An ab initio study of the electronic and dynamical properties of Si

167 Pages·2009·11.04 MB·English

UNIVERSITEIT ANTWERPEN
Faculteit Wetenschappen
Departement Fysica
An ab initio study
of the electronic and dynamical properties
of Si, Ge and Si/Ge nanowires
Een ab initio studie
van de elektronische en dynamische eigenschappen
van Si, Ge en Si/Ge nanodraden
Proefschrift voorgelegdtot het behalen van de graad van doctor in de
wetenschappen aan de Universiteit Antwerpen te verdedigen door
Hartwin Peelaers
Promotor: Prof. dr. B. Partoens
Co-promotor: Prof. dr. F. M. Peeters
Antwerpen 2009

Contents
Acknowledgements iv
List of abbreviations vii
1 Introduction 1
1.1 Nanowire growth . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Top-down approach . . . . . . . . . . . . . . . . . . . . 2
1.1.2 Bottom-up approach . . . . . . . . . . . . . . . . . . . 3
1.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.1 Nanowire FETs . . . . . . . . . . . . . . . . . . . . . . 8
1.2.2 Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.3 Logicdevices, solarcellsandnanoelectronicpowersources 10
1.3 Organization of the thesis . . . . . . . . . . . . . . . . . . . . 12
2 Density Functional Theory 15
2.1 Basic equations . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 The Hohenberg-Kohn theorems . . . . . . . . . . . . . . . . . 17
2.2.1 The first Hohenberg-Kohn theorem . . . . . . . . . . . 17
2.2.2 The second Hohenberg-Kohn theorem . . . . . . . . . . 19
2.2.3 The Levy-Lieb constrained search formulation . . . . . 20
2.3 Kohn-Sham equations . . . . . . . . . . . . . . . . . . . . . . 20
2.4 The exchange-correlation energy . . . . . . . . . . . . . . . . . 23
2.4.1 The exchange-correlation hole . . . . . . . . . . . . . . 23
2.4.2 Local Density Approximation . . . . . . . . . . . . . . 27
2.4.3 The Generalized Gradient Approximation . . . . . . . 29
2.5 Hellmann-Feynman theorem . . . . . . . . . . . . . . . . . . . 29
2.6 Practical calculations . . . . . . . . . . . . . . . . . . . . . . . 30
2.6.1 Crystal lattice and reciprocal space . . . . . . . . . . . 31
2.6.2 Bloch theorem . . . . . . . . . . . . . . . . . . . . . . . 32
2.6.3 Basis sets . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.6.4 Band structures and k-point grids . . . . . . . . . . . . 34
2.6.5 Pseudopotentials . . . . . . . . . . . . . . . . . . . . . 35
3 Electronic properties of B and P doped semiconductor nano-
wires 39
3.1 Silicon nanowires . . . . . . . . . . . . . . . . . . . . . . . . . 39
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CONTENTS
3.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 39
3.1.2 Computational details . . . . . . . . . . . . . . . . . . 41
3.1.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2 Germanium nanowires . . . . . . . . . . . . . . . . . . . . . . 47
3.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.3 Experimental verifications . . . . . . . . . . . . . . . . . . . . 51
3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4 Density Functional Perturbation Theory 55
4.1 Phonons: Introduction . . . . . . . . . . . . . . . . . . . . . . 55
4.2 Phonons from lattice dynamics . . . . . . . . . . . . . . . . . 56
4.3 Practical phonon calculations . . . . . . . . . . . . . . . . . . 57
4.3.1 Frozen phonons . . . . . . . . . . . . . . . . . . . . . . 57
4.3.2 Density functional perturbation theory . . . . . . . . . 57
5 Phonons in semiconductor nanowires 61
5.1 Silicon nanowires . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.2 Germanium nanowires . . . . . . . . . . . . . . . . . . . . . . 68
5.3 Core-shell nanowires . . . . . . . . . . . . . . . . . . . . . . . 73
5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6 GW approximation: theory 81
6.1 The bandgap problem . . . . . . . . . . . . . . . . . . . . . . 81
6.2 Green’s functions . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.3 Dyson equations and selfenergies . . . . . . . . . . . . . . . . 87
6.4 Hedin equations . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.5 Practical G W calculations . . . . . . . . . . . . . . . . . . . 98
0 0
7 Full quasi-particle corrected band structures of Si and Ge
nanowires 103
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7.2 LDA convergence . . . . . . . . . . . . . . . . . . . . . . . . . 104
7.3 GW convergence . . . . . . . . . . . . . . . . . . . . . . . . . 105
7.3.1 Convergence parameters . . . . . . . . . . . . . . . . . 105
7.3.2 Convergence study . . . . . . . . . . . . . . . . . . . . 107
7.4 Complete band structure . . . . . . . . . . . . . . . . . . . . . 113
7.4.1 Wannier functions . . . . . . . . . . . . . . . . . . . . . 114
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CONTENTS
7.4.2 GW corrected bands . . . . . . . . . . . . . . . . . . . 116
7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
8 Electronic structure of doped and undoped Si nanowires us-
ing a hybrid functional 127
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
8.2 Computational details . . . . . . . . . . . . . . . . . . . . . . 128
8.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
8.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
9 Conclusions and outlook 137
10 Samenvatting 141
Curriculum Vitae 144
List of publications 145
Bibliography 147
iii

On this page I wish to express my gratitude to my promotor Prof. B.
Partoens and to my co-promotor Prof. F.M. Peeters, as this thesis would not
be the thesis you are reading today without their support, their willingness to
answer all questions I had, and the fruitful discussions I had with them.
I also want to thank the people I collaborated with: those from Prof. X.
Gonze’s group in Louvain-la-Neuve and from Prof. Ph. Ghosez’s group in
Li`ege.
My thanks also go out to my colleagues from the CMT group in Antwerp,
to the visitors there, and especially to my roommates.
Finally, I also want to thank my parents and brothers for supporting me
during my studies.

List of abbreviations
Abbreviation or symbol
1D (2D, 3D) one-(two-, three-) dimensional
BZ Brillouin zone
CVD chemical vapor deposition
DB dangling bond
DFPT density functional perturbation theory
DFT density functional theory
DOS density of states
FET field effect transistor
GGA generalized gradient approximation
HOMO highest occupied molecular orbital
HK Hohenberg-Kohn
IBZ irreducible Brillouin zone
IFC interatomic force constant
KS Kohn-Sham
LDA local density approximation
LED light emitting diode
LSDA local spin density approximation
LUMO lowest unoccupied molecular orbital
MBE molecular beam epitaxy
MOSFET metal-oxide-semiconductor field effect transistor
MLWF maximally localized Wannier function
NW nanowire
PAW projected augmented wave
RPA random phase approximation
SEM scanning electron microscopy
TEM tunneling electron microscopy
VLS vapor-liquid-solid
VSS vapor-solid-solid
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