γA1 T2
eV eV
Figure 16. Difference Plot of the Local Density of States - Bandgap Region.
61
1NN DIFF LDOS (VAC-IDEAL)
-0.50
0.00
0.50
1.00
1.50
2.00
2.50
3.00
-1.11
-1.01
-0.90
-0.79
-0.68
-0.57
-0.46
-0.35
-0.24
-0.13
-0.03
0.08
0.19
0.30
0.41
0.52
0.63
0.74
0.84
0.95
Energy (eV)
Stat
es /
eV
pzpypxpsSi40
A1T2
2NN DIFF LDOS (VAC-IDEAL)
-0.40
-0.20
0.00
0.20
0.40
0.60
0.80
-1.11
-1.01
-0.90
-0.79
-0.68
-0.57
-0.46
-0.35
-0.24
-0.13
-0.03
0.08
0.19
0.30
0.41
0.52
0.63
0.74
0.84
0.95Energy (eV)
Sta
tes
/ eV
pzpypxps
Si31
DelocalizationT2
A1
3NN DIFF LDOS (VAC-IDEAL)
-0.40
-0.30
-0.20
-0.10
0.00
0.10
0.20
0.30
0.40
0.50
0.60
-1.11
-0.98
-0.84
-0.71
-0.57
-0.43
-0.30
-0.16
-0.03
0.11
0.25
0.38
0.52
0.65
0.79
0.93
Energy (eV)
Stat
es /
eV
pzpypxpsSi14
Delocalization
A1
T2
a.
b.
c.
62
induced states are dependent on the distance from the vacancy site; this is understood
to increase the DOS within the valence band edge.
For the data obtained corresponding to the plots in Figure 16, the localization
of the A1 and T2 states about the bandgap were dependent on the distance from the
vacancy. In addition, the delocalization of these states shifted farther into the
valence band edge. This result is very intuitive. The farther one gets from the
vacancy the more the DOS takes on properties of the bulk crystal silicon. In fact,
even for a larger supercell, the xNN vacancy LDOS would approach that of the
crystal silicon LDOS in Figure 13b.
From the outcome of the calculation of the supercell containing the vacancy,
it appears safe to conclude that this should be the scenario as one goes farther from
the vacancy site. This dependence, R distance from vacancy, vs. LDOS has been
examined in Figure 17. The A1 and T2 states appear to decrease almost
exponentially. The data is plotted from the calculated values in Table 4. The
distance from the vacancy is denoted R and is taken with respect to the xNN
distances. The fraction of the LDOS are normalized to the 1st NN LDOS in order to
project the decreasing magnitude of the A1 and T2 peaks in the LDOS from the
vacancy site.
Furthermore, via the data obtained from the supercell containing the vacancy
LDOS, the LDOS in the bandgap region is nearly zero at the next periodic vacancy
site; the vacancy image separation is 7.48 Å as indicated in Figure 17. Therefore,
Figure 17. R Dependence on the Vacancy Density of States.
64
Range of Intrinsic States A1 and T2
1.82
3.64
4.36
7.48
1.82
3.64
4.36 7.48
-0.20
0.00
0.20
0.40
0.60
0.80
1.00
1.20
0.00 2.00 4.00 6.00 8.00
R Distance from Vacancy (Å)
Fra
ctio
n D
OS
(xN
N/
1N
N)
T2
A1
65
Table 4
R Dependence of the Vacancy-Induced States in the Bandgap
xNN/1NN DOS R (Å) T2 A1
1NN 1.82 1.00 1.002NN 3.64 0.24 0.113NN 4.36 0.13 0.00
Vacancy-Vacancy 7.48 0.00 0.00Range of Vacancy DOS < 7.48 -- --
this calculation of the isolated neutral silicon vacancy is an excellent description of
the vacancy, which is isolated, and the vacancy image interaction is negligible.
Summary of Results
The results of the calculation, prepared using Wien2k, for the isolated neutral
silicon vacancy have been presented. The general properties of the vacancy have
been calculated for both quantitative values and qualitative results. Following the
initial SCF calculation, the forces due to the vacancy site were relaxed using the
PORT algorithm within Wien2k. Once the supercell containing the vacancy was
relaxed, a model of the vacancy including only its 1st NNs was developed. This
model represented the most dominant properties of the relaxation and DOS. The
initial purely tetrahedral symmetry (Td) of the vacancy relaxed into an effective Td
symmetry containing a minute amount of D2d character. The amount of D2d
character was calculated by examining the bond lengths and angles of the 1st NNs. It
66
was later verified in the DOS calculation that the amount of distortion (D2d character)
calculated was not sufficient to split the T2 state, and the final symmetry of the
vacancy was approximately Td (~Td). Next, the volume of the vacancy site was
plotted for the relaxation process and verified to be an inward relaxation. During this
process a typical value for the vacancy calculations, the formation energy, also
known as the defect energy, was obtained and compared with previous works.49 A
table of values and the calculated convergence of the formation energy were
included.
In order to estimate the delocalization effect of the vacancy, it was necessary
to include the presence of the 2nd and 3rd NNs into the model of the vacancy. First,
the variation in atomic position was plotted for successive xNN shells. The variation
in each shell was predominantly in the 1st NNs during convergence to the xNNs’
final atomic lattice positions.
The most intriguing piece of the vacancy calculation is the LDOS of the
vacancy site. It was shown that the vacancy-induced states lie about the forbidden
bandgap. In the calculation of the LDOS it was confirmed that the T2 state did not
split, implying that the final symmetry was in fact Td. The vacancy-induced states
for the Td point symmetry were the focus of the remainder of the calculation in order
to examine the dependence of the A1 and T2 states (delocalization of the LDOS from
the vacancy site). By analysis of the data obtained via the Wien2k calculation, an R
dependence, or range, of the LDOS was calculated with respect to the vacancy site.
49 Refer to the Comparative Works section in Chapter 4.
67
It was shown that the A1 and T2 states are predominant in the 1st NN LDOS and fall
off exponentially from the vacancy site.
Finally, through the examination of the supercell calculation done for an
isolated neutral silicon vacancy, the structural and electrical properties of the
vacancy were identified both quantitatively and qualitatively. This calculation
effectively represented excellent results for the vacancy; the vacancy was isolated
and the vacancy image interaction was negligible. In fact, the range of the vacancy
LDOS is estimated to be less than 7.48 Å, within error of the value reported by
Muller and colleagues’ estimated absolute value of the critical thickness of a SiO2
gate-oxide to be approximately 7 Å.50
50 (Muller et al., 1999)
CHAPTER IV
DISCUSSION
In order to establish the validity of the present work, the results of the present
vacancy calculation will be compared with selected published works.
First, it should be noted that in many computational schemes, especially those
within the DFT formalism, absolute quantitative results contain discrepancies from
one calculation to the next. Various dependences, i.e., assumptions, approximations,
and various computational parameters, involved in the calculation could hinder the
results. Some of the more prominent dependencies include pseudo-potential
methods, basis sets, supercell size, symmetry, and kmesh. Other factors may include
the variation in methods for calculating forces and algorithms for relaxation. The
details of the dependences related to discrepancies in various quantitative results
associated with the Wien2k code itself are beyond the scope of this thesis because
they may require modification of the code. Hence, various quantitative and
qualitative results that have accompanied the present calculated results for the
isolated neutral silicon vacancy will be further examined with respect to the
published work.
69
Comparative Works
The primary motivation for studying defects in this thesis was focused on
simulating its fundamental electronic and structural properties and its implications
for the performance of electronic devices. Kim et al. stated, “The role of defects is
essential to control and quality of semiconductor devices.”51 They used the tight
binding molecular dynamics (TB MD) method to calculate a fully relaxed point
defect in silicon. Using a 64-atom supercell model of the vacancy under 300 K
(about room temperature), they looked at 1st NN displacements and LDOS for the
vacancy. Their results showed various similarities to the present vacancy
calculation; the net displacement of the 1st NN ranged from 0.3 to 0.27 Å. The value
recorded for their relaxation energy was 1.42 eV, about 30% of the total formation
energy, which was 3.68 eV. They calculated the LDOS for the vacancy site;
although the exact position of the dominant peak within the gap was present at a
slightly different energy, they showed evidence of vacancy-induced states. Since the
methods for exchange correlation were different, the absolute locations of the A1 and
T2 peaks in the LDOS are undetermined. In addition, it was never specified that the
vacancy relaxes into D2d symmetry during their calculation. According to their
calculated LDOS, the T2 state did not split in the vacancy-induced states, implying
that the symmetry was most likely Td, as in the calculation done in this thesis. They
did mention that after relaxation the vacancy-induced states shifted toward the
51 (Song, Kim, & Lee, 1993, p. 1486)
70
valence band edge, which was verified in this thesis. The features in the LDOS
represent the formation of weaker bonds, reminiscent of the !-bonding structure
represented in the calculated LDOS.
In regards to the nature of the 1st NNs during relaxation, Kim and Lee also
studied the structure of vacancies in amorphous silicon (a-Si),52 as opposed to crystal
silicon (c-Si). Their studies set out to explore the effects of the vacancy-induced
states in the bandgap of a-Si. They discussed similarities between a-Si and c-Si
when the vacancy was created. The major difference between the two is that the
structure in a-Si is completely random. They categorized four sets of different
amorphous sites with average bond lengths and angles. Again Kim and Lee used the
TB MD scheme, like in their previous literature. Next, they sorted this random
network of vacancies, relaxed into two types: Type I corresponding to a decrease in
bond angle and Type II corresponding to an increase in bond angle. In Type I, the
vacancy-induced states are shifted toward the valence band edge while in Type II the
vacancy-induced states are shifted toward the conduction band edge. In Type II, the
sp2 + p hybrid bonding states are formed as a result of the increased bond angle. The
p states contributed to the unoccupied states, therefore the shift toward the
conduction band edge. For the Type I case, the bond angle had decreased and
therefore p3 (!-bonds) states formed. The resulting DOS led to states near the Fermi
energy (characteristic of !-bonds). These features are present in the relaxation
process of the vacancy calculation performed in this thesis, specifically in the
52 (Kim & Lee, 1995)
71
calculated distortion (D2d character) of the bonding angles. In the calculation in this
thesis there are four smaller angles < 109.5° and two larger angles > 109.5°. Hence,
for this one, would expect that, according to Kim and Lee the Type I case would
dominate the features in the DOS. Similarly, in the present thesis it was found that
the states were shifted to the valence band edge. Although the paper does not
indicate the existence of peaks exactly corresponding to the A1 and T2 states because
the a-Si has different point defect symmetry, the results imply that there is a
formation of weaker bonds in the calculated DOS that have a feature dependent on
the bonding angles.
In a more recent work by Puska et al., a more formal approach to the
calculation for the vacancy was considered.53 This calculation was done using a
supercell model of the isolated neutral silicon vacancy for large unit cells on the
order of 128-216 atoms. They used the LDA approach for exchange correlation
within the DFT formalism. The initial point defect symmetry was Td. In their
calculation, various-sized supercells and kmesh sampling of the brillouin zone were
chosen. This was a very detailed calculation in which various final structures of the
vacancy point symmetry were calculated. They obtained results for initial Td to final
D2d, C3v (trigonal distortion, stress related), and approximate Td like symmetry (~Td)
where the calculated bond lengths also showed minute evidence of D2d character.
This evidence verifies that final point symmetry of the vacancy is dependent on
initial conditions given to the supercell structure. In their calculation, under the LDA
53 (Puska et al., 1998)
72
approximation the bandgap width was underestimated about 60%; this is common in
DFT calculations of the bandgap. As mentioned previously in the results section,
corrections to this bandgap approximation are referred to as the generalized Kohn-
Sham scheme.54 Puska uses a dense kmesh in order to characterize the vacancy-
induced states, implying that smaller supercells may also produce sufficient results
for the vacancy calculation with a larger number of k-points (denser kmesh). They
have accumulated a table of results ranging from different supercell sizes, point
defect symmetries, and formation energies. The most relevant calculation to the
results presented in this thesis is their 128-atom supercell; the volume decreased
about 28%, the point defect symmetry is also approximately Td (~Td), and the
formation energy is 3.44 eV for the relaxed vacancy.
Probert and Payne did another recent calculation for the point defect
calculation of the silicon vacancy.55 They regard their research as being the best
converged ab-initio study of the isolated neutral silicon vacancy. A systematic
methodology for improving supercell point defect calculations, including possible
errors, fixes, and general schemes for ab-initio calculations, was presented.
The study of their calculated vacancy uses CASTEP, based on DFT, using
the GGA method for exchange correlation. The results obtained from their
calculation show that the initial tetrahedral symmetry Td relaxes to D2d symmetry. In
correspondence with the D2d point defect symmetry, they mentioned that the
54 (Stadele et al., 1997)
55 (Probert & Payne, 2003)
73
vacancy-induced states contain an A1 (singlet) near the valence band edge and T2
(triplet) within the energy bandgap. They fall short of mentioning how the T2 state
splits under orientational distortion, lowering the symmetry to D2d; in fact, they did
not calculate (or did not describe/present in the paper) the LDOS of the vacancy-
induced states in their calculation. They also commented that, depending on initial
conditions of the vacancy point symmetry and supercell structure, the final point
symmetry could vary. In accordance, various methods are discussed on how to
improve methods of calculations including supercell size and convergence; the
relaxation of forces (including atomic relaxation and defect symmetry) is discussed
in greater detail.
In this thesis, a corresponding result obtained in order to verify the presence
of a minute amount of D2d character in the defect symmetry was the convergence of
the 1st NN bond lengths versus iteration. The calculation of the D2d character in this
thesis intended to reproduce the plot by Probert and Payne shown in Figure 18.
Their results showed that the relaxation goes from Td to D2d point defect symmetry.
They calculated a significant amount of D2d character in the vacancy relaxation.
Their bond lengths differed by about 0.6 Å, whereas the results in this thesis show
bond lengths differ by only 0.05 Å. The results obtained for the 0.05 Å bond length
differences are not significant enough to split the T2 state, as mentioned previously.
Consequently, the calculation in this thesis therefore indicates the conservation of the
initial Td point symmetry of the vacancy. Their vacancy relaxed inward to 27% of its
Figure 18. D2d Character in 1st NN Bond Lengths.56
56 (Probert & Payne, 2003)
75
76
initial volume. The defect formation energy was estimated to be 3.17 eV with
relaxation energy of 1.19 eV.
Finally, a brief comment on the relationship between the isolated neutral
silicon vacancy calculation presented in this thesis and the charged states of a
vacancy introduced by Watkins. The last article is significant to the charged states of
a vacancy mentioned in the Watkins model. The charged states bring about electron
transport that was mentioned in Chapter I. Mueller et al.57 performed a study based
on the fundamental research of this aspect via the isolated neutral silicon vacancy. In
their studies they used arsenic (As) to deactivate vacancy sites in silicon. They
effectively found that the silicon vacancy can take on a charge state of up to – 4 and
act as an electron trap center. This process is termed “deactivation.” They
calculated results for different levels of As doping and vacancy dependence. This
paper is especially interesting for the ongoing research aspects regarding vacancy
calculations including structural and electronic properties. In regards to the present
calculation, it would be possible to introduce charged states within the vacancy, then
study the structural and electronic properties relating to the charged vacancy.
Various quantitative and qualitative results for the isolated neutral silicon
vacancy obtained in the present work have been examined and verified with respect
to the published work. This section served to bring together the primary aspects of
the vacancy calculation. It was finally concluded that the point defect symmetry of
the vacancy could vary depending on the initial conditions of the vacancy
57 (Mueller, Alonso, & Fichtner, 2003)
77
calculation. The point defect symmetry can relax from the initial Td to Td-like (~Td),
D2d, or C3v. The vacancy-induced states are A1 and T2 for the Td and ~Td point
symmetry which form bonding and anti-bonding states about the Fermi energy. In
effect this was comparative to the results section of the vacancy calculation
presented in this thesis.
CHAPTER V
CONCLUSION
The calculation of the isolated neutral silicon vacancy was presented in this
thesis. Both the quantitative and qualitative results were obtained in order that one
unfamiliar with the vacancy can become adept in the fundamental properties
included in the vacancy calculation. The research for developing more precise
methods of calculation is still underway. Currently the theoretical predictions within
the DFT formalism are honing in on more desirable results for material properties.
One of the goals of condensed matter theory is to extract the theoretical framework
applicable for developing calculations relating to electronic device technology. The
properties of the vacancy are ultimately implicated in device functionality. Muller et
al. found this in the study on the critical thickness of the SiO2 insulating barrier. It
was found that in order to identify with the structural and electronic properties of an
ultrathin gate-oxide one must investigate the contribution of vacancy properties.
The calculation performed in this thesis consisted of a vacancy in the bulk, an
ideal representation. In a realistic situation there are other types of vacancies
(defects) present in materials. These intrinsic defects tend to aggregate, form
dislocations, and be located at heterojunctions. The calculation does not set out to
explain all of these properties, but it represents the most ideal situation for
79
calculating the properties of a point defect, specifically the isolated neutral silicon
vacancy. The calculations that are done pertaining to device functionality set up a
framework for the description of electronic device properties. The vacancy
calculation gives one a perspective from which to view structural and electronic
properties on an exceptionally local scale.
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Blochl, P.E., Jepsen, O., & Andersen, O.K. (1994). Phys. Rev. B, 49, 16223.
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Lannoo, M. & Bourgoin, J. (1981). Point Defects in Semiconductors I: TheoreticalAspects. New York: Springer-Verlag.
Mueller, C.D, Alonso, E., & Fichtner, W. (2003). Arsenic deactivation in Si:Electronic structure and charge states of vacancy-impurity clusters. Phys. Rev. B, 68,045208, 1-8.
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APPENDIX A
DENSITY FUNCTIONAL THEORY - KOHN–SHAM EQUATIONS
84
The DFT formalism is based on solving the Kohn-Sham equations. The
Kohn-Sham equations58 are derived via an energy functional equation, E[ρ(r)] , in
terms of the electron density. The equations are derived assuming that the electrons
are noninteracting. The fictitious (noninteracting) fermions Ψ p represent the
electron density. The wavefunctions, Ψ p , can be expressed in terms of many-body
wavefunctions, as a Slater determinant. The new single-particle orbitals ϕ i (r)
appear in the Slater determinant and are solutions to the Kohn-Sham equations.
E[ρ(r)] = Ψ p H Ψ p = V (r)ρ(r)dr∫ + F[ρ(r)] (A.1)
ρ(r) = ϕ i (r)i∑ 2
(A.2)
F[ρ(r)]= ϕ i
i∑ −
2me
∇r2 ϕ i +
e2
2ρ(r)ρ( ′r )r − ′r∫ drd ′r + EXC[ρ(r)] (A.3)
The expression for the energy functional E[ρ(r)] contains all the pieces of
the many-body Hamiltonian. F[ρ(r)] is the piece of the Hamiltonian belonging to
the kinetic and electron-electron interactions. Therefore, the remaining piece found
inE[ρ(r)] is the ion-ion potential, V(r). When applying the variational principle to
the energy functional, E[ρ(r)] , the functional attains its minimum for the
corresponding total electron density, ρ(r) given V(r). V(r) is determined by the
local electronic structure; in general this would be just the potential due to the ions.
58 (Kohn & Sham, 1965)
85
A more elegant rearrangement of the energy functional E[ρ(r)] after a
variational simplification leads to the Kohn-Sham equations. In terms of Lagrange
multiplier, εi , here are the single-particle–like equations:59
[−
2me
∇r2 +V eff (r,ρ(r))]ϕ i (r) = εiϕ i (r) (A.4)
V eff (r,ρ(r)) = V (r) + e2 ρ( ′r )r − ′r∫ dr + δEXC[ρ(r)]
δρ(r)(A.5)
Now V eff contains all the terms for the total potential, including the ionic
potential, the Coulomb potential term, and the variational of the exchange correlation
potential. ϕ i (r) are the Kohn-Sham orbitals, which are the basis functions for the
Kohn-Sham equations. These equations are solved by iterations until self-
consistency.
Beyond the Coulomb interactions is the exchange correlation function,
EXC[ρ(r)] . The exchange correlation is needed to capture all the many-body effects
of the system. The nature of this functional remains an active area of research; the
Wien2k group has researched this area extensively. It is proposed that the
generalized gradient approximation (GGA) method is a successful improvement to
the local density approximation (LDA). GGA uses gradients of the local electron
density to evaluate the change in densities locally. In theoryEXC[ρ(r)] is a many-
body problem and should be locally dependent, for a finite system GGA gives the
best approximation for the exchange correlation.
59 (Kaxiras, 2003)
APPENDIX B
THE FULL POTENTIAL METHOD AND BASIS FUNCTIONS OF WIEN2K
87
Full Potential Method
For both methods described above the potentials expand in terms of spherical
harmonics inside the sphere (I) and plane waves (PW) outside the sphere boundary
(II). No shape approximations are made; therefore Wien2k uses the full potential
method, which serves as the basis for the calculation using Wien2k:
V (r) = VkeiKr , r > RMT (Interstitial region)K∑
VLM (r)ϒLM (r̂) , r < RMTLM∑{ (B.1)
The basis sets used by the Wien2k package correspond to sets of linearized
augmented plane wave (LAPW) and augmented plane wave (APW), plus the
addition of local orbitals, LAPW+lo and APW+lo.60
LAPW
The LAPW method is based on DFT theory for the treatment of the exchange
correlation potential found in LDA and GGA. Since the direct potential terms
including electron-electron and ion-ion are trivial, the basis set introduced by LAPW
method serves to solve the more nontrivial exchange correlation potential. In
accordance, the LAPW method is used to compute the electronic properties via
Kohn-Sham equations under self-consistency. The electronic properties calculated
are the total electron ground state energy E[ρ(r)] and density ρ(r) .
60 (Blaha et al., 2002)
88
An accurate model of the boundary conditions is needed when understanding
the application of LAPW used by Wien2k. A partitioning of the unit cell into atomic
spheres (I) and an interstitial region (II) is applied. Within the bounds of the atomic
spheres we define a muffin-tin radius (RMT); outside of this RMT is the interstitial
region. The interstitial region is the region between neighboring atoms with a
respective RMT.
(I) r < RMT inside the spheres (B.2)
ϕkn = [Alm,kn
l ,m∑ ηl (r,El ) + Blm,kn ηl (r,El )]ϒ lm (r̂)
El is the energy corresponding to band with l-like character
ηl (r,El ) is regular solution of the radial Schrodinger equation for El
ηl (r,El ) is the energy derivative evaluated at El
Alm,kn and Blm,kn are coefficients of the radial pieces, functions of kn
(II) r > RMT in the interstitial region (B.3)
ϕkn =1ωeknr
kn = k +Kn
kn are the wavevectors inside brillouin zone
Kn are the reciprocal lattice vectors
The solutions to the Kohn-Sham equations are expanded in this basis set of
LAPWs using a linear variation method. In order to increase the flexibility of the
89
basis, local orbital (LO) basis functions are added to ϕkn , which improve
linearization. These added basis functions are
ϕ lm
LO = [Alml ,m∑ ηl (r,E1,l ) + Blm ηl (r,E1,l ) + Clmη(r,E2,l )]ϒ lm (r̂) (B.4)
Alm , Blm , Clm require that ϕ lmLO be normalized and that slope and value be zero at the
sphere boundary. The superposition of the two basis sets is denoted by LAPW+LO.
APW+lo Method
In general the APW+lo method is similar to LAPW, and it has been shown
that they both converge to almost identical results. The coefficients Alm and Blm no
longer depend on kn so that the basis has “kinks” at the sphere boundary, but still the
total wavefunction is smooth and differentiable. The benefit to APW+lo over LAPW
(LAPW+LO) is that the former is used for atoms that are more difficult to converge.
ϕkn = [Alm,knl ,m∑ ηl (r,El )]ϒ lm (r̂) (B.5)
ϕ lmlo = [Almηl (r,El ) + Blm ηl (r,El )]ϒ lm (r̂) (B.6)
kn is determined by boundary condition, ϕ lmlo is normalized and zero at sphere
boundary. In order to distinguish the basis from LAPW method, the "lo" is used
instead of "LO". LAPW and "lo" orbitals (ϕ lmlo ) are almost identical except that the
coefficients Alm and Blm do not depend on kn .
APPENDIX C
HELLMAN – FEYNMAN THEOREM
91
The calculation of the forces about each atom is essential to the DFT
formalism. Since DFT can calculate the best value for the total energy, one can
easily calculate the forces. In any supercell calculation there will most likely be
forces due to local perturbations. Wien2k calculates these forces and then obtains
values for the force on each atom with respect to its coordinates in the supercell. In
order to calculate the local forces for each atom, Wien2k must first calculate the
Hellmann-Feynman forces directly via the total energy. The Hellmann–Feynman
forces are derived by classical methods cast in terms of R.
The typical Hamiltonian for an atomic system is of the form in
equation C.1:
HR = −12∇r2 +Ve−e(r) +Vion−e(r,R) +Vxc (r) +Vion− ion (R) (C.1)
R is the parameter of interest :
F ∝∂E∂R
= Ψ∂H∂R
Ψ (C.2)
Therefore the force F can be written in terms of an effective “scalar” potential, V eff ,
found in Appendix B.
F = −∇rVeff (R,ρ(r − R)) ; R = r − ′r (C.3)
V eff (R,ρ(r − R)) = V (r) + e2 ρ(r − R)R∫ dr + δEXC[ρ(r)]
δρ(r)(C.4)
V eff as in Appendix B includes all external potential terms.