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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
By
Sheikha Ahmed Awad Alnaeme
In Partial Fulfillment of the Degree of M. Sc. in
Materials Science and Engineering
December
2006
Thesis Supervisor
Prof. Dr. Mohamed Assaad Abdel-Raouf
Department 0/ Physics
Faculty a/Science, UAEU
Co-Supervisor
Dr. Noureddine Amrane
Department 0/ Physics
Faculty a/Science, UAEU
Examination Committee Members
Prof. Dr. Nacir Tit
Department 0/ Physics
Faculty a/Science, UAEU
Prof. Dr. Abdallah Qteish
Yarmuk University, Irbid-Jardan
11
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
1 1 1
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
TV
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.
\'
V1
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
Vll
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
1 1 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
IX
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
x
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.
Xl
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.
6
>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
2
\ 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] .
A. 04
> 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] .
3
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
4
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 .
5
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
( 1 )
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
(2)
7
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:
(3)
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-
8
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
" (4)
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),
(7)
9
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.
1 0
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[&" '
r-r
( 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
parts
( 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)
1 1
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 )
(22)
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).
1 2
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?
Yes
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 :
(23)
where, c...c: the exchange-corre lation energy per particle of a uniform electron gas of
1 3
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.
(24)
(25)
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
1 4
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
as:
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.
1 5
\ riting each e lectronic a e function a a sum of plane wa es [42]
'1/, (r) = I ,.k+(, exp[i(k + (i)J?] "
(30)
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
1 6
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
(32)
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
1 7
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
(34)
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] .
1 8
(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
(35)
(36)
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
1 9
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]
(37)
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,
(3 8)
(39)
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.
20
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
2 1
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.
(42)
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
22
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
(43)
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] .
23
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.
25
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-
50] .
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] .
26
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]
27
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 ] .
28
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] .
29
"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 .. . . . . . . . . . . . . . . . . . . . . . . . . . . .
I I I
· · · · , 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.
3 1
- 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] .
32
s�----�--�--�----�--�----r---��
3.5
1.5
·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=O.5
x=0.75 -- x=}
a
, "
t , j :', � f: 'i . :!
b. c. into S.
2
F ig. 1 0 : Variation of the total valence charge densities for ZnxCd l -xSe with x values [53] .
33
30
� l c Q U c 2 20 u " U OJ '6 15 "0 -@ '" a. c:- 10 '" c '0. co .� "0 5 c '" <ii '" a:
0
-5 0
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!
15
"
20
Fig. 1 2 : Calcu lated real and imaginary and real parts of the die lectric function for ZnSe [54] .
34
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 ] .
35
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.
37
(d) z H A
t> C
7-- K r y
.
(b)
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]
38
(45)
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]
(46)
\ 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]
(47)
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 .
(48)
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
39
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] .
40
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
4 1
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
42
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.
44
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
2.32b
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]
1 0
5 5' Q) � e> 0 Q) c
w
-5
- 1 0
- 1 5 [' M K A [' !'J.
wave vector
Fig. 1 5a: Band Structure of wurtzite GaN
45
- 10
-1 5
A wave vector
Fig. I Sb : Band Structure of wurtzite AIN
1 0
5 > a> 0 >: 0)
... a> c:
w -5
- 10
- 1 5 r M K A r tJ.
wave vector
Fig. I Sc : Band Structure of wurtzite I nN
46
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.
47
8
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 ... � �
2 f-
o t-
0
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
48
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
:::::--
Q �
� � =-
@ \�\ ---------- =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
49
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.
6
5
4 Q) � .....
3 VJ ......
0 .0 VJ 2 c Q)
0
o
- 15 - 10 -5 o 5 1 0 Energy[eV]
Fig. 1 8a: The density of states for w-GaN
50
'" E 5 '" '-0
.f' '" c: (!)
a
" � o .� g Cl
8 r---------------------------�
6
4
2
o
-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
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.
52
F ig. 1 9a: Variation of the imaginary part of the d ie lectric function for w-GaN
N W
1 0 ,-------------------------, AIN
8
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
53
8 InN
6
4
or
2
0
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
54
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
8
6
4
w- 2
0
-2
-4
0 10 20 30 40 Energy[eVI
Fig. 20a: Variation of the real part of the dielectric function for w-GaN
55
10
-2
10 20 energy(eV)
AIN
JO 40
Fig. 20b: Variation of the real part of the dielectric function for w-AIN
1 0 InN
8
6
4
w 2
0
-2
-4
0 1 0 20 30 40
Energy ! eVI
Fig. 20c : Variation of the real part of the d ielectric function for w-InN
56
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
57
2.50 GaN
2.45
... '" '" .= ... -£ 2AO
v '" ..: .. c::
l.JS
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
2.5
AIN
H
� ." 2.3 .s .� J1 .: .. " 2.2
2.1
Fig. 2 1 b: Variation of the refract ive index of AIN versus the energy
58
'" Q) "0 .E Q) > U £ Q) �
3.006
3 .004 I n N
3.002
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
59
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 -
60
1 0
:> <lJ 5 :;:: � <lJ C LU
0
-5 r M K A [' !1 A
wave vector
Fig. 22a: The band structure of AIN/GaN I x i superlatt ic
10
:> 5 <lJ :;:: � <lJ C
LU o
-5
r M K A r t:J. A wave vector
Fig. 22b: The band structure of InN/GaN I x l superlattic
6 1
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.
62
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
.77""
* Our work and * * Average ( I nN,AIN/GaN)
8 AIN/GaN
6
4 or
2
0
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
6
4 OJ'"
2
0
0 1 0 20 30 40
Energy[eV]
Fig. 23b: Variation of the imaginary part of the dielectric function for InN/GaN superlattice
63
1 0
AIN/GaN 8
6
4
.,- 2
0
-2
-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
6
4
.,- 2
0
-2
-4 0 1 0 20 30 40
Energy[eV]
Fig. 24b: Variation of the real part of the dielectric function for InN/GaN superlattice
64
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
2.40
� 2 35 " .5
u > B 2.30 £ �
2 25
2.20
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
2.58
2.57
� 2.56 " .5 " 2.55 > " u J; � 2.54
2.53
2.52
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
65
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
66
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.
67
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