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Investigations on the Origin of Brown Colouration in
Diamond
Luke Sebastian Hounsome
March 29, 2007
Submitted by Luke Sebastian Hounsome, to the University of Exeter as a thesis for the
degree of Doctor of Philosophy in Physics, March 2007.
This thesis is available for Library use on the understanding that it is copyright material
and that no quotation from the thesis may be published without proper acknowledgement.
I certify that all material in this thesis which is not my own work has been identified and
that no material has previously been submitted and approved for the award of a degree
by this or any other University.
— Luke Hounsome
i
Oh, Diamond! Diamond! thou little knowest what mischief thou hast done!
Isaac Newton (1642 – 1727)
ii
Abstract
This thesis documents theoretical abinitio calculations undertaken with the purpose of
discovering the source of brown colouration in diamond. Brown diamond has a unique
absorption continuum across the bandgap, with no detectable onset at low energies. The
source of colour is not known, but is vital for understanding the loss of colour which occurs
during HighPressure HighTemperature (HPHT) treatment. It can be surmised that the
defect responsible for the colour introduces a broad spectrum of energy levels into the
bandgap, allowing a range of electron transitions to occur.
Small vacancy clusters and chains are identified in diamond by Electron Paramagnetic
Resonance (EPR) studies. Modelling of these defects reveals that the vacancy clusters
are the more energetically favourable defects in all cases. In many of the defects the lowest energy spin state is S =1, which introduces bandgap levels due to unpaired electrons.
The levels are clustered at the midgap and so will not induce a broad optical absorption.
Dislocations are a prevalent defect in natural diamond, with optically active types proposed as a source of brown colour. The colour removal would come from transformation
to optically inactive types, with energy supplied by the HPHT treatment. Two dislocation
types are studied, the 90◦ glide and the 90◦ shuffle types. The formation energy of the
glide type is much lower that the shuffle type, indicating that it is a more stable defect. The
shuffle type has dangling bonds which introduce absorption throughout the gap. However
the density of shuffle dislocations required to match experimental observations is larger
than the observed density in natural diamond.
Networks of π bonds can form on the (111) surface of diamond, introducing π and π ∗
states into the bandgap. These states cause a wide range of electron transitions to be
available, potentially leading to broad optical absorption. The (111) surface is recreated
internally by removal of the α B double plane, leaving a vacancy disk. This gives a large
area of π bonds, and is found to be a stable defect, with an energy much lower than small
vacancy clusters. The optical absorption of the disk is a close match for experimental
measurements on type IIa diamond. There is also a close match to Raman frequencies
detected in brown diamond.
Two different mechanisms for removal of the brown colour are proposed. In CVD diamond
iii
there is a large amount of hydrogen incorporated during the growth process which can
move to the disk surface, passivating the π bonds and removing any optical absorption.
In natural diamond there is nothing to check the growth of the disk until it collapses into a
stacking fault bounded by a dislocation loop. These different mechanisms could account
for the different annealing temperatures required for natural and CVD diamond.
The optical absorption of brown type Ia diamond is characterised by additional absorption around 550 nm (2.25 eV), so the possibility of nitrogen interacting with the disk to
modify the absorption is investigated. The most stable configuration for nitrogen at the
disk surface is a nitrogen pair separated into a NCCN arrangement. If the density of
nitrogen at the surface is small enough to allow sizable chains of π bonded carbon to
form, then a broad absorption is induced, with a peak at 2.3 eV in excellent agreement
with experiment. It can be surmised that vacancy disks are a strong candidate for the
source of brown colour in both type IIa and type Ia diamond.
iv
Acknowledgements
This thesis, and indeed all the work undertaken for my Ph.D. would not have been possible without the support of many people. I must firstly give my thanks to my supervisor,
Prof. Bob Jones, who has guided and educated me over the last three years. His ability
to recall any measured value and tell you in what paper it was published continues to
impress. If it were not for his inspiration to publish and present my work I am sure my
studies would hold much less value. I hope he feels that the response to the ageold
question ‘any progress’, can now definitely be ‘yes’!
As well as guidance from my supervisor, my studies have been made easier by the help
of the rest of the AIMPRO group. Whether it be a deep discussion of formation of defects
in semiconductors, useful scripts to perform tasks (Steve!) or just how to change my
desktop background someone was always able to help. So thanks to Steve, James,
Thomas, Naomi, Colin, Alexandra, Patrick, Malcolm and all the others.
Aside from my immediate colleagues I must thank my friends inside and outside the
department for providing non workrelated support. Steve H’s dedication to teadrinking
always gave the opportunity for a useful break, and working in the Ram Bar took my
mind off endless calculations. Nomads CC will give me many fond memories of summer
evenings chasing a small red ball! In no particular order then, thanks Paul, John, Matt,
Martin, Tristan, Leigh, Prof. G.P. Srivastava, Nathan, Pete V, Ian, P.C, Bex, Gobby, Helen,
Claire, Tom and anyone I’ve overlooked.
Financial support for this work has been provided by the EPSRC and a CASE award from
DTC. In addition Philip Martineau and David Fisher from DTC have been very helpful in
providing experimental data and useful discussion.
Thanks must go to my family for their unending encouragement, and most importantly to
my fiance´ Catherine. She has always been there when I needed her, not least in proofreading all 180 pages! Completion of this thesis is only the secondmost important event
of this year. Thank you Catherine.
v
List of Publications
First author
• Optical properties of vacancy related defects in diamond
¨
L. S. Hounsome, R. Jones, P. M. Martineau, M. J. Shaw, P. R. Briddon, S. Oberg,
A. T. Blumenau, and N. Fujita
Physica Status Solidi (a) 202, 2182 (2005).
• Origin of brown colouration in diamond
L. S. Hounsome, R. Jones, P. M. Martineau, D. Fisher, M. J. Shaw, P. R. Briddon,
¨
and S. Oberg
Physical Review B 73, 125203 (2006).
• Photoelastic constants in diamond and silicon
L. S. Hounsome, R. Jones, M. J. Shaw, P. R. Briddon
Physica Status Solidi (a) 203, 3088 (2006).
Coauthor
• Combined TEM and STEM study of the brown colouration of natural diamonds
R. Barnes, U. Bangert, P. M. Martineau, D. Fisher, R. Jones and L. S. Hounsome
Journal of Physics: Conference Series 26, 157 (2006).
• Electron energy loss spectroscopic studies of brown diamonds
U. Bangert, R. Barnes, L. S. Hounsome, R. Jones, A. T. Blumenau, P. R. Briddon,
¨
M. J. Shaw, and S. Oberg
Philosophical Magazine A 86, 4757 (2006).
Contents
1 Introduction
1
1.1 Properties of Diamond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1.1 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.1.2 Electronic structure
. . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.1.3 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.2 Defects in Diamond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
1.2.1 Point defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
1.2.2 Extended defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
1.3 Brown Colouration in Diamond . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2 Theory of Modelling Crystals
23
2.1 Fundamental Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 First Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3 Wavefunction Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3.1 Hartree’s method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3.2 Hartree–Fock theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.4 Density Functional Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.4.1 The ThomasFermiDirac scheme . . . . . . . . . . . . . . . . . . . 33
2.4.2 The HohenbergKohnSham scheme . . . . . . . . . . . . . . . . . . 33
2.5 Application Using AIMPRO . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.5.1 Pseudopotentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.5.2 Basis sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.5.3 Brillouin zone sampling . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.6 Calculation of Material Properties . . . . . . . . . . . . . . . . . . . . . . . . 42
2.6.1 Total energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.6.2 Formation energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.6.3 Vibrational modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
vi
CONTENTS
vii
2.6.4 Mulliken bond population . . . . . . . . . . . . . . . . . . . . . . . . 47
2.6.5 Electron energyloss . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.7 Density Functional TightBinding Theory . . . . . . . . . . . . . . . . . . . . 50
2.8 Screened Exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3 Experimental Techniques
55
3.1 Transmission Electron Microscopy . . . . . . . . . . . . . . . . . . . . . . . 55
3.1.1 Fundamental concepts
. . . . . . . . . . . . . . . . . . . . . . . . . 55
3.1.2 Transmission Electron Microscopy in practice . . . . . . . . . . . . . 57
3.2 Electron Energy Loss Spectroscopy . . . . . . . . . . . . . . . . . . . . . . 60
3.2.1 Fundamental concepts
. . . . . . . . . . . . . . . . . . . . . . . . . 60
3.3 Optical Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.3.1 Fundamental concepts
. . . . . . . . . . . . . . . . . . . . . . . . . 62
3.3.2 Optical Absorption in practice . . . . . . . . . . . . . . . . . . . . . . 63
3.4 Raman Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.4.1 Fundamental concepts
. . . . . . . . . . . . . . . . . . . . . . . . . 65
3.4.2 Raman Spectroscopy in practice . . . . . . . . . . . . . . . . . . . . 68
3.5 Photoluminescence Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . 69
3.5.1 Fundamental concepts
. . . . . . . . . . . . . . . . . . . . . . . . . 69
3.5.2 Photoluminescence Spectroscopy in practice . . . . . . . . . . . . . 73
3.6 Positron Annihilation Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . 73
3.6.1 Fundamental concepts
. . . . . . . . . . . . . . . . . . . . . . . . . 74
3.6.2 Positron Annihilation Spectroscopy in practice . . . . . . . . . . . . . 75
3.7 Electron Paramagnetic Resonance . . . . . . . . . . . . . . . . . . . . . . . 76
3.7.1 Fundamental concepts
. . . . . . . . . . . . . . . . . . . . . . . . . 76
3.7.2 Electron Paramagnetic Resonance in practice . . . . . . . . . . . . . 78
3.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4 Small Vacancy Clusters
81
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.2 Structures Modelled . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.3.1 Energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.3.2 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.3.3 Electronic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.3.4 Optical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
CONTENTS
viii
4.3.5 π bonded carbon in multivacancies . . . . . . . . . . . . . . . . . . . 92
4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5 EELS of Bulk Diamond and Dislocations
99
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.2 Bulk Diamond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.2.1 Calculated results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.3 Experimental Results
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.4 Dislocations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.4.1 90◦ Glide dislocation . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.4.2 90◦ Shuffle dislocation . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6 (111) Plane Vacancy Disks
113
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.2 Simple Vacancy Disks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.2.1 Structures and energies . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.2.2 EELS and optical absorption . . . . . . . . . . . . . . . . . . . . . . 118
6.2.3 Electronic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.2.4 Removal of colour . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.3 Nitrogen at Vacancy Disks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.3.1 Structures and energies . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.3.2 Boron interaction with vacancy disks . . . . . . . . . . . . . . . . . . 138
6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
7 Photoelastic Constants in Diamond and Silicon
142
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
7.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
7.2.1 Silicon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
7.2.2 Diamond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
7.3 Strain of Dislocations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
7.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
8 Conclusions
153
8.1 Small Vacancy Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
8.2 EELS of Dislocations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
8.3 (111) Plane Vacancy Disks . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
CONTENTS
ix
8.4 Photoelastic Constants of Diamond . . . . . . . . . . . . . . . . . . . . . . . 157
8.5 Continuation Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
List of Tables
1.1 The types and subtypes of diamond. . . . . . . . . . . . . . . . . . . . . . .
7
1.2 Point defects in diamond, and their known properties. . . . . . . . . . . . .
8
4.1 Formation energies of n vacancy defects, using AIMPRO. . . . . . . . . . . . 86
4.2 Formation energies of chains of n vacancies calculated using DFTB. . . . . 87
4.3 Formation energies of clusters of n vacancies calculated using DFTB. . . . 87
6.1 Relative energies of hydrogen at diamond surfaces. . . . . . . . . . . . . . 123
6.2 Energies of nitrogenvacancy defects. . . . . . . . . . . . . . . . . . . . . . 127
6.3 Total energies of nitrogen configurations at the disk surface, compared to
the nitrogen Acentre. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
7.1 Effect of scissors shift on the calculated dielectric constant and the photoelastic constants for silicon.
. . . . . . . . . . . . . . . . . . . . . . . . . 144
7.2 Effect of scissors shift on the real part of the calculated dielectric constant
and the calculated photoelastic constants for diamond. . . . . . . . . . . . . 147
x
List of Figures
1.1 The FCC lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.2 The diamond structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.3 Bandstructure of bulk diamond. . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.4 Bandstructure of bulk diamond using screened exchange correction. . . . .
6
1.5 The Edge dislocation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1.6 The Screw dislocation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.7 Glide and Shuffle planes in diamond. . . . . . . . . . . . . . . . . . . . . . . 11
1.8 Dislocation structures in diamond I. . . . . . . . . . . . . . . . . . . . . . . . 12
1.9 Dislocation structures in diamond II. . . . . . . . . . . . . . . . . . . . . . . 13
1.10 Kink movement causing dislocation glide. . . . . . . . . . . . . . . . . . . . 14
1.11 Removal of colouration by HPHT treatment. . . . . . . . . . . . . . . . . . . 15
1.12 Vis/UV absorption of type IIa brown diamond. . . . . . . . . . . . . . . . . . 17
1.13 IR absorption of type IIa brown diamond. . . . . . . . . . . . . . . . . . . . 17
1.14 Vis/UV absorption of type IIa CVD diamond. . . . . . . . . . . . . . . . . . . 18
1.15 Vis/UV absorption of type Ia brown diamond. . . . . . . . . . . . . . . . . . 19
1.16 IR/Vis/UV absorption of a type IIa pink diamond. . . . . . . . . . . . . . . . 20
3.1 Schematic diagram of a highresolution TEM. . . . . . . . . . . . . . . . . . 58
3.2 Serial and parallel methods of EELS . . . . . . . . . . . . . . . . . . . . . . 61
3.3 Prism spectrograph and Grating monochromator. . . . . . . . . . . . . . . . 64
3.4 Vibrational energy level transitions . . . . . . . . . . . . . . . . . . . . . . . 68
3.5 Dispersive and FT Raman spectroscopy . . . . . . . . . . . . . . . . . . . . 70
3.6 Photoluminescence in a directgap semiconductor. . . . . . . . . . . . . . . 71
3.7 Photoluminescence in a indirectgap semiconductor. . . . . . . . . . . . . . 72
3.8 Exciton effects in Photoluminescence. . . . . . . . . . . . . . . . . . . . . . 72
3.9 Experimental measurement of PL. . . . . . . . . . . . . . . . . . . . . . . . 73
3.10 Positron annihilation reaction schematic. . . . . . . . . . . . . . . . . . . . . 75
3.11 PAS experimental setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
xi
LIST OF FIGURES
xii
3.12 Zeeman splitting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.13 Experimental EPR setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.1
110 vacancy chains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.2 Small vacancy clusters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.3 Formation energies of vacancy clusters. . . . . . . . . . . . . . . . . . . . . 85
4.4 Multivacancy KohnSham levels I . . . . . . . . . . . . . . . . . . . . . . . . 92
4.5 Multivacancy KohnSham levels II
. . . . . . . . . . . . . . . . . . . . . . . 93
4.6 Multivacancy KohnSham levels III . . . . . . . . . . . . . . . . . . . . . . . 94
4.7 Multivacancy KohnSham levels IV . . . . . . . . . . . . . . . . . . . . . . . 95
4.8 Multivacancy KohnSham levels V . . . . . . . . . . . . . . . . . . . . . . . 96
4.9 Absorption spectra for selected multivacancy defects . . . . . . . . . . . . . 97
5.1 Variation of dielectric function with MP sampling grid. . . . . . . . . . . . . . 101
5.2 Variation of dielectric function with number of basis set functions. . . . . . . 101
5.3 Variation of dielectric function with polynomial broadening. . . . . . . . . . . 102
5.4 The calculated dielectric functions for bulk diamond compared to experiment.102
5.5 Calculated and experimental absorption coefficient of bulk diamond. . . . . 104
5.6 Experimental EELS of colourless diamond. . . . . . . . . . . . . . . . . . . 104
5.7 Calculated EEL spectra of bulk diamond and graphite. . . . . . . . . . . . . 105
5.8 EEL spectra of brown, colourless and graphitic diamond. . . . . . . . . . . . 106
5.9 Structure of the 90◦ glide dislocation. . . . . . . . . . . . . . . . . . . . . . . 107
5.10 Bandstructure of the 90◦ glide dislocation. . . . . . . . . . . . . . . . . . . . 108
5.11 EELS spectrum and absorption of the 90 ◦ glide dislocation. . . . . . . . . . 108
5.12 Structure of the 90◦ shuffle dislocation. . . . . . . . . . . . . . . . . . . . . . 109
5.13 Bandstructure of the 90◦ shuffle dislocation. . . . . . . . . . . . . . . . . . . 110
5.14 EELS spectrum and absorption of the 90 ◦ shuffle dislocation. . . . . . . . . 111
6.1 Bandstructure of the (001)(2×1) surface of diamond. . . . . . . . . . . . . 114
6.2 Bandstructure of the (111)(2×1) surface of diamond. . . . . . . . . . . . . 115
6.3 Formation of the (111) plane vacancy disk in diamond. . . . . . . . . . . . . 116
6.4 Bandstructure of the (111) plane vacancy disk in diamond. . . . . . . . . . 117
6.5 Bandstructure of the (111) plane vacancy disk using screened exchange. . 118
6.6 EELS of the (111) vacancy disk and bulk diamond. . . . . . . . . . . . . . . 119
6.7 Optical absorption of the (111) vacancy disk. . . . . . . . . . . . . . . . . . 120
6.8 Optical absorption of the (111) vacancy disk, on a loglog scale.
. . . . . . 121
6.9 Bandstructure of the (111) vacancy disk with hydrogen termination. . . . . . 124
LIST OF FIGURES
xiii
6.10 Energy of dislocation loop compared to a (111) plane vacancy disk. . . . . 125
6.11 Bandstructure of an Acentre near to a vacancy disk. . . . . . . . . . . . . . 128
6.12 Bandstructure of a Ccentre near to a vacancy disk. . . . . . . . . . . . . . 129
6.13 Nitrogen pair at the (111) plane vacancy disk surface. . . . . . . . . . . . . 129
6.14 Nitrogen pair separated by one carbon atom, at the (111) plane vacancy
disk surface.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.15 Nitrogen pair separated by two carbon atoms, at the (111) plane vacancy
disk surface.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.16 Nitrogen pair separated by three carbon atoms, at the (111) plane vacancy
disk surface.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.17 Isolated nitrogen at the (111) plane vacancy disk surface. . . . . . . . . . . 130
6.18 Alternative reconstructions of a π bonded Pandey chain. . . . . . . . . . . . 132
6.19 Nitrogen pair at the (111) plane vacancy disk surface. . . . . . . . . . . . . 132
6.20 Nitrogen pair separated by one carbon atom, at the (111) plane vacancy
disk surface.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.21 Nitrogen pair separated by two carbon atoms, at the (111) plane vacancy
disk surface.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.22 Nitrogen pair separated by three carbon atoms, at the (111) plane vacancy
disk surface.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.23 Isolated nitrogen at the (111) plane vacancy disk surface. . . . . . . . . . . 133
6.24 Bandstructure of the NCCN defect at the surface of a (111) plane vacancy disk I. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.25 Bandstructure of the NCCN defect at the surface of a (111) plane vacancy disk II. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.26 Bandstructure of the NCCN defect with a 50% coverage on the (111)
plane vacancy disk. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.27 Absorption of the NCCN defect at the surface of a (111) plane vacancy
disk. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.28 Absorption of the NCCN defect at the (111) vacancy disk, at 50% density. 138
6.29 Bandstructure of single boron near to a (111) plane vacancy disk.
. . . . . 139
6.30 Bandstructure of single boron at the surface of a (111) plane vacancy disk. 140
7.1 Photoelastic constants for silicon. . . . . . . . . . . . . . . . . . . . . . . . . 145
7.2 Photoelastic constants for silicon III. . . . . . . . . . . . . . . . . . . . . . . 146
7.3 Photoelastic constants for silicon IV. . . . . . . . . . . . . . . . . . . . . . . 147
7.4 Photoelastic constants for diamond. . . . . . . . . . . . . . . . . . . . . . . 148
LIST OF FIGURES
xiv
7.5 Principal directions and polarizer relation. . . . . . . . . . . . . . . . . . . . 149
7.6 Polarizer and slip direction relation. . . . . . . . . . . . . . . . . . . . . . . . 150
7.7 Contours of constant diffraction intensity. . . . . . . . . . . . . . . . . . . . . 151
Chapter 1
Introduction
Carbon is a common element in nature, occurring naturally in three pure forms and in
many compounds such as carbon dioxide (CO 2 ), or methane (CH4 ). Its small size and
affinity for bonding with other elements mean it is found in solids, liquids and gases. This
prolificacy leads to the concept of carbonbased life, where all the elements that lifeforms
require are supplied in carboncontaining molecules. Indeed, no lifeform has ever been
observed that is not reliant on carbon in some way.
In nature, carbon is most commonly found in the allotropes of graphite, diamond and
amorphous carbon. Graphite is the most abundant allotrope as it is the most thermodynamically stable i.e. it has the lowest energy per atom at low temperatures and pressures.
Diamond is less stable, but once formed the barrier to transformation to graphite is prohibitively large, hence carbon can remain in the diamond form for millenia. Amorphous
carbon, such as soot, is generally just formed of small graphite particles. Further solid
allotropes may be artificially synthesised, notably the buckminsterfullerene C 60 [1, 2] and
carbon nanotubes [3, 4], but also C20 , C80 and others.
1.1
Properties of Diamond
The main properties of diamond are wellknown to nonscientists, even if the physical
causes are not. Diamond is extremely hard [5], with a value of 10 on the Moh scale,
due to the strong covalent bonding between carbon atoms. The actual value of hardness
1
CHAPTER 1. INTRODUCTION
2
depends on the crystal face, with a value of 137 GPa for the (100) face and a value of
167 GPa for the (111) face [6]. The bulk modulus is 442 GPa [7]. Diamond (especially
when cut) has a distinct sparkle arising from its large value of refractive index (∼2.4), and
exhibits fire; the display of a range of colours arising from a dispersion value of ∼0.04.
Diamond’s further unique properties are less wellknown outside its field of research. It
has a high thermal conductivity, which is unusual for electrical insulators. Diamond has
no free electrons, unlike metals, but the strong covalent bonds and low atomic mass
allow thermal energy to be transmitted quickly. This leads to a thermal conductivity of
895–2300 W m−1 K−1 at 300 K [8], depending on the purity. This value is to be compared
to the low value of silicon at 148 W m−1 K−1 and of copper at 401 W m−1 K−1 , both at
300 K (although copper has a higher conductivity at very low temperatures [9]).
The value of the thermal conductivity depends on the diamond type (section 1.1.3) and
the temperature [10]. Type I diamonds have the lowest thermal conductivity, which has
been linked to the nitrogen content [11] as phonons can be scattered by point defects [12].
This proposal is consistent with the observed thermal conductivity for type IIa, which
has the highest values. The conductivity is strongly dependent on temperature, with a
maximum value for type IIa reached at ∼70 K. The magnitude at 70 K is approximately 9
times larger than at 300 K, and approximately 17 times larger than at 10 K [9].
1.1.1
Structure
The structure of all crystals can be described by the combination of a lattice and a basis.
The lattice is a regular array of imaginary points in space, akin to a grid. The basis is the
relation of the real atomic positions to the lattice points. The diamond structure (which is
also the structure of silicon and germanium) consists of the Face Centred Cubic (FCC)
lattice (figure 1.1), with lattice vectors of (1/2 1/2 0), (1/2 0 1/2) and (0 1/2 1/2), plus a
twoatom basis of (0 0 0) and (1/4 1/4 1/4). The basis notation denotes that there is an
atom at each lattice point, and one at 1/4 x + 1/4 y + 1/4 z relative to each lattice point,
where x, y and z are the conventional unit cell vectors. This gives any atom 4 nearest
neighbours and 12 nextnearest neighbours.
Figure 1.2 shows the full diamond structure, obtained when the twoatom basis is added
˚
to the FCC lattice. The value a0 is the lattice constant, which has a value of 3.566 A
CHAPTER 1. INTRODUCTION
3
z
y
x
Figure 1.1: The Face Centred Cubic lattice, with lattice points represented by black
spheres. One of the facecentred points is denoted with the vector relative to the corner point.
a0
Figure 1.2: The diamond structure, with the basis atoms added to the FCC lattice. The
conventional unit cell vectors are shown in blue, and the primitive unit cell vectors in red.
at 300 K [9]. With a (0 0 0) and (1/4 1/4 1/4) basis the distance between atoms is then
√
˚ The blue arrows in the figure denote
given by 3/4 a0 , which yields a value of 1.544 A.
the vectors of the conventional unit cell of the FCC lattice, which is generally favoured
over the primitive unit cell for calculations of density etc. due to its cubic structure. The
CHAPTER 1. INTRODUCTION
4
atomic planes of diamond are referenced relative to these vectors. The primitive unit
ˆ A2 = 1/2ˆi + 0ˆj + 1/2kˆ and A3 =
cell is described by the vectors: A1 = 1/2ˆi + 1/2ˆj + 0k,
ˆ These represent the minimum volume that can be repeated (along those
0ˆi + 1/2ˆj + 1/2k.
vectors) to create the diamond structure, and are shown in red in figure 1.2.
The conventional unit cell contains eight atoms in total; each face atom is shared between two cells and each corner between eight. The volume of the unit cell is a 30 , which
˚ 3 or 45.346×10−30 m3 . Therefore there are 1.764×1029 atoms m−3 , or as it
is 45.346 A
is more usually expressed, 1.764×10 23 atoms cm−3 . This density is greater than germanium and silicon which have densities of ∼5×10 22 atoms cm−3 [9].
1.1.2
Electronic structure
60
Filled States
Empty States
50
40
Energy (eV)
30
20
10
0
10
20
30
L
Γ
X
Brillouin zone position
K
Γ
Figure 1.3: The electronic bandstructure of diamond, with the valence band top set to
0 eV. Conduction band energies are scaled up by 1.3 to compensate for the bandgap
underestimation of LDA.
Diamond is a semiconductor with an indirect bandgap of 5.5 eV. The indirect gap is located between the Γ point at the Brillouin zone centre and a position 3/4 towards X [9]
as seen in figure 1.3. The direct gap is at Γ and has a value of ∼7 eV. In figure 1.3, the
calculated energies of the levels above the valence band top have been scaled by a factor
1.3 which brings both the direct and indirect gaps into agreement with experiment. Such
CHAPTER 1. INTRODUCTION
5
a scaling is necessary due to the underestimation of the bandgap by the Local Density
Approximation (LDA), described in section 2.4.2. The gap to transitions covers the IRVisUV range, which is why pure diamond is colourless; colour is imparted by defects introducing gap levels, which allow photoexcited electron transitions. The indirect gap also
means that phonon assistance is required for electronic transitions, limiting diamond’s
usefulness in electronic applications unless it is doped to modify the bandstructure, as
described below.
As LDA is known to underestimate the bandgap, work has been undertaken to make
corrections using the screened exchange method [13], described in detail in section 2.8.
When the bandstructure of diamond is evaluated using this new method, the indirect gap
is found to be 5.30 eV, compared to 4.27 eV without correction (LDA), and 5.5 eV experimentally. The direct gap using screened exchange is 6.60 eV, compared with 5.63 eV in
LDA, and ∼7 eV experimentally. Using the previously mentioned 1.3 scaling factor for the
LDA bandgap gives an indirect gap of 5.55 eV and a direct gap of 7.32 eV. While such a
scaling corrects for the indirect gap it overestimates the direct gap, whereas the screened
exchange obtains 96% accuracy for the indirect gap and 94% for the direct, which are
more accurate relative to each other. A bandstructure using screened exchange correction is shown with an unscaled LDA bandstructure for comparison in figure 1.4.
As described above, the vast majority of diamond is electrically insulating, due to the
lack of free electrons. However type IIb diamonds are ptype semiconducting due to
the presence of boron impurities. Boron is a group 3 element, with one less electron
than carbon, and so introduces an acceptor level close to the diamond valence band. It
is energetically favourable for an electron from the valence band to occupy the acceptor
level, which leaves a hole in the valence band, thus maintaining electrical neutrality. However, while the electron is spatially confined to the boron atom, the hole is free to travel
through the diamond structure and thus there is an effective excess of positive charge
carrying quasiparticles. Such type IIb diamonds are rare in nature and are highly valued
due to their blue colour. Type IIb diamonds can be produced artificially by either Chemical Vapour Deposition (CVD) growth techniques [14], HighPressure HighTemperature
(HPHT) growth [15], or implantation [16].
Ntype semiconducting diamond does not occur in nature, but is required for electronics
applications; e.g. a transistor constructed from alternating layers of electronconductive
(ntype) or holeconductive (ptype) material in an npn (or pnp) arrangement. Thus both
CHAPTER 1. INTRODUCTION
25
6
Screened Exchange
LDA
20
15
Energy (eV)
10
5
0
5
10
15
20
25
L
Γ
X
K
Γ
Figure 1.4: Bandstructure of bulk diamond calculated using both the screened exchange
correction and with LDA. The screened exchange corrections recreates experimental results more accurately, without the need for any scaling. The valence band maxima are
set to 0 eV.
experimental [17, 18] and theoretical [19, 20] work has focused on the growth of ntype
diamond. Ntype growth has been partially successful, using phosphorus as a dopant.
1.1.3
Classification
Diamond can be split into two types [21], which can then be further subdivided. Type I encompasses diamond that contains nitrogen in high concentrations, while type II diamond
has little or no nitrogen. The various subdivisions are listed in table 1.1.
Type Ia diamonds are most commonly found, and type IIb the least, although pink type
IIa diamonds are the rarest specimens. Artificial diamonds have the same category system applied, however certain growth methods tend to produce certain types. Nitrogen is
usually added to the gas mixture in CVD growth and hence many of these diamonds are
type Ib, although as technology improves, less nitrogen needs to be used. HPHTgrown
diamonds usually have less inclusions when nitrogen is present.
CHAPTER 1. INTRODUCTION
7
Table 1.1: The types and subtypes of diamond, with their main defects and colours.
Type
I
Subtype
aA
N concentration (atoms cm −3 )
∼ 1020
Defect centres
A
I
aB
∼ 1020
N3, H3, B
I
b
∼ 1020
C
II
a
< 1017
Dislocations
II
b
< 1017
Boron
Colour
Colourless
Brown
Pink
Yellow
Brown
Pink
Intense Yellow
Brown
Colourless
Pink
Red
Brown
Blue
Grey
Certain colours can be associated with certain types of diamond, as listed in table 1.1.
This is due to the presence of defects which are associated with the colour, such as
nitrogen defects in type I diamond. Green diamonds are produced through irradiation
(either natural or artificial) to produce isolated vacancies, which induce absorption at
1.67 eV (740 nm) [22]. Hence the diamond type is less important.
1.2
Defects in Diamond
Defects in diamond can be of two types. Point defects consist of no more than a few
atoms or vacancies, so are the order of 1 nm in size. Extended defects are much larger
(at least in one direction) and may be hundreds of nanometres in extent, to the point
where they are visible optically.
1.2.1
Point defects
Point defects in diamond can be either intrinsic or extrinsic. Intrinsic or selfdefects, are
misalignments of the diamond structure, creating vacancies or interstitials, and involve no
foreign elements. Extrinsic defects involve foreign elements e.g. nitrogen, but many de
CHAPTER 1. INTRODUCTION
8
Table 1.2: Point defects in diamond, and their known properties [23].
Defect
Structure
Symmetry
D3d
Td
C3v
C3v
C2v
C2v
C3v
C3v
Optical
Absorption
∼4 eV
>2 eV
3 eV
2.463 eV
1.257 eV
2.156 eV
1.945 eV
Donor
/Acceptor
(+/0) Ec − 4 eV
(+/0) Ec − 1.7 eV
(+/0) Ec − 2 eV

A centre
B centre
C centre
N3 centre
H3 centre
H2 centre
VN centre
W15 centre
VN+ centre
VNH centre
Boron
Neutral vacancy
Negative vacancy
Divacancy
Interstitial
NN pair
VN4
N sub
VN3
NVN
NVN−
VN
VN−
VN+
B sub
V0
V−
VV
Ci
Td
Td
Td
C2h
D2
1.673 eV
3.150 eV
2.543 eV
1.685; 4 eV
(0/) Ev + 0.37 eV
(+/0) Ec − 3.2 eV
(+/0) Ec − 3.2 eV

fects are a combination of both intrinsic and extrinsic. Depending on the electron configuration point defects may be detectable by Electron Paramagnetic Resonance (EPR) [23],
or their electronic structure may produce photoluminescence (section 3.5). The main
defects and their known properties are listed in table 1.2.
1.2.2
Extended defects
Extended defects in crystals can, like point defects, either be intrinsic or extrinsic. Unlike other semiconductors however, diamond generally only exhibits intrinsic defects in
the form of dislocations, although the presence of nitrogen in carbon interstitial platelets
on (100) planes has been detected [24]. Dislocations are present at densities of 10 8 –
109 cm−2 in natural diamond [25], but at much lower densities in CVD or HPHTgrown
artificial diamond, often only 104 cm−2 . They generally exhibit photoluminescence in the
blue or orange regions, which can be used to differentiate between natural and artificial
stones [26].
Dislocations were first suggested in a material by Mugge
¨
and Ewinr in the late nineteenth
century, in relation to metals [27]. It was interpreted that observed slip was related to the
CHAPTER 1. INTRODUCTION
9
b
(a)
(b)
Figure 1.5: A perfect edge dislocation, with the Burgers circuit shown in (a) and the
equivalent in a perfect lattice in (b). The closure of the circuit yields the Burgers vector b
perpendicular to the line direction, which is out of the page.
shearing of one portion of a crystal with respect to another, upon a plane. A dislocation
is defined by two vectors, the Burgers vector b and the line vector t. The line vector is
directed along the dislocation core. The Burgers vector is defined in a more complex
manner by imagining a closed circuit surrounding the dislocation of interest, this circuit
must finish exactly where it starts but otherwise may consist of any combination of distances and directions. If this exact circuit is then reproduced in a section of perfect crystal
structure, there will be an offset between the start and end points of the circuit, the direction and length of the vector required to close the circuit is then defined as the Burgers
vector b. Examples of the loop required to construct the Burgers vector are shown in
figures 1.5 and 1.6.
The edge and screw dislocations are the simplest to visualise and construct. The edge
dislocation (figure 1.5) is created by the insertion (or removal) of a halfplane of atoms
causing stress perpendicular to the inserted plane. The line direction t in this instance is
perpendicular to the direction of b, so the edge dislocation is also called a 90 ◦ dislocation.
The screw dislocation (figure 1.6) is caused by the offset of one region of the lattice
with respect to another region along a plane, with the magnitude of the offset an integer
multiple of the lattice constant. In contrast to the edge dislocation, it can be seen that b
is parallel to t.
In reality, many dislocations are mixtures of the edge and screw types, with Burgers vector
CHAPTER 1. INTRODUCTION
10
b
Figure 1.6: The perfect screw dislocation, with the Burgers vector b parallel to the line
direction.
at e.g. 45◦ or 60◦ to the line direction. In diamond the main slip planes are the {111} set
of planes, and the dislocations usually lie in the 110 directions. The perfect dislocation
1
2
110 allowing two basic types to form. The screw
¯ which is parallel to its line direction,
dislocation has a minimum Burgers vector of 1 [110],
in diamond has a Burgers vector
2
whereas the
60◦
dislocation, which is a combination of screw and edge types, has a
¯
Burgers vector of 12 [011].
The two atom basis of diamond effectively causes two planes to be formed in the structure, with unequal spacing. If a dislocation is formed by insertion of a halfplane of atoms
(the 60◦ dislocation), then two further discrete types of dislocation can be defined, both
with the same Burgers vector described above. The glide type of dislocation terminates
between the narrowly spaced planes, as shown in 1.7, the extra half plane terminating on
the
a
12
111 plane. The shuffle type terminates on the widely spaced planes, the
a
4
111
plane. It is notable that conversion between glide and shuffle types is achieved by insertion or removal of a halfplane of atoms [28].
The screw dislocation, described above, can also be classified as a shuffle or glide type,
with a third type know as a mixed screw. In figure 1.7 the possibility that glide and shuffle
planes run diagonally, as well as horizontally/vertically, is evident.
In his thesis A. T. Blumeanau shows several possibilities for core structures in diamond [29],
reproduced in figures 1.8 and 1.9, with kind permission. These figures show a view along
CHAPTER 1. INTRODUCTION
11
1
3
2
c
Glide
[111]
C
b
B
a
Shuffle
A
[112]
[110]
Figure 1.7: The possible dislocation configurations in diamond. (1) is the shuffle type, (2)
is the glide type and (3) is mixed.
¯ direction and a view of the [111] glide plane for each disthe dislocation line in the [110]
location. Stacking fault regions accompanying the partials are shaded. In the case of the
90◦ partial the single and double period reconstructions are denoted in the labels by SP
and DP respectively.
The insertion or removal of pairs of layers, e.g. Aa or Bb in figure 1.7, creates a stacking
fault; a change in the stacking sequence of the crystal planes. Removal of a layer creates
an intrinsic stacking fault, with the addition of a layer leading to an extrinsic stacking fault.
Stacking faults are important when considering partial dislocations which, by definition,
have Burgers vectors which are not translation vectors of the lattice. These dislocations
necessarily border a twodimensional defect, which is usually a stacking fault.
In reality perfectly straight dislocations are almost nonexistent, they contain kinks and
jogs. The potentialenergy path that an atom would have to move through can be considered as a regular series of peaks and valleys, with each extreme separated by the
Burgers vector b of the dislocation. This layout is known as the Peierls potential. This
potential defines special low energy directions in which the dislocation can lie, the dislocation running in the minimum of the Peierls potential, with the crossing kinks as short as
possible.
12
[111]
CHAPTER 1. INTRODUCTION
[110]
[112]
(b) 60° glide
(c) 60° shuffle
(e) 90° glide (SP)
(f) 90° glide (DP)
[111]
(a) shuffle screw
[110]
[112]
(d) 30° glide
Figure 1.8: Dislocation structures in diamond I. Reproduced from [29].
A kink is a step in the dislocation line which is fully contained within the glide plane,
the plane of movement of a dislocation defined by the Burgers vector. Once a kink has
crossed into a neighbouring valley it may cross back to the original valley to form a double
kink. Subject to stresses the two kinks may then move apart, causing the entire dislocation line to move, as illustrated in figure 1.10. The movement of a dislocation without
assistance and within the glide plane is known simply as glide.
Jogs can be considered as all breaks or steps in a dislocation line, but it is customary to
use it to describe steps that are not contained in the glide plane, or move from one glide
plane to another. Jogs in dislocations are prime places for the emission and absorption of
point defects e.g vacancies and interstitials, and the effect of this is to cause movement of
CHAPTER 1. INTRODUCTION
[111]
A
13
B
[110]
[112]
(b) 30° shuffle (I)
(c) 30° shuffle (I’)
(e) 90° shuffle (I)
(f) 90° shuffle (I’)
[111]
(a) 30° shuffle (V)
[110]
[112]
(d) 90° shuffle (V)
Figure 1.9: Dislocation structures in diamond II. Reproduced from [29].
the dislocation. This is known as climb movement, it does not take place in the glide plane
of the dislocation. The necessity of movement of point defects to achieve climb motion
makes this process dependent on temperature, but it is a major mechanism in allowing
dislocation lines to pass otherwise insurmountable obstacles such as foreign elements in
the lattice. Many natural diamonds (and all brown ones) exhibit plastic deformation, which
is associated with dislocation movement and thought to be the major source of vacancies
in natural diamond.
CHAPTER 1. INTRODUCTION
(a)
14
(b)
(c)
Figure 1.10: Kink movement causing dislocation glide. Initially a double kink causes
the dislocation to move into an adjacent Peierls valley (a), the kinks separate causing
more of the dislocation to cross the potential (b), finally the process continues with a new
thermallyactivated double kink (c). Peierls valleys are indicated with dashed lines.
1.3
Brown Colouration in Diamond
Diamond occurs in a wide range of colours, as well as the classic colourless type. Yellow
stones are the most common of the ’fancy’ diamonds, with blue, green, pink and red being
less common. Brown coloured diamonds make up the majority of those mined [30],
although compared to the other coloured diamonds they are rarely found in jewellery.
Those stones that do make their way to the commercial market are often yellowbrown
rather than a deep brown colour.
The origin of colour in diamond is known in some cases. Yellow diamonds contain nitrogen in various defects, with substitutional nitrogen (Ccentre) causing colour on its own,
and the Acentre, Bcentre and NV defect combining to produce yellow hues. Blue colour
is caused by substitutional boron defects, hence all blue diamonds are type IIb and ptype semiconducting. Green diamonds have been irradiated to produce single vacancy
defects which have a zero phonon line (ZPL) at 1.67 eV [22], imparting the colour.
In the cases of red, pink and brown diamond the source of the colour is not known. All
three colours occur in both type I and II diamond, increasing the difficulty of linking them
to a specific impurity or defect. Due to its lack of commercial value, less research has
CHAPTER 1. INTRODUCTION
15
been conducted on brown diamond compared to other colours, so little has been known
until relatively recently. The catalyst for renewed interest in the origin of brown colour is
the discovery that by subjecting a diamond to HighPressure HighTemperature (HPHT)
annealing treatment the colour can be quickly removed [26, 30, 31, 32, 33]. The result
is a stone much increased in commercial value, either colourless or with a yellow colour,
which is also difficult to differentiate from an untreated diamond. It is therefore desirable
to understand both the origin of the brown colour and the process which occurs to remove
it when undergoing treatment. The removal of the colour is illustrated in figure 1.11, which
is taken from [34]. In their work a sample was grown by CVD, and then sectioned into
four parts. One part was left untreated and the other three subjected to HPHT annealing
of 1900◦ C for 1 hour, 2200◦ C for 1 hour and 2200◦ C for 10 hours. With the lower
temperature anneal there is a strong yellow tinge from incorporated nitrogen, but this is
removed at higher temperature.
Figure 1.11: The removal of brown colour by HPHT treatment, taken from Charles et
al. [34]. Proceeding clockwise from the top left are: (a) The asgrown diamond (b) Treated
at 1900◦ C (c) Treated at 2200◦ C and (d) Treated at 2200◦ C for a longer time.
Two main factors have influenced the previously proposed origins of brown colour. Firstly,
all brown diamonds have been subjected to plastic deformation during their formation [30],
although not all plastically deformed diamonds are brown. It has been suggested that this
deformation causes a large number of dislocations to be formed in the diamond, some of
which are known to be optically active [28]. These dislocations could then be transformed
into more energetically stable types which are optically inactive [29], with the energy provided by HPHT treatment. Such a transformation will release large numbers of vacancies
or interstitials, which may or may not annihilate depending on other defects or impurities
present [30].
CHAPTER 1. INTRODUCTION
16
The second area of interest is the presence of vacancies and vacancyrelated defects
in diamond. Positron Annihilation Spectroscopy (PAS) and Electron Paramagnetic Resonance (EPR) studies (described in sections 3.6 and 3.7) have both been undertaken
to discover the nature of these defects. Two distinct PAS lifetimes are seen in brown
diamonds, τ2 ∼ 130 ps and τ3 ∼ 430 ps [35, 36]. The shorter lifetime is also seen in
γ irradiated diamonds [37], but is not that of the monovacancy which is known to have a
lifetime of 142 ps [38]. It is hypothesised that τ 2 is similar in volume to the monovacancy
thus could be a modified vacancy structure, such as a vacancyinterstitial complex [37].
Modelling has predicted that, in a spherical arrangement, approximately 50 vacancies
would be required to produce a signal of the order of 400 ps [36]. When subjected to
annealing the τ3 lifetime drops to ∼ 350 ps, which would correspond to approximately 30
vacancies following the same spherical model.
EPR studies have focused on the small vacancy chains first proposed by Lomer et al. [39].
A range of centres denoted R5 – R12 and KUL11 – KUL15 (not necessarily inclusive)
have been identified in natural [40] and implanted [41] diamond. The majority of these
centres have C2v or monoclinicI symmetry which is compatible with a chain of vacancies
lying in the 110 direction. In addition, the value of the spinspin interaction matrix D
allows the distance between unpaired spins, i.e. at the ends of a chain, to be calculated.
The result is the assignment of several of these centres to 110 vacancy chains of 3
– 7 vacancies in size [40]. These centres are not necessarily linked directly to brown
colouration, as the samples studied were not themselves brown, but are further evidence
of the presence of a number of vacancy structures in diamond.
The absorption spectrum of brown diamond differs between type IIa and type Ia. Generally there is a continuum absorption from 0 eV until the indirect gap transitions start to
take effect at ∼5.5 eV. This is unlike the typical absorption from a point defect, which often
displays a distinct onset followed by vibronic bands e.g. N s [42]. In type IIa the absorption
is completely featureless with no onset, as shown on figure 1.12. This figure only displays
visiblerange and UV absorption, with figure 1.13 demonstrating the continuation of the
absorption to low energies.
The absorption in the IR region (< 1 eV) is continuous, but not featureless. There is a
distinct peak around 0.25 eV (4960 nm) for all five samples, the origin of which is not
known. It is clear that for a defect to be responsible for the brown colour in these samples
it must induce into the diamond bandgap a range of levels, with allowed transitions from
CHAPTER 1. INTRODUCTION
10
17
Natural Samples
9
1
Absorption coefficient(cm )
8
7
6
5
4
3
2
1
0
1
2
3
Energy (eV)
4
5
6
Figure 1.12: Visible and UV absorption of 15 natural type IIa brown diamonds, measured
by the Diamond Trading Company (DTC).
18
Natural Samples
16
1
Absorption coefficient (cm )
14
12
10
8
6
4
2
0
0
0.2
0.4
0.6
0.8
1
Energy (eV)
Figure 1.13: NIR and IR absorption of five type IIa diamonds, measured by DTC. These
five are included in figure 1.12.
∼0 eV up to the indirect gap of 5.5 eV.
CHAPTER 1. INTRODUCTION
18
CVD diamond often has nitrogen incorporated into the lattice, due to the growth conditions. This means that although nominally type IIa (nitrogen concentration is low overall)
nitrogen features can sometimes be seen. Figure 1.14 shows absorption data from two
different sets of CVD stones. In the lighter stones (red lines) the absorption profile is
similar in magnitude and shape to the natural IIa diamonds of figure 1.12. However, in
the darker stones (blue lines) the absorption magnitude is ∼5–10 times larger and there
are distinct peaks/broad bands at 4.55 eV (272 nm) and 2.4 eV (516 nm). The 4.55 eV
peak has been proposed as a nitrogenboron complex [42], which is possible even if no
borane is added to the growth gas as boron is nearimpossible to remove from a system if
it has ever been introduced. The 2.4 eV peak is very close to the 550 nm (2.25 eV) peak
which is observed in type Ia brown diamond, and is quite conceivably the same defect.
40
CVD set 1
CVD set 2
1
Absorption coefficient (cm )
35
30
25
20
15
10
5
0
0
1
2
3
4
5
6
Energy (eV)
Figure 1.14: Vis/UV absorption of four type IIa CVD grown diamonds, measured by DTC.
A darker brown colour is associated with greater nitrogen inclusion and faster growth.
This greater speed of growth may lead to disordered regions and defects being trapped
before they can reform or dissipate, which can influence the colour. During CVD growth
nitrogen is most often included as a substitutional defect. This is quite different from
natural type Ia diamond, in which nitrogen is mainly found in aggregated defects. The
lowest energy structure for nitrogen in diamond is predicted to be the B complex of four
nitrogen atoms surrounding a vacancy [43].
Brown type Ia diamond displays a different absorption profile from brown type IIa dia
CHAPTER 1. INTRODUCTION
19
mond. The main continuum of absorption exists down to low energies with no distinct
onset, however there are additional absorption features displayed. There is a peak at
2.25 eV (550 nm) [30] which is correlated with the Acentre concentration [44]. Brown
type Ia diamonds sometimes display a band at 3.18 eV (390 nm), the cause of which is
undetermined. Additionally, some brown diamonds display ‘amber centres’ which have
an associated absorption peak at 0.52 eV (2384 nm) [45].
Figure 1.15 shows three type Ia brown diamonds measured by F. De Weerdt at Hoge
Raad Voor Diamant (Diamond High Council), Antwerp. These samples have an absorption magnitude similar to the natural type IIa brown diamonds, but clearly display the
3.18 eV (390 nm) band. In addition, the absorption magnitude increases strongly from
∼4 eV towards the bandgap onset, this is unlike the type IIa absorption which remains
low until near 5 eV, then increases very rapidly.
20
18
1
Absorption coefficient(cm )
16
14
12
10
8
6
4
2
0
1.5
2
2.5
3
3.5
Energy (eV)
4
4.5
5
Figure 1.15: Vis/UV absorption of three type Ia brown natural diamonds, measured by
F. De Weerdt.
There are significant differences between the annealing characteristics of the different
kinds of diamond. In irradiated material [39] the absorption continuum disappears at
1400◦ C, and in the range 14001600◦C for brown singlecrystal CVD material [33]. For
brown natural type IIa diamond however, temperatures above 2200 ◦ C are required to
remove the absorption continuum [32]. This suggests that either the stabilities of the defects responsible for the absorption continuum in brown singlecrystal CVD material and
CHAPTER 1. INTRODUCTION
20
natural diamond are significantly different, or the mechanism for colour loss is different.
It is prudent to mention here the absorption profile of pink diamonds, as they bear a
number of similarities to type Ia brown diamond. Pink diamonds can occur in all types,
but in the vast majority of stones the pink colour has the same characteristics. The pink
colour is arranged in striations throughout the crystal and has a distinct spectrum, which
contains both the 2.25 eV (550 nm) and 3.18 eV (390 nm) peaks [44, 46, 47, 48]. This
spectrum is shown in figure 1.16, with the infrared and visible/ultraviolet spectra split.
The split arises as two different spectrometers were used to take measurements, and
they were not calibrated to the same levels. There is a photochromic link between the
absorption bands in pink diamond. Illumination with light of more than ∼2.6 eV (<475 nm)
will diminish the absorption bands concurrently by several cm −1 , while illumination at less
than 2.3 eV (>530 nm) will restore the bands and subsequently enhance them. This
behaviour suggests that the bands are associated with the same centre, perhaps as
excited states. The energy dependence suggests a barrier of ∼2.25 eV for excitation [48].
45
UV  Vis
IR
40
1
Absorption coefficient (cm )
35
30
25
20
15
10
5
0
0
1
2
3
4
Energy (eV)
5
6
7
Figure 1.16: IR/Vis/UV absorption of a type IIa pink natural diamond, measured by DTC.
The overall absorption magnitude and shape is close to the brown diamond type IIa
absorption, being only modified by the aforementioned peaks. Many of the fancy colours
of diamond, especially yellow, can have a brown tinge, so it may be the case that the
DTC sample is a dark pink approaching brown. The similarity between the pink diamond
spectral defects and the type Ia brown spectrum suggests that the same or related defects
CHAPTER 1. INTRODUCTION
21
may be responsible for both colours, perhaps the concentration or density of defects being
the difference between the two cases.
1.4
Summary
Diamond is a material with several extreme values. Its crystal structure gives it the largest
value of hardness in a natural material, and it has a larger thermal conductivity than many
metals, mainly due to very strong covalent bonding between atoms. The electronic structure is governed by the 5.5 eV indirect bandgap which renders diamond transparent to
visible light, allowing the high refractive index to impart diamond’s characteristic sparkle.
The large bandgap means that diamond needs to be doped to become electrically useful, which is an area of current research; although ptype semiconducting diamond does
occur naturally.
Classification of diamond is based on defect content and configuration. The largest split
is between diamond that contains significant amounts of nitrogen (type I) and that which
has little or no nitrogen (type II). Type I diamond is further split into several subtypes
depending on the configuration of nitrogenous defects within the structure, with type II
diamond split into two groups; boroncontaining stones and nonboroncontaining stones.
Diamond contains a variety of defects, both pointlike (small) and extended (large). Point
defects are generally related to vacancies and impurities, with nitrogen being important
due to its prevalence. Extended defects are most often dislocations; a region where the
regular crystal structure is disrupted along a line. These defects can alter the electronic
properties of the diamond to e.g. make it a useful semiconductor, or modify its colour.
Some type I diamonds exhibit a yellow tinge caused by the presence of nitrogen defects.
The colour of diamond varies from colourless to the ‘fancy’ colours of yellow, blue, green,
pink and red. Brown diamonds are common but few find their way to the jewellery market, for this reason little investigation on the nature of brown colour in diamonds has
been undertaken. Developments in the HPHT treatment of brown diamond to render it
colourless or fancycoloured have spurred interest in brown diamond and its properties.
The absorption spectrum of all types of brown diamond is broad and featureless, with a
continuum down to nearzero energies. Type Ia brown diamond does have additional ab
CHAPTER 1. INTRODUCTION
22
sorption features however, which may be linked to the nitrogen content. Several studies
have identified key areas of investigation for brown diamond. Vacancies are present, according to PAS, possibly in large clusters, the lifetime of which are modified by annealing.
Dislocations are prevalent in all natural diamonds, and may introduce absorption states in
the bandgap, but are absent in CVD material which can also be brown. Whatever defect
is responsible must be stable to high energies as evidenced by the temperatures required
for removal of brown colour; 1400–1600 ◦ C for CVD material and ∼2000◦ C for natural
diamond.
Chapter 2
Theory of Modelling Crystals
This chapter will describe the techniques used to study the properties of materials in the
context of this work. It will address the topic in terms of conceptual ideas, mathematical framework and practical application. General properties of a cluster of atoms are
assessed using the techniques of Density Functional Theory (DFT) and Density Functional TightBinding Theory (DFTB). Additionally, techniques to calculate specific properties such as vibrational modes or optical absorption are discussed.
2.1
Fundamental Concepts
A crystal can be considered as a collection of a large number of electrons and nuclei,
of the order 1023 cm−3 . The energy of the crystal can be evaluated by solution of the
¨
Schrodinger
equation:
Hˆ Ψ(R, r) = EΨ(R, r)
(2.1)
With Hˆ the Hamiltonian and R and r the collections of ions at R i and electrons at rk
respectively. Here also rk denotes both the electron coordinates and the spin. The Hamiltonian can then be expressed:
23
CHAPTER 2. THEORY OF MODELLING CRYSTALS
Hˆ = Tˆe + Tˆn + Uˆe−e + Uˆe−n + Uˆn−n
24
(2.2)
Where Tˆe and Tˆn are the kinetic energy operators for the electron and nuclei respectively. Uˆe−e , Uˆn−n and Uˆe−n yield the energy due to the electron only interactions, nuclei
only interactions and the electron–nuclei interactions. These terms can be written more
completely thus:
2 N
h¯
∇2k
Tˆe = −
2m ∑
k
(2.3)
with ∇k = ∇(rk ), for the kth electron.
h¯ 2 N
Tˆn = −
Mi ∇2i
2M ∑
i
(2.4)
where Mi is the mass of the ith nucleus and ∇i follows as before. The interaction terms
are given:
Uˆe−e =
e2
1 1
4πε0 2 ∑
kl rk − rl 
(2.5)
Uˆn−n =
Zi Z j e2
1 1
∑
4πε0 2 i j Ri − Rj 
(2.6)
(−Z j e2 )
1
4πε0 ∑
k j rk − Rj 
(2.7)
and
Uˆe−n =
In theory, knowledge of these terms allows a solution of the many–particle wavefunction
of a crystal.
CHAPTER 2. THEORY OF MODELLING CRYSTALS
2.2
25
First Approximations
Solution of equation 2.1 using 2.2 is computationally impossible for the number of nuclei
and electrons in a real crystal, therefore a range of approximations are applied to render
the problem soluble.
Translational symmetry
Considering a crystal as a repetition of identical parts, Bornvon Karman [49] periodic
boundary conditions are applied and the wavefunction ψ at a point can then be described:
ψ (r + Ni ai ) = ψ (r)
(2.8)
With Ni a set of integers (1,2,3) and ai the related translation vectors in threedimensional
space. The potential of a crystal with translational periodicity can now be expressed:
U(r + T) = U(r)
(2.9)
Where T is the direct space translation vector. Bloch’s theorem [49] now states that the
wave function of a particle with wavevector k in such a crystal can be expressed:
ψ (r + T) = e(k·T) ψ (r)
(2.10)
The system of atoms to be evaluated has now been reduced by ∼ 10 23 orders of magni
tude; the primitive unit cell of the system is now considered. However, even for just a few
atoms the equations are still difficult to solve and further reductions in complexity must
be found.
CHAPTER 2. THEORY OF MODELLING CRYSTALS
26
Adiabatic approximation
The large difference in mass of the nuclei and the electrons allows the motion of electrons
to be described as reacting instantaneously to any motion of the nuclei; in effect the ionic
nuclei are fixed to a particular configuration. The Born–Oppenheimer approximation [50]
proposes that the electron wavefunction can be modulated by a function which is only
dependent on the nuclei:
η (R, r) = χ (R)Ψ(R, r)
(2.11)
where η (R, r) is the total wavefunction, χ (R) is the ionic wavefunction and Ψ(R, r) is
the electronic wavefunction dependent on R in a continuous manner. This means that
∇Ψ(R, r) and ∇2 Ψ(R, r) can be assumed insignificant compared with ∇ 2 χ (R), allowing
the kinetic energy operator for the nuclei to be written:
Tˆn [χ (R)Ψ(R, r)]
Ψ(R, r)Tˆn χ (R)
(2.12)
[Tˆn + Uˆn−n + Ee (R)]χ (R) = En χ (R)
(2.13)
Thus, Hˆ η = E η can be expressed:
the solution of Ee (R) being provided by:
[Tˆe + Uˆe−e + Uˆe−n ]Ψ(R, r) = Ee (R)Ψ(R, r)
(2.14)
To gain a solution for the nuclear eigenvalue equation 2.12, the electronic eigenvalues
can be used as part of the potential energy:
Hˆe Ψ(r1 , r2 , . . . rN ) = Ee Ψ(r1 , r2 , . . . rN )
(2.15)
CHAPTER 2. THEORY OF MODELLING CRYSTALS
27
for N electrons. This equation is still highly complicated if N is greater than 2, so further approximations are required, to reduce equation 2.14 to single–particle eigenvalue
equations.
2.3
Wavefunction Theories
It can be recalled that Ψ ≡ Ψ(r1 , r2 , . . . rN ) where rk denotes both the spin and the position
of the k–th electron. This cannot be solved exactly so described here are two methods
to provide approximate solutions. The following must first be considered. The variational
principle states that the energy of an approximate wavefunction Ψ for a system is never
less than the ground state energy given by the true system wavefunction Ψ 0 :
E=
ΨHˆ Ψ
ΨΨ
≥ E0 =
Ψ0 Hˆ Ψ0
Ψ0 Ψ0
(2.16)
This principle allows the quality of the approximated wavefunction to be evaluated by
simply comparing energies, with the lowest energy expectation value denoting the closest
match to the true system wavefunction. With regard to the techniques described below,
the variational principle can be written:
δ
ΨHˆ Ψ
ΨΨ
=0
(2.17)
hence:
δ [ ΨHˆ Ψ − λ ΨΨ ] = 0
where λ is a Lagrange multiplier.
(2.18)
CHAPTER 2. THEORY OF MODELLING CRYSTALS
2.3.1
28
Hartree’s method
¨
The previously described Schrodinger
equation contains a term depending on the inˆ
teraction of pairs of electrons, Ue−e . It is not possible to separate the variables in a
secondorder differential equation, so Hartree’s method applies the approximation that
each electron interacts with an averaged electron density, rather than individually with
other electrons.
The electronic wavefunction Ψ(r1 , r2 , . . . rN ) for a N–electron system can be expressed as
a product of N orthonormal single–electron wavefunctions:
Ψ(r1 , r2 , . . . rN ) = ψ1 (r1 )ψ2 (r2 ), . . . ψN (rN ) = Πi ψk (rk )
(2.19)
Then, ΨΨ and ΨHˆ Ψ can be set:
ΨΨ =
d 3 r Ψ∗ (r )Ψ(r) = ∏
d3r
d 3 rk ψk∗ (rk )ψk (rk )
(2.20)
k
and:
ΨHˆ Ψ
+
=
∏
d 3 rk ψk∗ (rk )
k
1 1
4πε0 2 ∏
kl
dr3k
dr3l
−¯h2 2
∇ + Ue−n (rk ) ψk (rk )
2m k
e2
ψk (rk )2 ψl (rl )2
rk − rl 
(2.21)
The variational principle then reads:
δ [ ΨHˆ Ψ − εkH ΨΨ ] = 0
(2.22)
where εkH = λ i.e. a Lagrange multiplier, and ε kH are Hartree eigenvalues. The Hartree
Hamiltonian can then be written:
CHAPTER 2. THEORY OF MODELLING CRYSTALS
29
HˆH ψk (r) = εkH ψk
(2.23)
HˆH = Tˆe + Ue−n + UH
(2.24)
with:
explicitly:
Z j e2
1
e
−¯h2 2
∇ −
−
HˆH =
∑
2m
4πε0 R j r − R j  4πε0
d3r
ρ (r )
r − r 
(2.25)
where R j is an ionic position and ρ (r ) is the electronic charge density at r . Which is
given:
ρ (r ) = ∑ ρk (r ) = −e ∑ ψk (r )2
k
(2.26)
k
To use Hartree’s method in practise, some approximate electron orbitals ψ k are taken
and used to solve all N equations (where N is still the number of electrons). A set of new
orbitals ψk are obtained, giving a better approximation for the orbitals which can be used
as a new starting point for the process. When the ψ k ’s do not change, it is said that the
selfconsistent field orbitals have been obtained. The expectation value of energy can
then be found from the Hamiltonian given in equation 2.24.
The starting wavefunction for the Hartree method is unphysical. This results in a final
wavefunction which in not antisymmetric. That is to say, interchanging the electron labels
in the wavefunctions does not necessarily change the sign for the electrons. Without this
property the function will not correctly describe the system, in fact in all fermion systems
no two particles can be described by the same oneparticle wavefunction. Thus the
Hartree method is flawed in respect to this, the solution being given in the Hartree–Fock
method.
CHAPTER 2. THEORY OF MODELLING CRYSTALS
2.3.2
30
Hartree–Fock theory
The solution to the problems of the Hartree method were devised independently by
Fock [51, 52] and Slater [53]. Oneelectron functions are still used, but the total wavefunction for the system is an antisymmetrized sum of all the products which can be obtained
by the interchange of electron labels. The antisymmetric Hartree–Fock wavefunction can
be expressed as a Slater determinant:
1
ΨHF (r1 , r2 , . . . rN ) = √
N!
ψ1 (r1 )
..
.
· · · ψN (r1 )
..
..
.
.
ψ1 (rN ) · · · ψN (rN )
(2.27)
Where, as before, ri represents both space and spin. The Slater determinant is always
antisymmetric as exchanging two single electron spinorbitals changes Ψ(r) by a factor
of 1. Note also that if two identical orbitals are present Ψ(r) = 0.
The expectation value of the Hamiltonian then becomes:
ΨHˆ Ψ
=
∏
d 3 rψk∗ (r)
k
−¯h2 2
∇ + Ue−n (r) ψk (r)
2m
+
1 1
4πε0 2 ∏
kl
d3r
d3r
−
1 1
4πε0 2 ∏
kl
d3r
d3r
e2
ψk (r)2 ψl (r)2
r − r 
e2
ψ ∗ (r)ψk (r )ψl∗ (r )ψl (r)
r − r  k
(2.28)
where the second term of equation 2.28 is an electron repulsion term and the third term
is an exchange interaction, that is, an attractive interaction between an electron at r and
all other electrons with the same spin.
Hartree–Fock theory can be approached by a numerical technique, but this method is
computationally intense. It is more usually approached by representing the electron orbitals ψk as a linear expansion of a set of basis functions χ , which can be expressed:
CHAPTER 2. THEORY OF MODELLING CRYSTALS
31
n
ψk (r) = ∑ Cnk χk (r)
(2.29)
1
The coefficients Cnk are optimized rather than the basis functions.
Application of the variational principle:
δ
[ ΨHˆ Ψ − εk ΨΨ ] = 0
δ ψk∗ (r )
(2.30)
leads to the HartreeFock equations:
HˆHF ψk (r) = εkHF ψk (r)
(2.31)
Explicitly:
1
HˆHF ψk (r) = HˆH ψk (r) −
4πε0 ∑
l
d3r
e2
ψl (r)ψl∗ (r )ψk (r)
r − r 
(2.32)
where HˆH is the Hartree Hamiltonian. The last term in equation 2.32 is an attractive
exchange operator, the form of which is generally intractable. Local exchange forms
have been sought to make the equations soluble [54, 55, 56].
If the ψk ’s in equation 2.31 are now expanded into their basis functions the Hamiltonian
becomes:
n
n
1
1
HˆHF ∑ Cnk χk (r) = εkHF ∑ Cnk χk (r)
(2.33)
Multiplying by χl (r) and integrating over r:
n
∑ Cnk
1
n
χl∗ (r)HˆHF χk (r)d(r) = εk ∑ Cnk
1
χl∗ (r)χk (r)d(r)
(2.34)
CHAPTER 2. THEORY OF MODELLING CRYSTALS
32
these integrals are now represented as:
(F)lk =
χl∗ (r)HˆHF χk (r)d(r)
(2.35)
χl∗ (r)χk (r)d(r)
(2.36)
and:
(S)lk =
(F)lk is an element of an n × n Fock matrix F, and similarly (S) lk is an element of an n × n
overlap matrix S. There are n such equations where n is the number of basis functions.
Expressed in matrix form this is:
FC = SCε
(2.37)
where ε is the expectation value for the energy. The matrix C which diagonalises F can
be obtained by first finding a matrix X to diagonalise S:
X† SX = 1
(2.38)
C †F C = ε
(2.39)
so that:
The elements of F is this case depend on the orbitals ψ k , so must be solved through
an iterative process. Additionally the Fock matrix elements contain a large number of
twoelectron integrals, which are complex to solve.
To try and overcome the difficulties the problem can be approached by describing the
electron system using density rather than wavefunctions.
CHAPTER 2. THEORY OF MODELLING CRYSTALS
2.4
33
Density Functional Theories
Density functional schemes express the exchangecorrelation potential not as a function
of all electron orbitals, as in HartreeFock theory, but as a function only of the electron
charge density at a point. The number of variables to compute is reduced from 3n – where
n is the electron number – to just 3. The electronic charge density at a point r 1 caused by
all other electrons can be expressed:
ρ (r1 ) =
2.4.1
dr2 . . .
drN Ψ∗ (r1 . . . rN )Ψ(r1 . . . rN )
(2.40)
The ThomasFermiDirac scheme
The ThomasFermi model [57, 58] assumed that the electrons move independently in an
effective electrostatic potential determined by Poisson’s equation. The kinetic energy is
expressed in terms of the kinetic energy of a system of noninteracting electrons with
slowly varying density. This model did not attempt to represent the exchange energy
predicted by the HartreeFock technique, but this was later included by Dirac [59].
The neglect of electron correlation in the ThomasFermiDirac scheme, along with errors
in the exchange energy and kinetic energy representation, cause this method to be too
inaccurate for most situations.
2.4.2
The HohenbergKohnSham scheme
The HohenbergKohnSham scheme was formulated much later than the ThomasFermi¨
Dirac scheme. In essence the theory states that the Schr odinger
equation in 2.13 can
be reduced to a set of singleparticle equations to be solved selfconsistently. The
HohenbergKohnSham scheme is the method used in this work so will be described
more fully here.
There are two basic theorems laid down for the HohenbergKohnSham density functional
theory, each of which are expressed here.
CHAPTER 2. THEORY OF MODELLING CRYSTALS
34
Density as the central variable
It was demonstrated by Hohenberg and Kohn that the external potential affecting a number of electrons N, is a function of the electronic density in the ground state – plus an
additive constant [60]. Where the electronic density is:
n(r) = N
Ψ(r )2 d 3 r
(2.41)
Consequently, since the external potential fixes the Hamiltonian, the full manyparticle
ground state wavefunction is a function of n(r). Then the function is defined:
F[n] = Ψ(Te + Ue−e )Ψ
(2.42)
The total energy of the system is given by:
Ee [Ue−n , n] = F[n] +
Ue−n (r)n(r)dr
(2.43)
The potential Ue−n (r) is the external potential which defines the electron motion. F[n]
is an independent functional to account for electronic kinetic, electron correlation and
exchangecorrelation energies. The form of F[n] is generally unknown.
Energy variational principle
Subject to the correct electronic density, E e [Ue−n , n] is equal to the ground state energy E.
The energy functional of Ψ , given by:
εv [Ψ ] ≡ (Ψ ,V Ψ ) + (Ψ , (T +U)Ψ )
(2.44)
has a minimum at the correct ground state Ψ; relative to arbitrary variation of Ψ where
CHAPTER 2. THEORY OF MODELLING CRYSTALS
35
the number of particles N is kept constant. Applied to the ground state electronic energy:
Ee [Uext , ρ ] < Ee [Uext , n]
(2.45)
with n(r) = ρ (r) as the correct ground state charge density.
Electronic energy functional
The kinetic energy is treated differently in the HohenbergKohnSham theory compared to
the ThomasFermi approach, in which it has only an approximate form. Kohn and Sham
propose a different separation of the energy functional, so that the universal functional is
expressed:
F[n] = T0 [n] +
e2
8πε0
dr
dr
n(r)n(r )
+ Exc [n]
r − r 
(2.46)
where T0 [n] is the kinetic energy of noninteracting electrons with density n(r). The integral
terms are the classic meanfield Coulomb energy. E xc [n] is a functional of the density and
is the manybody exchangecorrelation energy functional; representing all corrections to
the independent electron model. Using equations 2.43 and 2.46 the ground state energy
of a manyelectron system is:
Ee [Uext , ρ ] = T0 [ρ ] +
dr Uext (r)ρ (r)
e2
8πε0
dr
dr
ρ (r)ρ (r )
+ Exc [ρ ]
r − r 
(2.47)
Three major problems arise from evaluation of E e [ρ ]:
• The ground state density ρ (r) must be evaluated selfconsistently.
• T0 [ρ ] cannot be evaluated as there is no wavefunction information if only ρ (r) is
known.
• Exc [ρ ] is only known for a few simple systems; such as a homogeneous electron
gas. It must be represented in a simplified but accurate form.
CHAPTER 2. THEORY OF MODELLING CRYSTALS
36
The KohnSham equations
Kohn and Sham’s approach was to take the approach of Hohenberg and Kohn [60] and
extend it to obtain a set of selfconsistent equations including approximate exchange and
correlation effects. These KohnSham equations [61] require a knowledge of the chemical
potential µ (ρ ) of a homogeneous interacting electron gas as a function of the density ρ .
It is defined as:
µ=
δ T0 [ρ ]
+ UKS [ρ ]
δ ρ (r)
(2.48)
where UKS [ρ ] is an effective KohnSham potential of three parts, given as:
UKS [ρ ] = Uext (r) + UH (ρ ) + Uxc (ρ )
(2.49)
The Hartree potential is defined as:
UH (ρ ) =
e2
4πε0
dr
ρ (r )
r − r 
(2.50)
and the exchangecorrelation potential is:
Uxc (ρ ) =
δ Exc [ρ ]
δ ρ (r)
(2.51)
With these equations Kohn and Sham then showed that the total system is described by
¨
a set of oneelectron Schrodinger
equations:
e2
−¯h2 2
∇ +
2m
4πε0
dr
ρ (r )
δ Exc [ρ ]
+
+ Uext (r) φ j (r) = ε j φ j (r)
r − r 
δ ρ (r)
(2.52)
where φ j and ε j are the singleparticle wavefunctions and eigenvalues of a noninteracting
CHAPTER 2. THEORY OF MODELLING CRYSTALS
37
system governed by the effective KohnSham potential. The charge density is obtained
by summing over the occupied states:
occ
ρ (r) = ∑ φ j (r)2
(2.53)
j
These equations 2.52 and 2.53 are the KohnSham equations, which can be solved selfconsistently to give the exact ground state charge density and total system energy to be
obtained; the major strength of Density Functional Theory. The total energy is then:
occ
E[n] = ∑ ε j − UH (ρ ) + Exc (ρ ) −
dr Uxc (ρ )ρ (r)
(2.54)
j
The KohnSham eigenvalues are real but are not representative of the occupied electron
energy states or excitation energies of the system; apart from the highest occupied KohnSham eigenvalue which is equal to the physical chemical potential.
The ExchangeCorrelation functional
The precise form of the exchangecorrelation functional E xc is unknown, but there are
several approximations devised. The approximation suggested by Hohenberg and Kohn
and Kohn and Sham, and used in this work, is the Local Density Approximation (LDA). It
is assumed that the electronic charge density in the system corresponds to that of a homogeneous electron gas. An exact solution for this is known, if ρ (r) is a sufficiently slowly
varying quantity in space. The exchangecorrelation energy under the LDA scheme can
then be expressed:
LDA
Exc
[ρ ] =
εxc [ρ (r)]ρ (r)d(r)
(2.55)
Here εxc [ρ (r)] is the exchangecorrelation energy per electron of a uniform electron gas
of density ρ (r). The exchangecorrelation potential can then be written [62]:
CHAPTER 2. THEORY OF MODELLING CRYSTALS
VxcLDA (r) =
d
εxc (ρ (r))ρ (r) ∼ µxc (ρ (r))
dρ
38
(2.56)
The exchangecorrelation potential can further be developed by expressing µ xc (ρ ) in
terms of the mean interelectronic spacing r s , thus:
Vxc (r) ∼ µxc [rs ] = εxc −
rs d εxc
3 drs
(2.57)
The requirement now is for an analytical expression of ε xc [rs ]. There are several expressions devised by KohnSham and G’asp’ar, Perdew and Zunger and Perdew and Wang
among others [62].
The main shortfall of the LDA is to underestimate the bandgap, which in diamond is
∼76 % of the true value. One reason for this is the LDA oneelectron effective potential contains an incorrect interaction of each electron with itself [63]. It is necessary to
consider this when estimating bandgap transitions, absorptions and emissions.
2.5
Application Using AIMPRO
In this work DFT calculations have been implemented using the Ab Initio Modelling
PROgram – AIMPRO [64]. To achieve satisfactory computational simplicity, further simplification of the equations are made – notably in the use of pseudopotentials and the
basis set chosen. This is a valence electron approach to the modelling of solids, as is the
tightbinding scheme; which is described later in this chapter.
2.5.1
Pseudopotentials
¨
When solving the selfconsistent Schr odinger
equation not all electrons need to be considered; the interaction of any atom with other atoms and its general environment is
mainly dependent on its valence electrons. The other electrons are closely bound to
the ionic core and hence are known as the core electrons. The core electron potential
CHAPTER 2. THEORY OF MODELLING CRYSTALS
39
can be considered inclusively with the ionic potential as one total potential acting upon
the valence electrons. This is the pseudopotential, which has the benefit of reducing the
number of functions required for all states – especially compared to the allelectron model
– allowing larger systems to be studied.
There are two constraints to be applied when constructing pseudopotentials for a system [65]:
• The pseudopotential has exactly the same valence energy levels as the true atomic
potentials; that is the s, p and d levels are the same. It follows that the lowest bound
state solutions are the valence energy eigenvalues – there being no core levels.
• The pseudowavefunctions must be equal to the true wavefunctions outside a given
core radius, which varies with atomic species.
Inside the core radius mentioned above, the pseudowavefunctions are not equal to the
true valence wavefunctions, as these wavefunctions often vary rapidly. Instead, a mathematically straightforward function is chosen which still satisfies the above conditions.
Throughout this work the pseudopotential of choice has been the HartwigsenGoedeckerHutter (HGH) type [66].
2.5.2
Basis sets
The KohnSham orbitals can be written as a product of a spinfunction and a spatial
function, which is expanded in terms of a basis set φ i (r), so that:
n
ψk (r, s) = χk (s) ∑ cnk φn (r)
(2.58)
1
where χk (s) is the spinup wavefunction for s =
1
2
and spin down for s = − 21 . AIMPRO
uses Cartesian Gaussian orbitals for φ i (r), the form of which depends on whether a full
or contracted basis set is chosen.
In the contracted basis set 13 functions per atom are used, which consist of 2 stype
CHAPTER 2. THEORY OF MODELLING CRYSTALS
40
functions, 6 ptype functions and 5 dtype functions. The s and p functions are linear
combinations of four Gaussians, and are premultiplied by x, y or z for ptype functions.
The dtype functions are necessary for polarisation effects and consist of single Gaussian
functions with premultipliers: 2z 2 − x2 − y2 , x2 − y2 , xy, xz and yz. The exponents of the
underlying Gaussians and the contraction coefficients are optimized for each material
under study.
The larger basis sets consist of at least 22 functions per atom, with the form:
φ (x, y, z) = xnx yny znz e(−α r
2)
(2.59)
The s and ptype functions are constructed using three different α exponents, with n x +
ny + nz ≤ 1. The dtype functions are formed using a further, different α and by setting
nx + ny + nz ≤ 2. As before, the exponents are chosen to optimize the bulk energy of each
particular system.
A more complex set of 28 functions can be formed by letting two of the four exponents
have nx + ny + nz ≤ 2. This is sometimes necessary for calculation of optical observables
as these often have a strong dependence on the dtype orbitals created in this basis set.
When the basis set has been constructed the equation to solve is reduced to:
H ∑ χλ (s)cλj φ j (r) = Eλ χλ (s) ∑ cλj φ j (r)
j
(2.60)
j
If this is multiplied through by χλ∗ (s)φi∗ (r), integrated over r and then summed over s the
equation is reduced to a set of matrix equations:
∑(Hi j − Eλ Si j )cλj = 0
(2.61)
j
where Hi j and Si j are the Hamiltonian and overlap matrix elements respectively. The
output spin density can then be generated:
CHAPTER 2. THEORY OF MODELLING CRYSTALS
41
no (r, s) = ∑ χλ (s)ψλ (r)2
(2.62)
λ
This can be iterated to obtain selfconsistency so that the selfconsistent density is obtained:
n(r, s) = ∑ ck (s)gk (r)
(2.63)
k
The selfconsistent density then allows the structural energy to be evaluated; and from
this other properties and observables.
2.5.3
Brillouin zone sampling
To calculate physical properties of a material the charge density integrated over the Brillouin zone is required. To reduce complexity a set of special kpoints can be taken, from
which an average value for the intergrand function f (k) is calculated. The average value
is given by:
f¯ =
Ω
(2π )3
f (k)dk ≈
1 Nk
f (ki )
Nk ∑
i
where the volume of the Brillouin zone is given by
Ω
(2π )3
(2.64)
and Nk is the number of special
kpoints.
This work has utilised the scheme of Monkhorst and Pack [67], where the special kpoints
are a grid of I × J × K points in reciprocal space given by:
k(i, j, k) = ui g1 + u j g2 + uk g3
where g1 , g2 and g3 are the reciprocal unit vectors and u i , u j and uk are defined:
(2.65)
CHAPTER 2. THEORY OF MODELLING CRYSTALS
42
ui =
(2i − I − 1)
, (i = 1, ....I)
2I
(2.66)
uj =
(2 j − J − 1)
, ( j = 1, ....J)
2J
(2.67)
uk =
(2k − K − 1)
, (k = 1, ....K)
2K
(2.68)
It is one of the advantages of the scheme that energy convergence of the system can be
ensured by increasing I,J and K to increase the accuracy of f (k).
It is necessary to ensure that appropriate values of I,J and K are chosen for the system
under study. There is an inverse relationship between the number of points chosen and
the size of the cell in that direction. This is due to the sampling being taken across the
Brillouin zone, in reciprocal space. A large realspace lattice is small in reciprocal space,
requiring less sampling points.
Computational complexity can be reduced in high symmetry cells as certain kpoints are
equivalent. In this case redundant kpoints are removed and a compensating weighting
factor applied to the remaining equivalent kpoint.
2.6
Calculation of Material Properties
This section will give overviews of several of the methods that are used in AIMPRO to
calculate material properties. AIMPRO has the ability to calculate many properties, which
is useful for comparison to experimental work. The methods discussed in this section are
ones that have been used in this work.
The KohnSham eigenstates are not electronwavefunctions but a set of basis functions
from which the charge density is expanded. The consequence of this is that DFT is a
groundstate theory; it does not describe excitedstate properties well.
CHAPTER 2. THEORY OF MODELLING CRYSTALS
2.6.1
43
Total energy
The force on an atom is given simply by:
Fi = −∇i E
(2.69)
It follows that the total energy of a system can be found by minimising the force on each
atom, structurally optimizing the system. In this technique each atom is moved a small
amount inducing an energy change δ E and the new forces are then calculated. The
atoms are then moved in accordance with these forces. This process can be repeated
iteratively until the change in total energy and forces are deemed insignificant.
AIMPRO uses the Conjugate Gradient Algorithm (CGA) method to attempt system relax
ation. In the case of M atoms in the cell, the force equation:
F[ρ , {τ }] = 0
(2.70)
can be solved for the equilibrium geometry {τ 0 }. The approach for solving this equation to
to assume the charge density remains fixed while the atomic configuration τ is variable.
In the case of an mth iteration in relaxation, then the atomic configuration is {τ (m) }. A
direction dm is chosen, then a minimization in this direction undertaken; a solution t 0 is
found to minimize:
E[ρ , {τ (m) + dmt}]
(2.71)
so that the geometry for the (m + 1)st iteration can be set:
{τ (m+1) } = {τ (m) + dmt0 }
(2.72)
E[ρ , {τ (m+1)] and F[ρ , {τ (m+1) ], the total energy and force can then be found. The process
is iterated as described above.
CHAPTER 2. THEORY OF MODELLING CRYSTALS
44
The key aspects of the CGA method are the choice of direction d and the minimization to
determine t.
Choice of d
If the initial choice for d of d= F[ρ , τ ] is maintained for all relaxation integration, this is
known as the steepestdescent technique. In the CGA method the {d (m) }, m = 1, 2, . . . , m
directions are chosen to be ’conjugate’, that is:
(d(i) )T Cd( j) ∝ δi j
(2.73)
where i < j and C is a symmetric matrix. The energy is related by:
E(x) = a − bx +
1
2xT Cx
+ O(x3 )
(2.74)
with a and b constants of the energy quadratic, and x the distance along the energy curve.
Minimization
When the results for the energy E(τ ) and the force F(τ ) have been calculated for the m th
relaxation integration, the values E(τ + σ d) and F(τ + σ d) for a small increment σ can
be found. These are fitted to a thirdorder (or greater) polynomial and an optimal value t 0
estimated to correspond to minimum energy in direction d.
The CGA approach is simple and effective but has one large factor to be taken into
account. The minimum energy that is found for a particular starting atomic geometry is
not guaranteed to be the global minimum. The atoms cannot be moved to escape from
a local minimum, so the start positions must be chosen with care for complex structures,
and ideally several start positions for optimization should be evaluated for the total energy.
CHAPTER 2. THEORY OF MODELLING CRYSTALS
2.6.2
45
Formation energy
Comparison of total energy when studying different defects can be misleading. It is often
the case that there are different numbers of atoms in each cell, or that different elements
are present. In this situation the formation energy is useful as it takes account of the
number and types of atoms in the cell.
The chemical potential of a species s is represented by µ s and is the derivative of the
Gibbs free energy with respect to the number of atoms of that species. In the case of
thermodynamic equilibrium µs is equal across the system. The Gibbs free energy is:
G = E + PV − T S
(2.75)
However, it is possible to neglect the second and third terms as PV is small for solid state
calculations and the −T S approximation is valid only at high temperatures. The formation
energy is then given by:
E f = E T = q µ e − ∑ ns µs
(2.76)
s
where ET is the total energy of the system, q is the net positive charge on the system
(with µe the fundamental electron charge) and n s and µs are the number and chemical
potential species of a species s. µe is usually taken as EF + Ev , where EF is the Fermi
level and Ev is the highest occupied energy level.
The formation energy can be used to compare energy convergence of a defect in differently sized supercells, or to check accuracy with different basis sets or pseudopotentials.
In addition the relative stability of defects can be compared, for example dislocations that
contain different numbers of atoms, or varying sizes of vacancy defects. The possible
transformation of defects at different temperatures can be inferred by looking at energies
required to dissolve defects; for example by vacancy/interstitial migration.
CHAPTER 2. THEORY OF MODELLING CRYSTALS
2.6.3
46
Vibrational modes
Calculation of a defect’s vibrational modes is highly important in relation to experimental
results as they can be compared to infrared, photoluminescent measurements and Raman spectra. As the vibrational modes are dependent on the structure of a defect and the
bonding of the atoms an agreement between calculated frequencies and those observed
in experiment is a strong indicator that the proposed defect structure is close to the real
situation.
The vibrational modes for a system of N atoms are given by solution of the eigenvalue
problem:
D · U = ω 2U
(2.77)
Where D is a dynamical matrix and the eigenvalues ω 2 are the squared frequencies associated with the normal modes U. The normal mode is a 3N dimensional vector describing
the motion of all atoms for that mode. Thus the dynamical matrix D has size 3N × 3N, with
each element:
Dab (i, j) =
∂ 2E
1
Mi M j ∂uia ∂u jb
(2.78)
here a and b are any of the Cartesian coordinates and u ia and u jb are the displacements
of atoms with mass Mi and M j respectively.
The energy double derivatives must now be found, starting from a relaxed cell. An atom i
is moved a small amount ε in direction a, thus affecting the charge density and resulting
in nonzero forces on the structure. The new force acting on atom j is in a direction b
and is denoted f b+j (a, i). Then atom i is moved by −ε a giving rise to a new force on b of
fb−j (a, i). Then the second derivative on the energy is given to second order in ε by:
fb+j (a, i) − fb−j (a, i)
∂ 2E
=
∂uia ∂u jb
2ε
(2.79)
CHAPTER 2. THEORY OF MODELLING CRYSTALS
47
The displacement ε is finite so the method includes a contribution from all even powers
of ε ; so some anharmocity is included. Thus the resultant frequencies are referred to as
quasiharmonic.
In practise only the atoms in or very near to a defect will have a significantly different
potential from those in the bulk atom. The energy second derivative of the bulk atoms
can be obtained using an empirical potential, in this work the MusgravePople potential is
used [68]. The use of this empirical potential reduces computational expense.
2.6.4
Mulliken bond population
Mulliken analysis allows the contribution of atomic bonds to a particular KohnSham level
to be calculated, by calculating the contribution from the basis functions. These results
can then to be compared to experimental EPR measurements.
The contribution to a KohnSham level λ from a basis function φ i is given by:
pλ (i) =
1
ckλ i Sikj (cλk j )∗
NL (∑
j,k)
(2.80)
The overlap matrix elements Sikj can be given by:
Sikj =
B∗ki e−ik·r Bk j dr
(2.81)
where Bki are basis functions, and ∑i pλ (i) = 1. The inclusion of basis functions in the
equations allows contributions to bond population from specific orbitals to be found, so
hybridisation of a state can be found.
2.6.5
Electron energyloss
Defects often affect the electronic structure of a crystal. The removal of atoms in a vacancy defect, for example, or a dislocation can introduce dangling bonds which may result
CHAPTER 2. THEORY OF MODELLING CRYSTALS
48
in deep bandgap states being introduced. Additionally, strain on bonds in the defect and
surrounding crystal can alter the density of states outside the bandgap, or cause shallow
gap states.
One of the most direct methods of measuring the electronic structure of extended defects
is Electron EnergyLoss Spectroscopy (EELS), first proposed by Hillier and Baker [69].
In this technique an electron microscope passes highenergy electrons through a crystal,
inducing electronic transitions between the valence and conduction bands, known as
LowLoss EELS. This is described more fully in section 3.2
There is a further area of measurement possible for bound atomic core state excitation,
known as CoreExcitation EELS. This requires knowledge of the core atom energy levels,
however these are removed through the use of pseudopotentials so cannot be modelled
in AIMPRO.
The highenergy electrons which pass through the crystal interact with the field of polarisation caused by the electric field of the crystal’s electrons. The quantity measured is the
fraction of electrons scattered into a solid angle dΩ having lost energy between ∆E and
∆E + dE. A differential crosssection d σ 2 /dΩdE can be related to the dielectric function
by:
dσ 2
−1
1
∝ 2 Im
dΩdE q
ε (q, ω )
(2.82)
Within the random phase approximation [70, 71], the macroscopic dielectric function is
given as:
ε (q, ω ) = 1 +
e2
ε0 Ω
(r)k v,k+q c 2
∑ Ek+q,c − Ek,v − h¯ ω
k
(2.83)
Here Ek,c and Ek,v denote the conduction (empty) and valence (filled) bands and Ω is
the volume of the crystal. The constant of proportionality linking the loss intensity with
the inverse dielectric function depends weakly on energy, thus making an exact correspondence with experiment difficult. The scalar expression for the longitudinal dielectric
function is, within the random phase approximation, the same as the expression for the
CHAPTER 2. THEORY OF MODELLING CRYSTALS
49
transverse optical dielectric function [72]. The optical absorption coefficient, α , is related
to the complex dielectric function by:
α=
4π −ε1 +
λ
2
ε12 + ε22
12
(2.84)
where ε1 and ε2 are the real and imaginary parts of the dielectric function, as described
below.
From a theoretical point of view, the imaginary part to ε (q, ω ) and written as ε 2 (q, ω ), is
first found and then its real part, ε1 (q, ω ), is given by the KramersKronig relation:
2
ε1 (q, ω ) = 1 + P
π
∞
0
ω ε2 (q, ω )
dω
ω 2 − ω2
(2.85)
A more general tensorial expression for the dielectric function appropriate for extended
defects is:
εi, j (q, ω ) = δi, j +
e2
ε0 Ω
∑
k
(r.i)k v,k+q c (r.j)k+q c,k v
Ek+q,c − Ek,v − h¯ ω
(2.86)
In the case where the momentum loss h¯ q lies in the beam direction and for sufficiently
high energies q = ω /v is be taken to be zero. The average of the diagonal components
of the tensor is taken in order to compare with experimental results. This is related to the
trace of the tensor and is independent of the orientation of the defect in the cell.
It is important to recognise that the expression for ε is limited and relies on perturbation
theory and the neglect of local fields, plasmons and exciton effects. As a consequence
the absorption of diamond in this theory begins strongly at the direct gap of 7 eV, as
opposed to the 5.5 eV of the indirect gap. More involved theories of dielectric functions
have been developed for semiconductors, to describe effects beyond the local density
approximation [73], beyond the longwavelength limit [74], and beyond the independent
particle limit [74, 75], but they are currently too computationally intensive to be applied to
anything other than bulk materials.
CHAPTER 2. THEORY OF MODELLING CRYSTALS
2.7
50
Density Functional TightBinding Theory
The DensityFunctional TightBinding theory (DFTB) [76, 77] is a more approximate method
of solving the KohnSham equations, which has application for studying defects in larger
cells, or to gain results quickly with less accuracy.
The DFT energy after introducing KohnSham orbitals [29] is expressed as:
occ
ε [n] = ∑ ni
i
φi∗ (r) −
∇2 1
+
2
2
n(r ) 3
d r + Uext (r) φi (r)d 3 r + Exc [n]
r − r 
(2.87)
where:
occ
n(r) = ∑ ni φi (r)2 , N = ∑occ
i ni
(2.88)
i
The numbers ni are the occupation numbers which describe how many electrons occupy
each KohnSham orbital.
The equations can be simplified by using a Taylor expansion around a reference density
n0 (r), then neglecting the higher order contributions. The energy is then [78]:
ε [n] =
occ
∑ ni
i
(
φi∗ (r)Hˆ [n0 ]φi r)d 3 r −
+ Exc [n0 ] −
+
1
2
1
2
n0 (r)n0 (r ) 3 3
d rd r
r − r 
Uxc [n0 r]n0 (r)d 3 r
1
δ 2 Exc [n]
+
r − r 
δ n2
∆n(r)∆n(r )d 3 r d 3 r
(2.89)
n0
(2.90)
CHAPTER 2. THEORY OF MODELLING CRYSTALS
51
The second to fifth terms here are expressible as the sum of twocentre integrals; which
can be written as E2cent . If threecentre and higher contributions are neglected the total
DFTB energy is:
occ
ε [n] = ∑ ni
(
φi∗ (r)Hˆ [n0 ]φi r)d 3 r + E2cent = εDFT B
(2.91)
i
In DFTB the starting density, n0 (r), is expressed as a superposition of weakly confined
neutral atoms; known as pseudoatoms. The wavefunctions of the pseudoatoms are
represented by Slater orbitals, φv (r). The KohnSham orbitals can then be expanded into
atom centred orbitals [77], yielding:
φi (r) = ∑ Cvi φv (r)
(2.92)
v
which is akin to a linear combination of atomic orbitals. The twocentre contributions can
ij
be approximated by a sum of shortrange repulsive pair potentials U rep
:
E2cent ≈
1
ij
(Ri − R j ) = Erep ({Ri })
∑ Urep
2 i=
j
(2.93)
The DFTB energy can then be written in terms of the expansion coefficients C vi . This
can then be varied with respect to the expansion coefficients and normalised to give the
DFTB secular equations:
∑ Cvi (Hµ0v − εiSµ v ) = 0
(2.94)
v
If Rv denotes the atom site the orbital is centred at, the Hamiltonian and overlap matrix
are given by:
Hµ v =
εµatom
:
µ =v
∗
3
ˆ
φµ (r)H [n0 ]φv (r)d r : Rv = Rµ
0
: otherwise
(2.95)
CHAPTER 2. THEORY OF MODELLING CRYSTALS
52
and:
φµ∗ (r)φv (r)d 3 r
Sµ v =
(2.96)
Once the pseudoatom orbitals have been calculated selfconsistently, the nondiagonal
elements of Hµ v and Sµ v can be tabulated as a function distance between the two centres.
The diagonal elements of Hµ v are taken as the atomic orbital energies of a free atom. It
is clear that both matrices can be calculated in advance so that equation 2.95 can be
solved directly and non selfconsistently for any coordinates of the nuclei. In this case
the DFTB energy becomes:
occ
εDFT B = ∑ ni εi + Erep ({R})
(2.97)
i
ij
with a repulsive energy Erep and a repulsive potential Urep
(Ri − R j ). Variation of distance
allows tabulation of the repulsive potential akin to H µ v and Sµ v . The use of the repulsive
potential in practise allows a minimal basis of Slatertype orbitals, reducing computational
complexity.
The ’inadvance’ calculations of DFTB reduce significantly the time taken to derive the
KohnSham eigenvalues and the total system energy. Although there is a penalty in
accuracy.
2.8
Screened Exchange
The local density approximation (LDA) and KohnSham approximations are known to underestimate the bandgap of semiconductors, leading to 0 eV bandgaps in some cases [79].
One technique which has been developed to compensate for this problem is the screened
exchange technique. The technique is described in full in [13], and only a summary is
provided here.
The KohnSham bandgap underestimation occurs in the main because the zeroth Fourier
CHAPTER 2. THEORY OF MODELLING CRYSTALS
53
component of the exchange potential changes discontinuously across the energy gap, i.e.
Vxc (G) is undefined. Bylander et al. separate the exchangecorrelation operator into two
parts. One part can be treated exactly, the other has a weak ψ dependent Vψ which can
be replaced by a density functional. The Hamiltonian then reads:
∂ Ex ∂ Esx ∂ Ec
Hˆ xc = Hˆ sx +
−
+
∂ρ
∂ρ
∂ρ
(2.98)
where Hˆ sx is an exchange operator calculated with the screened Coulomb interaction
(e−Ks r )/r. Ex [ρ ] is the LDA exchange energydensity functional in Rydberg units, E c [ρ ] is
one of the LDA correlation functionals, and E sx [ρ ] is the LDA screened exchange density
functional. Esx [ρ ] is dependent on a factor γ , and it is possible to choose γ to be dependent
−1/3
on the electron density as a function of position γ 2 ∼ ρ −1/3 , or not γ 2 ∼ ρ0
. The choice
of γ to be independent from ρ (r) is supported [13].
The LDA results in an exchange energy which is too small, and a correlation energy which
is too large, when used for materials with a finite energy gap. A modified correlation
functional is used:
Eˆ c [ρ ] = W Ec [ρ ]/(W + ∆)
(2.99)
Here Ec [ρ ] is a standard correlation density functional, W is the bandwidth and ∆ is some
average energy gap. In the freeelectrongas limit Eˆ c [ρ ] = Ec [ρ ].
The technique was tested on silicon, where an energy gap of 1.32 eV was found. This
compares to 0.44 eV using unmodified LDA and 1.17 eV for experiment. In sections 1.1
and 6.2.1 this theory is applied to AIMPRO and tested on diamond with very good results.
2.9
Summary
¨
This chapter has described methods of solving the Schr odinger
equation that reduce the
complexity of the problem with various approximations. Comparison has been made of
CHAPTER 2. THEORY OF MODELLING CRYSTALS
54
several techniques, and a special emphasis has been placed on the Density Functional
Theory used in this work. In all the techniques large changes are made to the original energy equations to render them soluble. However, the results that are obtained have a high
accuracy and have been proven to reproduce experimental measurements favourably.
The implementation of DFT using AIMPRO is detailed, and the concept of pseudopotentials and basis sets introduced. AIMPRO uses the Local Density Approximation to evaluate
electron density, but this leads to bandgap underestimation. A method to correct this, the
Screened Exchange technique, is briefly described.
The ability to compare theoretical predictions with experimental measurements is key to
modern research. As new technology allows calculations to be performed on ever larger
clusters of atoms, results should become more accurate and meaningful. To this end the
remainder of this thesis will describe the results obtained in this work and compare them
to experimental techniques and measurements.
Chapter 3
Experimental Techniques
The assignment of a particular experimental feature, such as a Raman line, to a defect is
one area where experimental and theoretical work can combine particularly well. Theory
can investigate the many possible configurations of a defect to find one that satisfies all
experimental data, or may suggest new experiments to confirm previous assumptions.
This chapter will review experimental techniques that are directly applicable to the results
of the theoretical work described subsequently.
3.1
Transmission Electron Microscopy
Transmission Electron Microscopy (TEM) is a highly important technique for studying
defects in crystals. Its strength lies in the accessibility of its results, which can be analysed
intuitively as they are the analogue of visiblelight microscopy. It is easy for even a casual
observer to observe and understand defects at a basic level.
3.1.1
Fundamental concepts
The electron microscope is similar in concept to the optical microscope, but uses the
unique wave properties of electrons to overcome the resolution barrier of light with wavelength of 100’s of nm.
55
CHAPTER 3. EXPERIMENTAL TECHNIQUES
56
The image from a point source is constrained by diffraction into a bright disc surrounded
by dark and light halos [80]. The radius of this disc is 1.22F λ /d, where F is the focal
length of the lens, d is the diameter of the lens and λ the wavelength of the light. Rayleigh
stated that if the maximum intensity arising from a point were to fall in the minimum of
intensity from a separate point, then those two points are resolvable. This occurs when
the separation is 1.22F λ /d, and is known as the resolving power.
The relation between the diffractionlimited resolving power and the angle subtended by
a point object at the objective lens, its apparent size, was given by Abbe in 1873 [81]. The
relation states that the minimum resolvable separation in the object being observed is:
d0 = 0.61λ /n sin α
(3.1)
where α is the halfangle subtended by the objective at the object, and n is the refractive
index of the space between the object and the lens. Clearly, in air the minimum resolvable
size is ∼ 0.6λ , which for visible light of 600 nm is 3.6 µ m.
Hence to increase the resolving power of the microscope λ must be reduced. Using ultraviolet light (λ ∼ 10nm) or Xrays (λ < 1nm) would theoretically produce orders of magnitude improvement in resolution, but lenses for this range of electromagnetic radiation
are difficult or impossible to produce. To achieve much greater resolution in microscopy
the wave nature of fundamental particles must be exploited.
Wave nature of the electron
De Broglie showed that a moving electron can be regarded as being both particlelike
and wavelike in its nature [82]. The wavelength of such a particle is expressed as λ =
h/mv. Where h is Planck’s constant and m and v are the mass and velocity of the particle
respectively. A stream of moving electrons can then be considered as a beam of radiation
in the same way as light. The other important factor to note is that by controlling the
energy given to the electron the velocity (and mass at high energy) are altered and hence
the resolution. As an example consider electrons accelerated through an electric field
of 10 kV. The resultant energy of each electron is 10 keV, and using the relation: pc =
˚ and the resolvable limit 7.2 pm which is less than
E 2 + 2Em0 c2 the wavelength is 0.12 A
CHAPTER 3. EXPERIMENTAL TECHNIQUES
57
the Bohr radius of the atom. Thus, in principle, it is possible to resolve individual atoms
using a stream of highly energetic electrons.
3.1.2
Transmission Electron Microscopy in practice
Control of the electron beam
The generation, manipulation and use of electron beams can only occur in a vacuum,
hence the TEM is a fully enclosed instrument evacuated to < 10 −4 mbar. In the simplest
form the TEM is analogous to a simple optical microscope with two lenses  enabling a
magnification of about 10,000 times [80]. Adding more lenses enables a larger magnification, so a modern highresolution TEM may have three or more lenses as well as extra
condensers for high accuracy incident electrons. This setup is shown schematically in
figure 3.1.
Control of the stream of electrons occurs through the use of electric and magnetic fields.
As electrons are charged particles and moving in a magnetic field they feel a force equal
to F = qv × B, with B the magnetic field and q the charge on the electron. The lenses in
a TEM consist of electromagnets which produce a variable field allowing focusing of the
electron stream.
The actual resolution limit of a TEM is dominated by aberration from the lenses. If the
˚
7 pm resolution calculated above is to be achieved with an electron wavelength of 0.12 A
then the beam width is 4.2 mm (assuming F = 2 mm). The lenses must control the
wavefront over this distance with accuracy better than a quarter of the wavelength [83],
˚ However, the shape of the lens is dictated by Maxwell’s equations which
that is, 0.3 A.
means that the ideal, circularly symmetric lens cannot be formed in free space [84] and
spherical aberration will always limit the resolution.
Scherzer, and later Feynman [85], realised that to correct spherical aberration a combination of octupole and quadropole fields was required. The octupole magnet has eight alternating poles, which produces a fourfold rotationally symmetric field. The field strength
increase as the cube of the distance from the centre, which is necessary as spherical
aberration is a thirdorder effect. In the octupole electrons are deflected away from the
CHAPTER 3. EXPERIMENTAL TECHNIQUES
58
10 − 100 kV
Electron source
Condensers
Specimen stage
Objective lens
First projector lens
Second projector lens
High vacuum pump
Viewing chamber
Figure 3.1: Schematic diagram of a highresolution TEM.
centre in two perpendicular directions but towards it in between. This aberration is precorrected with a quadropole, which diverges the stream in one direction aligned to the
divergent axis of the octupole (with resultant perpendicular convergence). In practice,
two such quadropoleoctupole combinations are used to correct in both perpendicular
directions.
Sample preparation and imaging
Preparation of a specimen to be observed in TEM must be undertaken carefully, and
satisfy several requirements:
• The sample must transmit a sufficient number of electrons and minimise energy loss
(inelastic scattering). This limits the sample thickness, with a maximum of 100 nm
CHAPTER 3. EXPERIMENTAL TECHNIQUES
59
generally being applicable.
• The sample must be able to withstand the energy of the incident electrons, as some
fraction will be inelastically scattered or absorbed; with resultant energy exchange.
• Any preparatory treatment should not affect the sample on a level observable with
the TEM.
Samples of crystalline defects (e.g. diamond) are often prepared by ionbeam milling or
chemical etching. The disadvantage of ionbeam milling is that it may impart significant
energies to atoms in the sample, which can then induce structural defects. An example
is patches of graphite or nondiamond carbon on the surface of diamond thin films. It is
necessary to exercise great care when preparing samples to ensure this does not occur.
Samples are often prepared with the faces perpendicular to some particular crystallographic direction to facilitate identification of diffracted directions.
Electrons incident onto the sample can interact in a number of ways, with their final trajectories determined by the process involved. The two groups of interactions are elastic
and inelastic, with zero and finite energy transfer respectively. These can then be subdivided into the groups of backscattered and forward transmitted electrons. The number
of elastic and inelastic electrons gives information on the variation of the atomic number throughout the sample. Elastic scattering is proportional to Z 4/3 , inelastic scattering
varies as Z 1/3 and hence the ratio is proportional to Z, the atomic number.
Transmitted electrons are focused onto a screen coated with a photoemitting substance.
If no sample is present then the image field is uniformly bright, when the sample is inserted darker areas indicate where electrons have been scattered by passing close to
atoms. In a regular crystalline material bright and dark images are formed depending
on whether the Bragg condition has been satisfied, so the sample is often mounted on a
tilting platform to allow adjustment.
Depending on whether the primary or diffracted beam is used for imaging gives rise to
the brightfield and darkfield conditions respectively. In a brightfield, the image is formed
by electrons that have not interacted with the sample, so defects or atoms appear dark
as electrons interacting with them have deflected away. It is useful to image a sample
using the diffracted beam of electrons, as they have undergone interactions with the
CHAPTER 3. EXPERIMENTAL TECHNIQUES
60
sample [80]. This is the darkfield condition, which has the advantage of showing lowcontrast detail more strongly.
In chapter 5, dislocations in diamond are studied, with reference to the Burgers vector.
TEM can accurately determine the Burgers vector and hence dislocation type using the
condition that g · b = 0, where b is the Burgers vector and g is the reciprocal lattice vec
tor. So, by tilting the sample and observing the same dislocation with various diffraction
vectors the angle of the Burgers vector to the line vector can be determined.
3.2
Electron Energy Loss Spectroscopy
Electron Energy Loss Spectroscopy (EELS) is a technique related to TEM, in that it uses
principally the same equipment setup, and the same method of analysing a sample with
a stream of high energy electrons. In fact, the two techniques can often be carried out in
the same instrument. EELS makes use of the energy lost by electrons through inelastic
collisions to determine the local electronic structure of bulk material or a defect.
3.2.1
Fundamental concepts
In terms of equipment, up until the collection of transmitted electrons the TEM and EELS
instruments are the same. In EELS, a magnetic spectrometer is used to separate electrons of different energies for analysis, shown in figure 3.2. Electrons which have suffered the greatest energy loss are moving the slowest and hence are deflected most. To
observe electrons having a particular energy a slit is placed in the beam of separated
electrons, which only allows electrons which have been deflected to that angle to pass
through. Electron numbers are counted by a scintillator or photomultiplier. In an alternative technique a lens focuses all paths of electrons with various energies onto an array of
detectors, similar to a CCD device. Thus, all energy ranges can be sampled simultaneously.
The detected energy loss of the electrons is generally displayed relative to the incident
beam energy. Therefore any electrons which have undergone only elastic collisions contribute to a peak at 0 eV, known as the zeroloss peak, which is seen in all EELS ex
CHAPTER 3. EXPERIMENTAL TECHNIQUES
61
Energy selecting
slit
E− E
EELS
E
Photomultiplier
Magnetic spectrometer
Parallel
EELS
E− E
E
Scintillator
&Detector Array
Quadropole lens
Figure 3.2: Diagram of serial and parallel methods of detecting energy loss of atoms
after interaction with a sample [80]. The dashed lines indicate the paths of electrons with
energy from E to E − ∆E.
periments. The rest of the energyloss spectrum can then be divided into two regions:
the lowloss and coreloss regions. The coreloss region relates to energy lost by the
electrons by transitions from the core to empty levels of the atoms in the sample, so
has higher energy loss of the order 300 – 800 eV. This requires a detailed knowledge
of the core electron wavefunctions, but these are eliminated in this work through the use
of pseudopotentials, thus coreloss EELS is beyond the scope of this work and is not
discussed further.
Lowloss EELS studies the energy losses occurring from the promotion of transitions from
occupied valence states to conduction states. This region is generally below 50 eV and
information can be deduced on the bandgap of a material as well as the valence and
conduction bands. In addition this is where the plasmon peak of a material often occurs,
which is observed experimentally at 33 eV in diamond [86], and can be predicted by the
free electron theory:
ωp =
4π ne e2
m
1/2
(3.2)
CHAPTER 3. EXPERIMENTAL TECHNIQUES
62
ω p is the plasmon frequency, ne is the electron density, e the electron charge and m the
effective mass of the electron in diamond.
3.3
Optical Absorption
Optical absorption simply measures the ratio of transmitted to incident power of a material, for a variety of wavelengths. In the range 400 nm – 700 nm (3.09 eV – 1.77 eV)
the absorption of the material will be directly visible as colour, but experiments will often
extend in to the ultraviolet and infrared regimes. The absorption spectrum of a material
is unique and can quickly classify that material. Defects may have a wellknown absorption ‘signature’ which can modify the host material absorption spectrum, sometimes with
visible results, allowing identification of defects and an estimate of their concentration.
3.3.1
Fundamental concepts
When a beam of electromagnetic radiation passes through a material photons may interact with the material atoms in a number of ways. There are two processes which are
relevant to absorption in the visible wavelength range. An incident photon may be absorbed causing an electron to jump from a ground to an excited state, such as a valance
band to conduction band transition, leaving a hole. Alternatively the incident photon may
give rise to a phonon or the local vibration of a defect. A dipole is required for onephonon
absorption which explains its absence in pure diamond, however twophonon and threephonon absorption in evident in the infrared region.
The intensity of an electromagnetic beam is described by:
I = (hν )N
(3.3)
where N is the photon flux [87]. When the beam traverses a material the flux is reduced,
by the processes described above. If photons are incident on a material of thickness
dx, and the number of transmitted photons is N − dN, then the absorbed photons can be
related to the thickness and incident flux by the relation:
CHAPTER 3. EXPERIMENTAL TECHNIQUES
dN = −µ Ndx
63
(3.4)
with a proportionality constant µ , known as the absorption coefficient. This constant is
unique to any given material, and depends on the density of the material (as this dictates
the number of atoms encountered by the incident photon beam). This relation can be
integrated for a material of total thickness x and for incident and transmitted fluxes N 0 and
N respectively. Therefore:
N = N0 e−µ x
(3.5)
I = I0 e−µ x
(3.6)
and from equation 3.3:
It is clear that the absorption increases with the thickness x and the absorption coefficient
µ . It is important to note that the absorption coefficient is dependent on the incident
photon energy (frequency), which allows the absorption spectrum to be determined.
3.3.2
Optical Absorption in practice
Optical absorption appears straightforward in principle, with simply a photon source shining through a sample and photon detectors measuring the incident and transmitted flux
at any given wavelength. In practice, modern experimental techniques use a variety of
techniques to vary the incident wavelength and measure it accurately. The emergence of
lasers as a suitable light source has depended on the ability to ‘tune’ the wavelength to
any desired value.
The simplest instruments, Spectrographs and Monochromators, both rely on the separation of transmitted light into its components by a prism or diffraction grating, but differ in
their detection of photon flux [88]. In a spectrograph a CCD array is used to sample all
CHAPTER 3. EXPERIMENTAL TECHNIQUES
64
offset wavelengths simultaneously, whilst a monochromator uses a photoelectric recording of a selected interval. In the monochromator the prism or grating is rotated to focus
the selected wavelength onto the detector, see figures 3.3(a) and 3.3(b).
Separated Wavelengths
Prism
Source
Sample
CCD Array
(a)
Mirror
Photoelectric Detector
Moveable
Grating
Mirror
Source
Sample
(b)
Figure 3.3: (a) Prism spectrograph and (b) Grating monochromator. The method of selection of wavelength is interchangeable between the devices, the difference is defined
by the detection technique.
Advanced instrumentation
A laser light source has the advantage of steady power (intensity), coherent light output
and ease of manipulation with optical components. The main disadvantage which must
be overcome is the single wavelength output, which can be modified by tuning a singlemode laser.
CHAPTER 3. EXPERIMENTAL TECHNIQUES
65
Singlemode lasers have an output wavelength determined by the size of the resonant
cavity [88], λ = 2nd/q. Here n, d and q are the refractive index of the gain medium, the
mirror separation and the quality factor of the cavity respectively. The output wavelength
is then altered by variation of n or d. If the gain medium is a gas then n may be altered by
changing the pressure. The mirror separation d can be accurately changed using a piezoelectric element as one of the mirror mounts. The variation of d alone presents problems
with mode hopping which can be solved using a tilting etalon inside the resonance cavity,
which is beyond the scope of this review but described in depth in [88].
3.4
Raman Spectroscopy
Raman spectroscopy makes use of the quantised rotational and vibrational energies of
a molecule or solid to deduce its properties. These excitations in the system absorb or
emit photons which causes a shift in energy of a light source incident on the system,
which is the basis of the technique. Raman spectroscopy is very sensitive to molecular
bonding and structure of a sample, which leads to a distinct spectral pattern for a material.
However it is a weak effect so must be carefully measured.
3.4.1
Fundamental concepts
When a photon of light is incident on an atom or molecule, it may be energetic enough to
induce an electronic transition, but if this is not the case it can be scattered in one of three
ways [89]. Rayleigh scattering, where there is no energy loss, is of no interest in Raman
spectroscopy. Inelastically scattered light can either lose or gain energy during interaction, giving rise to a shifted frequency in the emitted radiation. Such scattering of light (or
radiation generally) with an accompanying change of frequency is known as the Raman
effect [90]. The scattered radiation usually has different polarization characteristics from
the incident radiation [91].
A system at energy E1 can make a transition to a higher energy E 2 if supplied with energy
from incident radiation. It can be visualised as the annihilation of the incident photon, with
the simultaneous creation of a lower energy photon [91]; accompanied by the energetic
transition described. The newly created photon then has a wavelength longer then the
CHAPTER 3. EXPERIMENTAL TECHNIQUES
66
incident one. This is known as Stokes Raman scattering (figure 3.4). It is also possible for
the incident radiation to induce the reverse transition; lowering the system energy from E 2
to E1 . The emitted radiation has a shorter wavelength, known as AntiStokes scattering.
The Raman effect may be described both classically and quantum mechanically, with an
overview of each given here.
Classical approach
The classical approach regards the scattering molecule/system as a collection of atoms
undergoing simple harmonic motion [89]. To describe the scattering types requires knowledge of the dipole moment µi induced by the incident radiation, which for small fields is
proportional to the field strength:
µi = αε
(3.7)
where ε is the field strength and α is the polarizibility of the molecule. The electric field
generated by electromagnetic radiation is expressed:
ε = ε0 cos 2π t ν0
(3.8)
with ε0 the equilibrium field strength for the material and ν 0 the angular frequency (or
wavenumber) of the radiation. The incident radiation will induce a fluctuating dipole with
the same frequency in the system. If the dipole then emits radiation of the same frequency, there is no energy loss. This is Rayleigh scattering.
If the incident radiation interacts with the molecule the equations 3.7 and 3.8 combine to
give:
µi = αε0 cos 2π t ν0
(3.9)
CHAPTER 3. EXPERIMENTAL TECHNIQUES
67
If a molecule undertakes simple harmonic vibrations with a frequency ν 0 , its polarizibility
changes during vibration and is expressed:
α = α0 +
∂α
∂ qv
q0 cos 2π t ν0
(3.10)
0
where qv is a coordinate along the axis of vibration at time t. If this is then substituted into
equation 3.9, the total dipole moment can be written:
µi = α0 ε0 cos 2πν0 t +
∂α
∂ qv
ε0 q0
× [cos 2π (ν0 + νv )t + cos 2π (ν0 − νv )t]
0 2
(3.11)
The first term describes the Raleigh scattering with the second and third terms describing
the Stokes and antiStokes Raman scattering. It can been seen that the polarizibility of
the system must change during vibration, for that vibration to be Raman active.
Quantum approach
The vibrational and rotational energies of a molecule are quantised according to the relationship:
Ev = hν (n + 1/2)
(3.12)
Where n is the vibrational quantum number, having values 0,1,2,3. . .. The levels can be
visualised in the ideal model of figure 3.4. At room temperature most vibrations are in
the ground state (n = 0), which makes antiStokes transitions much less likely and hence
weaker in the scattering spectrum.
The transition from one energy level to another can be visualised using perturbation theory, with a perturbing wavefunction applied to the ground state which then produces a
new wavefunction corresponding to the excited state.
CHAPTER 3. EXPERIMENTAL TECHNIQUES
68
Virtual Levels
hν0
hν0
hν0
hν0
h( ν0
h( ν0 + νn)
νn)
n= 3
n=2
n= 1
n= 0
Rayleigh
Stokes
Anti − Stokes
Figure 3.4: The vibrational energy level transitions which can be induced by a photon of
energy hν0 .
3.4.2
Raman Spectroscopy in practice
The basic components needed to undertake Raman spectroscopy are a monochromatic
light source, a system to collect the scattered photons and separate them into individual
wavelengths (a monochromator) and a detector to measure intensity.
Monochromatic light is most easily obtained from a laser in either continuous wave or
pulsed configuration. It can be performed at various excitation energies, however sample
fluorescence is reduced by using nearinfrared excitation. Fluorescence is undesirable as
it can obscure the Raman signal. Red wavelength excitation is also less likely to cause
photodecomposition or thermal damage due to absorption.
The Raman scattered light is weak in magnitude compared to the elasticallyscattered
(Raleighscattered) light, so one of the critical tasks of the detection optics is to separate
out these components. There are two distinct methods to achieve this. Dispersive Raman
has become the most widely used method, consisting of a monochromator to separate
out the Raleigh wavelengths with a grating to further separate the light into its constituent
wavelengths (figure 3.5(a)). The spectrum is then detected by a CCD array. FourierTransform (FT) Raman utilises an interferometer to introduce a path difference between
the source and the signal beams (figure 3.5(b)). The resultant interference pattern can
CHAPTER 3. EXPERIMENTAL TECHNIQUES
69
then be used to reconstruct the Raman spectrum.
FT Raman generally uses a longer wavelength laser of 1064 nm which is necessary when
analysing samples which fluoresce when exposed to visible wavelength radiation [89].
This wavelength laser has poor illumination, and FT Raman suffers from poor spatial
resolution. Dispersive Raman spectroscopy can be used with several types of laser,
allowing selection of an appropriate wavelength for the material to be analysed. The
concept of the Raman Microscope, a nondestructive and noncontact analysis method,
can be realised only with dispersive Raman.
3.5
Photoluminescence Spectroscopy
Photoluminescence (PL) is the spontaneous emission of light from a material when optically excited. PL spectroscopy utilises this phenomena to investigate the electronic levels
of a material, and can classify bandgap and defect states as well as thermally activated
processes [92]. It has the advantage of being noncontact and nondestructive of any
sample material.
3.5.1
Fundamental concepts
When a photon of sufficient energy is incident upon an electron in a material, it may cause
that electron to be excited to a higher energy level. The excited electron can then directly
or indirectly relax back to its original state, emitting a photon in the process. In the case of
a directgap intrinsic semiconductor, the emission energy is the bandgap and hence the
frequency of the emitted light is dictated by E gap = hν0 = h2π /λ . If the material is doped,
and hence has a shifted Fermi level then the excited electron may move down to a lower
energy level before emission, and hence emit with a lower frequency (longer wavelength)
as shown in figure 3.6. Here hν1 will be equal to Egap in the intrinsic case and Ereduced in
the doped case.
There are modifications to these equations depending on phonon assistance and exciton
effects. Phonons allow transitions to take place between different positions in the Brillouin
zone, adding momentum when there is an indirect bandgap. This is the case for diamond.
CHAPTER 3. EXPERIMENTAL TECHNIQUES
Sample
70
Monochromator
Laser Source
Grating
Spectrometer
Dispersed Spectrum
CCD Array
(a)
Fixed Mirror
Sample
Laser Source
Moveable Mirror
Beamsplitter
Filter
Detector
(b)
Figure 3.5: (a) Dispersive Raman with a grating spectrometer and (b) Fourier Transform
Raman using a MichelsonMorely interferometer.
In this situation the incoming photon must have sufficient energy to both excite the electron across the gap, and create a phonon to assist the transition across kspace. Thus
the energy gained by the excited electron is hν 0 − hΩ where Ω is the phonon frequency.
When the electron relaxes through PL, it must sacrifice more energy to create an second
phonon with equal and opposite momentum to the first (thus satisfying conservation of
momentum), so the total possible energy of the emitted photon is hν 1 = hν0 − 2(hΩ) as
CHAPTER 3. EXPERIMENTAL TECHNIQUES
71
Conduction States
Dopant Level
Energy
h ν0
E gap
e−
Excitation
Ereduced
h ν1
Photoluminescence
Valance States
Brillouin Zone
Figure 3.6: Schematic of Photoluminescence in a directgap semiconductor.
shown in figure 3.7.
If there are defects with localised energy levels in the gap, then an electron or hole may
become trapped there. The PL emission is then characteristic of that defect. If both the
ground and the excited states are in the gap there will be a sharp line in the emission
spectrum.
When an electron is excited to the conduction band it leaves behind a hole of opposite
electric charge. These opposite charges are attracted to each other by the Coulomb
force, forming a quasiparticle known as an exciton. The exciton can move throughout
the material without feeling the effects of other electrons around it, known as screening.
There is an energy difference between the exciton and an unbound electron and hole,
due to the binding force. This results in a lesser amount of energy being available for
photon emission when recombination occurs. The modified emission energy is given by
hν1 = Egap − Eexciton in the case of direct transitions, and by hν 1 = Egap − Eexciton − 2(hΩ)
for an indirect transition. The reduced energy in exciton transitions can be visualised as
quasistates near the conduction band, shown in figure 3.8. A material must be very pure
for exciton emission to be prominent, so this can be used as a test of purity.
CHAPTER 3. EXPERIMENTAL TECHNIQUES
72
Conduction States
E
Energy
h ν0
Excitation
gap
−
h ν1
Phonon
Photoluminescence
hΩ
e
Valence States
K
Figure 3.7: Phononassisted PL in a indirectgap semiconductor. The conservation of
momentum is achieved by the emission of phonons in opposite directions during excitation and emission.
Conduction States
Exciton Band
Energy
h ν0
E gap − E
e−
exciton
h ν1
Photoluminescence
Excitation
Valence States
K
Figure 3.8: Exciton effects in Photoluminescence. The reduced energy due to electronhole binding can be visualised as a lowering the conduction band.
CHAPTER 3. EXPERIMENTAL TECHNIQUES
73
Laser
Exciting Beam
Sample
Photodetector
Spectrometer
Lens
Photoluminescence
Figure 3.9: Generalised experimental setup for measurement of Photoluminescence.
3.5.2
Photoluminescence Spectroscopy in practice
Depending on the number and position of dopant related levels in the material there
may be a wide range of frequencies emitted from any electron recombination. Clearly
then, there may also be a range of excitation energies required. The requirement for
the experimental setup is thus similar to the optical absorption in section 3.3, with a
tunable laser to deliver the range of energies, and a spectrometer to separate the emitted
wavelengths for analysis. The emitted photons may be spatially dispersed by virtue of
the location of the emission site in the sample as well as scattering from surrounding
material. This dispersion requires the positioning of a lens to direct emission onto the
detector as shown in figure 3.9. When exciting a material at room temperature bandgap
emission can mask luminescence from defect states. The experimental setup may make
provision to cool the sample to below 100 K to allow defectrelated luminescence to be
observed.
3.6
Positron Annihilation Spectroscopy
Positron Annihilation Spectroscopy (PAS) can be used to deduce information on vacancy
related defects in crystals. It utilises the positron, which is the electron antiparticle, by
measuring its lifetime of interaction in a crystal with the existing electrons. Vacancy related defects have a lower electron density and thus show up as having a longer positron
lifetime than the bulk material.
CHAPTER 3. EXPERIMENTAL TECHNIQUES
3.6.1
74
Fundamental concepts
PAS is based on the electromagnetic interaction between positrons (e + ) and electrons
(e− ), which makes possible annihilation of the two particles, with the resultant energy
transferred to gamma radiation. The principle reaction is the twophoton annihilation [93]:
e+ + e − → γ 1 + γ 2
(3.13)
the positron e+ annihilates with an electron e− to produce two photons, which emit in
directions differing by π radians i.e opposite, with an additional directional shift θ :
∆θ ≈ p−,T /m0 c
(3.14)
p−,T is the transverse momentum component of the annihilating electron. The annihilation
processes are characterised with an annihilation rate λ which is related to the positron
lifetime by λ = τ −1 . The annihilation rate is proportional to the effective electron density
ne and the classical electron radius r c :
λ ≈ π rc2 cne
(3.15)
In a metal the conduction electron density may be of order 10 −29 m−3 , which gives an expected positron lifetime of ∼ 600 ps. Experimentally, values of τ ∼ 200 ps are measured,
due to electron density enhancement at the positron site [93]. This occurs when the
Coulomb force between the positrons and electrons causes mobile conduction electrons
to move towards the positron site and increase the effective density.
As positrons are injected into the material, the majority interact with electrons in the bulk
and give rise to a dominant bulk lifetime τ b [38]. If monovacancies or larger vacancy
clusters are present, then some fraction of the injected positrons will become trapped
at them. These vacancy related defects have a lower electron density than the bulk, so
according to equation 3.15 the lifetime will be longer. In any given sample a range of
lifetimes τ1 , τ2 . . . etc. can be extracted from the total lifetime spectrum obtained over
CHAPTER 3. EXPERIMENTAL TECHNIQUES
75
p−, t
Photon Emission
Electron Momentum
e−
θ
e+
Photon Emission
Figure 3.10: Positron annihilation reaction schematic.
many annihilation events. These lifetimes can then be related to individual defect groups
in the sample.
The trapping rate of the positrons, κ is proportional to the vacancy concentration according to:
κ = µ Cv
(3.16)
where µ is the absolute specific trapping rate, and Cv the absolute vacancy concentration.
The trapping rate is a central parameter but cannot be directly determined by analyses
of the lifetime spectra [94], rather it must be calculated from experimentally obtained
parameters:
κ=
I2 (1/τb − 1/τ2 )
(1 − I2 )
(3.17)
with τ2 the lifetime for positrons trapped at vacancies, and I2 its intensity.
3.6.2
Positron Annihilation Spectroscopy in practice
The usual source of positrons for PAS experiments is a radioisotope emitting β + radiation. The energy range of the emitted positrons is typically of order 0.1 – 1 MeV [93],
which gives a penetration depth of about 10 – 100 µ m. Sources that simultaneously
emit gamma radiation are particularly useful, as the delay in detection between gamma
CHAPTER 3. EXPERIMENTAL TECHNIQUES
76
Detectors
Source Photon
Emission
e+
e+
Annihilation
Emission
Investigation
Sample
22
Na Source
Reference
Sample
Figure 3.11: PAS experimental setup. The double sample setup to provide a known
background count is illustrated.
emission from decay, and that from positron annihilation, gives the positron lifetime.
To ensure that all positrons emitted from the source are annihilated in a sample, the
source is placed between two sample plates [35]. If two identical samples are not available one can be replaced with a sample for which the properties are known, in the case
of diamond this could be a CVD or HPHT grown sample. The final lifetime spectrum then
contains a contribution from the sample of interest and the source contribution, which is
subtracted. Figure 3.11 demonstrates the sample setup.
3.7
Electron Paramagnetic Resonance
Electron Paramagnetic Resonance (EPR), also known as Electron Spin Resonance (ESR)
or Electron Magnetic Resonance (EMR), is technique utilising the reaction of unpaired
electrons to an applied magnetic field, the Zeeman effect. It has the capability to identify
the atomic species, structure and symmetry of many defects in a wide range of materials.
3.7.1
Fundamental concepts
The electron has a magnetic moment which, when subjected to an external field B 0 , can
align itself parallel or antiparallel to that field. These alignments differ in energy, with the
parallel state lower, hence the levels are split. This splitting is known as the Zeeman
CHAPTER 3. EXPERIMENTAL TECHNIQUES
77
m s = −1/2
Energy
Transition inducing
irradiation
m = +1/2
s
B0
Figure 3.12: Zeeman splitting for a single electron. Incident radiation may induce a transition between the parallel and antiparallel spin states.
effect, and the energy between the two states is given by:
∆E = E(ms = 1/2) − E(ms = −1/2) =
ge βe B0
h
(3.18)
where ms = ±1/2 is the spin quantum number for each spin state, g e is the gyromagnetic
ratio of the electron and βe denotes the Bohr magneton.
If electromagnetic radiation is applied to the electron, and the photon energy matches
the energy difference between the states, then a transition between the states can be
induced. The spin is flipped from one orientation to the other, and the resonance condition
is fulfilled. The electron splitting and spin states transition are illustrated in figure 3.12.
The single electron case is simple but can be applied to more complex cases such as
transition metal ions. In most other cases additional magnetic or electrical fields in the
material or the presence of more than one electron complicate the spectra. This will lead
to further energy level splitting. Interaction of the electron spin with the external magnetic
field yields an EPR spectrum with one single line which may be broadened due to g e
anisotropy. If nuclei with a spin quantum number of l > 0 are in the same region, the
electron spin will experience a local magnetic field built up by these nuclei. This is due to
the magnetic moment µ associated with the intrinsic spin angular momentum L, and is
CHAPTER 3. EXPERIMENTAL TECHNIQUES
78
called the hyperfine interaction. The nuclear spin is quantised in a magnetic field so, like
the electron spin, undergoes a nuclear Zeeman splitting, expressed by:
En = −
gn βn lB0
h
(3.19)
where gn is the nuclear gfactor, and βn the nuclear magneton. For a given nuclear quantum number l, there will be a splitting into (2l + 1) energy levels, each characterised by
a nuclear magnetic spin quantum number m i = l, l − 1, . . . , −l The hyperfine interaction
depends on the orientation of the nuclear magnetic moment with respect to the external
magnetic field, and results in a splitting of each electron state into (2l + 1) levels.
In practice, EPR observes a large number of centres throughout a material, which are
statistically described by the Boltzmann distribution:
nM j +1
∆E
= exp −
nM j
kb T
(3.20)
where nM j +1 is the number of centres in level M j .
3.7.2
Electron Paramagnetic Resonance in practice
Transitions from one energy level to another occur at certain combinations of the external
magnetic field B0 and the irradiation frequency ν . It is experimentally simpler to maintain
a steady irradiation and to sweep through a range of magnetic field strengths, rather
than viceversa. The excitation frequency is usually in one of four bands; the Sband
(24 GHz), Xband (810 GHz), Qband (∼35 GHz) or Wband (∼90 GHz). The swept
magnetic field varies in the range of 100 – 300×10 −3 T, for example in the Xband a
value of ge =2 corresponds to 320 mT.
It is the value of ge which describes the centre’s electronic structure. If the value of g e
measured at a resonance condition differs from the free electron value (g e = 2.0023),
then the electron has lost or gained angular momentum through spinorbit coupling. The
CHAPTER 3. EXPERIMENTAL TECHNIQUES
79
Load
Klystron
Waveguide
Circulator
A
Resonance
Chamber
Diode
Detector
Sample
Magnetic Coil
Magnetic Coil
Modulation
Coils
Figure 3.13: Experimental setup for EPR/ESR experiments. The resonance chamber
may be cooled to low temperatures to reduce noise
change in angular momentum is anisotropic and varies with orientation to the external
field, yielding information about the atomic orbitals containing the unpaired electron.
The sample to be studied is placed in the resonance chamber as shown in figure 3.13,
and the microwave radiation transferred via a waveguide (which can be as simple as a
brass tube). The illuminating power is low, typically 200 mW with a 2 MHz bandwidth.
The resonance chamber is designed to absorb the incident power completely, hence
any absorption of radiation by the sample will detune the chamber and radiation will
be reflected back up the waveguide to the diode detector. The measurement of this
radiation as a function of magnetic field yields the EPR spectrum. The signaltonoise
ratio is improved by applying an amplitude modulation to the magnetic field, typically at a
frequency of 100 kHz.
As well as the constant wave approach described above, EPR can be conducted in a
pulse configuration. In this arrangement a highpower microwave pulse at a single fre
CHAPTER 3. EXPERIMENTAL TECHNIQUES
80
quency ν , and fixed magnetic field B0 , excites a large range of frequencies in the sample.
Fourier transform leads to an excitation range for the pulse of ν ±1/2 (t p ), which typically
could be 100 MHz for a 10 ns pulse. Reduction of the pulse time increases the excitation
range, however due to technical limitations it is difficult to excite the whole spectrum by a
single EPR pulse.
Continuous wave EPR has the advantage of higher sensitivity, plus spectra can be recorded
at room temperature unlike pulse EPR. Pulse EPR does allow additional information
about weakly coupled nuclei and relaxation properties of the spin system to be gathered, via manipulation of spin with a sequence of microwave pulses. A combination of
both techniques is necessary for full details of a paramagnetic centre to emerge.
3.8
Summary
There are a range of experimental techniques that can be applied to diamond, or any
other material. It is possible to use them to deduce mechanical, electronic and optical
properties, or to discover point and extended defects. The main importance of experimental techniques to theoretical modelling work is to compare known results to those
calculated by theory, for example the absorption spectrum of bulk diamond, to gauge the
accuracy of the model. In turn this accuracy will provide the degree of confidence in the
model’s prediction of unknown quantities.
The main techniques relevant to this work are those that deal with the optical properties of a material, which can be modelled as described in section 2.6.5. Therefore most
reference will be made to the Electron Energy Loss spectroscopy (EELS), on which collaboration with experimental work [95] has been undertaken, and to optical absorption
studies. In addition, the predictions of this work have been studied using Transmission
Electron Microscopy [96].
Chapter 4
Small Vacancy Clusters
4.1
Introduction
Elements of this work have been published in Physica Status Solidi (a) 202, 2182 (2005).
Electron paramagnetic resonance (EPR) studies of multivacancy centres have been conducted on chemically pure type IIa diamond which has been irradiated and subsequently
annealed [39, 41]. Eight defects are detected following anneals between 700 and 1400 ◦ C.
These are labelled O1, R5, R6, R7, R8, R9, R10 and R11. The first three possess spin
S =1 and R7 is described as a S =3/2 centre [39]. For the remainder of centres it was
uncertain whether they possessed spin 1 or 3/2. More recent work indicates a S =1 for
R10 and R11 [41], and also for R7 [40]. Centres O1,R5,R6,R10 and R11 possess C 2v
symmetry, and are thought to be composed of vacancy chains along 110 [39, 40]. Centres R7 and R8 along with a new centre R7a have been tentatively linked with vacancy
clusters possessing C2v (R8/R7a) and monI (R7) symmetry [40]. This is further to the
trigonal R12 defect with a 111 axis previously proposed [39].
In the case of longer vacancy chains, V n , n >2, it is possible that the spindensity does
not lie on interior atoms having dangling bonds but on the dangling bonds associated with
the end atoms of the chain. If this is the case then the fine structure term D in the S =1
multivacancy centres can be used to determine the length of the chain. The interaction
√
of two spin dipoles, separated by (n − 2)a 0 / 2 in Vn , at the ends of a [110] vacancy chain
is D = 3/2(g2 β 2 r−3 ). Such an analysis indicates that O1, R7, R8, R6, R10 and R11 are
81
CHAPTER 4. SMALL VACANCY CLUSTERS
82
due to 110 chains of 4, 4, 45, 5, 6 and 7 vacancies respectively [39]. It is important to
reiterate that the spin designation of R8  R11 is not certain, but is assumed to be 1 for
this analysis. If the centres were to have spin 3/2 then this assignment may be invalid.
More recent work [41, 40] assigns R5, O1, R6, R10 and R11 to chains of 3, 4, 5, 6 and 7
vacancies and an additional EPR centre (KUL11) to a chain of eight vacancies. However,
this analysis also assumes that the spin is localized at the chain ends and not on dangling
bonds along its length.
The optical properties of annealed irradiated type IIa diamonds have been reported [39].
With heat treatments of 600◦ C there is a loss of the GR1 peak and a correlated growth of
TH5. After the 700◦ C anneal there is the loss of TH5 and the growth of a continuum. The
absorption spectrum simply scales the spectrum seen in unirradiated type IIa diamond
and may signify graphitic or disordered regions. A comparison of the absorption produced
by electron and neutron irradiation followed by heat treatments at 450 ◦ C suggests that
the latter produces a set of thermally stable defects giving rise to a continuum extending
right up to the band edge of diamond. These defects were attributed to disordered carbon
network produced by endofrange neutron damage.
The initial irradiation rendered the diamond opaque but an anneal beyond 1050 ◦ C revealed an absorption edge starting at 1.5 eV and cutting off at 2.2 eV with lesser features
at around 1 eV. The introduction rates of the EPR centres following a 1000 ◦ C anneal
are very low. For a dose of almost 1020 cm−2 electrons with energy 2 MeV, only about
1018 cm−3 O1 defects are created.
It could be expected that the centres associated with vacancy chains would appear sequentially. The dissolution of the monovacancy and divacancy would occur at similar
temperatures so there would be a clear startpoint for the formation of vacancy chains.
With the increase of time there would be a steady appearance of larger chains. In the
experiments referenced the annealing time was in the order of 12 hours [41], so that
all possible centres would have formed by the time of observation. Hence the centres
appear to form simultaneously.
A study of possible vacancy defects is necessary to examine the validity of the assignment of these EPR centres. In addition, the prevalence of vacancy defects in brown diamond means that they must be investigated to see if they could be acting as the source
of brown colour.
CHAPTER 4. SMALL VACANCY CLUSTERS
4.2
83
Structures Modelled
Vacancy chains and clusters are created by the removal of atoms from the bulk diamond
structure. Chains of vacancies were created by the removal of atoms in the 110 direction as shown in figure 4.1. Candidate vacancy clusters, where not all vacancies lie along
[110], were investigated from models minimizing the number of dangling bonds and are
shown schematically in figure 4.2. Here the atoms are labelled numerically in order of removal. Such an approach has been used previously to describe multivacancy structures
in silicon [97, 98].
Vacancy clusters up to 14 vacancies and chains of up to four vacancies were inserted
into cubic unit cells of 216 atoms. As well as AIMPRO studies DFTB calculations were
¨ Paderborn/Exeter University). Initial investigations
performed by N. Fujita (Universitat
for V3 chains were carried out in cells of 64 atoms using both AIMPRO and DFTB. The
formation energies in AIMPRO were calculated with a contracted basis set of 13 functions,
and found to be 12.908 eV in a 64 atom cell and 13.129 eV in a 216 atom cell; a difference
of 1.7%. Calculations with DFTB showed the formation energy in a 64 atom cell to be
12.607 eV, and 14.175 eV in the 216 atom cell; a difference of 11%. The large differences
in formation energy with cell size indicate that a 216 atom cell is the minimum size that
should be used for reliable results. DFTB calculations in 512 and 1000 atom cells show
only marginal changes in formation energy from a 216 atom cell, supporting the use of
216 atoms cells for modelling. A 23 set of MonkhorstPack (MP) [67] kpoints was used
to sample the Brillouin zone.
v
vi
iii
10
[1
iv
]
ii
i
−−
[112]
Figure 4.1: Diagram of order of removal of atoms to create the 110 vacancy chain.
Longer vacancy chains of five and six vacancies were also placed in cells of 216 atoms,
CHAPTER 4. SMALL VACANCY CLUSTERS
84
12
14
11
[111]
13
5
4
1
6
3
2
7
9
8
10
−−
[112]
Figure 4.2: Diagram of order of removal of atoms to create vacancy clusters of up to 14
vacancies.
but these are noncubic. These cells were longer in the [110] direction to ensure the
terminating atoms were not too close to one another upon repetition. A vacancy chain
of infinite length was investigated using a thin slab of diamond, whose surface normal is
parallel to [110] and contains 140 atoms. This approximates the energy of a long loop of
vacancies.
It has previously been reported how a distortion to C2h symmetry for the divacancy could
be energetically favourable if the reduction in oneelectron energies caused by such a
distortion exceeds the additional strain energy [99]. The vacancy chains studied here
have been subjected to inward and outward distortion to C2h (even numbers) and C2v (odd
numbers) symmetry, and the results compared to undistorted models.
4.3
Results
4.3.1
Energies
Figure 4.3 shows the calculated formation energies. The formation energy per vacancy
decreases with n due an increasing binding in the larger defects. However for the vacancy
CHAPTER 4. SMALL VACANCY CLUSTERS
85
clusters, as found for other semiconductors, there are sets of ‘magic’ numbers showing
exceptional stability. These occur at 6, 10 and 14 vacancies and are the same numbers
as found in silicon [98] where they are attributed to bond reconstruction and hence a
minimization in the number of dangling bonds. The shape of these clusters is shown in
figure 4.2. V6 in silicon has been linked with a defect found in neutron irradiated material
and detected in luminescence [100]. The bond reconstruction explanation for the stability of these magic numbered vacancy clusters holds only for the 6 vacancy cluster, as
explained below.
5.5
S=0 Chains
S=1 Chains
S=0 Clusters
S=1 Clusters
Formation Energy per Vacancy (eV)
5
4.5
4
3.5
3
2.5
2
2
3
4
5
6
7
8
9
10
11
12
13
14
Inf
Number of Vacancies
Figure 4.3: Formation energies of vacancy clusters calculated with AIMPRO.
An alternative structure for the V4 vacancy cluster in silicon has been suggested by
Makhov et al. [101]. Here, one, two or three selfinterstitials are added to the six vacancy model of Staab, with the extra atoms being added between atoms neighbouring
a vacancy. These interstitials eliminate dangling bonds and lower the energy. However,
for diamond the formation energy of this structure for V 4 is 4.12 eV per vacancy which is
greater than the Staab model of V4 of 3.88 eV per vacancy. A model of 11 vacancies in
an arrangement which differs from this work has been proposed for the R7 centre [40].
This model satisfies the apparent monoclinicI symmetry of the R7 centre. It was found
that the formation energy per vacancy was higher at 2.88 eV compared to 2.67 eV for the
Staab model  both defects being in the S =1 state. This can be explained by the number
of dangling bonds caused by removal of atoms in each case. In the Staab model for 11
CHAPTER 4. SMALL VACANCY CLUSTERS
86
Table 4.1: Formation energies, in eV, of defects with n vacancies calculated using AIMPRO. The lowest energy spin state is indicated.
Defect
Divacancy
3 vacancy chain
4 vacancy chain
5 vacancy chain
6 vacancy chain
4 vacancy cluster
5 vacancy cluster
6 vacancy cluster
7 vacancy cluster
8 vacancy cluster
9 vacancy cluster
10 vacancy cluster
11 vacancy cluster
12 vacancy cluster
13 vacancy cluster
14 vacancy cluster
∞ vacancy chain
E f orm
Lowest
9.89
13.13
16.21
19.19
22.19
15.54
18.85
17.64
21.73
24.59
25.16
25.41
29.14
32.02
32.68
32.93
12.58
E f orm /n
Lowest
4.94
4.37
4.04
3.83
3.70
3.88
3.77
2.94
3.10
3.07
2.80
2.54
2.67
2.71
2.51
2.35
3.15
Spin
Lowest
1
0
0
1
1
0
1
0
1
1
1
1
1
1
1
1
0
E f orm
Highest
10.11
13.15
16.23
19.44
22.38
15.90
18.86
18.94
22.02
24.85
25.51
25.69
29.58
32.67
33.22
33.34

E f orm /n
Highest
5.06
4.38
4.06
3.88
3.73
3.98
3.77
3.16
3.15
3.11
2.83
2.57
2.69
2.72
2.55
2.38

Spin
Highest
0
1
1
0
0
1
0
1
0
0
0
0
0
0
0
0

vacancies, 18 dangling bonds are left, but 19 are formed by the model for R7.
For both 110 vacancy chains and vacancy clusters, spinpolarized calculations were
also performed to determine the most favourable spin state. These energies are also
given in table 4.1, where it is seen that there are relatively small differences in the energies of S =1 and S =0 defects. It is noticeable that the V 4 vacancy chain is not as stable
as the V4 cluster. This is the same in silicon [98].
It should be noted that the prediction of the three and four vacancy chains to be at
their lowest energy in a spin state S =0 is inconsistent with their apparent detection by
EPR [41]. Detection in EPR relies on S being nonzero, so the divacancy and chains of
five/six vacancies should be detectable, as should all clusters bar those of four and six vacancies. This discrepancy is unresolved as yet, but there are two main possibilities. The
formation energies (per vacancy) of many of the defects are very close, but it is possible
that the output error of AIMPRO is larger that the energy differences. This would lead to
misidentification of the favourable spin states of the defects. It is also possible that there
has been an incorrect interpretation of EPR data, and hence assignment of signals to
structures.
CHAPTER 4. SMALL VACANCY CLUSTERS
87
Table 4.2: Formation energies, in eV, of chain defects of n vacancies calculated using
¨ Paderborn.
DFTB. Calculations performed by N. Fujita at Universit at
Defect
Divacancy
3 vacancy chain
4 vacancy chain
5 vacancy chain
6 vacancy chain
7 vacancy chain
E f orm
216 atom
10.87
14.18
17.61

E f orm /n
216 atom
5.43
4.73
4.40

E f orm
512 atom
11.06
14.41
17.63
20.77
23.84
27.58
E f orm /n
512 atom
5.53
4.80
4.40
4.15
3.97
3.94
Table 4.3: Formation energies, in eV, of cluster defects of n vacancies calculated using
¨ Paderborn.
DFTB. Calculations performed by N. Fujita at Universit at
Defect
4 vacancy cluster
5 vacancy cluster
6 vacancy cluster
7 vacancy cluster
8 vacancy cluster
9 vacancy cluster
10 vacancy cluster
11 vacancy cluster
12 vacancy cluster
13 vacancy cluster
14 vacancy cluster
E f orm
216 atom
16.78
18.50
18.79
23.02
24.07
25.66
25.09
29.75
31.21
32.31

E f orm /n
216 atom
4.19
3.70
3.13
3.29
3.01
2.85
2.51
2.71
2.60
2.49

E f orm
512 atom
16.90
18.56
18.87
23.16
24.12
25.21
25.16
29.63
31.30
32.42
32.45
E f orm /n
512 atom
4.23
3.71
3.14
3.31
3.01
2.80
2.52
2.69
2.61
2.49
2.32
E f orm
1000 atom
16.92
19.43
18.86
23.17
24.11
25.21
25.16
29.64
31.31
32.43
32.44
E f orm /n
1000 atom
4.23
3.89
3.14
3.31
3.01
2.80
2.52
2.70
2.61
2.50
2.32
These energies are to be compared to the values calculated for the same systems in
DFTB, shown in tables 4.2 and 4.3. DFTB also finds that the clusters of 6, 10 and 14
vacancies are the most stable, and the cluster is more stable than the chain configuration
for four vacancies. The energy values for differing size unit cells support the use of 216
atom cells to model the defects.
4.3.2
Structure
The two atoms forming the ends of the 110 chains possess dangling bonds and these
relax back onto the plane of their neighbours thus adopting sp 2 character. This relaxation increases with the chain length. The separation of the atoms at the chain ends
CHAPTER 4. SMALL VACANCY CLUSTERS
88
˚ for the divacancy and V3 – V6 chains
increases by 0.13, 0.142, 0.152, 0.165 and 0.190 A
respectively.
The atoms which surround the finite chains of vacancies along its length are paired with
other atoms which have also been left with dangling bonds; the unrelaxed separation of
˚ The magnitude and direction of movement under relaxation varies
these atoms is 2.499 A.
between the divacancy, V3 and V4 chains. The divacancy shows a symmetric movement
˚ for the surrounding atoms. This is in contrast to the V 3 and V4 chains
outwards of 0.102 A
where the surrounding atoms move both towards and apart from each other. In the V 3
˚ apart while the other moves 0.069 A
˚ together. In V4 four
chain two pairs move 0.094 A
˚ with the remaining two moving 0.119 A
˚ closer to
pairs of atoms are repelled by 0.1 A
each other.
The ideal infinite vacancy chain along 110 has four atoms surrounding the line of removed atoms, arranged so that each atom with a dangling bond is 2.52 A˚ away from
another. Upon relaxation, these atoms move closer together and form a reconstructed
˚
bond of length 2.16 A.
When six atoms in a chair configuration are removed to form an ideal hexavacancy clus˚ Upon structural
ter, the distance between pairs of atoms with dangling bonds is 2.49 A.
optimization two pairs of these atoms move closer together to form a very weak bond of
˚ The others move apart with a separation of 2.63 A.
˚ The neighbours to the
length 2.30 A.
hexavacancy lying along {111} are pulled closer to the hexavacancy resulting in a bond
˚ corresponding to approximately 19% strain. It can be seen in figure 4.2
length of 1.83 A,
that the 10 and 14 vacancy clusters consist of closed ‘cages’ of vacancies adjoining the
hexavacancy. Formation energy calculations predict that these are especially stable defects. Atoms with dangling bonds in the 10 vacancy defect exhibit a larger movement
˚ apart from two
than those in the six vacancy cluster. The resultant separation is 2.26 A,
˚ To allow
atoms which remain unpaired and move away from the vacancy group by 0.26 A.
˚ a strain of just less than
this movement to occur several bonds are stretched to 1.81 A,
18%. As a larger amount of atoms are removed for the 14 vacancy cluster, the separation of atoms with dangling bonds is increased. There are just four pairs of atoms that
display signs of rebonding occurring, moving closer together with a resultant separation
˚ Atoms surrounding the vacancy move inwards with a range of lengths of moveof 2.32 A.
ment. The bonds of four of the surrounding atoms are stretched, but to a lesser degree
˚ a
that that seen in the 6 and 10 vacancy clusters; the bonds being stretched to 1.77 A,
CHAPTER 4. SMALL VACANCY CLUSTERS
89
strain of 15%.
As reported above, the predicted stable cluster sizes are 6,10 and 14 vacancies, but only
the 6 vacancy cluster exhibits full bond reconstruction. The reason for this is clear if the
structure of the 10 and 14 vacancy clusters is considered, both being based on the 6
vacancy ring. The 6 vacancy ring has atoms with dangling bonds close enough together
to allow reconstruction, but to make larger clusters, some of these must be removed.
Thus there are atoms which are too far apart to reconstruct and the appearance of filled
bandgap energy levels is linked to this. For the 10 vacancy cluster, those atoms which
rebonded in the 6 vacancy ring and still have a close dangling bond become even closer,
˚ To allow optimization of energy several bonds of
with resultant separation of 2.26 A.
˚ slightly shorter than in the
atoms surrounding the vacancies have to stretch to 1.81 A,
6 vacancy ring. The 14 vacancy cluster has all the atoms which were bonding partners
for atoms surrounding the 6 vacancy ring removed. Therefore it is not obvious by simply
analysing the optimized structure where rebonding – if any – occurs. It is possible to find
further explanation for the stability of 6,10 and 14 vacancy clusters by considering the
number of dangling bonds formed by removal of each atom. The formation of 7 and 11
vacancy clusters by removal of an atom causes 2 more dangling bonds, and removal of
any atom adjacent to the 14 vacancy cluster creates 3 further dangling bonds. However,
the creation of 6,10 and 14 vacancy clusters from smaller clusters does not require the
creation of further dangling bonds.
4.3.3
Electronic properties
Assignment of centres to the family of 110 vacancy chains has assumed that the spin
density is localized on the atoms terminating the chain. It is possible to deduce the
wavefunction localization by performing a Mulliken analysis on the optimized structures,
and this has been calculated for all the 110 vacancy chain defects. In no instance is
it observed that the spin density is exclusively localized on the terminating atoms of the
vacancy chains.
The three vacancy chain has the spin density located on three pairs of atoms, separated
˚ In the four vacancy chains, just two pairs of atoms have the spin
by 2.45 and 2.65 A.
˚ This trend of pairs of atoms
located on them, both having an atom separation of 2.64 A.
differs for the five vacancy chain, here three atoms have spin density upon them. One of
CHAPTER 4. SMALL VACANCY CLUSTERS
90
these three is a terminating atom of the chain, and the other two are nearest neighbours to
˚ The six vacancy
the vacancy at the opposite end of the chain, with a separation of 2.63 A.
chain exhibits spin density located on the terminating atoms of the chain but also on four
further atoms, which are arranged in nearest neighbour pairs to the end vacancies of the
˚
chain. The pairs are at opposite ends of the chain, with an atom separation of 2.56 A.
Assumptions in other work [41, 40] about the spin density being located at the chain end
is unsupported by this analysis, as is the assignment of defects based on this assumption.
In the consideration of a defect’s effect upon the optical properties, it is desirable to know
what transitions are possible between energy levels as this will indicate what optical absorption takes place. Previous measurements of optical absorption on natural brown
diamond have shown the existence of a broad and largely featureless absorption band,
with the absorption coefficient increasing with photon energy [30, 44]. It is expected that
if the multivacancy defects studied here are related to brown colour we will see a range
of energy levels throughout the bandgaps of these defects. This range of energy levels
allows a broad distribution of transitions and hence absorption. The KohnSham levels
for the defects considered here are shown in figures 4.4–4.8. The KohnSham levels are
scaled by a factor 1.31 to compensate for the underestimation of the bandgap by LDA and
are taken at the gamma point. In the case of defects in an S =1 spin state, two spectra
are produced.
It is seen in figure 4.4 that the six vacancy cluster has no occupied bandgap states, indicating that it is an S =0 defect and should exhibit reconstruction of any dangling bonds.
The divacancy is more stable in a S =1 state. This can be understood from the energy
levels, shown in figures 4.4 and 4.6, and described in more detail below. The C 2h divacancy symmetry leads to a configuration occupying levels in the lower half of the band
gap.
The divacancy having C2h symmetry leads to both filled and empty levels in the band
gap. The lower is occupied with two electrons. The optical absorption band TH5 linked
with divacancies has a threshold at 2.45 eV. The KohnSham levels for the lower energy
S =1 state are shown in figure 4.6, and it can be seen that two lowest transitions are of
0.96 and 2.14 eV. The other defects have a range of threshold absorption energies, from
0.10 eV for V12 to 2.28eV for V6 .
It is interesting to note that the bandstructures of the various defects studied are similar,
CHAPTER 4. SMALL VACANCY CLUSTERS
91
despite the greatly differing structures. All defects studied showed several deep bandgap
energy states, which were grouped together in, or below, the middle of the bandgap. It is
seen in figures 4.4 – 4.8 that as the clusters get larger the groups of empty energy states
become nearer to Ec , although there are still shallow acceptor levels. The exact number
of states varies between structures, depending on the number of dangling bonds left by
the removal of atoms. For all defects possessing terminal dangling bonds the deep gap
levels are occupied. However, for the infinite chain and hexavacancy where there are no
terminal dangling bonds the occupied states are resonant with bulk states.
The 6 vacancy cluster shape allows a high reconstruction of the dangling bonds [102].
This means that there are no unpaired electrons and hence this defect has a spin of
S =0. Conversely, the other defects studied have nonzero spin, due to the presence of
unpaired electrons. These unpaired electrons arise in the other clusters and finite length
110 chains due to the structure not allowing dangling bonds to reconstruct. These give
rise to filled bandgap energy levels, which in turn dictate allowable energy transitions.
4.3.4
Optical properties
The small range of bandgap energy levels displayed by the multivacancy defects is expected to give rise to an absorption peak, at an energy below that of the absorption onset
at ∼5.5 eV from the diamond indirect bandgap.
Studies of the optical transitions have been undertaken by modelling the Electron EnergyLoss Spectroscopy (EELS) spectrum. This is described in detail in section 2.6.5. This
technique only takes into account direct transitions at any particular kpoint, so the absorption rise at 5.5 eV is not reproduced. The energies at which transitions occur must be
scaled due to the DFT bandgap underestimation. Transitions are subjected to a Gaussian
broadening of 0.8 eV.
The expected absorption peak is clearly seen for selected defects in figure 4.9.
CHAPTER 4. SMALL VACANCY CLUSTERS
92
7.0
6.0
5.0
Energy (eV)
4.0
3.0
2.0
1.0
0.0
1.0
2.0
(a) (b) (c) (d) (e) (f) (g) (h) (i)
Figure 4.4: The KohnSham levels (scaled) of various multivacancy defects, taken at
Γ. (a) Infinite vacancy chain (b) Divacancy (c) 3 vacancy chain (d) 4 vacancy chain (e)
5 vacancy chain (f) 6 vacancy chain (g) 4 vacancy cluster (h) 5 vacancy cluster (i) 6
vacancy cluster. The filled boxes denote occupied states and the empty boxes denote
empty states. The shaded regions indicate the position of the bands in bulk diamond.
4.3.5
π bonded carbon in multivacancies
The {111} surface in diamond can reconstruct into the Pandey π bonded chain for
mation [103], which has been shown to be energetically favourable [104] This surface
structure gives rise to π and π ∗ bandgap levels and is described in section 6. The energy
CHAPTER 4. SMALL VACANCY CLUSTERS
93
7.0
6.0
5.0
Energy (eV)
4.0
3.0
2.0
1.0
0.0
1.0
2.0
(a) (b) (c) (d) (e) (f) (g) (h)
Figure 4.5: Further KohnSham levels of multivacancy defects, taken at Γ. (a) 7 vacancy
cluster (b) 8 vacancy cluster (c) 9 vacancy cluster (d) 10 vacancy cluster (e) 11 vacancy
cluster (f) 12 vacancy cluster (g) 13 vacancy cluster (h) 14 vacancy cluster. The filled
boxes denote occupied states and the empty boxes denote empty states. The shaded
regions indicate the position of the bands in bulk diamond.
transitions between these levels have a range from zero to the bandgap energy [104].
This indicates that the optical absorption properties of π bonded carbon may be comparable to that observed in brown diamond [34]. If multivacancy defects contained such
types of carbon then their experimental detection in natural brown diamond [35] could be
associated with the colouration.
CHAPTER 4. SMALL VACANCY CLUSTERS
94
7.0
6.0
5.0
Energy (eV)
4.0
3.0
2.0
1.0
0.0
1.0
(a)
(b)
(c)
(d)
(e)
Figure 4.6: The KohnSham levels (scaled) of multivacancy chains in S =1 spin state,
taken at Γ. The spinup and spindown electrons are separated for each defect. (a)
Divacancy (b) 3 vacancy chain (c) 4 vacancy chain (d) 5 vacancy chain (e) 6 vacancy
chain. The arrows denote occupied spin up/down levels and the empty boxes denote
empty levels. The shaded regions indicate the position of the bands in bulk diamond.
Models were constructed for the 110 vacancy chains to attempt to introduce sp 2 bonded
carbon atoms, while still retaining the symmetry of the defect. When analysed the total
formation energies were considerably higher than those of the original models. Two adjacent chains of vacancies in the 110 direction were removed from the 140 atom cell
used in the infinite vacancy chain, to attempt to introduce π bonding. As with the other
studies, the formation energy was larger.
CHAPTER 4. SMALL VACANCY CLUSTERS
95
7.0
6.0
5.0
Energy (eV)
4.0
3.0
2.0
1.0
0.0
1.0
2.0
(a)
(b)
(c)
(d)
(e)
(f)
Figure 4.7: KohnSham levels (scaled) of multivacancy clusters in S =1 spin state, taken
at Γ. Spinup and spindown electrons are separate. (a) 4 vacancy (b) 5 vacancy (c) 6
vacancy (d) 7 vacancy (e) 8 vacancy (f) 9 vacancy. The arrows denote occupied spin
up/down levels and the empty boxes denote empty levels. The shaded regions indicate
the position of the bands in bulk diamond.
It is possible to introduce π bonding into diamond by removing every third atom in the
110 direction, the dangling bonds left on the pairs of adjacent atoms should be able to
reconstruct to give them sp2 nature. Analysis of such a defect was made in a 188 atom
unit cell, similar in construct to the cell used to model the infinite 110 vacancy chain.
Here though, the cell was elongated in the 110 direction to allow correct reproduction
of the defect when the cell is repeated during analysis. The formation energy of this
CHAPTER 4. SMALL VACANCY CLUSTERS
96
7.0
6.0
5.0
Energy (eV)
4.0
3.0
2.0
1.0
0.0
1.0
2.0
(a)
(b)
(c)
(d)
(e)
Figure 4.8: Further KohnSham levels of multivacancy clusters in S =1 spin state, taken
at Γ. Spinup and spindown electrons are separate. (a) 10 vacancy (b) 11 vacancy (c) 12
vacancy (d) 13 vacancy (e) 14 vacancy. The arrows denote occupied spin up/down levels
and the empty boxes denote empty levels. The shaded regions indicate the position of
the bands in bulk diamond.
defect was found to be 20.85 eV, which equates to 6.95 eV per vacancy; over double the
formation energy of the infinite vacancy chain. The results of these preliminary studies
suggest that π bonded carbon cannot be a feature of 110 vacancy chains or clusters in
diamond.
CHAPTER 4. SMALL VACANCY CLUSTERS
200000
97
Divacancy
3 vacancy Chain
6 vacancy Cluster
180000
Absorption coefficient (cm1)
160000
140000
120000
100000
80000
60000
40000
20000
0
1
2
3
4
5
6
7
8
Energy (eV)
Figure 4.9: Absorption spectra for selected multivacancy defects. The absorption peak
caused by the group of levels in the midgap is evident. Energy values are scaled by 1.3
to account for bandgap underestimation.
4.4
Conclusion
Studies were carried out on 110 orientated vacancy chains and clusters of vacancies.
It is seen that for four, five and six vacancies the cluster formation is more energetically
favourable than the 110 chain, and it can be speculated that this will be the case for
larger numbers of vacancies also. The most stable sizes were the 6, 10 and 14 vacancy
clusters, which all consist of closed structures minimising the number of dangling bonds
and thus reducing the energy.
Calculated absorption onset for the divacancy is 2.14 eV, this is in fair agreement with
the assignment of the S =1 C2h R4/W6 centre to this defect, and the absorption band at
2.54 eV.
The energies of defects calculated indicate that there will not be a range of optical absorption, but rather a distinct peak of absorption in the 3.5 – 4.5 eV range. Positron anni
CHAPTER 4. SMALL VACANCY CLUSTERS
98
hilation studies on brown diamond before and after annealing have shown the presence,
and then loss, of vacancy defects [35]. This has lead to suggestions of a link between
multivacancy structures and brown diamond. The absorption spectrum of brown diamond
is broad and featureless, with an approximate square or cube dependence [34] on energy. It is clear that the defects studied are not able to account for such an absorption
spectrum, so it must be concluded that small multivacancy structures are not the cause
of brown colouration in diamond.
Chapter 5
EELS of Bulk Diamond and
Dislocations
5.1
Introduction
This work has been published in part in Philosophical Magazine 86, 4757 (2006).
As described in section 3.2, lowloss Electron Energy Loss Spectroscopy (EELS) offers
the possibility of revealing the local electronic structure of extended defects in semiconductors or insulators with ∼10 nm spatial resolution and with energy resolution down to
0.3 eV. Diamond is an ideal prototype material to investigate by lowloss EELS as the
band gap of 5.5 eV is large enough to minimise the influence of the zeroloss peak.
Any defect having levels in the bandgap is of immediate interest given the importance of
colouration in diamond for the gem industry.
Lowloss EELS involves measuring the loss of energy of a electron beam in promoting
transitions between occupied and empty levels where the energy separation is less than
about ∼50 eV. Coreloss EELS is due to transitions between core levels and empty levels,
and is much harder to investigate by modern theoretical methods. It requires the detailed
knowledge of the core wavefunctions which are not available to calculations using pseudopotentials as the core electrons are removed.
It has been shown [96, 105] that brown diamond exhibits a different EEL spectrum to
99
CHAPTER 5. EELS OF BULK DIAMOND AND DISLOCATIONS
100
colourless diamond. Lowloss studies do not show a clear relationship between dislocation cores and enhanced densityofstates (JDOS) which is a measure of available
transitions, but rather an increase in sp 2 content in the vicinity of dislocations and stacking
faults. This suggests the investigation of extended defects as a possible source of brown
colour is required. An increase of the EEL signal in the energy region around 6 eV at
dislocations in brown diamond has been noted, compared to the colourless variety [106].
These investigations have been extended to regions remote from dislocations [96] and it
is observed that there is a generally higher ‘background’ in states giving rise to intensity
at 6 eV in brown diamond than in colourless diamond. This is linked to sp 2 related defects.
5.2
5.2.1
Bulk Diamond
Calculated results
In order to understand the change, if any, that defects have on the EELS spectrum of
diamond, a complete calculation of the properties of pure bulk diamond is necessary.
Diamond is modelled in a minimal twoatom cell, repeated along the FCC lattice vectors,
with the dielectric function calculated as described in section 2.6.5. The variation of
the real part of the dielectric function with the number of Brillouin zone sampling points
is shown in figure 5.1. The variation with the number of basis set functions and the
polynomial broadening of data points is illustrated in figures 5.2 and 5.3 respectively.
From these figures it can be seen that a MonkhorstPack grid of at least 20 3 points is
required, with 22 functions in the basis set. A polynomial broadening of 0.8 eV is used to
smooth the points into a easily readable curve.
Figure 5.4 shows the real and imaginary parts of the dielectric function of diamond compared with experimental data [107, 108]. The experimental spectrum for ε 2 shows a
threshold around the direct band gap at 7 eV and a peak at 12 eV due to bandband transitions. It then falls slowly with increasing energy. Unusually, and in contrast with other
semiconductors, diamond does not exhibit additional sharp peaks in this region, due to
its bandstructure.
The theoretical curve displays a threshold at 5.4 eV, and a peak around 12 eV, which
are shifted from the experimental values due to the underestimation of the gap and lower
CHAPTER 5. EELS OF BULK DIAMOND AND DISLOCATIONS
15
403 rid
303 Grid
203 Grid
103 Grid
10
Real dielectric constant
101
5
0
5
10
0
5
10
Energy (eV)
15
20
Figure 5.1: Variation of the real part of the dielectric function of bulk diamond with the
MonkhorstPack sampling grid. 22 functions were used in the basis set, with a broadening
of 0.8 eV.
20
40 Functions
22 Functions
13 Functions
Imaginary dielectric constant
15
10
5
0
5
10
0
5
10
15
20
Energy (eV)
Figure 5.2: Variation of the real part of the dielectric function of bulk diamond with the
number of functions in the basis set. A 40 3 grid of sampling points was used, with a
broadening of 0.8 eV.
CHAPTER 5. EELS OF BULK DIAMOND AND DISLOCATIONS
20
102
0.8 eV
0.5 eV
0.1 eV
Imaginary dielectric constant
15
10
5
0
5
10
0
5
10
Energy (eV)
15
20
Figure 5.3: Variation of the real part of the dielectric function of bulk diamond with the
polynomial broadening applied. A 40 3 grid of sampling points was used, with 22 functions
in the basis set.
25
Calculated Real
Experimental Real
Calculated Imaginary
Experimental Imaginary
20
Dielectric Constant
15
10
5
0
5
10
0
5
10
15
20
Energy (eV)
Figure 5.4: Real and imaginary parts of the calculated dielectric function for bulk diamond
compared to experiment. Experimental values from [107].
CHAPTER 5. EELS OF BULK DIAMOND AND DISLOCATIONS
103
conduction bands by LDA. It has been described when modelling diamond using AIMPRO
how both the direct and indirect gap can be bought into agreement with experimental
values, by scaling the levels from the valence band top upwards by a factor of 1.3. This
technique has been applied to the defect related gap absorption in chapter 6. The method
of scaling energy levels is quite distinct from a rigid energy shift, also known as a ‘scissors’
shift. Here all levels above the valence band top have the same value shift added, but
this has the effect of changing the relative position of gap levels and hence related energy
transitions. Due to this it is felt unsuitable for this work.
The real part of the dielectric function is also shown in figure 5.4 where it is compared
with experiment. Note that in the long wavelength limit, the calculated value of ε 1 =5.89
agrees well with the experimental value of the static dielectric function of 5.82 [107]. The
experimental peak at 7 eV is much sharper than the calculated one due to the neglect of
excitonic effects. However the energy where ε 2 =0 is in good agreement.
The calculated and experimental absorption coefficients are shown in figure 5.5. The
magnitude of the absorption agrees well with experiment but the threshold around 7 eV
clearly reflects the underestimation of the direct gap by the theory. The peak in absorption
at ∼12.5 eV is related to the vanishing of ε 1 and is reproduced in the theoretical spectrum.
However, the sharp rise of the absorption near the threshold is not reproduced due to the
lack of excitons in the theory, and the absorption is ∼13% too large in the region beyond
the peak.
5.3
Experimental Results
Figure 5.6 shows the experimental EEL spectrum for diamond, measured by U. Bangert
(University of Manchester). The apparent electron energy loss in the bandgap energy region is due to background noise. This is normally accounted for by a subtraction method,
but this has not been completed here. In addition the xaxis range commences at 1 eV as
there is a 0 eV singularity which arises from elastically scattered electrons, the socalled
zeroloss peak. The peak around 35 eV is due to plasmons with frequency
(ne2 /ε0 m)
where the valence electrons with density n behave collectively. The calculated EEL spectrum for bulk diamond shown in figure 5.7 shows excellent agreement with experimental
data. The calculated peak occurs at 35.6 eV, with the experimental peak at 33.5 eV. No
CHAPTER 5. EELS OF BULK DIAMOND AND DISLOCATIONS
4.5e+06
104
Calculated
Experiment
4e+06
1
Absorption coefficient(cm )
3.5e+06
3e+06
2.5e+06
2e+06
1.5e+06
1e+06
500000
0
0
5
10
15
Energy (eV)
20
25
Figure 5.5: Calculated and experimental absorption coefficient of bulk diamond. Experimental values from [107].
compensation for LDA bandgap underestimation is used as the freeelectron approximation becomes more accurate at these higher energies.
25000
Type IIa Diamond
EELS count (arbitary units)
20000
15000
10000
5000
0
5
10
15
20
25
30
35
40
45
50
Energy (eV)
Figure 5.6: Experimental EELS of colourless diamond. The bandgapenergy counts arise
from background noise which is not corrected for.
CHAPTER 5. EELS OF BULK DIAMOND AND DISLOCATIONS
5
105
Bulk Diamond
Bulk Graphite
4.5
EELS count (arbitary units)
4
3.5
3
2.5
2
1.5
1
0.5
0
0
5
10
15
20
25
30
Energy (eV)
35
40
45
50
Figure 5.7: Calculated lowloss EEL spectra for bulk diamond and graphite.
The EEL spectrum for graphite has also been calculated and is shown with the calculated
diamond spectrum in figure 5.7. It is clear that graphite has energy transitions down to
zero, which is to be expected from its semimetallic nature. There are peaks centred
about 8 eV and 33 eV which agree well with those seen experimentally around 7 eV and
31 eV [109, 110].
If the raw EEL spectra of brown and colourless diamond is compared to measurements
on a graphitic region of a brown type IIa diamond (figure 5.8), an increase in EELS
count around 7 eV is seen for the brown diamond, with a much stronger increase for
the graphitic region. This correlation indicates that the defect causing the brown colour
must have some graphitelike component i.e. dangling bonds or sp 2 bonded carbon,
which allows many electrons to be free to interact optically.
5.4
Dislocations
Dislocations are found in all types of natural diamond, with densities of up to ∼ 10 9 cm−2 .
They have been modelled previously [111], and some were found to have a bandstructure
which could be consistent with a broad absorption. The absorption and relative stability
CHAPTER 5. EELS OF BULK DIAMOND AND DISLOCATIONS
25000
106
Graphite Region
Brown Diamond
Colourless Diamond
Energy loss
20000
15000
10000
5000
0
0
5
10
15
20
25
30
35
40
Energy (eV)
Figure 5.8: Raw EEL spectra of colourless diamond, brown diamond at fringes of graphitized region and within graphitic region. Measured by U. Bangert (University of Manchester).
of such dislocations is now described for two types of partial dislocation. These are the
90◦ glide (single period) and 90◦ shuffle (vacancy type) dislocations [27]. Both lie along
√
¯ and are periodic with a repeat distance of a/ 2. Both types border intrinsic stacking
[110]
faults. The glide partial has only reconstructed bonds which do not introduce states into
the gap. However, the shuffle dislocation has a line of dangling bonds and can be formed
from a glide partial by removing a line of atoms at the core of the latter.
5.4.1
90◦ Glide dislocation
The 90◦ glide (single period) dislocation, hereafter referred to as the glide dislocation, is
modelled in a cell of 440 atoms with a pair of glide partials inserted. The core construction
is shown in figure 5.9. The pairing of partials is necessary as each dislocation introduces
a Burgers vector which must be matched by another with equal magnitude but opposite
direction. This requires the second dislocation to be inverted with respect to the first.
˚ which
The reconstructed core bonds of the glide dislocation have a length of 1.65 A
is 7% longer than those of bulk diamond. The formation energies of the two types of
CHAPTER 5. EELS OF BULK DIAMOND AND DISLOCATIONS
[110]
[111]
[112]
[112]
[110]
107
[111]
Figure 5.9: Structure of the 90◦ glide (single period) type dislocation. The reconstruction
across the core can be seen, leading to a reduction in energy and eliminating bandgap
states.
dislocations are calculated as follows. Defining core atoms as atoms that are no longer
tetrahedrally bonded in the diamond structure, then there are four core atoms per repeat
√
length (a0 / 2) for the 90◦ glide dislocation, and four core atoms per repeat length for the
90◦ shuffle type. The formation energy of the glide dislocation is then found to be 9.03 eV
per core atom.
Figure 5.10 shows the bandstructure of the glide dislocation. The glide partial has an
almost totally clear bandgap with no electrons available for optical interactions, indicating
that optical absorption will not occur until in excess of 4 eV.
Analysis is now extended to the EEL and absorption spectra. The large size of the cell
means the electon wavefunction is less dispersed and fewer kpoints are needed for
accurate sampling. Here 216 points are used compared to nearly 6000 for bulk diamond
in a 2 atom cell. A broadening of 0.8 eV is used, and the EELS count is scaled for clarity
by a factor of 1 × 107 . The resultant spectra are shown in figure 5.11. It is clear that the
glide dislocation has little bandgap absorption or EELS count, as expected from the lack
of any shallow or deep bandgap levels. The absorption onset is around 4 eV, as predicted
from the energy gap.
CHAPTER 5. EELS OF BULK DIAMOND AND DISLOCATIONS
7
108
Filled States
Empty States
6
5
Energy (eV)
4
3
2
1
0
1
(110)/2
(111)/2
(011)/2
Γ
(001)/2
2
Γ
Brillouin zone position
Figure 5.10: Bandstructure of the 90 ◦ glide type dislocation. The clear bandgap indicates that no absorption is expected until ∼4 eV. Empty states have had their energies
multiplied by 1.3, and the valence band top is set to 0 eV.
500000
Absorption
EELS
400000
300000
200000
100000
0
0
2
4
6
8
10
Energy (eV)
Figure 5.11: EELS spectrum and absorption of 90 ◦ glide type dislocation, using unscaled
energy values. As expected from the bandstructure there is no bandgap EELS count or
absorption.
CHAPTER 5. EELS OF BULK DIAMOND AND DISLOCATIONS
109
[110]
[111]
[112]
[112]
[110]
[111]
Figure 5.12: Structure of the 90◦ shuffle (vacancy) type dislocation. Here there are dangling bonds along the dislocation core leading to available bandgap transitions.
5.4.2
90◦ Shuffle dislocation
The 90◦ shuffle (vacancy type) dislocation, hereafter referred to as the shuffle dislocation, is modelled in a cell of 448 atoms. As with the glide dislocation model a pair of
dislocations is required, but here the shuffle dislocation is paired with a glide type. The
advantage of using a mixed pair of types is that since the glide partial does not lead to
dangling bonds, the electronic levels lying in the gap will only arise from the shuffle partial.
Figure 5.12 shows the core structure of the shuffle dislocation, with the unreconstructed
core atoms leading to dangling bonds.
In contrast to the glide dislocation, there is an asymmetric separation across the core of
the shuffle dislocation, between one atom having sp 2 coordination and one with only two
˚ or 16% elongation, and 1.77 A˚ or 15% elongation.
bonds. The separations are 1.80 A
Therefore many dangling bonds are present, leading to a high formation energy and to
a wide range of optical transitions. As before, there are four core atoms for the shuffle
dislocation, with the formation energy found to be 16.03 eV per core atom, much higher
than the glide partial. This demonstrates the increased stability for the glide dislocation,
indicating that many of the dislocations in diamond are likely to be of the glide type.
However, the shuffle type could arise as a grownin defect or when vacancies are present
and are trapped at the dislocation core.
The bandstructure of the shuffle type dislocation shown in figure 5.13 reveals a half
CHAPTER 5. EELS OF BULK DIAMOND AND DISLOCATIONS
18
110
Filled States
Empty States
17
16
Energy (eV)
15
14
13
12
11
10
(110)/2
(111)/2
(011)/2
Γ
(001)/2
9
Γ
Brillouin zone position
Figure 5.13: Bandstructure of the 90 ◦ shuffle type dislocation. There is a range of transitions from 0 to ∼2.5 eV, so a broad lowenergy peak in EELS/absorption is expected.
Energy levels above the valence band top at 0 eV have been multiplied by 1.3
filled band of states in the gap. This implies a range of optical transitions, but with a
possible gap at higher energies. This would not provide a featureless absorption but
would be stronger at lower energies. Along with the reduced stability compared to the
glide type, this would suggest that the shuffle dislocation also does not have a link to
brown colouration in diamond.
The EEL and optical absorption spectra of the shuffle dislocation are shown in figure 5.14,
with the EEL spectrum scaled by a factor of 2.5×10 6. Bandgap absorption is evident but
it does not correspond to the experimentally observed absorption of brown diamond,
displaying a dip in absorption around 1 eV and a strong onset that is below the bandgap
edge, even with scaling. The EEL spectrum also shows the predicted lowenergy peak
due to a concentration of levels in the lower bandgap.
A further check as to whether shuffle dislocations can be responsible for brown colour
in diamond is to calculate the expected number of core atoms required to produce an
absorption magnitude comparable to that seen experimentally. There are four core atoms
per cell which contribute to the absorption magnitude in a cell of 438 atoms, i.e. 0.009%.
The model has an absorption coefficient of 2475 cm −1 at 2.5 eV, compared to ∼1 cm−1
CHAPTER 5. EELS OF BULK DIAMOND AND DISLOCATIONS
50000
111
Absorption
EELS
40000
30000
20000
10000
0
0
1
2
3
4
5
6
Energy (eV)
Figure 5.14: EELS spectrum and absorption of 90 ◦ shuffle type dislocation EELS spectrum. The energy values are unscaled. There is clear bandgap absorption but it does not
correspond to the the characteristic pattern seen in brown diamond.
in most natural brown diamonds. Thus the fractional core atom density must be scaled
to 3.6 × 10−6 to account for this, which corresponds to an atomic density of 6.49 × 10 17 .
The maximum density of dislocations observed in natural diamond is ∼10 9 cm−2 , which
corresponds to 1 core atom in about 10 8 carbon atoms, or an atomic density of 1 × 10 15 .
Thus it is clear that there are insufficient core atoms to account for observed absorption
even if all dislocations were of the shuffle type, which seems unlikely. In CVD diamond
the dislocation density is much lower, around 10 4 cm−2 , which makes the assignment of
brown colour to dislocations in this material even more difficult to accept.
5.5
Conclusion
The EEL and absorption spectra have been calculated for both bulk diamond and for
defects. The modelling requirements have been explored for bulk diamond, in terms of the
sampling grid and the wavefunction bases needed. Excellent agreement for bulk diamond
has been demonstrated for the real and imaginary parts of the dielectric function, and the
EELS spectrum. Graphite has also been modelled to demonstrate that the enhanced
EELS count around 7 eV in brown diamond can be linked to sp 2 bonded carbon.
CHAPTER 5. EELS OF BULK DIAMOND AND DISLOCATIONS
112
Two different dislocations have been modelled to assess their link to brown diamond. The
90◦ glide type dislocation is found to be optically inactive through most of the bandgap
energy range, due to core atom reconstruction of dangling bonds. The 90 ◦ shuffle type
dislocation has a core of dangling bonds which induces absorption across the bandgap,
however there are several reasons why these dislocations cannot be responsible for the
brown colour of diamond. Firstly the shuffle dislocation is much higher in energy that
the glide dislocation, meaning it is less likely to naturally occur. Secondly the absorption
profile does not match experiment. Most critically however, it has been shown that the
dislocation density required by this model to account for the experimentally observed
magnitude far exceeds the dislocation densities of both natural and CVD diamond.
Chapter 6
(111) Plane Vacancy Disks
6.1
Introduction
This work has been published in Physical Review B 73, 125203 (2006).
It has been shown in chapters 4 and 5 how small point defects and dislocations cannot
be linked to the origin of brown colour in diamond. The presence of vacancies is known
from PAS experiments on brown, colourless and transformed diamonds [30, 35, 112], but
a single multivacancy cluster cannot explain the brown colouration as it is expected to
induce a peak in absorption. On the other hand, an ensemble of clusters of different sizes
may not exhibit a sharp peak but would be expected to give an absorption pattern which
varied from sample to sample reflecting a different relative concentration of the clusters.
Moreover, during the anneal, the less stable clusters would dissolve leaving the more
stable ones. The latter would lead to an absorption spectrum with structure contrary to
observations.
Carbon atoms in diamond prefer to form four covalent bonds in a tetrahedral fashion, as
described in chapter 1. If this is not possible then a double bond may be formed between
two adjacent carbon atoms. Such a bond will introduce π and π * states in to the bandgap,
as observed in graphite [113], and in disordered carbon at grain boundaries [114]. Such
π bonds can form both on the (001)(2×1) surface [115] and the (111)(2×1) surface of
diamond [103]. If the π bonds are delocalized, i.e. they are connected to many other
π bonds, then the π and π * states in the bandgap will be dispersed, varying in separation
113
CHAPTER 6. (111) PLANE VACANCY DISKS
114
at different points in the Brillouin zone. This is clearly seen in the bandstructures for the
(001)(2×1) surface in figure 6.1, and the (111)(2×1) surface in figure 6.2.
10
Filled States
Empty States
8
Energy (eV)
6
4
2
0
2
(010)/2
(111)/2
(011)/2
Γ
(001)/2
(100)/2
4
Γ
Brillouin zone position
Figure 6.1: The bandstructure of the (001)(2×1) surface of diamond, showing the occupied π and unoccupied π * states in the gap. There is little dispersion, as the π bonded
atoms are not connected. Energies are scaled by 1.3 to compensate for the bandgap
underestimation by LDA.
The dispersed bandgap levels allow a range of transitions to occur, which in principle will
lead to an optical absorption across a wide range. The occupied and empty levels meet
in the bandgap, leading to a semimetallic nature, which indicates transitions will start
close to 0 eV, i.e. there will be no threshold of absorption.
If a high density of π bonded atoms are to be incorporated into diamond then the most
promising method is the creation of surfaces on the (111) set of planes inside the material
as small defects, dislocations and grain boundaries cannot support an extended network
of such bonds. A (111) surface can be created by the removal of two planes of atoms,
as shown in figure 6.3. These lie on α B planes of the normal Aα Bβ Cγ (111) stacking
sequence of diamond and their removal leads to dangling bonds lying parallel to [111]
on each internal surface. Subsequently all dangling bonds are eliminated through the
¯ The length
Pandey reconstruction [103] which leads to lines of π bonds lying along [1 10].
˚ a similar length to that found in graphite. The disk of vacancies
of the π bond is 1.426 A,
CHAPTER 6. (111) PLANE VACANCY DISKS
12
115
Filled States
Empty States
10
8
Energy (eV)
6
4
2
0
2
4
(010)/2
(111)/2
(011)/2
Γ
(001)/2
(100)/2
6
Γ
Brillouin zone position
Figure 6.2: The bandstructure of the (111)(2×1) surface of diamond, showing the occupied π and unoccupied π * states in the gap. There is strong dispersion of these levels,
due to the long chains of π bonded atoms. Energies are scaled by 1.3 to compensate for
the bandgap underestimation by LDA.
which is subsequently created should display a bandstructure similar to that of the plain
(111)(2×1) surface of diamond, as will be described in the following sections.
6.2
Simple Vacancy Disks
6.2.1
Structures and energies
Vacancy disks are modelled in 40 and 80 atom cells. The separation between the disks in
different cells is then 20 and 40 atomic layers respectively. The use of two differently sized
cells helps determine the interaction range of the disks and will also allow the absorption
density of vacancy disk defects to be deduced. The cells are structurally optimized using
a 2 × 2 × 1 grid of MonkhorstPack kpoints.
The formation energy per vacancy is 1.46 eV in both the 40 atom and 80 atom cells,
CHAPTER 6. (111) PLANE VACANCY DISKS
116
[111]
[112]
[110]
Figure 6.3: Illustration of the removal of a {111} double plane (boxed) from bulk diamond
and the subsequent surface rebonding leading to a vacancy disk with chains of π bonded
¯ . The π bonded atoms are shown in white for clarity.
atoms along 110
demonstrating that a separation of 20 layers is sufficient for modelling purposes. The
formation energy of a disk with (111)(1×1) unreconstructed surfaces is 2.21 eV per vacancy. These energies are to be compared with 5.96 eV for an isolated vacancy studied
using a Quantum Monte Carlo technique [116], and 7.17 eV for the vacancy using AIMPRO. The most stable multivacancy cluster found previously is a cluster of 14 vacancies
with formation energy 2.35 eV per vacancy [117]. Comparison of these formation energies allows prediction of which is more likely to occur. It is clear that, given the ability to
overcome any migration barrier, it is preferable for vacancies to form a disk lying on the
(111) plane over any other defect.
The band structure, shown in figure 6.4, demonstrates that the band gap is completely
filled with states and that the top of the occupied π band is degenerate with the bottom
of the empty π ∗ band. The similarity to the bandstructure of the (111)(2×1) is clear,
but here there are two sets of bandgap levels arising from the dual surfaces. It can be
anticipated from the arrangement of levels that the absorption will be continuous without
any threshold.
The bandstructure can also be evaluated using the screened exchange technique de
CHAPTER 6. (111) PLANE VACANCY DISKS
12
117
Filled States
Empty States
10
8
Energy (eV)
6
4
2
0
2
4
(010)/2
(111)/2
(011)/2
Γ
(001)/2
(100)/2
6
Γ
Brillouin zone position
Figure 6.4: The bandstructure of a vacancy disk in a 40 atom cell. Note the absence of
any pronounced gap in the spectrum suggesting a featureless absorption spectrum. The
centre of the Brillouin zone is marked Γ, and other kpoints denoted with their positions
relative to the primitive unit vectors of the Brillouin zone of the cell. Energies are scaled
by 1.3 to compensate for the bandgap underestimation by LDA.
scribed in section 2.8. Figure 6.5 compares the LDA values, which are scaled by 1.3,
with those produced by the screened exchange functional. When the valence maxima
are set to 0 eV for both methods there is very good agreement for valence and conduction states, with a fair agreement for the π /π ∗ crossing point. As the screened exchange
functional has been shown to reproduce the bulk diamond bandstructure correctly, it can
be assumed to be accurate in this case. This agreement justifies the use of the 1.3 scaling factor in the bandgap energy range. It is still computationally difficult to use screened
exchange for large cells or dielectric analysis, hence no comparison can be made for
optical absorption.
Due to its small crosssectional area, the vacancy disk can be expected to have a PAS
¨
lifetime comparable to the monovacancy (140 ps). Calculations perfomed by JM. M aki
show the disk to have a lifetime of 271 ps, which is similar to a cluster of 14 vacancies. Although larger than expected, there is a clear discrepancy between the observed ∼400 ps
lifetime and that produced by the vacancy disk.
CHAPTER 6. (111) PLANE VACANCY DISKS
12
10
118
Screened Exchange
LDA x 1.3
8
Energy (eV)
6
4
2
0
2
4
6
5
10
15
20
Brillouin zone sampling point
25
30
Figure 6.5: The bandstructure of a vacancy disk in a 40 atom cell using LDA and screened
exchange. The LDA energy values are scaled by 1.3 above the VBM, which is set to 0 eV.
Good agreement for valence and conduction bands is evident.
6.2.2
EELS and optical absorption
Initially the diagonal components of the dielectric tensor for the planar defect along the
¯ [21¯1]
¯ directions are calculated. The largest component lies in the (111)
[111], [110],
plane as is the case for graphite. The average values of the dielectric function and the
corresponding optical absorption coefficient are calculated.
The EELS spectrum of the vacancy disk is shown in 6.6, compared to the spectrum of
bulk diamond. The effect of the filled and empty bandgap states on the magnitude of
the EELS signal is clear. Above ∼8 eV the two curves are essentially the same as the
transitions here are mainly due to the bulk, but below this point the bandgap transitions
dominate. It is especially significant that there is an increase in signal for the disk around
6 eV, as this agrees with experimental findings on brown diamond [95] These transitions
maintain the EELS magnitude down to 0 eV. It is described below how the calculation
technique can lead to a nonphysical peak in the 0 – 1 eV range, which is evident for the
vacancy disk. Hence the 0 – 1.5 eV region is omitted from the graph for clarity of viewing.
Figures 6.7 and 6.8 show the calculated optical absorption compared to experimental
CHAPTER 6. (111) PLANE VACANCY DISKS
0.06
119
Vacancy Disk
Bulk Diamond
0.05
EELS count
0.04
0.03
0.02
0.01
0
2
4
6
8
10
12
14
Energy (eV)
Figure 6.6: The calculated EELS spectrum of the (111) vacancy disk in an 80 atom cell,
with the EELS spectrum of bulk diamond shown for comparison. It is clear that there are
electron transitions for the vacancy disk far below those for bulk diamond, which indicates
bandgap absorption. The 0 – 1.5 eV region is omitted as described in the text. Spectra
are scaled on the energy axis to account for LDA bandgap underestimation.
data, in standard and loglog plots. The absorption is featureless in the range 1 to 5 eV
as indeed is the experimental spectrum. The energy dependence of the absorption is
very similar to the more transparent CVD1 sample in the midgap region. Supplementary
absorption in CVD4 is due to additional defects which are likely to be nitrogen related,
due to its presence in such samples. The expression for the dielectric function above
takes only direct electronhole transitions into account. Hence the large rise in absorption
at 5.5 eV for diamond, due to indirect excitations and exciton effects, is not reproduced. In
addition, the semimetallic character of the infinite vacancy disk results in an absorption
coefficient diverging at low energies [118]. A finite disk can be expected to possess a
small gap between occupied π bonding and empty π ∗ states leading to an absorption
coefficient which vanishes with energy. However the vacancy disk leads to an absorption
spectrum consistent with experiment on CVD1, although the natural diamond may have
additional centres.
An important finding is that the absorption coefficient calculated in the 80 atom cell is
roughly a factor of half that found in the 40 atom cell where the density of vacancies
CHAPTER 6. (111) PLANE VACANCY DISKS
9
120
Natural 31
CVD 1
CVD 4 (Scaled)
Vacancy Disk (Scaled)
8
6
1
Absorption (cm )
7
5
4
3
2
1
0
2
3
4
Energy (eV)
5
6
Figure 6.7: Absorption of natural and CVDgrown brown singlecrystal diamond compared to absorption of (111) vacancy disk in an 80 atom cell. The absorption of the
vacancy disk has been multiplied by a factor of 0.00025, and the CVD4 sample by 0.01
for clarity.
is twice as large. Thus the absorption coefficient scales with this fraction. This allows
calibration of the absorption with the sp 2 /sp3 fraction or vacancy density. An absorption
coefficient at 2.5 eV of 0.05 cm−1 for a vacancy concentration of ∼ 1 ppm or 1.52×10 17
cm−3 is found. This is comparable with an extrapolation of experimental data on poor
quality CVD material [119]. Thus one can deduce that there would be about 10 17 vacancies in the CVD1 diamond whose absorption is shown in figure 6.8 if all the absorption
were due to disks.
The Raman active mode of bulk diamond is 1332 cm −1 , and of graphite is ∼1600 cm−1 [120].
Amorphous carbon displays shifted peaks, or a mixture of the two. In nanocrystalline
diamond a 1490 cm−1 band has been observed and linked to diamond undergoing structural modification due to high pressure [121], which is likely at grain boundaries where
sp2 bonding can occur. It is to be expected that π bonds could lead to vibrational modes
in the 1400 – 1600 cm−1 region. A Raman active mode at 1494 cm −1 is calculated, which
is close to one observed around 1540 cm −1 in CVD material [26]. Similar modes must be
present to some extent in brown diamond if the vacancy disk model is correct.
CHAPTER 6. (111) PLANE VACANCY DISKS
1000
1
Absorption (cm )
100
121
Energy3
Energy2
Calculated (scaled)
Experimental
Experimental
Experimental
10
1
0.1
0.01
2
3
Energy (eV)
4
5
Figure 6.8: Comparison of the calculated absorption coefficient for a (111) vacancy disk
with experimental data from natural and CVDgrown brown diamonds. The vacancy disk
absorption is calculated in an 80atom cell, and scaled by a factor of 2.5×10 −4. The lines
E 2 and E 3 indicate the energy dependence of the absorption.
6.2.3
Electronic properties
It is possible to consider the (111) plane vacancy disk as a charge trap. There are a large
amount of sp2 bonds which are free to interact with mobile electrons, if such an interaction
reduces the system energy, then a charge trap may be formed. This is calculated by
comparing the energy change of the vacancy disk when an electron is added with the
energy change of a known defect, in this case substitutional boron.
An 80 atom unit cell containing a vacancy disk was structurally optimized with one electron added, to mimic charge transfer. The energy of the neutral vacancy disk is taken
as zero, and the extra energy gained by the negative negative disk is then 14.42 eV.
In order to calculate any energy saving the substitutional boron defect was modelled in
neutral and singly negative charge states. The neutral defect was found to have C 3v
symmetry and the negative case Td symmetry, with an increase in energy of 13.58 eV
for the negative defect. Taking the energy gain for the negative disk and subtracting the
boron energy gain yields 0.837 eV, which is added to the acceptor level of substitutional
boron (Ev + 0.37 eV [122]) resulting in an acceptor level for the negative vacancy disk of
CHAPTER 6. (111) PLANE VACANCY DISKS
122
Ev + 1.137 eV.
The energy saved by charge transfer depends on the defect from which an electron is
supplied. In the case of the Acentre with a donor level at E c − 4 eV [122, 123] the saving
is 0.293 eV.
If vacancy disks do act as a charge trap they could be a critical defect in the field of
diamond electronics, as they will have a significant effect on the conduction properties
of any diamond that they are present in. Recent results from the EPR group at the
University of Warwick indicate that the density of acceptors in CVD diamond is 10 times
greater than the density of known point defects [124]. This large margin means that there
must be some unidentified defect responsible for the acceptor level.
6.2.4
Removal of colour
There are significant differences between the annealing characteristics of the different
types of diamond. The absorption continuum disappears at 1400 ◦ C for irradiated material [39] and in the range 14001600 ◦C for brown singlecrystal CVD material [33]. For
brown natural type IIa diamond however, temperatures above 2200 ◦ C are required to
remove the absorption continuum [32]. This suggests that either the stabilities of the defects responsible for the absorption continuum in brown singlecrystal CVD material and
natural diamond are significantly different, or the mechanism for its loss are different. In
this section the latter argument is examined, as it has been postulated that the vacancy
disk model can explain the brown colour in both natural and CVD diamond.
Recent experiments [34] show that the transformation of brown to colourless CVD brown
diamond is accompanied by the growth of a broad CH band around 2900 cm −1 . This
frequency is close to the CH stretch mode on a (111) surface, which occurs at 2838 cm −1
[125]. The question arises as to whether hydrogen could passivate the optical activity of
the disk.
The vacancy disk is modelled in the same 40 atom configuration previously described,
but with 4 hydrogen atoms added; a number equivalent to the number of vacancies. This
cell was structurally optimized with two surface arrangements, the (111)(2×1) surface
(the Pandey chain) and the (111)(1×1) surface, which is equivalent to the bulk (111)
CHAPTER 6. (111) PLANE VACANCY DISKS
123
plane. As both systems contain the same number of each species of atoms they can
be compared directly, with the (111)(1×1) surface configuration being 3.08 eV lower in
energy.
Hydrogen has also been modelled on the other major crystallographic planes, with the
relative energies shown in table 6.1. As there are differing numbers of carbon and hydrogen atoms in the cells a comparison is made of the energy per carbonhydrogen bond at
the surface. The formation energy of the cell is calculated, and then divided by the number of carbonhydrogen bonds. The lowest energy configuration here is (111)(1×1):H
which releases energy during the bonding process. This supports the view that mobile
hydrogen will become trapped by vacancy disks.
Table 6.1: Relative energies of hydrogen at diamond surfaces.
Surface
(111)(2×1):H
(111)(1×1):H
(110)(1×1):2H
(100)(2×1):H
(100)(1×1):H
(100)(1×1):2H
Formation energy
per CH bond (eV)
0.41
0.36
0.29
0.21
0.21 (relaxes to (100)(2×1):H)
0.55
Hydrogen requires two electrons in its S shell to be stably bonded, hence hydrogen forms
H2 naturally. If hydrogen is near the (111) disk surface it will effectively gain an extra
electron by sharing one from the π bonded carbon in the Pandey chains. This means
that the carbon is now fourfold coordinated and there are no free electrons for optical
or electronic interaction. Figure 6.9 shows the bandstructure of a (111)(1×1):H plane
vacancy disk with hydrogen saturation at the surface. It is evident from the figure that
all absorption up to an energy value of 4.35 eV is eliminated, which covers the visible
spectrum and would render the diamond colourless.
Calculation of the vibrational modes for the hydrogenated (111)(1×1):H plane disk indicates that there are modes localized on the surface hydrogen atoms at 2923, 2927, 2973
and 2976 cm−1 . These values are in excellent agreement with those observed when
transforming CVD diamond from brown to colourless with HPHT annealing [34]. This
supports the proposition that hydrogen passivation of disk surface bonds is the method
of removal of colour in CVD diamond.
CHAPTER 6. (111) PLANE VACANCY DISKS
14
124
Filled States
Empty States
12
10
Energy (eV)
8
6
4
2
0
2
4
(010)/2
(111)/2
(011)/2
Γ
(001)/2
(100)/2
6
Γ
Brillouin zone position
Figure 6.9: Bandstructure of the (111) vacancy disk with hydrogen termination. The
bandgap is cleared of states and hence optical absorption in that energy range is removed. Energy states are scaled by 1.3 to compensate for LDA underestimation.
Natural type IIa diamond is very pure and contains little hydrogen, so here another process must account for the majority of the colour removal. Monovacancies and divacancies are mobile at temperatures around 700 ◦ C [126, 127] and 900◦ C [128] respectively,
hence under annealing conditions they can move through the crystal and become trapped
by existing disks  causing them to grow. There is nothing to check the growth of the disks
until they become unstable against collapse. During collapse one of the surfaces will be
displaced towards the other by a(110)/2, eliminating the fault but forming a perfect dislocation loop with a Burgers vector b equal to this displacement. The energy of such a
collapsed disk of radius r, according to elasticity theory [27] is:
∼
µ b2 r
ln(4r/b) − 1
2(1 − ν )
(6.1)
here µ and ν are the shear modulus and Poisson ratio of diamond. Plotting the surface
area energy of the disk versus the energy from equation 6.1 leads to the dislocation loop
˚ which contain
being more stable than the disk for loops of radius greater than about 12 A
CHAPTER 6. (111) PLANE VACANCY DISKS
125
about 200 vacancies. This is shown in figure 6.10. This process for removal of the colour
would require much higher temperatures to remove the colour in a reasonable space of
time, hence the higher annealing temperature for natural diamonds [31, 32].
1e16
Vacancy Disk
Dislocation Loop
9e17
8e17
Energy (J)
7e17
6e17
5e17
4e17
3e17
2e17
1e17
0
0
2e10
4e10
6e10
8e10 1e09 1.2e09 1.4e09 1.6e09 1.8e09
Radius (m)
Figure 6.10: Comparison of the energy of a vacancy disk to a dislocation loop as described in the text. The crossing of the curves indicates the maximum stable loop radius
˚
to be just less than 12 A.
6.3
Nitrogen at Vacancy Disks
The previous section considered type II diamond, which contains little nitrogen. Brown
coloured diamonds exist in all types, and as type I diamond contains nitrogen to densities
in excess of 1 partpermillion (ppm) the interaction of the vacancy disks with existing
nitrogen defects is of high importance.
As described in chapter 1 the optical absorption of type I brown diamond is quite different
from type II brown diamond. Type I diamond exhibits extra absorption peaks, notably at
∼2.25 eV (550 nm) [44], over the background continuum. It is known that nitrogen in
diamond can form several centres; the Acentre consisting of an NN pair, the Bcentre
consisting of four nitrogen atoms surrounding a vacancy and the N3centre of three nitrogen neighbouring a vacancy being the main defects [123, 22]. However, none of these
CHAPTER 6. (111) PLANE VACANCY DISKS
126
defects are known to produce an absorption band at 2.25 eV (table 1.2). It has been observed that electronirradiation of type I diamond with subsequent annealing creates an
unknown defect with absorption in the 536 and 575 nm areas (2.31 and 2.15 eV) [129].
This is attributed to the H4 centre dissociating to form other defects.
It is known from CVD growth studies that nitrogen disrupts growth to a greater extent
when the growth direction is (111), due to easier incorporation on these surfaces [130,
131]. Abinitio studies suggest that nitrogen incorporated into Pandey chains on the (111)
surface are less energetically favourable [132], but this may differ for a disk. This section
will examine the energetically favourable configurations in which nitrogen may be able
to exist at the surface of a (111) vacancy disk, and the optical and electronic properties
which arise from this.
Initially a study of point vacancynitrogen defects is made. This allows the lowest energy defect to be identified. The chemical potential of nitrogen in diamond can be determined from this lowenergy defect, allowing further formation and binding energies to
be calculated. The defects are all modelled in 216atom cubic supercells, with a 2×2×2
MonkhorstPack sampling grid.
Table 6.2 shows the relative formation energies of the defects studied with energies of
other defects for comparison. The Bcentre formed of four nitrogen atoms surrounding a
vacancy is found to be lowest in energy, in agreement with previous theoretical work [43].
The lower formation energy of the N3, A, B, C and H3 centres indicates that these are
more likely to be found in diamond containing nitrogen. Measurements by DTC of brown
type Ia diamond show low concentrations of H3 (NVN) and H2 (NVN − ) centres. When
such diamonds are subjected to HPHT treatment the concentration of these defects, as
well as N3, is seen to increase [30]. This indicates that free nitrogen and vacancies are
released from a source under these conditions.
The defect concentration based on these energies is able to account for the defect concentration observed in treated brown type Ia diamond, but is in contrast to the population in irradiated material. It is likely that irradiated brown diamond is in a strong nonequilibrium state, which would change the relative concentrations of any defects.
CHAPTER 6. (111) PLANE VACANCY DISKS
127
Table 6.2: The formation energies of various nitrogenvacancy defects, compared to the
Bcentre.
6.3.1
Defect
Label
VN4
VN3
NN
N
NVN
VN
V2 N4
V
VV
Bcentre
N3
Acentre
Ccentre
H3
H4
GR1
TH5
Relative Formation
Energy (eV)
0
1.8
1.9
3.1
3.5
5.6
6.1
7
9.9
Structures and energies
The vacancy disks are created in unit cells consisting of a total of 80, 120 or 160 atoms
depending on the configuration of nitrogen and the surface and in the bulk. Removal
of the α B (111) planes allows Pandey chain reconstruction to lower the surface energy [103], as shown in figure 6.3 with the π bonded atoms marked in white.
The binding energy of a single nitrogen (Ccentre) and the nitrogen pair (Acentre), to the
vacancy disk, are calculated using the formation energies of the individual defects:
f orm
Ebinding = (EA
f orm
+ EB
f orm
) − EAB
(6.2)
where Exf orm is the formation energy of defects A and B separately, and together AB. The
formation energy of the Ccentre is found to be 3.12 eV, and of the Acentre to be 1.9 eV.
These energies are calculated using a chemical potential for nitrogen derived from the energy of the B centre, which is the most favourable configuration for nitrogen in diamond.
In comparison the formation energy for the vacancy disk is 1.46 eV/vacancy [133]. Knowing these values the formation energies of the combined systems can be calculated. The
formation energy of the Ccentre at eight atomic layers from the surface, in a 120 atom
cell, is 3.25 eV, The Acentre at the same distance has a formation energy of 1.93 eV.
These values lead to a binding energy of the Ccentre to the disk of 1.0 eV and a value
for the Acentre of 2.35 eV.
CHAPTER 6. (111) PLANE VACANCY DISKS
12
128
Filled States
Empty States
Fermi Level
10
Energy (eV)
8
6
4
2
0
2
(110)/2
(111)/2
(011)/2
Γ
(001)/2
(100)/2
4
Γ
Brillouin zone position
Figure 6.11: Bandstructure of an Acentre near to a vacancy disk. The centre is 6 atomic
layers from the disk surface.
The bandstructures of an Acentre and a Ccentre near the disk are displayed in figures 6.11 and 6.12. Single defects have been placed at 6 and 7 atomic layers from the
disk surface for the A and C centres. In both cases it can be seen that the π and π ∗
states induced by the disk surfaces are still evident, and cross to give a continuous range
of energy transitions. The gap level associated with the nitrogen defect in each case is
superimposed over the disk bandstructure. In the case of the Acentre it is noticable that
the nitrogenrelated level is mostly below the Fermi level, and hence filled. This suggests
that charge transfer to the disk is unlikely, even though calculations indicate that this
would lower the system energy. The level associated with the Ccentre is much closer to
the Fermi level and crosses it at several points. This suggest that electron transfer here
is more feasible.
The binding energies of further bulk vacancynitrogen defects have been calculated for
comparison, with the binding energy of two single vacancies to form a divacancy being
4.45 eV. The binding energy of a vacancy to a nitrogen is 4.69 eV and the binding energy
of a vacancy to an Acentre to form the H3 centre is 5.53 eV.
If an Acentre migrates to the surface of a (111) vacancy disk then it may arrange itself
CHAPTER 6. (111) PLANE VACANCY DISKS
12
129
Filled States
Empty States
Fermi Level
10
Energy (eV)
8
6
4
2
0
(110)/2
(111)/2
(011)/2
Γ
(001)/2
(100)/2
2
Γ
Brillouin zone position
Figure 6.12: Bandstructure of a Ccentre near to a vacancy disk. The centre is 7 atomic
layers from the disk surface.
Figure 6.13: Nitrogen pair at the (111) plane vacancy disk surface.
into several configurations, which are shown in figures 6.13 – 6.17. Here the nitrogen
atoms are coloured white for clarity. The presence of the nitrogen in the Pandey chains
˚ bond length of the carbon. The symmetry of the
causes disruption to the usual 1.42 A
system is broken due to varying bond lengths. The nitrogen pair (figure 6.13) has a
˚ Nitrogen has 1.41 – 1.44 A
˚ bond lengths with neighbouring carbon
separation of 1.47 A.
˚ bond lengths. This
atoms. Carboncarbon bonds next to the nitrogen have 1.40 – 1.47 A
disruption of the bond lengths will affect the optical absorption properties of the Pandey
chain.
CHAPTER 6. (111) PLANE VACANCY DISKS
130
Figure 6.14: Nitrogen pair separated by one carbon atom, at the (111) plane vacancy
disk surface.
Figure 6.15: Nitrogen pair separated by two carbon atoms, at the (111) plane vacancy
disk surface.
Figure 6.16: Nitrogen pair separated by three carbon atoms, at the (111) plane vacancy
disk surface.
Figure 6.17: Isolated nitrogen at the (111) plane vacancy disk surface.
CHAPTER 6. (111) PLANE VACANCY DISKS
131
Table 6.3: Total energies of nitrogen configurations at the disk surface, compared to the
nitrogen Acentre.
Configuration
NN
NCN
NCCN
N3CN
N
Energy difference (eV)
4.709
5.177
6.310
5.707
2.067
Energy per nitrogen (eV)
1.177
1.294
1.577
1.427
1.033
The favourable configuration for nitrogen atoms at a disk surface is deduced by simply
comparing the total energy of each defect in a supercell with the supercell containing the
Acentre several layers from the disk. The cells containing isolated nitrogen consist of
118 carbon + 2 nitrogen atoms and the N3CN cells have 156 carbon + 4 nitrogen. All
other cells consist of 116 carbon + 4 nitrogen. Table 6.3 displays the energies of the
various configurations of nitrogen compared to the Acentre.
It is shown that the most stable configuration for nitrogen to be in at the disk surface is
for two nitrogen atoms to be separated by two carbon atoms. This configuration has the
˚ length, and nitrogencarbon
most regular symmetry with carboncarbon bonds of 1.40 A
˚ length. Using figures 6.19 – 6.23, the ratio of π bonds to singlebonds
bonds of 1.44 A
along the chain when nitrogen is included can be compared. The ratio of 2:6 π bonds to
singlebonds is the same for the NN and NCCN configurations, but the energy is much
lower in the latter case. The nitrogen pair has many varyinglength bonds and hence low
symmetry, which may be reflected in its higher energy. This can also be understood by
thinking of the π bonded chain as a resonance between the two reconstructions shown in
figure 6.18. The effect of N is to disrupt the bonding leading to a lone pair on each N atom
together with three NC bonds. Replacing C=C by NN is less stable than replacing
C=CC=C by NC=CN. This shows that A centres will split apart at the surface of the
disk.
It has been suggested by M. I. Heggie [134] that each nitrogen atom disrupts the system
of π bonds around it, weakening the three neighbouring bonds (but principally the two
bonds lying on the surface). At the first neighbour positions this disruption is a minimum,
but the nitrogen atoms repel due to a mixture of electrostatics and filling of antibonding
orbitals with the extra two electrons. At the third neighbour positions the disruption of
the π system is least because the carbon atoms having weakened bonds due to nitrogen
CHAPTER 6. (111) PLANE VACANCY DISKS
C
C
C
C
132
C
C
(a)
C
C
C
C
C
C
(b)
Figure 6.18: Alternative reconstructions of a π bonded Pandey chain. The horizontal axis
¯ (see figure 6.3) and the vertical axis is along [ 1¯12].
¯ Only the π bonded atoms are
is [110]
shown.
N
C
C
C
N
C
Figure 6.19: Bonding configuration of a nitrogen pair at the (111) plane vacancy disk
surface.
now neighbour each other and can enhance their bond strength with each other. This can
be summarised as the nitrogen atoms creating a π radical on the neighbouring carbon
atoms. In NCCN these radicals neighbour each other and form a stronger π bond.
The bonding configuration for nitrogen in Pandey chain is shown in figures 6.19 – 6.23.
Single and double bonds are denoted by single and double lines, and those carbon atoms
which have a dangling bond are circled. The cell repetition is such that there will be only a
short length of Pandey chain between nitrogen sites. This high nitrogen density may result
in higher energy due to interference between the nitrogen defects, which is investigated
in larger cells. Cells of 240 and 360 atoms are created by removing the nitrogen atoms to
CHAPTER 6. (111) PLANE VACANCY DISKS
N
C
133
N
C
C
C
Figure 6.20: Bonding configuration of a nitrogen pair separated by one carbon atom, at
the (111) plane vacancy disk surface.
C
C
N
N
C
C
Figure 6.21: Bonding configuration of a nitrogen pair separated by two carbon atoms, at
the (111) plane vacancy disk surface.
N
C
C
N
C
C
C
C
Figure 6.22: Bonding configuration of a nitrogen pair separated by three carbon atoms,
at the (111) plane vacancy disk surface.
N
C
C
C
C
C
Figure 6.23: Bonding configuration of isolated nitrogen at the (111) plane vacancy disk
surface.
create a cell of only carbon, then appending this cell onto the original one to create a cell
with a longer chain of carbon atoms between the nitrogen atoms. For the 240 atom cell,
both the NN and NCCN configuration are tested. It is found that the NCCN system
is 0.17 eV lower in energy when there is a larger gap between the nitrogen atoms. The
CHAPTER 6. (111) PLANE VACANCY DISKS
134
NN system is also lower in energy, when this configuration is tested, by 0.1 eV. These
lowering in energies can be attributed to the lower disruption of the Pandey chains and
reduced interaction between nitrogen atoms.
The bandstructure of the NCCN configuration in the 120 atom cell is shown in figure 6.24. Compared to the simple (111) vacancy disk [133], the semimetallic bandstructure is lost, indicating a absorption onset will occur in the region of 2 eV. A broad continuum absorption like that seen experimentally in brown diamond is not to be expected
from such a bandstructure.
12
Filled States
Empty States
10
Energy (eV)
8
6
4
2
0
(010)/2
(111)/2
(011)/2
Γ
(001)/2
(100)/2
2
Γ
Brillouin zone position
Figure 6.24: Bandstructure of the NCCN defect at the surface of a (111) plane vacancy
disk. This is modelled in a 120 atom cell with a small separation between the nitrogen
atoms. This introduces a gap of ∼2 eV into the bandstructure. The top of the valence
band is set to 0 eV, and states above it have their energies scaled by 1.3.
The bandstructure of the NCCN configuration in a 240 atom cell is shown in figure 6.25.
With a larger separation of the nitrogen atoms the gap in levels is narrowed to ∼1 eV,
so it is expected that a much lower absorption onset will occur. This demonstrates a
significant interaction between NCCN defects, but in reality the density of nitrogen at
the disk surface would be low enough that almost no defects will be at all close to any
other.
CHAPTER 6. (111) PLANE VACANCY DISKS
135
Filled States
Empty States
8
Energy (eV)
6
4
2
0
(010)/2
(111)/2
Γ
(001)/2
(100)/2
2
Γ
Brillouin Zone Position
Figure 6.25: Bandstructure of the NCCN defect at the surface of a (111) plane vacancy
disk. This is modelled in a 240 atom cell with an increased separation between the
nitrogen atoms. The gap in levels is reduced to ∼1 eV. The top of the valence band is set
to 0 eV, and states above it have their energies scaled by 1.3.
Using the 120 atom supercell, the nitrogen atoms are removed from one surface and
the bandstructure recalculated with half of the Pandey chains undisturbed. This approximates a lower density of NCCN defects, in a system that is computationally more
simple. As described above, this is a lower energy system due to less interaction between
nitrogen atoms. The bandstructure of such a system is a combination of the semimetallic
structure of an uninterrupted Pandey chain combined with that of the nitrogen decorated
disk. The bandstructure is shown in figure 6.26, with the same scaling as described previously. Such a semimetallic bandstructure may be expected to return an absorption
continuum, like that seen in brown diamond.
EELS and optical absorption
The absorption spectrum for the highdensity NCCN configuration is shown in figure 6.27, with a 0.5 eV polynomial broadening applied, and a scaling factor of 1.3 applied
to the energy to compensate for LDA bandgap underestimation. The absorption displays
CHAPTER 6. (111) PLANE VACANCY DISKS
12
136
Filled States
Empty States
10
Energy (eV)
8
6
4
2
0
2
(010)/2
(111)/2
(011)/2
Γ
(001)/2
(100)/2
4
Γ
Brillouin zone position
Figure 6.26: Bandstructure of the NCCN defect with a 50% coverage on the (111)
plane vacancy disk. The combination of π bonded chains and nitrogen defects has given
a modfied but still semimetallic system. The top of the valence band is set to 0 eV, and
states above it have their energies scaled by 1.3.
an onset at ∼1.7 eV (729 nm) and a peak at 2.5 eV (495 nm). Type I brown diamond
shows a peak at 2.25 eV (500 nm) which is not inconsistent with the calculated peak, but
brown and pink diamonds do not exhibit the absorption onset seen here, rather they have
a finite absorption to low energies. The precise value of the onset cannot be predicted
due to the broadening applied.
The density of nitrogen to carbon modelled for the absorption in figure 6.27 is high. There
are four nitrogen atoms and 116 carbon atoms in the cell, giving a density of 3.45 % which
is equivalent to 5.8×1021 cm−3 of nitrogen atoms. However, the calculated absorption can
be scaled at 4 eV from 16 358 cm−1 to a value agreeing with one typical for brown type Ia
diamonds. A more realistic absorption magnitude is 4 cm −1 at 4 eV (310 nm), and using
this one obtains a revised nitrogen density of 1.5×10 17 cm−3 . Though this assumes most
nitrogen is at the disk surface, which is unrealistic.
As described above, the concentration of nitrogen at the disk surface can be reduced
by half, leading to a bandstructure which has a wider range of available transitions (fig
CHAPTER 6. (111) PLANE VACANCY DISKS
137
600000
1
Absorption coefficient (cm )
500000
400000
300000
200000
100000
0
0
2
4
6
8
10
Energy (eV)
Figure 6.27: Optical absorption of the NCCN defect at the surface of a (111) plane
vacancy disk. This is modelled in the 120 atom cell with a high density of nitrogen. The
energy values are scaled by a factor 1.3.
ure 6.26). The absorption profile for this defect is shown in figure 6.28. The higherenergy
peak has now shifted to 2.4 eV (516 nm) and is 59 % of the magnitude when at full density. The absorption threshold has been replaced by a continuum absorption down to
0 eV, which is more alike that observed experimentally in brown diamond. Additionally
there is now a peak in absorption at 0.7 eV (1770 nm), which is close to that observed in
amber diamonds [45].
It is possible to get an idea of the density of nitrogen required for the observed absorption by scaling the calculated magnitude to match the experimental values, as described
above. This requires density be ∼4000 times lower. The modelled nitrogen density is
2/118 or 1.69 %, this equates to 2.87×10 21 cm−3 , which is reduced to 7×1017 cm−3 upon
scaling. The fraction of the total nitrogen available to migrate to the disk surface must
be considered, if it is 10 % then a total density of 7×10 18 cm−3 is required, for 1 % it
is 7×1019 cm−3 and at 0.1 % it is 7×1020 cm−3 , which is just less than 1000 parts per
million (ppm). Nitrogen is known to be present in diamond at densities up to, and just
over, 100 ppm [11] so these values are not unreasonable assuming 1 % of nitrogen is at
a disk surface.
CHAPTER 6. (111) PLANE VACANCY DISKS
138
600000
1
Absorption coefficient (cm )
500000
400000
300000
200000
100000
0
0
2
4
6
8
10
Energy (eV)
Figure 6.28: Optical absorption of the NCCN defect at the surface of a (111) plane
vacancy disk, at half the initial density. This is calculated in a 120 atom cell, with one
surface free of nitrogen. The energy values are scaled by a factor 1.3.
The model described here fails to account for the 3.17 eV (390 nm) band exhibited in pink
diamond as described in section 1.3. However it accurately reproduces the absorption
spectrum of type Ia brown diamond observed experimentally. Both brown and pink diamonds have a similar magnitude of absorption, which will correlate to defect density, and
hence the difference between the two may be more than just density of colour centres. It
appears that another defect may be involved, which could be related to the ability of the
disk to act as a charge trap.
6.3.2
Boron interaction with vacancy disks
In order to provide further comparison to these results, and for completeness, single
boron has been modelled at the vacancy disk. In this model boron replaces one carbon
atom in each chain. The binding energy is determined by modelling two single boron
atoms away from the disk and is found to be 1.44 eV. This binding energy is slightly more
than a single nitrogen which has a binding energy of 1.0 eV. If we look at the bandstructure
of a single boron atom in a bulk configuration but close to the disk, we see that the Fermi
level lies below the crossing of the π π ∗ levels. The bandstructure is shown in figure 6.29.
CHAPTER 6. (111) PLANE VACANCY DISKS
139
The Fermi level position is such that rather than charge transfer to the disk, the reverse
is expected to occur. Electrons from the disk could passivate the boron acceptor level,
leaving an excess of holes in the π bonded chains.
12
Filled States
Empty States
Fermi Level
10
Energy (eV)
8
6
4
2
0
2
(110)/2
(111)/2
(011)/2
Γ
(001)/2
(100)/2
4
Γ
Brillouin zone position
Figure 6.29: Bandstructure of single boron in bulk configuration, but near the surface of
a (111) plane vacancy disk. The Fermi level lies below the crossing of the π π ∗ levels,
suggesting that no electron transfer occurs. The top of the valence band is set to 0 eV,
and states above it have their energies scaled by 1.3.
As with the nitrogen case, the boron disrupts the π bonding configuration, which perturbs
the bandstructure. However, as boron has fewer electrons, states lower in the gap are
unoccupied. This means that there is only a small gap available for transitions, evident in
figure 6.30.
It is possible that the energy of substitutional boron could be lowered by moving to a
position nearer to the disk. Here, instead of being surrounded by bulk diamond it could
bond to sp2 carbon in the Pandey chains. It has been shown for boron in silicon that the
secondnearest neighbour site to a vacancy is energetically favourable [135]. This would
require further work to clarify.
CHAPTER 6. (111) PLANE VACANCY DISKS
12
140
Filled States
Empty States
10
Energy (eV)
8
6
4
2
0
(110)/2
(111)/2
(011)/2
Γ
(001)/2
(100)/2
2
Γ
Brillouin zone position
Figure 6.30: Bandstructure of single boron at the surface of a (111) plane vacancy disk.
A small gap is introduced, but it is much less than the gap in the nitrogenrelated defects.
The top of the valence band is set to 0 eV, and states above it have their energies scaled
by 1.3.
6.4
Conclusion
It has been shown that a disk of vacancies lying on the (111) plane is the most stable
configuration for large vacancy clusters in diamond. The stability of the disk is due to
the elimination of dangling bonds through the formation of π bonded chains similar to the
(2 × 1) reconstruction of the (111) surface.
The band structure of the (111) vacancy disk is quite distinct from that of the ideal 2 × 1
(001) diamond surface, where again a reconstruction leads to π bonding. However, in
this case the π bonds do not overlap appreciably, and lead to only narrow π bands and
π ∗ bands lying within the gap, of width less than 2 eV and separated by ∼1.5 eV [136].
To achieve very broad π and π ∗ bands, separated by a narrow gap, it is necessary for the
π bonds to be connected together in, for example, chains or as in a graphene sheet. Only
in this case, will the optical properties approach those of brown diamond. Disks lying on
(110) planes also do not exhibit the broad π and π ∗ bands which completely fill the gap
CHAPTER 6. (111) PLANE VACANCY DISKS
141
and hence would not show the characteristic absorption of brown diamond.
The absorption coefficient of the disk exhibits the same featureless absorption profile as
seen experimentally for brown diamond. Moreover, the highest local vibrational mode of
the disk at 1496 cm−1 is close to a band detected experimentally at 1540 cm −1 in brown
CVD diamond. Vacancy disks in the (111) plane are identified as a likely candidate for
the origin of the brown colouration in diamond. In natural diamond, the experimental
absorption bands vary more dramatically with energy which may reflect the presence of
other vacancy centres. It may be that octahedral voids of vacancies having (111) surfaces
could also explain the brown colour and these could also account for the very long 400 ps
lifetime detected by positron annihilation in brown diamond [35, 112].
Two mechanisms for the loss of the brown colouration upon annealing are investigated.
In CVD diamond, the surface of the disk is passivated with hydrogen, possibly explaining
the growth of vibrational bands around 2900 cm −1 detected after the transformation of
brown CVD diamond [34]. In natural type IIa diamond, the collapse of a large disk, and
the formation of an optically inactive dislocation, is found to be energetically favourable
for disks containing more than about 200 vacancies. Such a mechanism might account
for the transformation of brown natural diamond, although it is unclear whether the disks
are growing and then become unstable, or the activation barrier to the collapse and the
formation of a dislocation loop are overcome at temperatures around 2000 ◦ C.
The interaction of nitrogen (specifically the Acentre) with a disk of vacancies lying on
the (111) plane has been evaluated. The most stable configuration for a pair of nitrogen atoms at the disk surface is for a carbon pair to be between them, disrupting the
Pandey chain. This leads to a gap in the bandstructure and an optical absorption onset
at ∼1.7 eV. At lower densities of nitrogen the continuum absorption observed in type Ia
brown diamond returns, with absorption bands at 2.4 eV (516 nm) and 0.7 eV (1770 nm),
which are close matches to experiment. The total nitrogen density required is estimated
at ∼1020 cm−3 , which is a realistic figure. This model is proposed as a possible cause of
brown colour in type Ia diamond.
The model fails to completely explain the origin of pink colour in diamond, as no band at
3.17 eV (390 nm) is produced, but could be part of a system of defects which combine to
produce the pink colour.
Chapter 7
Photoelastic Constants in Diamond
and Silicon
7.1
Introduction
This work has been published in Physica Status Solidi (a) 203, 3088 (2006).
When a strain is applied to a crystal there is a corresponding change in the refractive
index, arising from a change in the dielectric function [137]. The photoelastic tensors
relate the change of the inverse dielectric function with the applied strain. The presence
of a dislocation or extended defect in a crystal causes a local strain field which in turn
can cause a nominally isotropic material to become anisotropic and induce birefringence
in light passing through crosspolarizers [138]. The birefringence due to dislocations
in silicon has been observed [139] and modelled theoretically [140]. However, these
calculations require the photoelastic tensor which is not well understood in diamond,
compared to silicon [141].
The change in the inverse dielectric function due to strain is given by:
δ ε −1
ij
= pi jkl µkl ≈ −
1
δ εi j
εb2
(7.1)
where pi jkl is the fourth rank photoelastic tensor, µ kl is the strain tensor and εb is the
142
CHAPTER 7. PHOTOELASTIC CONSTANTS IN DIAMOND AND SILICON
143
value of the dielectric function in unstrained material. Adopting a contracted notation
where 11 → 1, 22 → 2, 33 → 3 and 23,32 → 4, p i jkl reduces to a symmetric matrix with
three nonzero elements for silicon and diamond. These are p 11 ,p12 and p44 and they can
be found individually and in combination using various strains. For hydrostatic strain:
−
1
εb2
δ εi j =
δa
a
p11 + 2p12
0
0
p11 + 2p12
0
0
0
p11 + 2p12
(7.2)
where a is the lattice constant, so δ a/a is the percentage strain. Similarly for uniaxial
strain along [001]:
1
− 2 δ εi j =
εb
p
δ a 12
0
a
0
0
p12
0
0
0
p11
(7.3)
For uniaxial strain along [111], instead of a change in lattice parameter as a measure of
strain, the change in a length l is considered:
1
− 2
εb
δ εi j =
δl
3l
p11 + 2p12
2p44
2p44
2p44
p11 + 2p12
2p44
2p44
2p44
p11 + 2p12
(7.4)
These three strains are thus sufficient to determine p 11 , p12 and p44 .
7.2
Results
The photoelastic constants are calculated in twoatom unit cells, with a high MonkhorstPack sampling of 83 kpoints required. The unit cells are taken and the lattice vectors
changed to apply strain in the [001] and [111] directions and hydrostatically. The values
of strain applied are ±0.5% for hydrostatic and [001] strain, and ±0.25% for strain along
[111]. The values of p11 , p12 and p44 are found for an average of these compressive and
tensile strains.
CHAPTER 7. PHOTOELASTIC CONSTANTS IN DIAMOND AND SILICON
7.2.1
144
Silicon
Initially, values of p11 , p12 and p44 are calculated for silicon. This demonstrates the reliability of the method described, as the photoelastic constants for silicon are better studied [142, 143, 144, 145]. Previous abinitio studies on the dielectric response of silicon [142] have reviewed the available experimental data. In only one case [143] were
all the constants of interest found experimentally, so reference will only be made to the
results from that work.
The use of a rigid shift to compensate for the LDA underestimation of the bandgap was
tested. This shift affects all levels above the valence band top, to the same extent. It
was used with some success by Levine et al. [142], and is often referred to as a ’scissors
shift’. The scissors shift becomes important for defective crystals where states are introduced into the band gap and lead to additional optical transitions. To correct the AIMPRO
bandgap requires a shift of 0.52 eV. Such a shift changes the dielectric constant so evaluation is made of the changes to the photoelastic constants when a scissors operator is
used.
Table 7.1: Effect of scissors shift on the real part of the calculated dielectric constant ε b ,
and the photoelastic constants for silicon. Experimental and previous abinitio work is
shown for comparison. All values refer to a wavelength of 3542 nm (0.35 eV).
εb
p11 + 2p12
p11 − p12
p44
13.670
12.195
11.764
11.327
11.830
0.058
0.041
0.039
0.034
0.054
0.109
0.116
0.119
0.121
0.112
0.051
0.054
0.055
0.055
0.051
10.900
0.062
0.118
0.051
AIMPRO
0 eV Scissor shift
0.52 eV Scissor shift
0.7 eV Scissor shift
0.9 eV Scissor shift
Experiment [143]
Abinitio [142]
0.9 eV Scissor shift
The photoelastic constants have been found using the calculated values of ε b . It is shown
in table 7.1 that increasing the scissor shift to compensate for the bandgap underestimation gives a better agreement for the real part of the dielectric constant with experiment.
In contrast, the agreement for the low frequency values of the photoelastic constants becomes worse. An averaged combination of tensile and compressive strain is used for
accuracy.
CHAPTER 7. PHOTOELASTIC CONSTANTS IN DIAMOND AND SILICON
0.2
145
p11 + 2p12 (001)
p12  p11
p44
p11 + 2p12 (Hydrostatic)
0.15
Photoelastic constants
0.1
0.05
0
0.05
0.1
0.15
0.2
0
0.5
1
1.5
2
Energy (eV)
Figure 7.1: Photoelastic constants for silicon calculated from the average of compressive
and tensile strain.
Figure 7.1 shows the variation of the calculated photoelastic constants with frequency for
silicon. No scissors shift is applied, as described above. Comparison with experimental values [143] is shown in figure 7.2, and with previous abinitio calculations [142] in
figure 7.3. The values of p11 − p12 are slightly larger than experiment by about 3% but
the low dispersion with frequency is in good agreement. Excellent agreement is shown
for p44 . The calculated values for p11 + 2p12 are in good agreement with experiment at
∼ 0.5 eV but disperse quite strongly with increasing frequency. As stated above, this is
due to the small band gap and is lessened when a shift is used.
The calculated constants for silicon display relatively little dispersion except for the calculated values of p11 + 2p12 . If a scissors shift of 0.9 eV is used, p 11 + 2p12 becomes less
sensitive to frequency but the overall values are increased. As described in the introduction, p11 + 2p12 can be derived from hydrostatic strain or from strain imposed along [001].
Both methods lead to the same value of p 11 + 2p12 .
CHAPTER 7. PHOTOELASTIC CONSTANTS IN DIAMOND AND SILICON
0.1
p11 + 2p12
p11  p12
p44
p11 + 2p12 ([11])
p11  p12 ([11])
p44 ([11])
0.05
Photoelastic constants
146
0
0.05
0.1
0.15
0.2
0
0.5
1
1.5
2
Energy (eV)
Figure 7.2: Calculated photoelastic constants for silicon using a zero scissors shift, compared to experimental data from [143].
7.2.2
Diamond
The values of the photoelastic constants for diamond in the literature differ widely and
have been reviewed previously [141], where there is very little information on the variation
of the constants with wavelength. Grimsditch et al. [141] evaluate one set of results as
being ”most reliable”, these results are from [146]. For these reasons it is of interest to
present a set of calculated values.
Table 7.2 shows the variation of the real part of the dielectric constant and photoelastic
constants with scissor shift. A shift of 1.33 eV is required to correct the bandgap for diamond. As described for silicon the increase in shift causes the calculated photoelastic
constants to diverge from the experimental results. It is notable that in the case of diamond, when the bandgap is corrected by a scissors shift of 1.33 eV the value of the
dielectric constant εb is just less than 95% of the experimental value. In silicon, the required shift of 0.52 eV gives an εb of 103% of experiment. These results highlight the
inconsistency of using a scissors shift, which cannot be relied upon to improve both the
CHAPTER 7. PHOTOELASTIC CONSTANTS IN DIAMOND AND SILICON
0.1
p11 + 2p12
p11  p12
p44
p11 + 2p12 ([10])
p11  p12 ([10])
p44 ([10])
0.05
Photoelastic constants
147
0
0.05
0.1
0.15
0.2
0
0.5
1
1.5
2
Energy (eV)
Figure 7.3: Calculated photoelastic constants for silicon using a zero scissors shift, compared to previous abinitio work from [142].
dielectric constant and photoelastic constants for a given material.
Table 7.2: Effect of scissors shift on the real part of the calculated dielectric constant ε b ,
and the calculated photoelastic constants for diamond. Experimental work is shown for
comparison.
Frequency (eV)
εb
p11 + 2p12
p11 − p12
p44
2.27
2.27
2.27
2.27
2.27
2.27
2.27
2.30
6.172
5.755
5.662
5.526
5.819

0.160
0.124
0.121
0.122
0.164
+0.12
0.1
0.160
0.335
0.331
0.331
0.330
0.292
0.668
0.668
0.292
0.171
0.156
0.156
0.155
0.172
0.162
0.162
0.172
AIMPRO
0 eV Scissor shift
0.78 eV Scissor shift
1.0 eV Scissor shift
1.33 eV Scissor shift
Experiment [107, 146]
Experiment [147]
Experiment [147, 141]
Experiment [148]
It is evident that p11 − p12 , is underestimated by approximately 14%, but p 11 + 2p12 and
p44 show very good agreement, with less than 3% and 1% difference respectively. Fig
CHAPTER 7. PHOTOELASTIC CONSTANTS IN DIAMOND AND SILICON
148
ure 7.4 shows the experimental data for diamond along with the calculated values for the
photoelastic constants found for one frequency. The figure is in three parts for clarity.
0.4
0.3
Photoelastic constants
0.1
p11 + 2p12
p11 + 2p12 [15]
p11 + 2p12 [16,5]
p11 + 2p12 [16]
p11 + 2p12 [17]
0.2
0.2
0.1
p11  p12
p11  p12 [15]
p11  p12 [16]
p11  p12 [17]
p44
p44 [15]
p44 [16]
p44 [17]
0.12
0.3
0.14
0.1
0.4
0
0.5
0.1
0.6
0.16
0.2
0.18
0.7
2
2.2
2.4
2.6
Energy (eV)
2.8
3
0.2
2
2.2
2.4
2.6
2.8
Frequency (eV)
3
2
2.2
2.4
2.6
2.8
Frequency (eV)
3
Figure 7.4: Photoelastic constants for diamond calculated with a zero value scissors shift.
Experimental data is shown for comparison, taken from [146, 147, 148].
7.3
Strain of Dislocations
In a tetragonal solid such as diamond, the velocity of light along the three principal directions ([100], [110] and [111]) is different. Irradiating with polarized light then leads to
a shift in the two polarized parts. This shift induces a phase difference at the exit point,
which leads to birefringence.
If the sample to be analysed is placed between a polarizer and analyser which are perpendicular, the intensity of the transmitted light is given by [140]:
T = a2 sin2 (2γ ) sin2 (δ /2)
(7.5)
where δ is the phase difference in the two principal directions of the crystal plane, a is
the amplitude of the incident light and γ is the angle between the incident light vector and
a principal direction. The relationship between these directions is shown more clearly in
figure 7.5.
CHAPTER 7. PHOTOELASTIC CONSTANTS IN DIAMOND AND SILICON
149
y
A
P
γ
γ
x
π/2 − γ
Figure 7.5: Relationship between the principal directions of the crystal and the polarizer/analyser directions P and A respectively.
If a dislocation is present in a crystal it will cause a strain effect in the area surrounding
it, which can be characterised using the equations:
A
[x1 − (1 − 2ν )x2 ]
r2
A
µ22 = − 2 [x1 + (1 − 2ν )x2 ]
r
x2 − x21
1
θ = tanh 2
2
2x1 x2
µ11 =
(7.6)
(7.7)
(7.8)
where µ11 and µ22 are the strain values along the principal directions. θ can then be
defined as the angle between the principal axis and the slip direction. The prefactor A is
itself expandable to:
A = b/[4π (1 − ν )]
(7.9)
with b the Burgers vector magnitude. The angle γ between the principal x axis and the
polarizer direction can now be redefined as γ = β + θ as shown in figure 7.6.
It can thus be shown [140] that the transmitted intensity can be expressed:
CHAPTER 7. PHOTOELASTIC CONSTANTS IN DIAMOND AND SILICON
150
y
P
Slip
β
θ
x
Figure 7.6: Relationship between the principal directions of the crystal and the polarizer
direction. The slip plane of the dislocations is marked.
T (r, ψ ) = (B2 /r2 ) cos2 (2φ ) cos 2 (φ − B)
(7.10)
Where φ = γ + π /4. The contours of constant intensity are given by the polar equation:
r2 = (B2 /T ) cos2 (2φ ) cos 2 (φ − B)
(7.11)
These contours are plotted in figures 7.7(a) – 7.7(g), for several different angles between
the polarizer and the slip direction (β ). It is clear that when β = 0 ◦ and 90◦ the plots are
the same, and these plots are in exact agreement with previous work in silicon [140].
The ratio of the transmitted intensity to the incident intensity is defined:
Tr ∼ A2C2 4π t 2 /λ 2 r2
(7.12)
Where t is the thickness of the material; in this case a typical thickness of 1 mm is assumed. C is defined as p11 − 2p12 , and is found from the values in table 7.2 to be −0.392,
hence C2 is 0.153. The magnitude of the Burgers vector for a pure edge dislocation in
˚ hence A has a value of 2.48× 10−11 m. Inserting these values into
diamond is 2.5 A
equation 7.12 give the radial dependence of the strain:
CHAPTER 7. PHOTOELASTIC CONSTANTS IN DIAMOND AND SILICON
0.8
0.8
1
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1
0.4
y
0.2
0
0.2
0.4
0.2
0
0.2
0.4
0.4
0.6
0.6
0.6
0.4
0.2
0
x
0.2
0.4
0.6
1
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1
0.6
y
0.6
0.8
0.8
0.8
0.8
0.8
0.6
0.4
(a) β = 0◦
0.2
y
0.2
0.4
0.6
0.8
0
0.2
0.4
0.2
0
0.2
0.4
0.4
0.6
0.6
0.6
0.4
0.2
0
x
0.2
0.4
0.6
0.6
0.4
0.2
0
0.2
0.4
0.6
0.6
y
0.4
0.8
0.8
0.8
0.6
0.4
(c) β = 30◦
0.2
0
x
0.2
0.4
0.6
0.8
(d) β = 45◦
0.8
0.8
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
0.4
0.2
0
0.2
0.4
0.2
0
0.2
0.4
0.4
0.6
0.6
0.6
0.4
0.2
0
x
0.2
0.4
0.6
1
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1
0.6
y
0.6
y
0
x
0.8
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
0.6
0.8
0.8
0.2
(b) β = 15◦
0.8
0.8
0.8
151
0.8
0.8
0.8
0.6
0.4
(e) β = 60◦
0.2
0
x
0.2
0.4
0.6
0.8
(f) β = 75◦
0.8
1
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1
0.6
0.4
y
0.2
0
0.2
0.4
0.6
0.8
0.8
0.6
0.4
0.2
0
x
0.2
0.4
0.6
0.8
(g) β = 90◦
Figure 7.7: Contours of constant diffraction intensity from equation 7.11, plotted with
varying angles between the polarizer and the slip direction (β ).
CHAPTER 7. PHOTOELASTIC CONSTANTS IN DIAMOND AND SILICON
Tr ∼ 1.81 × 10−16 /r2
152
(7.13)
Thus at a radius of 1 µ m from the dislocation core, the strain is 1.809× 10 −4 the value at
the centre  about 0.2%.
7.4
Conclusions
The photoelastic constants for diamond and silicon have been calculated using density
functional theory applied to strained unit cells. Experimental data for silicon is accurately
reproduced with absolute values and variation with frequency matched for almost all the
constants. It is has been shown that the scissors shift technique of correcting the LDA
underestimation of the bandgap does not uniformly improve the calculated photoelastic constants. The agreement for silicon gives confidence that the diamond values will
accurately predict experimental results.
Experimental data for diamond is limited. However, the values calculated here agree well
with one particular experimental set [142], which has previously been regarded as ’most
reliable’ [141]. In addition, the calculated results agree with the sign of the constants in
the experimental set, confirming that p 11 + 2p12 should be a negative value.
It has been shown how the strain of dislocations can induce birefringence in polarized
light, with the intensity caused by a pure edge dislocation used as an example. This
strain falls off as r 2 , to a low value at ∼1 µ m from the dislocation core.
Chapter 8
Conclusions
Diamond is a scientifically interesting material with unique electrical, mechanical and
thermal properties. These properties make diamond both cosmetically attractive and
potentially important in nextgeneration electrical devices. Many features of diamond
have been previously studied, such as point defects and dislocations, and these were
overviewed in chapter 1. This thesis has focused on the optical properties of defects in
an effort to come closer to understanding the origin of brown colour in diamond. Brown
diamond has a unique absorption continuum across the bandgap, with no detectable
onset at low energies. The source of colour is not known, but is vital for understanding the
loss of colour which occurs during HighPressure HighTemperature (HPHT) treatment.
The theoretical method used (AIMPRO) has been used on a range of materials, and is
proven to be a good match for experimental results. A review of the inherent theory and
its method of application is presented in chapter 2. In addition, a summary of relevant
experimental techniques is given in chapter 3, as results from these techniques are referred to throughout this work. Full understanding of any scientific problem is made easier
through a combination of theoretical and experimental work, hence as much of this work
as possible is linked to known results.
153
CHAPTER 8. CONCLUSIONS
8.1
154
Small Vacancy Clusters
Electron paramagnetic resonance (EPR) studies on diamond have revealed a range of
small vacancy defects, which are identified as clusters and chains along 110 . Due to
their prevalence, a study is made of the properties of such defects. Vacancy chains of up
to six vacancies are modelled, and vacancy clusters of up to 14 vacancies. The clusters
are constructed using a bondcounting model previously demonstrated for clusters in
silicon. DFTB theory (chapter 2) is used to check convergence of energy with cell size,
with a cell of 216 atoms required to accurately model such defects.
Analysis of the formation energies of these defects in both S =0 and S =1 states indicates
that the vacancy clusters are more stable than vacancy chains in all cases. In addition,
the clusters of 6, 10 and 14 vacancies are extrastable, due to their closed structure, a
property also predicted for silicon. The 14 vacancy cluster has the lowest overall formation energy of 2.35 eV per vacancy. Apart from the four and six vacancy clusters, and
the three and four vacancy chains, all defects are in the S =1 state for lowest energy,
with unpaired electrons. These unpaired electrons will be expected to introduce bandgap
energy levels, and an analysis of the KohnSham energy states for each defect confirms
this.
With the introduction of bandgap states a peak in absorption is observed around 3.5 –
4.5 eV range, with the energy of the peak maximum increasing with cluster size. This
is unlike the absorption of brown diamond, so small vacancy clusters are discounted as
a possibility for the source of brown colour, even as a combination. Initial work on the
possibility of including π bonded carbon in small vacancy defects indicates that this will
not be feasible.
8.2
EELS of Dislocations
Dislocations are a prevalent defect in natural diamond, with a density of ∼10 9 cm−2 ob
served by TEM. However the density in artificial CVD diamonds is five orders of magnitude lower. Previous abinitio studies of dislocations in diamond predict that some types
will have deep gap levels which could introduce optical absorption. The density of disloca
CHAPTER 8. CONCLUSIONS
155
tions which is observed in HPHT treated brown (now colourless) diamond is not markedly
different from untreated samples, so it can be concluded that few dislocations are created
or destroyed during the process. If this is the case then the proposal that dislocations giving rise to absorption are converting to optically inactive types is considered.
The dielectric function, Electron Energy Loss (EEL) and absorption spectra are calculated
for bulk diamond as a test of the technique and to verify if scaling is required to compensate for LDA bandgap underestimation. A good agreement is demonstrated for peaks
in EEL and dielectric spectra. The static value of the real part of the dielectric function
shows excellent agreement, with a calculated value of ε 1 =5.89 compared to 5.82 experimentally. The value where ε2 falls to 0 is not so closely matched, with a value of 5.29 eV
calculated, compared to 6.81 eV experimentally. This mismatch can be attributed to the
broadening applied to the calculated points, which will lower the threshold of onset. In
the case of the EEL spectra for diamond and graphite the main peaks at 35 eV (diamond)
and 22 eV (graphite) are a very good match for experiment.
Two dislocation types are studied, the 90 ◦ glide and the 90◦ shuffle types. There is a
potential that the shuffle type could convert to the glide type if supplied with sufficient
energy. The formation energy of the glide type is much lower that the shuffle type, indicating that such a process could take place, but that it can be expected that there will
be more glide type dislocations present. The glide type is optically inactive due to its
reconstructed core atoms, with no absorption below ∼4 eV. In contrast the shuffle type
has dangling bonds which introduce absorption throughout the gap, but the spectrum is
not like that seen experimentally in brown diamond. A density of shuffle dislocations of
∼1011 cm−1 is calculated to be required to account for brown colour, however natural
diamonds have a dislocation density of ∼10 9 cm−1 and CVD diamonds even less. This
eliminates dislocations as a possible source of brown colouration, but they may still have
an important part to play as a source of vacancies or as defect traps.
8.3
(111) Plane Vacancy Disks
If a double bond is formed between two carbon atoms, it will introduce π and π ∗ states into
the bandgap, hence is known as a π bond. Previous abinitio studies of diamond surfaces
show that networks of π bonds can form on the (111) surface, and the delocalization of
CHAPTER 8. CONCLUSIONS
156
the electrons over the network gives rise to dispersion of the π and π ∗ states. This
dispersion causes a wide range of electron transitions to be available, potentially leading
to broad optical absorption.
The (111) surface can be recreated internally by removal of the α B double plane in the
(111) plane, with subsequent surface reconstruction into a Pandey chain configuration.
Such a vacancy disk has a formation energy per vacancy of 1.46 eV, much lower that
any of the vacancy clusters or chains previously modelled. This indicates that the disk
will be a stable structure. The bandstructure of the disk displays the dispersed π and π ∗
states, as expected. These states give rise to a continuum bandgap absorption which is
very like the experimentally observed brown diamond absorption, although our method
does not take into account indirect transitions so the 5.5 eV onset is not reproduced.
The density of vacancies required to match experimental magnitude is ∼10 17 cm−3 . The
Raman frequency of the disk is calculated to be 1494 cm −1 , which compares well with
the 1540 cm−1 frequency observed in some brown diamonds.
If the (111) plane vacancy disk is taken to be the source of colour in brown diamond,
a method for removal of colour must also be proposed. There is a major difference in
the temperature required during HPHT treatment for natural and CVD material, which a
temperature of ∼2100 ◦ C required for the former but the lower temperature of ∼1500 ◦ C
needed for CVD material. This suggests that two different mechanisms are responsible.
In CVD diamond there is a large amount of hydrogen from the growth process which can
move to the disk surface, passivating the π bonds and removing any optical absorption.
In natural diamond there is nothing to check the growth of the disk until it will collapse into
a stacking fault bounded by a dislocation loop. This is predicted to happen at a radius of
˚
12 A.
Type Ia diamond contains nitrogen in aggregated forms such as the VN defect, NN
pair (A centre) and VN4 (B centre). The optical absorption of brown type Ia diamond
is characterised by additional absorption around 550 nm (2.25 eV), so the possibility of
nitrogen interacting with the disk to modify the absorption is investigated. The most stable
configuration for nitrogen to be in at the disk surface is for a nitrogen pair to separate into
a NCCN arrangement. The bandstructure of this system is perturbed by the nitrogen,
with an energy gap of 2 eV being opened up. This is attributed to the nitrogen being too
close to one another, which was investigated with larger cells. These studies proved that
the NCCN defects needed to be sufficiently separated to allow a chain of >10 carbon
CHAPTER 8. CONCLUSIONS
157
atoms to form. Analysis of the absorption induced by the nitrogen perturbation reveals
a peak at 2.3 eV, in excellent agreement with experiment. The total nitrogen density
required to produce an absorption magnitude matching experiment is ∼10 20 cm−3 .
8.4
Photoelastic Constants of Diamond
When strain is introduced into diamond, it changes the dielectric function both parallel and
perpendicular to the direction of strain. This change is characterised by the photoelastic
constants p11 , p12 and p44 . The constants are derived by inducing strain through variation
of the lattice constants in the cell, and recalculating the dielectric function along x, y and
z.
This procedure is first undertaken for bulk silicon, as there is more experimental data for
comparison. It is also an opportunity to test the efficiacy of using a rigid ‘scissor’ shift to
correct for LDA bandgap underestimation, as used in other abinitio work. It is found that
use of such a shift improves the values of the dielectric function somewhat, but that the
values of the photoelastic constants have their best match with experimental data when
no shift is used. Very good agreement for frequency dependence is also found.
The results from the silicon work give confidence in our values of the photoelastic constants for diamond. Experimental data for diamond is much more limited, and some
values are markedly different from others. Good agreement is found for absolute values
and frequency dependence with one particular set of results, the results also support the
assignment of a negative value to p11 + p12 .
Strain around a dislocation has been shown to induce birefringence in incident polarised
light. This birefringence can thus give an indication of the magnitude of the strain. The
value is calculated for a pure edge dislocation in diamond and the radial dependence of
the strain found to be Tr ∼ 1.81 × 10−16 /r2 .
CHAPTER 8. CONCLUSIONS
8.5
158
Continuation Work
This work has investigated several ideas relating to brown colour in diamond. The concept of π bonded carbon in vacancy disks is proposed as a colour source. However, there
are further aspects of this work to be investigated to gain a complete understanding, and
extensions to the work to give insight into new areas are also possible. Suggestions for
further topics are elucidated upon below.
• Colour removal in type Ia brown diamond In chapter 6 two methods for removal
of brown colour in type IIa diamond are proposed. In type Ia diamond, it is pro
posed that nitrogen is present at a disk surface, which may alter the colour removal
process. In particular the stability of the disk and the formation of nitrogenvacancy
defects are of interest. As nitrogen disrupts the chains of π bonded carbon atoms
and introduces further free electrons, it can be thought to lower the disk stability.
This needs to be checked against experimental data, and perhaps migration barrier
calculations performed. If the disks are disintegrating then it is likely that nitrogenvacancy defects may form. The stable configurations and expected defect densities
can be calculated for comparison to experimental data on treated type Ia brown
diamond.
• π bonded carbon in voids One problem with the vacancy disk model is that it
fails to account for the ∼400 ps lifetime seen in PAS. This lifetime extrapolates to
a cluster of ∼50 vacancies if roughly spherical, although could be as much as 100
vacancies. The narrow crosssection of the disk gives it a lifetime comparable to a
small vacancy cluster (∼14 vacancies). It is possible to create an void by removing atoms in an octahedral pattern, which leaves a vacancy cluster with surfaces in
the {111} set of surfaces. These surfaces can conceivably form π bonded chains,
which can be expected to have a similar broad absorption as a vacancy disk. Careful investigation of the formation and energetics of such a void are required, but
would require large cells in excess of 1000 atoms. This could only be approached
though a more approximate method such as DFTB (chapter 2).
• Vacancy disk generation The formation of vacancy disks (or indeed voids) requires the creation of large numbers of vacancies, probably in areas of high density
rather than an even distribution. There is also an observed link between brown
colour and strain, with the brown colour stronger in areas that have reduced strain.
CHAPTER 8. CONCLUSIONS
159
There is a suggestion that movement of dislocations (glide and climb) will reduce
strain and release vacancies. As dislocations are often grouped together, the released vacancies will be close to each other allowing them to form vacancy defects.
The plausibility of this proposal must be considered. The generation process in
CVD material will be different as it has low strain and dislocation concentration, so
another method must be postulated here.
• Charge transfer to disk Preliminary investigations on the vacancy disk model in
dicate the system energy can be lowered by transfer of an electron from a nearby
defect to the disk. This may have implications for diamond electronics, as it will
affect conduction and charge states on other defects. Investigations must be made
on the density of electrons that a disk may hold, and the required defect density
to support such an arrangement. It will be interesting to compare these results to
experimental EPR work which detects high acceptor levels in nitrogendoped CVD
diamond. The source of these acceptor levels is as yet unknown.
In summary, there remain several areas of potential investigation related to brown diamond. Abinitio modelling techniques are ideally placed to consolidate available experimental data, and help explain observations. As new developments in highperformance
computing occur, and software is improved, ever more realistic models can be investigated and new areas of research become accessible. Continued research into the brown
diamond problem, and diamond in general, is sure to lead to new and exciting discoveries
and applications.
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