Investigation of the band structures and optical properties of ...

United Arab Emirates UniversityScholarworks@UAEU

Theses Electronic Theses and Dissertations

12-2006

Investigation of the band structures and opticalproperties of super latticesSheikha Ahmed Awad Alnaeme

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Recommended CitationAwad Alnaeme, Sheikha Ahmed, "Investigation of the band structures and optical properties of super lattices" (2006). Theses. 572.https://scholarworks.uaeu.ac.ae/all_theses/572

INVESTIGATION OF THE BAND STRUCTURES AND OPTICAL PROPERTIES OF

SUPERLATTICES

A Thesis Submitted to

The Dean of Graduate Studies

of the United Arab Emirates University

Sheikha Ahmed Awad Alnaeme

In Partial Fulfillment of the Degree of M. Sc. in

Materials Science and Engineering

December

Thesis Supervisor

Prof. Dr. Mohamed Assaad Abdel-Raouf

Department 0/ Physics

Faculty a/Science, UAEU

Co-Supervisor

Dr. Noureddine Amrane

UAEU Library

I 11111111111111 111111111 1 1000442323

AQAM LIBRARIES

Examination Committee Members

Prof. Dr. Nacir Tit

Prof. Dr. Abdallah Qteish

Yarmuk University, Irbid-Jardan

Contents

The i up rvi or. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

Examination ommi ttee Members . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1

cknowledgement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

bbreviation & ymbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

L i t of F igures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v i i

L ist of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

Ab tract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

Chapter 1: I ntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Chapter 2: Theoretical background of the computat ional technique . . . . . . . . . . . 6

2 . 1 Density Functional Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 . 1 . 1 The many body problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 . 1 .2 The Born-Oppenheimer approximat ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 .1.3 The theorems of Hohenberg and Kohn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 . 1 .4 The Kohn-Sham equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0

2 . 1 .5 The exchange correlation functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2

2 .2 The m in imizations of the Kohn-Sham energy functional . . . . . . . . . . . . . . . . . . . . . . . . 1 5

2 .2 .1 k-poi nt samp l i ng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 5

2 .2 .2 Plane wave basi s sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 6

2 .2 .3 Plane wave representation of the Kohn-Sham . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 7

2 .2 .4 D i rect m in im izat ion of the Kohn-Sham energy funct ional . . . . . . . . . . . . . 1 7

2 .2 .5 The He l lmann-Feynman theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2 .6 The Augmented Plane Wave (APW) and Linearized Augmented

Plane Wave (LAPW) methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1

2 .2 .6. 1 The A PW method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1

2 .2 .6 . 1 The regu lar LAPW and the LAPW+lo methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Chapter 3: omputat ional techn ique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3 . 1 Computation al aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3 . 2 Prior step o f c alcu l ations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Chapter 4: Pre ent i nvestigations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4 . 1 Detai l of calcu lations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4 . 1 . 1 B inary compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4 . 1 .2 uperlattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1

Chapter 5: Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5 . 1 B inary compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5 . 1 . 1 Electronic Propert ies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5 . 1 . 1 . 1 Band structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5 . l. 1 . 2 Total charge density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 47

5 . l. l.3 Density of states (DOS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5 . l. 2 Optical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2

5 . 2 Superlattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5 .2. 1 Band structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5 .2.2 Optical propert ies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5 .3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

Acknowledgment

Fir t of all, I would like to expres s my gratitude and thanks to Allah, who gave me

health and s trength to finis h this thes is . Second, I would like to expres s my great

appreciation to my s upervis or Prof. Dr. Mohan1ed As s aad Abdel-Raouf and my co

s upervi or Dr. Noureddine Amrane for their guidance, as s is tance and encouragement

during this work. Many thanks are due to the United Arab Emirates Univers ity , which

offered me this chance to get enrolled in the Mas ter Program. Finally , I would like to

expres s my warmes t gratitude to all my family , es pecially my mother, father, s is ters

and brothers for their moral s upport and patience.

Abbreviations & Symbols

Symbol Abbreviation

APW ugmented Plane Wa e

BZ Bri l lou in zone

YO Chemical Vapor Deposit ion

OFT Density Functional Theory

DO Density of States

EPM Empirical-Pseudopotential Method

FPLAPW Fu l l Potential L inearized Augmented Plane Wave Method

GGA General ized Gradient Approximation

HCP Hexagonal C lose Packed

HLYB H ighest Laying Valence Band

HYPE Hydride Vapor Phase Epitaxy

K Kohn-Sham

LAPW L inearized Augmented P lane Wave

LLCB Lowest Laying Conduction Bands

LO Local Orbitals

L DA Local Spin Density Approximation

MBE Molecular Beam Epi taxy

MOCVD Metalorganic Chemical Vapor Deposit ion

PW Plane Wave

SOC Spin Orbit Coupl ing

SL (s) Superlattice (s)

TBM Tight -B ind ing Method

List of Figures

Figure Title Page

Fig. I: Band gap energies and latt ice constants of group I I I n itrides

(he "agonal phase) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Fig 2: chematic i l lu trat ion of a GaAs/AlAs alternati ng structure,

including the correspond ing potential profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Fig. 3: elf consistent loop for solution of Kohn-Sham equat ions . . . . . . . . . . . . . . . . 1 3

Fig.4: The teepest descent and conj ugate-grad ients methods . . . . . . . . . . . . . . . . . . . . 1 9

Fig. 5: The crystal structure of TiC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Fig. 6: The graph ical user i nterface w2web in : Structure Generator for TiC . . . . 29

Fig. 7: Data flow dur ing a SCF cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Fig. 8: Band structure of TiC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Fig. 9: The density of states (DOS) for TiC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Fig. 1 0: Variat ion of the total valence charge density for ZnxCd l-xSe

Ternary a l loys with x va lues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Fig. 1 1 : Energy vs . volume curve for TiC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Fig. 1 2: Calculated rea l and i maginary parts of the d ie lectric

function of ZnSe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Fig. 1 3: Refract ive i ndex of S iC, GeC, SnC and GeSn . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Fig. 1 4: (a)Wurtzite crysta l structure and (b) Bri l lou in zone for hexagonal

unit cel l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Fig. 1 5a: Band structure of wurtzite GaN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Fig. ISb: Band structure of wurtzite AIN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Fig. 1 5c: Band structure of wurtzite I nN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Fig. 16a: Total alence charge density of wurtzite Ga . . . . . . . . .. . . . . . " ......... .48

Fig. 16b: Total alence charge density of wurtzite A I . . . . . . . . . . . . . . . . . . . . . . . . . . .48

Fig. 16e: Total alence charge density of wurtzite I nN . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Fig. 1 7a: The charge dens ity d istribution of wurtzite GaN . . . . . . . . . . . . . . . . . . . . . . . .48

Fig. 1 7b: The charge dens ity d i stribution of wurtzite A IN . . . . . . . . . . . . . . . . . . . . . . .48

Fig. 1 7e: The charge density d i stribution of wurtzite l nN . . . . . . . . . . . . . . . . . . . . . . . . .49

Fig. 1 8a: The densi ty of states for wurtzite GaN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 50

Fig. 1 8b: The density of states for wurtzite AI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1

Fig. 1 8e: The density of states for wurtzite I n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1

Fig.1 9a: Variation of the i maginary part of the d ielectric function

ofv,rurtzite GaN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Fig. 1 9b: Variat ion of the imaginary part of the d ielectric function

of wurtzite AIN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Fig. 1 ge: Variat ion of the i maginary part of the d ie lectric function

of wurtzite [n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

Fig. 20a: Variat ion of the real part of the d ie lectric function of

\vurtzite GaN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 5

Fig. 20b: Variat ion o f the real part o f the d ielectric function of

wurtzite AIN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

Fig. 20e: Variat ion of the real part of the d ie lectric function of

wurtzite I nN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

Fig. 2 1 a: Variat ion of the refract ive i ndex of wurtzite GaN with the energy . . . 5 8

Fig. 2 1 b: Variat ion o f the refracti ve i ndex of wurtzite A IN with the energy . . . . 5 8

Fig. 2 1 e: Variat ion o f the refract ive index of wurtzite I nN with the energy . . . . 59

Fig. 22a: The band structure of A IN/GaN Superlatt ices . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1

Fig. 22b: The band structure of I nN/GaN Superlatt ices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1

Fig. 23a: Variation of the imaginary part of the dielectric function

of liGaN 1 I uperlattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Fig. 23b: Variation of the imaginary part of the dielectric function

of I nN/GaN 1 x I uperlattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Fig. 24a: Variation of the real part of the dielectric function

of AIN/Ga L 1 uperlattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Fig. 24b: Variation of the real part of the dielectric function

oflnN/GaN Ixl uperlattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Fig. 25a: Variation of the refractive index of AIN/GaN Ix! uperlattices

\vith the energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Fig. 25b: Variation of the refractive index of I nN/GaN 1 x 1 Superlattices

\vith the energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

List of Tables

Table Title

Table 1 : Lattice constants a, c and parameter u of the wurtzite structure Ga , AI and I n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1

Table 2: Lattice constants a c and parameter u of the wurtzite structure AI /Ga and I n /GaN superJattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Table 3: Band energies (eV) of the wurtzite GaN, AIN and InN . . . . . . . . . . . . . . . . . .45

Table 4: The Parameters of Peng-P iprek's formula of GaN, A IN and I nN . . . . . . 57

Table 5: Band gaps of AIN/GaN and I nN/Ga superJattices . . . . . . . . . . . . . . . . . . . . . . . 60

Table 6: The calcu lated d ie lectric constants of d i fferent superlatt ices . . . . . . . . . . . . . 63

Table 7: The parameters of Peng-Piprek's formu la of A IN/GaN

and I n /Ga superlattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Abstract

Group - I I I n itrides are h igh ly attractive materials because of their great potent ial for the

de elopment of optoelectronic dev ices in short-wavelength, semiconductor lasers and

optical l ight-emi tter and detectors. GaN, A IN and InN are considered as the most

important compounds in this c lass of materials. I n this work we present a theoretical

i nvestigation of the e lectronic properties (band structure, dens i ty of state and contour

maps of the charge dens ity d istribution ) and optical propert ies (refract ive index,

absorption and d ie lectric constants) . A lso, we i nvestigate the band structures and optical

properties of A l I Ga and InN/Ga superlatt ices. A ful l -potential l i nearized augmented

plane-wa e (FPLAPW) method is employed within the framework of the computer

package WIEN2k. Comparisons between the properties of each system and those of its

constituents are made. The values of the energy gaps of these compounds obtained by

LDA d iffer s ign ificantly from the experimental values and genera l ly agree more close ly

with other theoretical values. I n add it ion, the charge densities have been presented and

pro ide addi tional evidence for the s imi larity of the bonding of GaN, A l and InN . Also,

we ha e shown that the optical properties are excel lently reproduced using LDA

calcu lat ions if we a l low for a rigid shift of the band structure, namely us ing the so cal led

scissors operator.

CHAPTERl

Introduction

Group-Il l n itride (AI , I n , Ga ) ha e recent ly become one of the most interesting and hot

topic of re earch for man ph ic i t and engineers. Th is is mainly attributed to their usage

in optoelectronic de ice operating in green, blue, violet and u l traviolet spectral regions [ 1 ] .

The e materials, in he agonal phase, ha e direct band gap ranging from 0.7 eV for InN to 3 .4

e for Ga and to 6 .2 eV for AIN. Also, they have other properties l ike h igh thermal and

chemical tabi l ity, 10\: compressibi l ity and h igh melting temperatures [2] . Therefore, these

material , together with the ir temar a l loys could in princ iple cover almost a l l the visible and

near-ultra iolet regions of the spectrum, as shown in Fig. 1 [3a] . The n itrides(see [3b]) are

u ed in man} application , for example, surface acoustic wave devices [4], UV detectors [5-

8], � a eguides, UV and vis ible l ight emitting d iodes (LEDs) [9- 1 1 ] , and laser d iodes (LDs)

[ 1 2] for d igital data read-write appl ications.

>5 Q) -

�4 '-Q) c Q) 3

0.32 0.34 0.36 lattice parameter a (nm)

Fig. 1 Bandgap energies and lattice constants of I II-n itrides material system (hexagonal phase)[3a]

The quest for l ight emitters and detectors operat ing from the blue to UV regions of the

electromagnetic spectrum has tantal ized researchers for several decades. In the 1 960's and

1 970's, much effort was expended to grow and characterize GaN, AIN and I nN. Those

\ orker encount red sign ificant d ifficult ies in obtain ing h igh qual ity materials. Among the

difficultie , fr m our ide i due to the fact that there was no avai lable su itable substrate

material with a rea onably c lo e lattice match and from the other side was the difficulty to

grO\\ the p-type cry ta I [ 1 3 ] .

Many techn iques have been used to grow Group- I ll nitrides thin films ,however, the most

ucces ful in growing high -qual ity materials, which are reported today are molecu lar beam

epitax (MBE) and chemical vapor deposition (CYD) techniques [13] .

emiconductor uperlattices consisting of alternating layers of d ifferent materials- as shown

in Fig. 2- pro ide extra d imensions for tai loring material properties. The combination of

control led variat ions in the composition, strain, and th ickness of the layers provides

fascinating electron ic and optical properties [14- 1 6] , unl ike those of any ord inary bulk

material that, might lead to important appl ications in optoelectron ics such as switching,

ampl ification and signal processing [ 1 7-20] .

> I •

! Z /lJAs G.A. 0 Q.

0 I I'JAs

Go /i� As As

/lJ Ga Ga ,

I GaA'

F ig.2. Schematic i l lustration of a GaAsI AlAs alternati ng structure, incl uding the corresponding potential profile [2 1 ] .

orne techniques have been used to grow semiconducting superlattices, such as molecular

beam epitaxy (MBE), metalorganic chemical vapor deposition (MOCYD) and hydride vapor

phase epitaxy (HYPE) [22] .

Much of the theoretical work done to explain these interesting physical properties of

superlatt ices ha been largely concerned with the understanding of the electronic band

structure. hort-period uperlatt ices ( Ls) made of binary constituents are often treated by

atomi tic approaches uch as the density functional theory (OFT) and the tight- binding

method (TBM). For example, Franceschetti et a l . [23] have calcu lated the pressure

coefficient of optical transitions in InP/GaP SLs. The pseudopotent ial method within the

local den ity approximat ion (LDA) has been used by Kobayash i et a l . [24] to calculate the

optical tran ition strengths of the (AIP)n/(GaP)m type multi-quantum-we l l SLs. Simi lar

calcu lation for GaAs/AIAs has been done by Vee et a l . [25 ] . Confinement effects on the

optical transitions for various superlattice periods in GaAs/ AlAs have been d iscussed by

chmid et a l . [26]. Moreover, Botti et a l . [27] have employed the l i near combination of bulk

band method to determine the optical propert ies of GaAs/AIAs Ls. The electroabsorption

properties of GaAs/AIAs SLs have been studied by Kawashima et a l . [28 ] . In addition,

hibata et a l . [29] used the pseudopotentia l method to determine the osci l lator strength and

the band structures of the A lP/GaP SLs.

The e lectronic band-structures of the strained-layer ZnS/ZnSe (00 1 ) (SLs) have been

invest igated recently by Tit [30] using the Sp3sD tight-binding method, which includes the

strain and spin-orbit effects. The SL band-structures were stud ied versus the biaxial strain,

layer thickness, and band offsets. Also, the electronic structures of the ZnSe/Si, ZnSe/Ge

and ZnSe/SiGe superlattices were performed by Laref et a l . [3 1] via a semi-empirical spV

tight-binding method .

For n itride SLs there are several stud ies to investigate their electronic properties and most

work has been done for hexagonal structures. The band structures of short -period

superlattices a n d ordered a l l oys of AIN and G aN were ca l cu l ated by Rub io et a l .

[32]. T h e ca lcu l ated gaps range from 3 . 1 t o 5 . 8 eV . F o r wurtzite structures they

found d i rect band gaps w i th aJues between 3 . 5 and 5 . 8 eV . A lso, the 1 x 1 and 1 x 2

A l /Ga z inc -b lende s u perlatt ices were treated . It was found that they have d i rect

gaps of 4.4 and 4 . 3 eV re pecti ely [ 3 2] .

The aim of the present the i is to investigate the electronic and optical propert ies of GaN,

and In and the uperlattices (In /GaN and Al /GaN), using ful l potential l inearized

augmented plane wave method (FPLAPW) .

The thesis contain , beside this chapter other four Chapters and one Append ix.

In Chapter 2 the theoretical background of the computational technique IS reviewed

beginning from the discussion of the density funct ional theory, the semi-empirical methods

and final ly the min imizat ion technique of Kohn-Sham energy functiona l .

Chapter 3 is de oted to the description of the computational technique as wel l as the

computer code (W IEN2k) employed in the thesis .

Chapter 4 i s concerned \ ith the research problems under the scope of this thesis, in addit ion

to e laborate more deta i l s about the computational deta i l s of our calculation.

Chapter 5 contains the results and discuss ions of the e lectronic and optical properties of the

considered compounds and their related superlattices.

The thesis ends with a summary of conclusions and a complete l i st of references appearing in

the text as wel l as a translational of the abstract into Arabic .

CHAPTER 2

Theoretical Background of The

Computational Technique

Man computational methods nowadays used in material science for the electronic structure

calculations are d iverse . These methods span from very accurate quantum chemistry

technique appl ied to mal l molecules up to tight-binding semi-empirical schemes which

make it pos ible to imulate systems composed of about one thousand atoms.

In thi chapter we " i l l d iscuss the theoretical background of the computat ional techniques

emplo ed in the present thesis in three sections. I n section 1 , we wi l l d iscuss the density

functional theory for treating the many body problems and reducing them into the Kohn-

ham equations" ith the exchange correlation functional, and specifical ly the local density

approximation: as wel l as the general ized grad ient approximation (GGA). I n section 2, we

\ i l l explore the minimization of Kohn-Sham functional and discuss the p lane wave basis sets

employed in the present thesis' also the methods for min imizing the Kohn-Sham energy

functional and the APW and LAPW methods.

2.1 Density Functional Theory

2.1.1 The many body problem

The chrod inger equation of a time - independent non - relat ivistic system is described by

Where, \{' is the total wave function of many e lectrons system, E denotes as the total energy

of the system, and H is the total Hami ltonian of the system.

The exact many - particle Hami ltonian for N nuclei is defined as

where rl is the po ition of the electron i, rJ is the position of the electron j, R, is the posit ion of

the nuclei i, RJ i the posit ion of the nucleus j me is the mass of the electron and M, is the

rna of the nucleus i .

The fi r t term represents the k inetic energy operator o f the nuclei . The second term i s the

kinetic energ} operator for the e lectrons. The third term is the coulomb interaction between

electron and nuclei . The fourth term repre ents the electron-electron interaction. The last

term represents the interaction between a pair of nuclei [33] . It is d ifficult to solve problem

( I ) exactly, 0 we must introduce some approximations; that preserve the physics and reduce

mathematical chal lenges: that can be described in the next sections.

2.1.2 The Born - Oppenheimer approximation

Born - Oppenheimer [33] were the first who made an approximat ion to obtain solut ions to

the Schrod inger equat ion for many particles. Because the nuclei are much heavier, and

hence, much slower than the electrons , one can consider the electrons in a molecule to be

moving in the field of fixed nuclei . This leads to neglect the k inetic energy operator of the

nuclei and the i nteraction energy between the nuclei becomes constant as a background.

So, we can \ rite the Hami ltonian (2) as:

where, T is the k inet ic energy operator of the electrons Vee i s the potential energy operator

due to electron- e lectron i nteractions and Vex! stands for the potential energy operator of the

electrons in the field of the nuclei . We see from this that the Born - Oppenheimer

appro imation al lows us to concentrates on the electronic system, but the resultant many-

bod problem is t i l l far too d ifficult to sol e. 0, we must simpl ify for the above problem to

make it easy to sol e by search ing for further methods.

2. 1 .3 The Hoheoberg - Koho theorems

In the year 1 964 Hohenberg and Kohn [34] introduced two fundamental theorems which

gave birth to modem density functional theory. The first Hohenberg - Kohn theorem states

that the e 'temal potential V.ir) is a unique functional of the e lectron density (p) . So, every

ob er able quant ity of a stationary quantum system is determined by the ground - state

density alone. To proof the first Hohenberg and Kohn theorem let us assume that there are

h\ 0 different e ternal potential Vexl r ) and V'ei r ) which differ by more than a constant and

give rise to the same density p( r ) associated with the (non degenerated ) ground state \jIo .

ince both external potentials d iffer, they lead to two d ifferent Hami lton ians H and H', \ ith

hvo different ground state wave functions \jI and \jI'. Thus we have

Eo = (If I HI If) with H = T + V

E� = (If' I H'IIf') with H' = T' + V

Therefore \jI and \jI' are d ifferent N particle wave functions, and by using the variational

princip le one can write the i nequal ity

And also,

Eo < (If I H'IIf) = (If I HI If) + (If I H' - HIIf) = Eo + f p(;)[Vex/' - Vex/ ]dr (6)

By add ing (5) and (6),

0, we can conclude that there cannot be t\ 0 different e ternal potent ials V and V' that yield

the ame den it} p (r).

The electron den ity p (r) determines all properties of ground states such as the kinetic energy

T [p] , the potential energy V [p] and the total energy E [p] . 0 we can write the total energy

as a function of densit} :

E[p] = V .... Ap] + T[p] + VeJp] (8)

Where,

FHA [p] = T[p] + Vee [p] (9)

Fm.. is Hohenberg - Kohn functional .

I f we know Fm.. we can solve the Schrod inger equation exactly [34] .

The second Hohenberg-Kohn theorem [34] states that the density that minimizes the total

energy is the exact ground state density, and th is can be obtained variational ly.

For any trial density per) associated to an N electron system with external potential, one can

state that

Eo � E[p] = T[p] + Vext[p] + VeJp]

and th is val id if only p = p [34]

( 1 0)

The two theorems do not g ive any idea as to the nature of the functional or how to calculate

the ground state density.

2. 1 .4 Tbe Kobn - Sham equations

The equations of Kohn and Sham [35] were publ ished in 1 965 to obtain the ground state

density of electron system.

The Kohn - ham equations tate that the ground state density of the interacting particle

tern can be calculated via the ground state density of non interact ing systems.

We know before that the Hohenberg - Kohn functional can be written in the form:

where,

e2 pC;) � Vf< = -4 - f � d r + E NCI. [p] = J[p] + E NCL [p] 1[&" '

( 1 1 )

( 1 2)

J (p) stands for the c lassical interactions of charge density and ENCL contains al l non -

c1as ical parts.

We v"rite no\ the complete energy functional to define the known and unknown terms.

( 1 3)

unknown known

To solve the unknown kinetic energy the functional of the kinet ic energy is spl it into two

( 1 4)

Where, Ts is the k inetic energy of a non-interacting electron and Tc is the unknown part

\ hich equals to the d ifference between the real functional T[p] and Ts [p] .

Ts =-lI(¢,r21¢,) ,

( 1 5)

¢, is a one partic le wave functions

0, we can write the total energy functional

E[p] = Ts [p] + J[p] + V .. ,Jp] + ENCL[P] + Te ( 1 6)

ett ing, E,Jp] = E�'Cl [p] + T(

Pro ide u .. ith

E[p] = T. [p] + J[p] + V.<t [p] + E xc [p]

( 1 7)

( 1 8)

where, E\c[p] refer to the exchange-correlation functional. This exchange and correlation

functional contains everything that is unknown [35] . Applying the variat ional princ iple leads

to the Kohn - ham equations

H Ks '" =£ '" Y'J , '1',

\\-here, the Kohn- ham Hami ltonian is defined by:

where,

and _ N _ _

per) = I ¢, (r) * ¢, (r) ,=1

( 1 9)

(2 1 )

ince K potential depends on the density, equations (20)-(22) have to be solved Self -

consistently [36], as shown in Fig. 3 . From this Figure, we conclude that the Kohn - Sham

equat ions provide also a way for calculat ing the density p (r) and energy.

2. 1 .5 The exchange correlation functional

In this section we w i l l d iscuss two approximations, for exchange-corre lation functional.

The first one is the local density approximation (LDA), and the second one is the general ized

gradient approximation (GGA).

Initial Guess

n (r),n (r)

'. Calculate Effective Potential

V<T(r) = Vczt(r) + VHart[n] + V:C[nt,n�]

! Solve KS Equation

( - �\72 + V<T(r)) 'I,l1f(r) = Ef'I/J'[('r)

! Calculate Electron Density

nIT(r) = L::i ftI'l/Jf(r)12

No Self-consistent?

Output Quantities

Compute Energy, Forces, Stresses

Fig. 3 : Sel f consistent loop for solving Kohn-Sham's equations [36] .

(i) The local density approximation

This approximation is the simplest and most widely used for the exchange-correlation

functional :

where, c...c: the exchange-corre lation energy per particle of a uniform electron gas of

den ity per) [35 ] .

Th exchange correlation potential x c may be written as

The quant ity v,c can be spl it into two parts:

Where, rs i the W inger- eitz rad ius.

0\\ the exchange part V;DA can be derived analytical ly [37] and is equal to

( 9 ) ' 3 1 \x(r,)=- 4Jr2

rs (26)

The correlation part can be calcu lated numerical ly using Monte Carlo s imulations [38-39] .

The local density approximat ion is val id i n the l imit of slowly varying density per).

There are a number of features of the local density approximation that can be summed up i n

the fol lowing points:

1 - I t has a tendency to favour more homogeneous systems and overbinds molecules and

sol ids.

2- The errors are exaggerated and bond lengths are too short in weekly bonded systems.

3- The local density approximat ion can fai l in systems, l i ke heavy fermions systems, so

dominated by e lectron - electron i nteraction [40] .

(ii) The generalized gradient approximation (GGA)

I n this approximation the grad ient of the e lectron density is taken to overcome the fai l ure of

local density approximation i n the field where density undergoes rapid changes such as

molecules. The generalized gradient approximation (GGA) can be written as

ETC = Exc[p(r), per)] (27)

The GG functional gi es a good description of several systems. It possesses the fol lowing

ad antage :

1 - It impro e the total energies of atoms, binding energies, bond lengths and angles.

2- It impro e the geometrie ,dynamical and energetic of water, ice and water c luster.

However, it has been shm n [40] that the GGA is less accurate than LDA in some cases such

1 - I t i s better to describe semiconductors within the LDA than GGA.

2- The improvement of the GGA over LDA is not c lear for 4d-5d transition metals.

2.2 The minimizations of the Kohn-Sham energy

functional

2.2.1 k - point sampling

Many calcu lations i n crystals such as the calcu lation of the die lectric constant and

general ized susceptibi l i l i t ies involve integrating period ic functions of a B loch wave vector

over the Bri l louin zone [4 1 ] . B loch 's theorem states that each electronic wave function can

be written as product of a function that has the periodicity of the latt ice and a wave l ike part

If, (r) = exp (ik .r)J, (r) (28)

The second term is the ce l l periodic part of the wave function. It can be expanded using

plane waves basis sets.

J, (r) = I C,.G exp[iG .r] , (29) G where, G is the reciprocal lattice vectors. I t is defined by C.R = 2 1t m for a l l R where R is

a lattice vector of the crystal and m is an integer and C is the expansion coefficients.

\ riting each e lectronic a e function a a sum of plane wa es [42]

'1/, (r) = I ,.k+(, exp[i(k + (i)J?] "

and u ing the B loch theorem, the problem of calculat ing the infin ite number of electronic

wa e function can be changed to one of calcu lating a fin ite number of electronic wave

function at an infinite number of k points. This can be done by sampl ing the Bri l louin zone

at pec ial sets of k points.

0\ , \ e can \ rite an integrated function f (r) over the Bri l louin zone as

(3 1 )

\ here, F (k) is the Fourier transform of f (r), n is the cel l volume and wJ i s the weighting

factor.

To choose a set of special k - points for sampl ing the Bri l louin zone some approximation

methods can be used to obtain an accurate approximation for the electronic potential and total

energy. The magnitude of the error in the total energy due to using k-point sampl ing can be

decreased by using a denser set of k- points [4 1 , 43] .

2.2.2 Plane wave basis sets

I n the preceding section, we discussed B loch's theorem and how it can be used to overcome

the problem of calculat ing the wave functions at infin ite number of e lectrons by expanding

each function in terms of a d iscrete plane wave basis set. So, the electronic wave functions at

each k-point can be expressed in terms a discrete plane wave basis set.

I n principle, for expanding the electronic wave functions the infinite basis set is needed. But,

the coefficients for the plane wave C1,k+G have a kinetic energy (1l2/2m)lk + GI2 and the plane

wave ith smal l ki netic energy are more important than the very h igh kinetic energy. The

introduction of a plane \ a e energ cut-off produces a fin ite basis set and this leads to an

error in the total energ) of the ystem but we can make th is error smal l by increas ing the

alue of the cut-off energy. There are some difficulties associated with the use of a plane

\ ave ba i et, for example, the number of basis states changes d iscontinuously with the cut-

off energy. Reducing this problem can be done by using denser-k point sets [44] .

2.2.3 Plane wave representation of Kohn-Sham equations

Kohn- ham equat ions take a s imple form when plane waves used as a basis set for the

electronic wave functions.

Ih e substitute equation (3 1 ) in the Kohn-Sham equation (20) we w i l l have the s imple form

In this form, the rec iprocal space representat ion of the kinetic energy is d iagonal and the

various potentials can be described in terms of their Fourier transforms. The method of

solving plane wave expansion of Kohn-Sham equations is by diagonal ization of the

Hamiltonian matrix whose matrix e lements Hk+G,k+G' are given by the term in the square

brackets above. The size of the matrix is determined by the energy cut-off [44]

2.2.4 Direct minimization of the Kohn-Sham energy functional

I n th is section we w i l l d iscus the methods that can be used for direct min imization of the

Kohn- ham energy functional . There are hi 0 general methods the first one is cal led the

teepe t de cents and the econd one is cal led the conjugate-grad ient.

(i) teepe t descents method

If 'Ii e have not an informat ion about the function F(x), the best choice for a d isplacement

from x to reach the min imum is the steepest descent direction

· aF I g =--ax x • x ' (33)

The \ alue of the function F(x) can be reduced by movtng from the point x to the

point x·

+ bg' , 'Ii here the function is minimum. I f we determine the value of b by sampl ing

the function F(x) at a number of points a long the l ine x·

+ bg' then the min imum of the

function can be found by locat ing the point where the grad ient of the function is orthogonal

to the search direction so that

where, G is the grad ient operator.

A series of such l ine min imizations must be done to find the absolute m in imum of the

function F( ). The vector x·

+ bg' is used as the starting vector for the next iteration. The

next point is x2 and by repeat ing this process the function F(x) decreases at each iteration

unt i l it reaches the minimum value. Fig. 4 shows the schematic i l lustration of the steepest

descent method. This method is simple but does not guarantee convergence to a minimum in

a fin ite number of steps and we need a large number of steepest-descent iterations to get c lose

to the minimum of the function [44] .

(ii) The conjugate-gradient method

Thi method i b tter than the previous method because it combines the information from all

pre ious earch d ir ction, such that a subsequent search d irection is independent from al l

pre 10US earch d irections. If gm is the steepest descent vector associated with iteration m,

then the conjugate grad ient direction dm is determined by

where.

m gm .gm r = m-I m-I g .g and

Becau e the conjugate d irections are independent each minimization step reduces the

dimen ional i ty of the problem by 1 , so after n iterations, the dimensional i ty of the

problem wi l l be zero and the minimum is reduced. So, the method is a good

technique for locating the minimum of the Kohn-Sham energy functional [44] .

STEEPEST DESCENTS

CONJUGATE GI?ADIENT

Fig. 4: the Steepest descents method and the conjugate-grad ients method

2.2.5 Hellmann-Feynman theorem

Hel lmann-Fe nman theorem is usual ly used to calcu late the force fJ act ing on ion I . This

force i mtnu the deri ati e of the total energy with respect to the position of the ion [45]

The force on an nucleu is determined by the wave function, depend ing on the fact that the

wave function changes a the atom moves. So, the fu l l derivative can be expanded out in

terms of the \' ave functions

where, \f'1 is the eigenfunctions of the system.

By using d ifferentiation we can write the above equation as

dE _ " ( IdH I )+,, /dl//, HI )+" ( IHdl//, ) dR, -7 1//, dR, 1//, 7 \ dR, 1//, 7 1//, dR,

The second and th ird terms in the derivative vanish due to the variat ional property of the

ground state. So, the force w i l l be equal to

B('II,IHI'II,) = _( I BH I ) BR 'II, BR 'II, , , (40)

This shows that when each \f'1 is an eigenstate of the Hami l ton ian the part ia l derivative of the

energy with respect to the position of an ion gives the real physical force on the ion. This

result is usual ly referred to as the Hel lmann-Feynman theorem [45-46] .

I t is easier to calcu late an accurate total energy than to calcu late an accurate force because the

error in the force is fi rst order with respect to errors in the wave functions and the error in the

energy is second order with respect to errors in the wave functions.

2.2.6 The Augmented Plane Waves (APW) and Linearized

Augmented Plane Waves (LAPW) methods

A \ e di cussed before, the plane wa es (PW's) form a natural basis for calcu lat ing the k-

dependent one electron wave functions in periodic solids. However, they are a very

inefficient ba is et for de cribing the rapidly varying wave function c lose to the nuclei . To

o ercome thi problem one can either e l iminate these osc i l lations, due to the presence of the

core electron , a done in the pseudopotential calcu lations or one can augment the PW basis

et [47, 48] .

- Augmented Plane Waves (APW)

I n 1 937, the augmented plane" aves (APW) were employed as basis functions for so lving the

one-e lectron equat ions, which now correspond to the Kohn-Sham equations with in the

framework of the density functional theory (DFT). In the APW scheme the un it cel l is

divided into n 0 regions:

( i ) Non-overlapping atomic spheres (centered at the atomic sites)

( i i ) An i nterstitial region.

In the nvo types of regions d ifferent basis wi l l be used. Outside the muffin t in spheres, the

basis functions are simple plane waves and inside the spheres they are l inear combinations of

the solutions to the Schrodinger equation [47-48] . {L>� li,a (r', £)YL ( ) ¢ ( ) = L

-I fP exp(i( + . ) . ) rEI

(4 1 )

Where, L is a short for 1m, n is un it cel l volume, r = r - ra is the position inside sphere a.

with the polar coord inates r' and;, k is a wave vector in the irreduc ible Bri l louin zone, K is

a rec iprocal latt ice ector, u� is the numerical solution to the rad ial Schrod inger equation at

energ and Y1 (P') are the spherical hannonics.

The coeffic ient a';" are cho n such that the atomic functions, for a l l L components, match

the PW's \ ith k+K at the sphere boundary. In order to describe Kohn-Sham

orbitallf/, (r) accurately with APW, one has to set E equal to the eigenenergy EI oflf/, (r) .

Therefore a d ifferent energy dependent set of APW basis funct ions must be found for each

eigenenergy. This leads to a non l inear eigenvalue problem that is computational ly very

demanding [47-48 ] .

- The regular Linearized Augmented Plane Waves

LAPW+Local Orbitals methods

(LAPW) and

There are se eral improvements made to avoid the energy-dependence problem in the APW

calcu lations but the first rea l ly successfu l one was the l inearizat ion APW method . In the

LAPW method, we l i nearize the energy dependence of rad ial functions ins ide each sphere by

adding a second tenn to the radial part of the basi s functions [47-48] .

_{� [a� u�(r')+bf �(r')]YL()

¢. ( ) - -I Q 2 exp[i(. +. ). ) rEI

Where, !&is the energy derivat ive computed at the same energy.

From LAPW's we can obtain a l l eigenenergies with a single d iagonal ization. However, i t i s

d ifficult to treat states that l ie far from the l inearization energy, for example, semi-core states

by LAPW's these states have one principle quantum number less than the corresponding

valence state. So, i t needs a special attention. Consequently the local orbitals (LOs) have

been introduced for treati ng the semi-core states . An LO is constructed by LAPW radial

funct ion u and kt one energy EI in the valence band region and a third radial function at E2

[47-49] .

{�)arWu� (r') + b,aJ0llf, (r') + cf'wu�, (r')]Yl ( )

¢ ( = /. O rE I

For determin ing the three coefficients the LO's should have zero value and s lope at the

Muffin tin (MT) sphere boundary and the normal ization. From this we can increase the size

of the ba i s set so a l l electrons can be treated accurate ly [47-48] .

CHAPTER 3

The Computational Technique

In thi chapter \ e w i l l d i cribe our computat ional technique which is based on ful l potential

L PW method for 01 ing the Kohn- ham equation for the many body electron system. We

u ed the WIE 2k. program for calcu lat ing the e lectronic and optical properties of the wurtzite

pha e of Ga , A I and I nN, and thei r related superiatt ices. Therefore, d ifferent aspects of

th i program \ i l l be d i cussed in this chapter.

nder tand ing of uch materials requires a quantum mechanical description of the related

o l id and thu re l ies on the calcu lation of the electronic structure. Correspond ing first

pri nc iples calcu lat ions are mainly done within Density Functional Theory (DFT), accord ing

to which the man bod problems of interacting electrons and nuclei is mapped to a set of

one-electron equations, the so-cal led Kohn-Sham (KS) equations. There are several methods

for solving the Kohn-Sham (KS) equations. The most accurate method is the L inearized

Augmented-Plane-Wave (LAPW) method which enables accurate calculations of e lectronic

and magnetic properties of poly-atomic systems using density functional theory (DFT). One

successful implementat ion of the ful l -potent ial LAPW (FP-LAPW) method is the program

package WIEN, a code developed by Blaha, Schwarz and co-workers over a period of more

than twenty years [47 49-5 1 ] . Other versions were developed in recent years which were

cal led WIE 93, WIE 95 and W I EN97. Now a new version i s avai lable, which i s based on

an alternative bas i s set. Th is al lows a s ign ificant improvement, especial ly in terms of speed,

universal i ty, user-friend l iness and other features. This program package has been

successful ly appl ied to a wide range of problems such as electric field grad ients and systems

such as h igh-temperature superconductors, minerals, surfaces of transition metals, or anti

ferromagnetic oxides and even molecules. Minim izing the total energy of a system by

relaxing the atomic coord inates for complex systems became poss ible by the implementation

of atomic forces, and even molecular dynamics became feas ible.

3. 1 Com putational aspects

The alt rnat ive ba i s et (APW+lo) is used in the ne\ est vers ion inside the atomic spheres

for the chem ical ly important orbitals, whereas LAPW is used for the others orbitals [47, 49-

There are se era l features guaranteed in modern computer code W I EN2k package, which are

ummed up in the fol lo\ ing points:

- Accuracy

This feature i s ach ieved by the wel l -balanced basis set which contains numerical radial

functions that are adapted i n each iteration to changes due to charge transfer or hybrid ization.

The basis set i s accurate near the nuc leus and satisfies the cusp cond ition. The PW

convergence can be contro l led by one parameter, namely the cutoff energy. There is no

dependence on selecting atomic orbita ls or pseudopotentials . It is a fu l l -potent ial and al l

electron method . Relat iv ist ic effects ( includ ing spin orbit coupl ing SOC) can also be treated

with by solving Dirac 's equation [47, 49-50] .

- Efficiency and performance

By using the new m ixed bas is APW+loILAPW, the program sat isfies this criterion . This

al lows us to save computer t ime and study larger quantum mechan ical systems [47, 49-50].

- Parallelization

The program can run in paral le l , e ither if the memory requirement is larger than that avai lable

on one CPU or in a coarse grain version where each k-point is computed on a single

processor [47, 49-50] .

ser friendliness

Thi feature can be achie ed b a web based graphical user interface (w2web) [47, 49-50) .

3.2 Prior steps of calculations

There are man teps \: h ich should be taken before calcu lat ing the properties of any

compound u ing W I EN2k. I n the fol lowing paragraphs the most important steps w i l l be

di cussed .

- Step ( 1 ) : Creating the structure file

A modem software package needs a good graphical user interface (GU I) . This i s

implemented a s a web-based GUI , cal led w2web. As an example, the structure generator i s

i l lu trated below in F ig . 6 for the case of Titani um carbide T iC as shown in F ig. 5 .

The calcu lation for a crystal structure needed to fi l l the data in the empty structure template

as shown in fig. 6. Accord ing the fol lowing prior: ( J )One chooses a t i t le for the compound

(2) One selects the treatment of core electrons (relat ivistic); and (3) One defines the lattice

type (face centered cubic) and lattice parameters (a, b c, a p, y). In a cubic system such as

in TiC, there is only one i ndependent lattice constant and a l l angles are 90°. Then the atoms,

Ti and C, need to be specified, with the atomic number Z; the muffin t in radi us (RMT) of the

sphere and the corresponding position in the unit cel l .

F ig. 5 : The structure of T iC [52]

tep (2) : In itial ization and run SCF

- The CF calculation

The CF cyc le of the W I EN code cons ists of fi e independent programs as shown in

fig. 7 :

( 1 ) L PWO: generates the potential from a given charge density

(2) LAPW I : computes the e igenvalues and eigenvectors

(3) LAPW2: computes the alence charge dens i ty from the eigenvectors

(4) CORE: computes the core states and densit ies

(5) MIXER: m i es the densities generated by LAPW2 and CORE w ith the density of the

previous iteration to generate a new charge density [5 1 ] .

Execution » <:'tnlctG�n!l:! inlti<lli::e calc. run SCF SIngle prog opt'mizeN.c a I mini poslfions

Tasks »

Rles » stnlct file's input fill?s cuiput f:les SCF files

Session gmt » c�anoe !'oeSSlon change dlr chanoe Info

Configuration

Usersgul de html-IJ ersi on rdf-Ve�on

TUE lIay 24 00 \70c ;l)J5 STATUS Idl-

Session: nc IsusiJpblahailapwmC rdrf,h 11" fwh

StruclEdlt isusllpblaha!lapwmC '*

StructGennA

You have to click "Save Structure" for changes to take effect!

Save Structure

TItle : TIC

Lattice: Spacegroup: 225Jm-3m_

p '* F B cxv ca C z R H lJl • Space groups from Bilbao Cryst Server

Lattice parameters in a .., a=4 328000036 b=.J.32S00003S C=.j 32SDOO[J38

a= 90.000000 b= 90.000000 g= 90.000000

I nequivalent Atoms: 2 Atom 1 . TI 1 Z= 22.0 RMT= .0000 remove atom

Pos 1 . x= 0.00000000 y= (1 00000(100 z= 0.00000000 <- ed� only thiS position I

Atom 2: C Z= 6.0 RMT = 1.9000 remo Ie atom

Pos l ' x= 0.50000�OO y= 0.50000000 z= 0.50000000 <- edit only this

positionl

add an atom

Number of symmetry operations: 4S ..,

You have to click "Save Structure" for c/Janges to take effect!

Save Structure •

F ig. 6 : The graphical user interface w2web in WIEN2k: tructure generator for TiC [52] .

"Tl qq

-....l

CJ � £' ::!I 0 � 0-

V-l s:: ...

0 S· OQ � C/) () 'Tj (l '< (l �

,......., VI N ......

LCORE • •

• • opbonal - ncccmry

' cunsum L.............. f al1'� L--....I . - . L • • • 1

I 811llSC .. . . . . . . . . . . . . . . . . . . . . . . . . . . .

· · · · , e I

• • I I I

I . broyd2 . ,

8 : : e

I I I I I I

kg c n • • • ' ; I I . . . broydl 'i I I I I I I I

. . .. . . .. . .. . . . . . . . . . . . . . .. .. . .. .. ... .. .. . .. .. .. - . ... .. .. .. .. . .. . .. .. .. . .. .. . .. . . .. .. .. .. . .. . . . . . . .. . . . . . .. .. .. . . .. . . . . .. .. .. .. ... . .. . .. _ .. . .. . .. .. ... .. .. .. - - _ .. . .. ... .. . _ _ • _ _ . .. .. .. . I 1 . .. . .. .. - _ .. . . .. .. .. ... .. . .. . . . . .. . . . . .

I _ . .. .. .. _ . .. . ... _ _ . .. .. . _ _ _ .. _ .. .. .. .. _ . .. .. .. . 1

- Step (3) : Calculating the properties

nce the CF C) cle has con erged, one can launch calcu lat ions of various properties l i ke

electronic, mechan ical and optical properties. There are several uses of the programs' for

in tance the most common tasks l i ke band structure, electron density, or dens ity of states. A

guided menu i s avai lable to simpl ify their generat ion.

- Electronic properties

- Band structure and density of states (DOS)

The energy band tructure and the corresponding density of states are the dominant quantities

that determine the e lectronic structure of a system. The band structure and the density of

states are ery important because they determine the electronic properties, optical properties

and a ariety of other properties of the material under investigation. This al lows for the

cia s ification of the materia l as, either insu lator or semiconductor or metal ; and gives ins ight

into the chemical bond ing of the material [49-50] . In Figs. 8 and 9, we find, as examples, the

band structure and the density of states for the TiC compound as calcu lated using the

WIE 2k package .

- Electron density

The electron density i s a useful quantity for analysis and interpretation. I t can be calculated

for a l l occupied states through total DOS or local DOS. This al lows to visual ize the nature of

the bond character and to explain the charge transfer and the bonding properties [50] .

F ig . 1 0 shows the Variation of the total valence charge densit ies for ZnxCd l .xSe with x values

as determined us ing WIEN2k package.

- Mechanical properties

For the e propert ie • the calculation needs h igh accuracy and a fu l l-potential treatment and

these are a ai lable in W I E 2k. The total energy of a system is a quantity that is used to

calcu late the mechanical properties. F ig. 1 1 shows volume Optimization for TiC.

- Optical properties

Dielectric functions and conductivity (see section 4 . 1 ) are the most important response

function in condensed matter physics, because they determine the optical properties of

material , e lectrical conduct ivity, and a host of technological appl ications. In addition, optical

spectra are perhaps the most widespread tool for studying the electronic exc itations

themsel es. Fig. 1 2 shows the calculated real and imaginary parts of the d ielectric function

for Zn e, and fig. 1 3 shows the refract ive index of SiC, GeC, SnC and GeSn obtained using

WIEN2k.

:> �

� � '"

LO 10 �O S.D ... '0 �O 1.0 DO ·It .,. .1.0 .... .,.. .. 0 -7.0 ·S.O -OJ)

·10.0 -1 1 0 ·12.0 llO W

...... L A � X Z W K

Fig. 8: Bandstructure of TiC [52] .

s�----�--�--�----�--�----r---��

·IZ ·to .v

Fig. 9: The density of states COOS) for TiC [52] .

I .-... 1 5 l!3 '§ � '--' 0 .� 10 v

'"d v �

..t:: U

Zn Cd Se x l·x I

. . . . . · · x=O -- -- - - x=0.25 - - - - -

_._.- -

x=0.75 -- x=}

t , j :', � f: 'i . :!

b. c. into S.

F ig. 1 0 : Variation of the total valence charge densities for ZnxCd l -xSe with x values [53] .

� l c Q U c 2 20 u " U OJ '6 15 "0 -@ '" a. c:- 10 '" c '0. co .� "0 5 c '" <ii '" a:

Toe � f 783 l3eo

.... maQh&n VII.8(GPo).BP.EO 13&1 132252.991 Z 4.2216 -17B3_t57OG7 ·1783 !o&OO

l no. ... � i

·1783 944Q \ i · 1 783 9480

\ .•. , f � 1183 ..... � ... .., 1 78:1 .... ,..

,.-· t783 9520 / .'

';83 950&0 " " / .' , 1783 9S6O .... -. .... ---.-"--.�.�

-1783 9SItG 1:20 1 25 :30 135 144 145 150 I. VnWm. r. u lQI

Fig. II: Energy vs. volume curve for TiC [52] .

$ , l.-A/�\ -,'" . . ' + .. + ... � *J "

5 10 Energy(eV)

Real part(Exp\.) Imaglna�part(ExPt)

Real part eore�cal Imaginary part heoretocal!

Fig. 1 2 : Calcu lated real and imaginary and real parts of the die lectric function for ZnSe [54] .

4.5 - - · GeC

4 0 SiC SnC

3 5 / "

3.0 - - -)( - - - - - - - - -'" - - - - - - - -� 2 5 -= '" 2.0 > 1:; 4.390 0 2 3 5 6 E 4 386 _ • . - GeSn - I ...

a:: 4 386 4 384 I 4 382 4 380 I 4.378 I 4 376 4.374 / 4 372 . . - � .

4 370 0.1 0.2 0.3 0 4 0.5 0.6

Photonenergy (eV) Fig. 1 3 : Refractive index for SiC, GeC, SnC and Ge n [55 ] .

CHAPTER 4

Present investigations

I n thi chapter \ e focu a gal l ium n itride (Ga ), aluminum n itride (A IN) and ind ium nitride

( In ) and al 0 the ir related superlatt ices; namely: In IGa and AI IGa . We wi l l d iscuss

the tructure of these compounds and then we \ i l l d iscuss the electronic and optical

propert ie of other to general ize the trends. Final ly, the computational deta i l s w i l l also be

demon trated; including the input data.

It i wel l known that group- I I I nitrides, such as AIN, Ga and In are very promising

material for their potential use in optoelectron ic devices (both emitters and detectors) and

h igh power/temperature electronic devices. In this \ ork, we study the e lectronic and optical

properties these ompounds and their superlattices ( lnN/Ga and A IN/GaN) .

There are three common crystal structures shared by the group- I I I n itrides: the wurtzite, zinc

blende and rock alt structures. The most thermodynam ical ly stable phase at ambient

conditions i wurtzite structure. In this work, all the calcu lations have been done on the

wurtzite structure of the above compounds. The wurtzite structure has a hexagonal unit cel l

and thus two lattice constants, c and G, as could be seen from F ig. 1 4 . I t contains 6 atoms of

each type. The space group of the wurtzite structure is P6)mc (C46v) [56-5 7 ] .

I n thi s work, we study the electronic properties (band structure, density of states, e lectrons

density and contour p lots of charge d istribution) of GaN, AIN, I nN and the optical properties

(real and imaginary parts of d ielectric functions and refractive ind ices) of GaN, A IN and InN.

Concerning the superlattices I nN/GaN and A IN/GaN, we study the band structure, real and

imaginary parts of d ie lectric functions and refract ive index.

(d) z H A

7-- K r y

Fig. 1 4 : (a)Wurtzite crystal structure [58] and (b) Bri l louin zone for hexagonal unit cel l [36] .

4.1 Details of calculations

The fir t-principles calcu lations are performed by employing a ful l-potential l inear

augmented p lane wave approach based on density functional theory and implemented in the

code. The Kohn-Sham equations are solved self-consistently using the FPLAPW method.

The local density approximation (LDA) is used. In the FPLAPW method framework, the

unit cel l is di ided i nto two parts: ( J ) non-overlapping atomic spheres (centered at the atomic

sites) and ( I I ) an interstitial region. The FPLAPW method expands the potential in the

fol lowing form [52] :

inside sphere

outside sphere (44)

where, Y LM (r) : is a l i near combination of rad ial functions times spherical harmonics.

The optical properties of a solid are usual ly described in terms of the complex dielectric

function [ 59]

The imaginary part of the die lectric function in the long wavelength l imit has been obtained

direct l from the electronic structure calculation, using joint density of states (JOO ) and the

optical matrix element . The joint density of states between the in itial and the final state

P,/'lW) is define as [36]

\ here, PI) (TJw) i the number of states per unit volume per unit energy range wh ich occur

\ ith an energy d ifference ben een the in itial and the final state equal to the photon energy.

The imaginary part of the d ielectric function (Im[&(w)]) = &2 (w) is given by [ 59]

where, M is the d ipole matrix, i , j refer to the in itial and final states, /, and E, are the Fermi

distribution function and the energy of the electron for the in it ial state, respectively. / r and Er

stand for the Fermi d i stribution function and the energy of the electron for the final state,

respectively.

The real part of the d ielectric function can be derived from the imaginary part using the

Kramers-Kronig Relationsh ip, which is given in the form

R [ ( )] = = 1 � pCOfWI &2 (WI )dWI e & w &) + 2 2 '

7l 0 (W' -w ) \ here, P is the principle value of the integral .

The knowledge of both the real and the imaginary parts of the dielectric function al lows the

calcu lation of important optical funct ions. The optical spectra, for example, refractive index

of a emiconductor , new) can be eas i ly calcu lated In terms of the components of the

complex d ielectric function as fol lo� s :

n(l1» � [i + �6, (11)) ' ;6, (11)) ' r (49)

4. 1 . 1 Binary compounds

We \ i l l d i cuss the input data of our treated binary compounds (GaN, AIN and InN). The

electronic configuration of Ga, AI , In and atom are taken as

Ga: Ar 3d 1o 4s2 4p l , A I : Ne 3s23pl

In : Kr 4d 1 o SS2Sp i and N: He 2S22p3.

In the fol lowing calcu lation, we have d i stinguished the Al ( l s22s22p6), Ga ( l s22s22p63s23p6)

In ( I s22s22p63s23p63d 104s24p6) and N ( I S2) inner-shel l electrons, taken as core states, from the

alence band e lectrons of the Al (3s23pl) Ga (3d l O 4s2 4p l ), In (4d 1o SS2Spl ) and (2s22p3).

For a l l the three compounds, we take the same atomic positions in our calculat ions. The

atomic positions of wurtzite Ga , AIN and InN are:

Ga Al and I n (0, 0, 0), ( 1 /3 , 2/3, 1 12);

(0, 0, u) ( 1 /3 , 2/3 Yz+u) .

For the WZ structure the i nternal parameter u, which is the relat ive d isp lacement between AI,

Ga, In and sublatt ices along the z- direction. Table I shows the value of the lattice

constants a, c and the u parameter of the GaN, A IN and InN in the wurtzite structure.

The calcu lations are performed using a Ceperly-A lder parametrization of the exchange

corre lation potential in the local density approximation (LDA) [38] .

Table 1: Lattice constants a, c and u parameter of the Ga , Al and I n in the \i urtzite structure.

Compounds a(nm) c(nm) u

GaN 0.3 1 89[60] 0 .5 1 85 [60] 0 .3762

AIN 0.3 1 1 1 [60] 0 .4978[60] 0 .385

I nN 0.3544[60] 0 .57 1 8[6 1 ] 0 .3794

Ba i functions, electron densities and potentials were expanded inside the muffin-tin spheres

in combination \ ith spherical harmonic functions with a cut-off Imax = 1 2 and in Fourier

serie in the intersti t ia l region. Here, 1 529 plane waves were used for the expansion of the

charge den ity and potent ia l . In the calcu lations reported here, we use a parameter RMTKmax

= 7; which determines the matrix s ize (convergence), where Kmax is the plane wave cutoff and

R IT is the smal lest of a l l atomic sphere rad i i . We choose the muffin-t in radi i of Ga, Al I n to

be 1 . 82 a .u and for N to be 1 . 72 a .u . We point out that in a fu l l -potential calculation the

choice of the MT rad i i should not affect the final results, provided that the calculations are

wel l converged .

4. 1 .2 Superlattices

I n the fol lowing calcu lations, the subshel ls of the noble gas cores have been d istinguished

from the subshel l s of valence electrons. This is an implementation of a ful l potential l inear

augmented plane-wave (FP-LAPW) method within the density functional theory. We have

used the l ocal dens i ty approximation (LOA) with a parameterization of Ceperly-Adler

data [38] .

The valence wave functions inside the spheres are expanded up to I = 1 0 . A plane-wave

expansion with RMT · KMAx of 7. We have adopted the values of 2.0 (bohr) for both gal l ium

and aluminum and 1 .562 (bohr) for nitrogen, as the muffin-tin rad i i for AIN/GaN and also the

alue for the muffin-t in rad i i are 2.0 (bohr) for both indium and gal l ium and 1 .562 (bohr) for

nitrogen for I n IGa . The latt ice structure of 1 x 1 systems is obtained by alternat ing cation

atom in the group- I l l atomic positions within the wurtzite structure. For the two

uperlatt ice , the atomic position are taken from [62] .

The symmetry of wurtzite (A I )m(Ga )n SL_s grown along the symmetry axis depends on

the number of mono layers m and n: the appropriate space groups are C:v and C�v for odd

and even values of m + n, respectively. For odd values of m +n, the nonsymmorphic space

group C:, of the SL is that of the bulk. The symmetry of the latter fami ly is described by the

yrnmorphic C�, space group. Also, for the lattice constants a, c and u parameter we take

the averaged alues as shown in Table 2.

Table 2: Lattice constants a, c and u parameter of AIN/GaN and I nN/GaN the in the wurtzite structure.

Superiatt ices a(nm) c(nm) u

AIN/GaN 0.3 1 5 0 .508 1 49 0 .38

I nN/GaN 0.33665 0.53479 0.3778

CHAPTER S

Results and Discussions

Th i chapter conta ins the final re ults of our comprehensi e computational work on binary

compound and uperlatt ice . Thus, it is d ivided into two main sections. The first (5 . 1 ) is

about the b inar compounds and the econd section (5 .2) is dea l ing with the superlattices

re pecti e l} .

5.1 Binary compo u nds

The binary compounds stud ied in this thesis are GaN, AIN and I nN. We were mainly

intere ted in the inve t igation of the electron ic properties as wel l as the optical properties of

the e compounds.

5. 1 . 1 Electronic properties

5. 1 . 1 . 1 Electronic band structures

Due to the fact that the spin-orbit interaction and relativist ic effects are important in the

description of the n itride electronic properties, particu larly for those involving the heavier

element Ga, AI and In, we analyze the role played by these effects on the band structure of

the three binary compounds GaN AIN and InN.

The electronic band structures of Ga , AIN and InN along h igh symmetry l ines of the

irreducible \ edge of the Br i l lou in zone are shown in F igs. 1 5a, 1 5b and 1 5c, respectively.

The calculated band energy gaps at h igh symmetry points are given in Table 3, the bandgaps

are found to be d i rect for Ga , AlN and InN and equal to 2 .096 eV, 4.33 eV and 0 .096 eV

respecti e ly, which are in c lose agreement with other s imi lar theoretical results, methods but

sign ificantl} d i fferent from experimental data as shown in Table 3 . Th is is due to the fact

that the LDA underest imates the gap since electronic excitations are not taken into account.

If we add the sc issors-operator, this lead to correct the energy gaps of the three compounds

which become c lose to the experimental val ues.

Our re u lt how that the InN had the smal lest band gap and this is due to the fact that the In-

4d electron influence strongly the size of the band gap in i nN. Also, there is a considerable

h brid ization of the -p orbita ls \: ith both Ga-3d and In-4d orbitals. This p-d coupl ing

result in a level repu lsion, moving the YBM upwards and lead ing to sign ificant band gap

reduction .

Table 3. Band energies (eV) for wurtzite GaN, AIN and InN

Compounds Present Other Experiment work calculations

GaN 2.096 2.0a, 1 .970 3 .2c

2 .0 I a, 2 .08a 3 .4c

AIN 4 .33 4.25°, 4 .2a 6 . 1 c

4.3a 4 .37c

I nN 0 .096 -0.270a 0.780c

O . I 92a

• Reference [63 ] , b Reference [64]. c Reference [65]

5 5' Q) � e> 0 Q) c

- 1 5 [' M K A [' !'J.

wave vector

Fig. 1 5a: Band Structure of wurtzite GaN

A wave vector

Fig. I Sb : Band Structure of wurtzite AIN

5 > a> 0 >: 0)

... a> c:

- 1 5 r M K A r tJ.

wave vector

Fig. I Sc : Band Structure of wurtzite I nN

5. 1 . 1 .2 Total charge density

To vi uaJ ize the nature of the bond character and to explain the charge transfer and the

bond ing propertie of w-GaN w-AIN and w-I nN, we calcu late the total charge density.

The electronic charge density is obtained for each band n by summing over the k-states in the

band .

PII (r) = IJI'''k (rt (50) k and the total charge density is obtained by summing over the occupied bands.

P (r) = L Pn (r) (5 1 ) II

The total valence charge densities for the three binary compounds are d isp layed respectively

along the Ga- , Al-N and I n-N bonds (Fig. 1 6a, 1 6b and 1 6c).

Figs. 1 7a, 1 7b, and 1 7c show the charge density distribution in the ( 1 ] 0) plane (contour

maps) . From the figures we concl ude the fol lowing points:

( 1 ) The calcu lated e lectron charge distributions indicate that there I S a strong JODIC

character for a l l the three compounds as can be seen along the Ga,AI , ln-bonds.

(2) The overa l l shapes of the charge distributions are simi lar for GaN, AIN and InN.

(3) The charge densities around the atoms have asymmetric forms which are s imilar to

those given in previous reports using the ab in it io pseudopotential method [66] .

(4) The charge transfer gives rise to the ionic character in the three n itrides.

(5) The driving force behind the d isp lacement of the bonding charge is the greater abil ity

of N to attract e lectrons towards it due to the d ifference in the e lectronegativity

between Ga, AI, I n and N.

7 ,.-... 6 .�

c ;::l 5 ..ci ..... � 4 '-" .£

3 '" 5 "0 2 r: g C,) Q) i:i3 0 Ga

• • -1

0 20 40 60 80 1 00 1 20 1 40

Position(Atomic unit)

Fig. 1 6a: Total valence charge densities for w- GaN

1 0 �------------�

� 8 c =

.ci � 6 -'-' C-.;;:;

4 f-5 "0 = o ... � �

A� N I • I I . I I

20 40 60 80 1 00 1 20 1 40

Position(atom ic u nits)

Fig. 1 6b:Total valence charge densities for w-AIN

1 20 =-

1 00 ..,.r-""' Ga ,........,

80 "0 -'" E '--' 60 Q)

"So � �

J .< 40 � ) ( � Ga

I \ c 20 (

20 40 60 80 1 00 1 20

Ang\e[mrad]

Fig. 1 7a: The charge density distribution for woGaN

i ln the ( I 1 0) plane

120 ---- ------�

1 00 I �) � r I r Al � I ----

80 � "0 '" --------..... E --------.. ........, 60 U) ..--------. Q)

00 1 ) ) c 40 <r:: ( -= .N' ( Ir I G Al

20 �� ------- � � �

20 40 60 80 1 00 120

Angles (m rad) Fig. 1 7b: The charge density distribution for w-AIN

in the ( 1 1 0) plane

,........ "0 E E ........ Q,) 'S.o �

o W � W W 100 IW I�

Position (Atomic units)

Fig. 1 6c: Total valence charge densities for w-InN

5.1.1.3 Density of States (DOS)

1 20 1 00

1 1\ 80 60 40 20

:::::--

� � =-

@ \�\ ---------- =3

20 40 60 80 1 00 1 20 AngJe [ m rad]

Fig. 1 7c: The charge density distribution for w-InN

In the plane ( 1 1 0)

The total densities of states for a l l the three compounds are shown in F igs. 1 8a, I 8b and 1 8c .

From the figures, we see that the total densities of states are quite s imi lar for a l l three

compounds, w ith some smal ler d ifferences in the fine detai ls . The fust structure encountered

in the DOS i f we start from the left, is a narrow peak centered around - 1 1 eV for GaN, - l 3 eV

for AIN, and - 1 5 eV for I nN. This peak consists almost ent irely of n itride s states which is

very local ized. We notice the s imi lar behavior of GaN and AIN as they have ionicit ies that

are c lose; we could see that from the behavior of the charge densities which provide an

intuitive look at the e lectronegat iv ity of the materials. InN has a more pronounced ionic

behavior and hence a d ifferent trend in the DOS.

Furthermore, this peak corresponds to the lowest laying band, and its width originates mainly

from the region around the r point in the Bri l louin zone, since only there the d ispersion of

this band is appreciable. The narrow structure centered at 5 eV below the Fermi level consists

predominant ly of n itride s and p states as is apparent from the figure. This structure is highest

in InN and smal lest in GaN .

The min imum of the valence band density of states occurs at fl at -6. 1 eV for GaN, at -4 . 8

eV for ] n and - 5 . 0 eV for A IN . The large peak comes primarily from the onset of the second

valence band. The conduction bands are wel l separated from the valence bands by an energy

gap of 2 .096 eV 4.33 eV and 0.096 eV for GaN, AlN and I nN, respectively.

4 Q) � .....

3 VJ ......

0 .0 VJ 2 c Q)

- 15 - 10 -5 o 5 1 0 Energy[eV]

Fig. 1 8a: The density of states for w-GaN

'" E 5 '" '-0

.f' '" c: (!)

" � o .� g Cl

8 r---------------------------�

-1 5 -1 0 -5 o 5 1 0 1 5 Energy (eV)

Fig. 1 8b: The density of states for w-A1N

- 1 5 - 1 0 -5 o 5 1 0 Energy[eV]

Fig 1 8c : The density of states for w-InN

5. 1 .2 Optical properties

Our theoret ical ly calcu lated absorpt ive part of the dielectric functions for GaN, AIN and I nN

(u ing equat ions (46-48) see section 4. 1 in chapter 4) are shown in figs [ 1 9-2 1 ] . For GaN,

several measurements of the reflect ivity have been publ ished, but no e l l ipsometry

measurements have been performed yet. Of course, we could have chosen to compare our

calcu lated spectra a lso with resu lts from reflectiv ity measurements. Our reasons for not doing

so are as fo l lows. Obvious ly, the d ielectric functions obtained from reflectivity measurements

and from e l l ipsometry d i ffer in important respects. Compare for instance, the absolute

ampl i tudes of the d ielectric functions measured with e l l ipsometry to the d ielectric functions

calcu lated from reflect iv ity experiments [67-70] . First of a l l , the latter ampl itudes are much

smal ler. The smal l ampl itudes could be due to the uncontrol led intensity losses in the

reflect ivity measurements originating from surface roughness, or they could be due to the

extrapolations necessary for the Kramers-Kronig transformation.

The calcu lated spectra given in F igs. 1 9a, 1 9b and 1 9c have been rigidly shifted to correct the

underestimations of the band gaps produced by the density functional theory. This is due to

the fact that the quasi particle exc itations are not taken i nto account in DFT. The calculated

spectra are not broadened, and thus have more structure than the experimental spectra.

It is seen that £2 is rather s imi lar for a l l the three nitrides; the main feature is a broad peak

with maximum around 7 .49 eV for GaN, 7.63 eV for AIN and 9.64 eV for InN. Also, a

shoulder around 3 eV for GaN and I nN. For A lN, no shoulder i s vis ible . The peak as wel l as

the shoulder are excel lently reproduced in the calcu lations, as are the general trend of the

spectra.

F ig. 1 9a: Variation of the imaginary part of the d ie lectric function for w-GaN

1 0 ,-------------------------, AIN

o 1 0 20 30 40

Energy (eV)

Fig. 1 9b : Variation of the imaginary part of the d ielectric funct ion for w-AIN

0 1 0 20 30 40

Energy leV )

Fig. 1 9c : Variation of the imaginary part of the d ie lectric function for w-InN

The maximum ampl itudes of the calcu lated spectra are higher than the experimental spectra.

Th is is so because the calcu lated spectra are unbroadened . In order to compare ampl itudes of

experimental and calcu lated spectra, it is better to compare integral s of the spectra instead of

maximal ampl itudes, if we do th is, we wi l l find that a lso the ampl itudes are excel lently

reproduced in our calculat ions. Our calculated spectra are also very s imi lar to the spectra in

references [67-70] .

The trends in £2 may be l inked to the trends observed in the DOS and band structures.

Compare the h ighest laying valence band (HVB) for the three systems and the lowest laying

conduction bands (LCB) . Our optical calcu lations show that transit ions between these two

bands account for almost a l l structure in the optical spectra at energies below 20 eV.

In InN, these bands have c learly less d ispersive than in GaN . This is the reason why the main

peak moves to lower energy and becomes sharper as we move from Ga to In .

ext we consider the d ispers ive part of the d ielectric function, £1 for the three binary

n itrides, GaN, A IN and I nN as d isplayed in F igs. 20a, 20b and 20c. The calculated spectra

ha e been obta ined by Kramers-Kronig transformation of the shifted £2 spectra. The main

features are the existence of a shoulder at lower energies, a rather steep decrease between 4

and 5 eV, after which £ 1 becomes a negative minimum, and then a s low increase toward zero

at h igher energies. A l l these features are very wel l reproduced in the calculated spectra. The

structures are more pronounced in the calculated spectra s ince these are not broadened.

1 0 GaN

0 10 20 30 40 Energy[eVI

Fig. 20a: Variation of the real part of the dielectric function for w-GaN

10 20 energy(eV)

Fig. 20b: Variation of the real part of the dielectric function for w-AIN

1 0 InN

0 1 0 20 30 40

Energy ! eVI

Fig. 20c : Variation of the real part of the d ielectric function for w-InN

Final ly e \: ould briefly d i scuss some of the spectra obtainable from the d ielectric function.

A i l l ustrated in F igs 20a, 20b and 20c, we find local minima for all the three n itrides.

The refractive index \ as computed using both real and imaginary parts of the d ielectric

function as sho\ n in Figs. 2 1 a, 2 1 b and 2 1 c . The values obtained were about 3% lower than

those reported in reference [7 J ] . We have fitted our calculated refractive index neE) using

the empirical formula of Peng and Piprek [72], which is given by the fol lowing re lation:

Where the values of the d i rect energy gaps (EJ are obtained from our optical spectra, and E

is the photon energy. The parameters a and b are obtained from the fit of the calculated

refractive i ndex spectra for the three binary compounds. A l l parameters used or obtained in

the fitting are l i sted in Table 4 .

Table 4: The Parameters of Peng-Piprek' s formula for the GaN, AIN and I nN compounds

Compou nds Eg (eV) a b

GaN 2 .096 eV 2.5 6.87

AIN 4.33 eV 8.7 9 .3

I nN 0.096 eV 6.57 5 .5

2.50 GaN

... '" '" .= ... -£ 2AO

v '" ..: .. c::

0.0 0.5 1.0 I .S 2.0 2.5

EnergyleV)

Fig. 2 1 a: Variation of the refractive index of GaN versus the energy

� ." 2.3 .s .� J1 .: .. " 2.2

Fig. 2 1 b: Variation of the refract ive index of AIN versus the energy

'" Q) "0 .E Q) > U £ Q) �

3 .004 I n N

3 .000

2 .998

2 .996

2 .994

0 .00 0 .02 0 .04 0.06 0 .08 0 . 1 0 0 . 1 2 0 . 1 4

Energy(eV )

Fig. 2 1 c : Variation o f the refractive index of lnN versus the energy

5.2 S uperlattices

The superlattices stud ied in this thesis are AIN/GaN and lnN/GaN. We were interested in

studying their band structure (see subsection 5 .2 . 1 ) and optical properties (see subsection

5 .2.2) .

5.2. 1 Band Structure

Figs. 22a and 22b show the E-k relation (the energy band d iagram) of A IN/GaN and

I nN/GaN respectively. The most important features of the band structures are represented by

the min ima Ec of the lowest conduction band and by the maxima Ev of the highest valence

band. The min ima and maxima are the places where the free electrons and holes are most

l i kely to be found where thermal equ i l ibrium is establ ished . The valence band in each of

these materials peaks at the zone center (k=O) and is actual ly composed of three subbands

have the same a l lowed energy at k=O (when spin is included). In the two superlattices, the

two upper bands are indi st inguishable, and the maximum of the third band is barely

somewhat below the maximum of the other two bands. Consistent with the effective mass

concept, the band with a smal ler curvature around K=O is cal led the heavy-hole band and the

band , ith a larger curvature around k=0 is cal led the l ight-hole band. The conduction bands

are also composed of a number of subbands.

The trend in the band gap shows 3. I 8 (eV) for AIN/GaN and 0 .46 (eV) for InN/GaN. For

In IGa SL's (see Table 5) . Standard LDA calcu lations, in general , fa i ls to correct ly

describe energy gaps and semicore d bands originating from highly local ized states. This

shortcoming i s wel l known in the case of group-I l l n itrides where the band gap is

underestimated up to more than 1 00% for InN. Our results show that In-4d electrons

influence strongly the size of the band gap in InN/GaN Superlatt ice. For I nN/GaN, there is a

considerable hybrid ization of the N-p orbita ls with both Ga-3d and In-4d orbitals. This p-d

coupl ing results in a level repuls ion, moving the YBM upwards and leading to s ign ificant

band gap reductions.

I t is interesting to compare our calcu lated gaps with experimental data. S ince quasi -particle

exc itat ions are not taken i nto account in density functional theory (DFT), the energy gap

calcu lated from DFT often cal led the Kohn-Sham gap, tends to be smal ler than the

experimental one.

Table 5: Band gaps (eV) of A IN/G aN and I nN/GaN superiattices.

Compounds Band gaps (Eg)

Our work Ref [ 73] Ref [32]

AIN/GaN 3 . 1 8 3 . 1 3 .0 I nN/GaN 0.46 0.4 -

:> <lJ 5 :;:: � <lJ C LU

-5 r M K A [' !1 A

wave vector

Fig. 22a: The band structure of AIN/GaN I x i superlatt ic

:> 5 <lJ :;:: � <lJ C

r M K A r t:J. A wave vector

Fig. 22b: The band structure of InN/GaN I x l superlattic

5.2.2 Optical properties

The d ie lectric funct ions of A IN/GaN and I nN/GaN superlattices are resolved i nto two

components 8x)(W) average of the spectra for the polarizat ion along the x and y

d i rect ions and 5; the polarization paral lel to the z-di rection.

Figs. 23a and 23b shows the variat ion of the imaginary part of the e lectron ic d ie lectric

funct ion for the I I I -n itr ide based superlattices, for rad iation up to 30 eV. The calculated

resu l ts are r igid ly sh ifted upwards by 1 .64 eV in AIN/GaN and 1 .23 in I nN/GaN

superlattices. I t i s seen that 82 i s rather s im i lar for the two SLs. The main feature is a

broad peak with a maximum around 7 .5 eV for AIN/GaN and 5 .94 eV for I nN/GaN .

Some features are i n common. There are three groups of peaks, the first group i s in (6.3

e V - 8 . 1 e V) photon energy range, they are mainly due to transit ions i n the v ic in ity of M

point, th i s is usual ly assoc iated with E1 transit ion. However the L3v-L 1 c transit ion occurs

at 7 .5 eV in A I /GaN . S im i larly, the main peak i n the spectra of I nN/GaN i s s ituated at

5 .94eV. The second group of peaks (9.02 eV- I O .6 eV in A IN/GaN) and (7 .4 eV- 9. 1 eV in

I nN/Ga ) . These peaks account for the transit ion at P and L ' -point i n GaN . The l ast

group of peaks (up to 1 1 eV) is connected main ly to transit ions at r, M-point ( in GaN,

I nN) and T, M ( in AlN) .

Next, we consider the d ispersive part of the d ie lectric function, 81 , for the two

superlatt ices, see Figs . 24a and 24b. The calcu lated spectra have been obtained by

Kramers-Kron ig ' s transformation of the sh ifted 82 spectra. The main features are a

shoulder at l ower energies, a rather steep decrease between 4 and 5 eV, after which E 1

becomes negat ive, a m in imum and a s low i ncrease toward zero at h igher energies.

Final ly, we a l so brlyief d i scuss the spectra obtainable from the d ie lectric funct ion. The

calcu lated d ie lectric constants are shown in Table 6.

Table 6: The calcu lated d ie lectric constants of d i fferent superlatt ices

Compounds Cxy Cz 4 .86 4.40

AIN/GaN 4 .7 1 "" 4.62"

4. 1 9 4.2 1 InN/GaN 3.9 1"" 3

* Our work and * * Average ( I nN,AIN/GaN)

8 AIN/GaN

0 1 0 20 30 40

Energy [eV]

Fig. 23a: Variation of the imaginary part of the dielectric function for AIN/GaN superlattice

8 InN/GaN

4 OJ'"

0 1 0 20 30 40

Energy[eV]

Fig. 23b: Variation of the imaginary part of the dielectric function for InN/GaN superlattice

AIN/GaN 8

-4 0 1 0 20 30 40

Energy[eV]

Fig. 24a: Variation of the real part of the dielectric function for AIN/GaN superlattice

8 InN/GaN

-4 0 1 0 20 30 40

Energy[eV]

Fig. 24b: Variation of the real part of the dielectric function for InN/GaN superlattice

As seen in figs. 25a and 25b the refractive index was computed us ing both real and

imaginary parts of the d i e lectric function.

The parameters a and b (see equation 52) are obtained from the fit of the calcu lated

refract ive i ndex spectra for the two superlattices. A l l parameters are l isted in table 7 :

Table 7 : The Parameters u sed o r obtained by fitt ing t o thePeng-P iprek's formula for

A lN/GaN and I nN/GaN superlattices

Compounds Eg (eV) a b A lN/GaN 3 . 1 8 4 .6 8 .77 I nN/GaN 0.46 9. 1 1 2 .4

2.45 AIN/GaN

� 2 35 " .5

u > B 2.30 £ �

0.0 0.5 1 0 1 5 2.0 2.5 3 0 3 5 Energy[eV]

Fig . 25a : Variat ion of the refract ive i ndex of A lN/GaN SL versus with the energy

2.59 [nNIGaN

� 2.56 " .5 " 2.55 > " u J; � 2.54

0.0 0.2 0.4 0.6 0.8 Energy[eV]

Fig . 25b : Variat ion of the refract ive index of I nN/GaN S L versus with the energy

5.3 Concl usion

I n th i s work, we have u sed the a l l -e lectron fu l l -potential l inear-augmented p lane wave

method \ ith i n the local density approximat ion to i nvestigate the e lectron ic and optical

propert ies of the b inary compounds (GaN, AIN and I nN) and their related superIatt ices

AI /Ga and I n /GaN.

Our resu lts of the e lectron ic structure investigation of the wurtzite phase of GaN, AIN

and I n have provided add it ional deta i l s on the group- I I I n i trides. The L DA-eigenvalue

pectrum gives an exce l l ent descript ion of the band structures. The charge densit ies have

been presented and provide addit ional evidence of the s im i l arity of the bonds i n GaN,

A l and I nN as far as bond ing characters are concerned. As a resu l t of the strong ionic

character of these n i trides, i t was noticed that they share many s imi lar propert ies . On the

other hand, the opt ica l propert ies of wurtzi te n itrides have a l so been investigated. The

real and imaginary parts of the d ie lectric functions were calcu lated for polarizat ion in the

xy p l ane and a long the z-axi s as wel l . Moreover, o u r results on the band structure

invest igat ion of A IN/GaN and I nN/GaN superIattices provided add it ional deta i l s on the

I I I -V n itride- based super lattices. From the comparison between the band gaps of the

compounds and thei r related properties we notice that GaN has reduced the band gap of

AIN in superlatti ces and a l so I nN has reduced the band gap of GaN in the superlattice and

th i s leads to improve the conduction properties of these compounds.

The resul ts presented here would give substant ia l improvements on those of previous

calcu l at ions . The ca lcu l ated energy gaps agree wel l with some other theoretical

techn iques. On the other hand, the optical propert ies of AIN/GaN and I nN/GaN

superIatt ices have been investigated. We have shown that the optical properties are

excel lently reproduced us ing L DA calcu lations, if we al low for a rigid sh i ft of the band

structure, the so cal led scissors operator. Not only the general form but rather the absolute

ampl i tudes are very wel l reproduced in our calcu lat ions.

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