ABSTRACT Name: Wesley J. Fabella Department: Physics Title: Ab

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


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


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


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


-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


-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)


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


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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Crystal Maker Software. (2005). Retrieved May 29, 2005, fromhttp://www.crystalmaker.com/

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Kaxiras, E. (2003). Atomic and Electronic Structure of Solids. Cambridge, U.K:University of Cambridge.

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Kroumova, E., Aroyo, M.I., Perez Mato, J.M., Kiroz, A., Capillas, C., Ivantchev, S.,& Wondratschek, H. (2003). Bilbao crystallographic server: Useful databases andtools for phase transition studies. Phase Transitions, 76, 1-2, 155-170. RetrievedMay 29, 2005, from http://www.cryst.ehu.es/

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.

Muller, D.A, Sorsch, T., Moccio, S., Baumann, F.H., Evans-Lutterodt, K., & Timp,G. (1999). The electronic structure at the atomic scale of ultrathin gate oxides.Nature, Vol. 399, 758-761.

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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.

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ABSTRACT Name: Wesley J. Fabella Department: Physics Title: Ab