The Modelling of Dislocations
in Semiconductor Crystals
Alexander Thorsten Blumenau
The Modelling of Dislocations
in Semiconductor Crystals
Dissertation
zur Erlangung des akademischen Grades
Doktor der Naturwissenschaften (Dr. rer. nat.)
vorgelegt dem
Department Physik der Fakult¨at f¨ur Naturwissenschaften an
der Universit¨at Paderborn
Alexander Thorsten Blumenau
Paderborn, 2002
Von der Fakult¨at f¨ur Naturwissenschaften der Universit¨at Paderborn als Dissertati-
on genehmigt.
Tag der Einreichung: 22. Oktober 2002
Tag der m¨undlichen Pr¨ufung: 13. Dezember 2002
Promotionskommission
Vorsitzender Prof. Dr. rer. nat. Artur Zrenner
Erstgutachter Prof. Dr. rer. nat. Thomas Frauenheim
Zweitgutachter Prof. Dr. rer. nat. Harald Overhof
Beisitzer PD Dr. rer. nat. Siegmund Greulich-Weber
Archiv
Elektronische Dissertationen und Habilitationen der Universit¨at Paderborn
http://www.ub.upb.de/volltext/ediss
Version: 13. Dezember 2002
Alexander Thorsten Blumenau, The Modelling of Dislocations in Semiconductor Crystals.
PhD Thesis (English), Department of Physics, Faculty of Science, University of Paderborn,
Germany (2002).
140 pages, 70 figures, 14 tables.
Abstract
This thesis studies dislocations in semiconductor crystals by means of theoretical modelling.
In particular, the work presented is focussed on dislocations in diamond and silicon carbide
and addresses technologically relevant effects.
A thorough description of dislocation effects covers a wide range of length scales and re-
quires a rather precise modelling of the electronic structure of a small core region at one end
to long range elastic effects at the other end of the scale. This can hardly be achieved with
one method only. Therefore, it is one of the main objectives of this work, to give a more com-
plete description of dislocations in semiconductor crystals by combining different theoretical methods:
Density functional theory (DFT) forms the basis for quantum mechanical atomistic calcula-
tions. Within DFT a pseudopotential approach is used to obtain electronic structures. A
more approximate and far less computationally expensive DFT-based tight-binding method
allows the prediction of core structures and energies embedded in larger models, represent-
ing a more extended region of the crystal. And finally, linear elasticity theory enables one to
describe long range elastic effects at almost no computational costs.
After introducing the methods to be used and presenting the basics of dislocation theory,
the dislocations of the {111}h110islip system in diamond serve as an example of how
the different methods are combined in this work. Straight perfect as well as dissociated
dislocations are investigated. The resulting low energy core geometries can be used as input
coordinates for the simulation of high-resolution transmission electron microscopy images,
and the calculation of the electronic structure allows the modelling of electron energy-loss
spectra. Both are compared with experimental data.
Furthermore, the thermally activated glide motion of Shockley partial dislocations in di-
amond is modelled in a process of kink formation and migration.
Summarising, the results obtained support an annealing scenario for natural brown di-
amond involving a transition from shuffle to glide character at the dislocation core, which
might explain the decolouring observed under high-pressure, high-temperature treatment
— a process still not understood, but of utmost interest to the international diamond trade.
In modern silicon carbide technology a mechanism of recombination-enhanced disloca-
tion glide is believed to play the key role in the observed disastrous degradation of bipolar
devices under forward bias. To shed some light on this mechanism, in this work the different
partial dislocations involved and their glide motion are modelled. The resulting electronic
structures and glide activation energies are then directly related to recent experimental ob-
servations.
Keywords
dislocations, diamond, silicon carbide, SiC, density functional theory, elasticity theory
PACS
61.72.Lk Linear defects: dislocations, disclinations
61.72.Bb Theories and models of crystal defects
61.72.-y Defects and impurities in crystals; microstructure
71.15.Nc Total energy and cohesive energy calculations
71.55.-i Impurity and defect levels
61.72.Ff Direct observation of dislocations and other defects
Alexander Thorsten Blumenau, Modellierung von Versetzungen in Halbleiterkristallen.
Dissertation (in englischer Sprache), Department Physik, Fakult¨at f¨ur Naturwissenschaften,
Universit¨at Paderborn (2002).
140 Seiten, 70 Abbildungen, 14 Tabellen.
Kurzfassung
Diese Dissertation befaßt sich mit der theoretischen Modellierung von Versetzungen in
Halbleiterkristallen. Ein besonderes Augenmerk gilt dabei den technologierelevanten Aus-
wirkungen von Versetzungen in Diamant und Siliziumkarbid.
Eine umfassende Beschreibung der durch Versetzungen hervorgerufenen Effekte er-
streckt sich ¨uber mehrere Gr¨oßenordnungen auf der L¨angenskala — angefangen von der
elektronischen Struktur einer kleinen Core-Region, bis hin zu langreichweitigen elastischen
Effekten. Da dies alles mit einer einzigen Methode nicht in jeweils ausreichender Genau-
igkeit erreicht werden kann, ist es ein Ziel dieser Arbeit, durch die Kombination unterschied-
licher theoretischer Methoden eine umfassendere Beschreibung von Versetzungen in Halbleiterkri-
stallen zu erreichen: Die Dichtefunktionaltheorie (DFT) bildet die Grundlage f¨ur atomisti-
sche quantenmechanische Rechnungen, in denen elektronische Strukturen in einem DFT-
Pseudopotentialansatz bestimmt werden. Im Gegensatz dazu erfolgt die Vorhersage von
atomaren Strukturen und deren Energien unter Verwendung einer DFT-basierten Tight-
Binding Methode. Diese ist zwar approximativer, aber deshalb auch rechentechnisch we-
sentlich effizienter und erlaubt somit die Einbettung von Versetzungen in Modelle mit ei-
ner gr¨oßeren Anzahl von Atomen. Die Beschreibung der langreichweitigen elastischen Ef-
fekte ist schließlich mit Hilfe linearer Elastizit¨atstheorie unter minimalem Rechenaufwand
m¨oglich.
Nach einer Einf¨uhrung in die verwendeten Methoden und die Grundlagen der Verset-
zungslehre wird anhand des {111}h110i-Versetzungssystems in Diamant die Kombinati-
on der verschiedenen Methoden demonstriert. Dabei werden sowohl geradlinige perfekte
als auch dissoziierte Versetzungen betrachtet. In beiden F¨allen erfolgt ein direkter Vergleich
mit experimentellen Daten: Die resultierenden Core-Geometrien k¨onnen zur Simulation von
transmissions-elektronenmikroskopischen Bildern verwendet werden und die Berechnung
der elektronischen Struktur erm¨oglicht die Modellierung von Elektronen-Energieverlust-
Spektren.
Desweiteren wird das thermisch aktivierte Gleiten von Shockley-Partialversetzungen in
Diamant durch einen Prozeß modelliert, dessen wesentliche Schritte durch die Bildung von
Knicken (“kinks”) in der Versetzunglinie und deren Migration gegeben sind.
Insgesamt unterst¨utzen die erhaltenen Ergebnisse schließlich ein Ausheil-Szenario f¨ur
braunen Naturdiamant, in dem Versetzungen mit “Shuffle”-Charakter in solche mit
“Glide”-Charakter ¨uberf¨uhrt werden. Ein solcher Mechanismus k¨onnte die beobachtete
Entf¨arbung brauner Diamanten unter extrem hohem Druck und hoher Temperatur erkl¨aren.
Dieser bisher unverstandene Prozeß ist von gr¨oßtem Interesse im internationalen Diamant-
handel.
Ein aktuelles Problem der modernen Siliziumkarbid-Technologie stellt die beobachtbare
katastrophale Degradation von unter Vorw¨artsspannung betriebenen bipolaren Bauelemen-
ten dar. Diese ist vermutlich auf ein sogenanntes rekombinationsverst¨arktes Versetzungs-
gleiten zur¨uckzuf¨uhren. Um diesen Mechanismus besser zu verstehen, werden in dieser
Arbeit die verschiedenen beteiligten Partialversetzungen und ihre Gleitbewegung model-
liert. Die resultierenden elektronischen Strukturen und Gleitaktivierungsenergien werden
im Anschluß direkt mit aktuellen experimentellen Beobachtungen in Verbindung gebracht.
Schlagw¨orter
Versetzungen, Diamant, Siliziumkarbid, SiC, Dichtefunktionaltheorie, Elastizit¨atstheorie
Contents
Contents v
List of Figures ................................... viii
List of Tables .................................... xi
Introduction 1
1 Modelling the Crystal: Theories and Methods 3
1.1 Density functional theory ......................... 3
1.1.1 Fundamental concepts and equations . . . . . . . . . . . . . . 4
1.1.2 The pseudopotential approach .................. 6
1.1.3 The density-functional based tight-binding approach . . . . . 8
1.2 Linear elasticity theory ........................... 12
1.2.1 Fundamental concepts and equations . . . . . . . . . . . . . . 12
1.2.2 The contracted matrix notation . . . . . . . . . . . . . . . . . . 13
1.2.3 The link to isotropic elasticity theory . . . . . . . . . . . . . . . 14
1.2.4 The Voigt average of elastic constants . . . . . . . . . . . . . . 15
2 An Introduction to Dislocation Theory 17
2.1 The Burgers vector ............................. 17
2.2 Edge and screw dislocations . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Straight dislocations in linear elasticity theory . . . . . . . . . . . . . 19
2.3.1 The elastic strain energy of a straight dislocation . . . . . . . 19
2.3.2 The elastic interaction between two straight dislocations . . . 20
2.4 Dislocation motion ............................. 21
2.5 The dissociation of dislocations . . . . . . . . . . . . . . . . . . . . . . 22
2.6 The influence of lattice periodicity: Kinks and jogs . . . . . . . . . . . 23
2.6.1 The periodic displacement potential of a crystal . . . . . . . . 23
2.6.2 Dislocation kinks .......................... 23
2.6.3 Dislocation jogs ........................... 25
v
vi
Contents
3 Dislocations in Tetrahedrally Bonded Semiconductors 27
3.1 Crystal slip systems and perfect dislocations . . . . . . . . . . . . . . 28
3.1.1 The main slip systems . . . . . . . . . . . . . . . . . . . . . . . 28
3.1.2 Glide and shuffle structures . . . . . . . . . . . . . . . . . . . . 29
3.2 Crystal stacking and partial dislocations . . . . . . . . . . . . . . . . . 31
3.3 Further classes of dislocations . . . . . . . . . . . . . . . . . . . . . . . 33
4 Dislocations in Diamond 35
4.1 Introduction and background . . . . . . . . . . . . . . . . . . . . . . . 35
4.1.1 Experimental evidence for dislocations in diamond . . . . . . 36
4.1.2 HPHT treatment — a threat to the international gem trade . . 36
4.1.3 Earlier theoretical work . . . . . . . . . . . . . . . . . . . . . . 37
4.2 The atomic scale modelling of dislocations . . . . . . . . . . . . . . . 38
4.2.1 The supercell-cluster hybrid as the model of choice . . . . . . 39
4.2.2 The elastic energy as a size-convergence criterion . . . . . . . 41
4.3 Core structures and energies . . . . . . . . . . . . . . . . . . . . . . . . 44
4.3.1 Core structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.3.2 Core energies ............................ 48
4.3.3 The screw dislocation — a special case . . . . . . . . . . . . . . 52
4.4 The dissociation of dislocations in diamond . . . . . . . . . . . . . . . 54
4.4.1 The equilibrium separation of partials . . . . . . . . . . . . . . 55
4.4.2 Modelling the first stages of dissociation atomistically . . . . 56
4.5 Kinked Shockley partials and dislocation glide . . . . . . . . . . . . . 58
4.5.1 Dislocation glide by kink formation and migration . . . . . . 58
4.5.2 The 90◦glide partial . . . . . . . . . . . . . . . . . . . . . . . . 59
4.5.3 The 30◦glide partial . . . . . . . . . . . . . . . . . . . . . . . . 61
4.6 Electron microscopy — a first link to experiments . . . . . . . . . . . 65
4.7 Electronic structure calculations and electron energy-loss . . . . . . . 69
4.7.1 The computational approach . . . . . . . . . . . . . . . . . . . 70
4.7.2 Calculated band structures and EEL spectra . . . . . . . . . . 71
4.7.3 Experimental EELS . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.7.4 Electronic structure calculations — conclusions . . . . . . . . 76
4.8 Summary and conclusions (diamond) . . . . . . . . . . . . . . . . . . 77
4.8.1 Selected results . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.8.2 The decolouring of brown diamonds by HPHT treatment . . 79
4.8.3 Outlook ............................... 80
Contents
vii
5 Dislocations in Silicon Carbide 81
5.1 Introduction and background ....................... 81
5.1.1 The different polytypes of SiC . . . . . . . . . . . . . . . . . . 82
5.1.2 The degradation of SiC PiN diodes under forward-bias . . . 84
5.1.3 Earlier theoretical work ...................... 85
5.2 Modelling bulk SiC — the elastic constants . . . . . . . . . . . . . . . 86
5.3 Straight Shockley partials in the basal plane . . . . . . . . . . . . . . 88
5.3.1 Core structures ........................... 88
5.3.2 Core energies ............................ 92
5.4 Dislocation glide motion .......................... 95
5.4.1 The glide motion of 90◦partial dislocations . . . . . . . . . . . 95
5.4.2 The glide motion of 30◦partial dislocations . . . . . . . . . . . 99
5.4.3 Dislocation glide motion — summary . . . . . . . . . . . . . . 101
5.5 Electronic structure calculations ...................... 102
5.6 Summary and conclusions (SiC) ...................... 108
5.6.1 Selected results ........................... 108
5.6.2 Recombination-enhanced dislocation glide . . . . . . . . . . . 109
5.6.3 Outlook — an alternative model . . . . . . . . . . . . . . . . . 110
6 Summary and Outlook 111
A Straight Dislocations in Elasticity Theory 115
A.1 Screw dislocations in isotropic media . . . . . . . . . . . . . . . . . . . 115
A.2 General straight dislocations in isotropic media . . . . . . . . . . . . . 117
A.3 Straight dislocations in anisotropic media . . . . . . . . . . . . . . . . 118
B Transmission Electron Microscopy 119
B.1 Conventional transmission electron microscopy . . . . . . . . . . . . 119
B.2 High-resolution transmission electron microscopy . . . . . . . . . . . 122
B.3 Alternative techniques ........................... 123
C Electron Energy-Loss Spectroscopy 125
C.1 The basic principles ............................. 125
C.2 The simulation of EELS ........................... 127
Bibliography 129
Acknowledgements 139
viii
List of Figures
List of Figures
1.1 Stress distribution on an infinitesimal volume element . . . . . . . . . 12
2.1 The Burgers integration circuit . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Edge and screw dislocation in a simple cubic lattice . . . . . . . . . . 18
2.3 The elastic strain energy of a dislocation . . . . . . . . . . . . . . . . . 20
2.4 Dislocation glide in a simple cubic lattice . . . . . . . . . . . . . . . . 21
2.5 Dislocation climb in a simple cubic lattice . . . . . . . . . . . . . . . . 21
2.6 The glide plane of a dislocation . . . . . . . . . . . . . . . . . . . . . . 21
2.7 Dislocation dissociation in a simple cubic lattice . . . . . . . . . . . . 22
2.8 A kinked dislocation in the glide plane . . . . . . . . . . . . . . . . . . 24
2.9 A dislocation kink pair ........................... 24
2.10 A dislocation jog .............................. 25
3.1 Unit cells of tetrahedrally bonded semiconductors . . . . . . . . . . . 28
3.2 The glide and the shuffle structure of the perfect 60◦dislocation . . . 30
3.3 The Stacking sequence in the cubic and the hexagonal lattice . . . . . 30
3.4 Intrinsic stacking faults in the cubic and hexagonal lattice . . . . . . . 30
3.5 The offset between bulk and faulted region . . . . . . . . . . . . . . . 31
3.6 The dissociation of the perfect 60◦dislocation . . . . . . . . . . . . . . 32
3.7 The dissociation of the perfect 60◦glide dislocation seen atomistically 33
4.1 A weak-beam image of a typical dislocation distribution in diamond 37
4.2 Schematic sketch of a dislocation in a cluster . . . . . . . . . . . . . . 39
4.3 Schematic sketch of a dislocation dipole in a supercell . . . . . . . . . 40
4.4 Schematic sketch of a dislocation in a supercell-cluster hybrid model 40
4.5 The core reconstruction of the 30◦glide partial . . . . . . . . . . . . . 42
4.6 The radial formation energy of the 30◦glide partial . . . . . . . . . . 43
4.7 Dislocation core structures of the {111}h110islip system in diamond 45
4.8 Dislocation core structures of the {111}h110islip system in diamond 46
4.9 The three unique types of the screw dislocation . . . . . . . . . . . . . 52
4.10 The glide screw dislocation . . . . . . . . . . . . . . . . . . . . . . . . 53
4.11 The competing energy contributions for dissociation into partials . . 55
4.12 The first stages of dissociation of the screw and the 60◦dislocation . 56
4.13 The dissociation energy of the screw and the 60◦dislocation . . . . . 57
4.14 Dislocation glide of partials by kink formation and migration . . . . 58
List of Figures
ix
4.15 A double-kink in the 90◦partial (SP) . . . . . . . . . . . . . . . . . . . 59
4.16 Kink migration at the 90◦glide partial . . . . . . . . . . . . . . . . . . 60
4.17 The parametrisation of a diffusing atom . . . . . . . . . . . . . . . . . 61
4.18 Double-kinks in the 30◦partial . . . . . . . . . . . . . . . . . . . . . . 62
4.19 Kink migration at the 30◦glide partial . . . . . . . . . . . . . . . . . . 63
4.20 Cross sectional HRTEM image of a dissociated 60◦dislocation . . . . 65
4.21 Simulated HRTEM image of the 30◦glide partial . . . . . . . . . . . . 66
4.22 Simulated HRTEM images of low energy dislocation core structures . 66
4.23 HRTEM image of an undissociated 60◦dislocation . . . . . . . . . . . 67
4.24 HRTEM image of a dissociated 60◦dislocation . . . . . . . . . . . . . 68
4.25 Band structures and simulated EEL spectra of the 60◦dislocation . . 72
4.26 Band structures and simulated EEL spectra of partial dislocations . . 73
4.27 Low-loss EELS and STEM on dislocations in CVD diamond . . . . . 75
5.1 The Stacking sequences of 3C-, 4H- and 6H-SiC . . . . . . . . . . . . 82
5.2 Weak-beam contrast experiments on an extended dislocation node . 83
5.3 A time sequence of EL images of partial dislocation motion in 4H-SiC 84
5.4 HRTEM images of partial dislocations in 4H- and 6H-SiC diodes . . 85
5.5 The relaxed core structures of the Shockley partials in 3C-SiC . . . . . 90
5.6 The relaxed core structure of a 30◦C Shockley partial in 2H-SiC . . . 91
5.7 The radial formation energy of the 90◦(DP) glide partial in 3C-SiC . 93
5.8 Kink migration at the 90◦(SP) glide partial . . . . . . . . . . . . . . . 96
5.9 The energies and barriers of dislocation glide of the 90◦(SP) partial . 97
5.10 The kink migration steps of left kinks at the 30◦glide partial . . . . . 98
5.11 The kink migration steps of right kinks at the 30◦glide partial . . . . 98
5.12 The energies and barriers of dislocation glide of the 30◦partial . . . . 100
5.13 The projected band structures of bulk and faulted SiC . . . . . . . . . 102
5.14 The projected band structures of glide partials in 3C-SiC . . . . . . . 103
5.15 The projected band structures of glide partials in 2H-SiC . . . . . . . 104
5.16 The Kohn-Sham eigenvalue spectra of kinked 30◦partials in SiC . . . 106
5.17 The Kohn-Sham eigenvalue spectra of kinked 90◦partials in SiC . . . 107
A.1 A screw dislocation oriented along the z-axis . . . . . . . . . . . . . . 116
B.1 Schematic sketch of a transmission electron microscope . . . . . . . . 120
B.2 Bragg diffraction in a TEM sample . . . . . . . . . . . . . . . . . . . . 121
B.3 Illustration of the invisibility criterion for an edge dislocation . . . . 121
x
List of Figures
B.4 Schematic X-ray topography setup . . . . . . . . . . . . . . . . . . . . 123
B.5 Sketch of an etched surface . . . . . . . . . . . . . . . . . . . . . . . . 124
C.1 A simplified EELS setup and EELS excitations . . . . . . . . . . . . . 126
C.2 Theoretical and experimental EELS on diamond . . . . . . . . . . . . 128
List of Tables
xi
List of Tables
4.1 Some elastic properties of diamond . . . . . . . . . . . . . . . . . . . . 38
4.2 The calculated intrinsic stacking fault energy in diamond . . . . . . . 41
4.3 The calculated energy factors, core radii and core energies . . . . . . 49
4.4 A comparison of DFTB results with independent calculations . . . . 51
4.5 The calculated equilibrium stacking fault widths . . . . . . . . . . . 55
5.1 The lattice constants and the band gap of different SiC polytypes . . 83
5.2 Elastic properties of 3C-SiC . . . . . . . . . . . . . . . . . . . . . . . . 87
5.3 Elastic properties of hexagonal SiC . . . . . . . . . . . . . . . . . . . . 87
5.4 Reconstruction bond lengths of the Shockley partials in 3C-SiC . . . . 89
5.5 Differences in core bond lengths of the Shockley partial dislocations 91
5.6 The calculated energy factors for 30◦and 90◦partials . . . . . . . . . 92
5.7 The calculated energy factors, core radii and core energies . . . . . . 93
5.8 Kink formation energies and migration barriers for the 90◦partials . 97
5.9 Kink formation energies and migration barriers for the 30◦partials . 99
Introduction
Real semiconductor crystals contain numerous defects which perturb the perfect lat-
tice periodicity and influence its electrical and mechanical properties. Thus, these
defects play an important role in semiconductor technology: Some defects have use-
ful effects and are crucial for device design (e.g. doping to achieve n- or p-type
conductivity), while others have a disastrous or destructive effect on devices and
their performance. With regard to dimensionality, one has to distinguish between
point-like defects (lattice vacancies, interstitial atoms or impurities), line defects (e.g.
dislocations), planar defects (e.g. stacking faults or domain boundaries) and volume
defects (e.g. voids or precipitates).
Since experiments alone often cannot yield information about the exact origin of
defect-related effects or about how to suppress unwanted effects or promote those
which are desirable, we need theoretical models and calculations to interpret the
experimental data.
This thesis studies the atomic structures, energies and electronic properties of dis-
locations in semiconductor crystals by means of theoretical calculations and mod-
elling. Whenever possible (and valuable) results are compared with experimental
data. The prediction of the mechanical properties of dislocated crystals, however, is
clearly outside the scope of this work.
Objectives
In the current literature on dislocations in semiconductors it is not uncommon to
find investigations to be restricted to only one or two aspects of the defect. This
very often stems from the wide range of length scales and precisions needed for
a thorough description. The range spans from the very precise modelling of the
electronic structure of a rather small core region at one end to the calculation of long
range elastic lattice effects at the other end of the scale. For the former, quantum
mechanical approaches are crucial — the latter however barely needs any quantum
mechanics.
This wide range cannot be covered by one method only. Therefore it is one of the
main objectives of this work, to give a more complete description of dislocations in semi-
conductor crystals by means of combining different theoretical methods and discussing their
overlap: Density functional theory (DFT) forms the basis for quantum mechanical
electronic and atomistic calculations. Within DFT a pseudopotential approach (im-
plemented with a localised basis in the AIMPRO code [17]) is used to obtain precise
electronic structures. A more approximate and far less computationally expensive
1
2
Introduction
DFT-based tight-binding method (DFTB [18,19]) allows the prediction of core struc-
tures and energies embedded in larger models, representing a more extended region
of the crystal. Finally, linear elasticity theory enables us to describe long range elas-
tic effects at almost no computational costs. In the course of this work, the afore
mentioned combination of methods has proven very effective.
Outline
Chapter 1introduces the different theoretical methods used in this work to model
defects and dislocations in particular. Here we have to distinguish between DFT-
based methods, namely the local density pseudopotential approach (AIMPRO) as
well as DFT-based tight-binding (DFTB), and elasticity theory as a continuum the-
ory. As mentioned, these methods represent different levels of approximation to the
description of real semiconductor crystals and their defects.
Some readers might not be very familiar with the theory of dislocations. Thus, to
provide the necessary background, the basic concepts of dislocation theory are in-
troduced in Chapter 2. This includes selected results from linear elasticity theory
which either help in the general understanding or will be useful in later chapters.
To prepare for the subsequent examples, Chapter 3gives an overview of the low
energy types of dislocations in tetrahedrally bonded semiconductors.
With Chapter 4the introduction of methods and dislocation basics is followed by the
first example: Dislocations in diamond. In terms of methodology this is surely the
central and most important chapter, as it shows in detail how the three theoretical
methods mentioned above are combined to describe the different aspects of a dis-
location on different length scales. With the {111}h110islip system in diamond as
an example, the core structure, core energy, elastic long range energy terms, the dis-
sociation into partial dislocations and dislocation glide motion are modelled. The
resulting geometries can be used as input coordinates for the simulation of high-
resolution transmission electron microscopy (HRTEM) images and the calculation of
the electronic structure allows the modelling of electron energy-loss spectra (EELS)
which can be compared with experimental data. The results obtained support a spe-
cific scenario explaining the high-pressure, high-temperature decolouring of brown
diamond.
After this very detailed example of a homonuclear system Chapter 5discusses dis-
locations in the compound semiconductor silicon carbide. In particular the role of
Shockley partial dislocations and stacking faults in the technology-relevant degra-
dation of silicon carbide PiN-diodes is examined.
The appendices mainly give additional considerations and calculations that would
disturb the flow of the main text if included there. In Appendix Alinear elastic-
ity theory is applied explicitly to some simple dislocation-related problems. This
should give the reader a feeling for the general procedure. The Appendices Band C
are rather specific as they contain a brief presentation of the important experimental
techniques — especially (HR)TEM and EELS — which yield information about the
atomic and the electronic structure of dislocations in semiconductors and thus are
directly related to the properties calculated in this work.
Chapter 1
Modelling the Crystal: Theories
and Methods
This chapter will give a brief overview of the two DFT-based methods used in this
work on the one hand, and linear elasticity theory on the other. As will be shown
from Chapter 4onwards, the combination of the three methods allows us to cover
the description of dislocation properties on the whole length scale — ranging from
below interatomic distances to the continuum elastic limit.
1.1 Density functional theory
To model a semiconductor crystal on the atomic scale, a wide variety of classical
methods is known — ranging from simple ball and spring models to more so-
phisticated empirical interatomic potentials with two- or even three-body contribu-
tions [20,21,22,23]. These methods are more or less successful in describing perfect
and strained crystals of tetrahedrally bonded semiconductors. However, when it
comes to defects with considerable deviations from the standard bonding configu-
ration (full four-fold coordination of atoms and tetrahedral bond angles), then quan-
tum mechanical effects, or effects involving charge transfer between atoms, might
dominate. Very often this is the case in the very core region of point or line defects
in semiconductors. Hence here the use of quantum mechanics is essential.
To describe a crystal in terms of non-relativistic and stationary quantum mechan-
ics1with no further approximations would mean to solve the time independent
Schr¨odinger equation for Matomic nuclei and Nelectrons. This is an impossible
task if one considers the corresponding antisymmetric wavefunction, which de-
pends on the 3M+3Ncoordinates of all atoms and electrons. A first reduction of
complexity is given by the Born-Oppenheimer approximation [24], where the motion of
electrons and nuclei are decoupled — inspired by their large difference in masses.
The remaining problem is now the determination of the electronic wavefunction for
a given configuration of nuclei. Since the modelling of defects involves tens or even
1In this work relativistic effects are not taken into consideration explicitly, even spin-related effects
are mostly ignored, as they play no crucial role in the discussed problems.
3
4
Chapter 1. Modelling the Crystal: Theories and Methods
hundreds of atoms, the full many-body wavefunction is still by far too complex to
be handled. The method of choice to further reduce the number of parameters is
given with density functional theory (DFT), where the electron density is the central
variable. This implies a reduction from the 3Ncoordinates of the electron wave-
function to only three coordinates (if we ignore spin-polarisation), as shown in the
next section. The reader interested in a more detailed and more exact description
may read the original work by Hohenberg and Kohn [25] and Kohn and Sham [26]
or the rather mathematical treatment of Lieb [27], to give just a few examples.
1.1.1 Fundamental concepts and equations
In the investigation of defects in semiconductors, it is one central goal to identify
the stable structures of the defect: Structures where the forces between atoms are in
equilibrium. To achieve this, one needs to calculate the total energy of the model
representing the defect. An equilibrium of forces is then given as a minimum on the
energy surface.
The measurable quantities of a quantum mechanical ensemble are determined via a
state function, which is a solution of the Schr¨odinger equation, e. g. the total energy
Eof a many-electron system is determined by its electronic wavefunction
:
E[
] =
|b
H|
(1.1)
Here b
His the corresponding Hamiltonian and the right hand side represents a scalar
product in Hilbert space. It is the basic idea of DFT, to introduce the electron density
n(r)as the central variable, and determine the measurable quantities in terms of this
density:
n(r) =
|X
i
δ
(r−r0
i)|
(1.2)
As Hohenberg and Kohn have shown, the total energy of the (non-degenerate)
ground state of a quantum mechanical ensemble of electrons is a unique functional
of its electron density [25]:
E=E[n(r)] (1.3)
Therefore the mere knowledge of n(r)is sufficient to determine the corresponding
total energy.
Now we will turn towards the explicit form of E[n(r)]. We separate E[n(r)] into
different components as follows2:
E[n] = T[n] + 1
2ZZ n(r)n(r0)
|r−r0|d3r0d3r+Zn(r)vext(r)d3r(1.4)
Here the second term denotes the classical Coulomb-interaction of all electrons and
the last term gives the energy arising from the external potential vext(r). The latter
2Quantities are given in atomic units, and the energy given in Hartree.
1.1. Density functional theory
5
in addition depends on the atomic coordinates. Unfortunately the kinetic contribu-
tion T[n]is generally unknown. However, in the specific case of Nnon-interacting
electrons a simple solution can be given:
T0[n] =
N
X
i=1Z
φ
∗
i(r)−∇2
2
φ
i(r)d3r(1.5)
Here the density n(r)has been expanded into normalised and orthogonal one-
electron wavefunctions
φ
i(r):
n(r) =
N
X
i=1|
φ
i(r)|2(1.6)
To make use of Eq. (1.5), Kohn and Sham suggested to expand n(r)accordingly to
Eq. (1.6) for the case of interacting electrons also [26]. This approximation means
that for the system of interacting electrons the
φ
i(r)are not true one-electron wave-
functions. In the following we shall call them Kohn-Sham orbitals. To still include
the many-electron effects in T[n], we have to add the so called exchange-correlation
functional Exc[n]3:
T[n] = T0[n] + Exc[n](1.7)
A variation of E[n]in Eq. (1.4) with
δφ
∗
i(r)and subject to the normalisation con-
straint delivers the Kohn-Sham orbitals
φ
i(r), which yield a minimal total energy.
We obtain the Kohn-Sham equations [26]:
−∇2
2+veff[n(r)]
|{z }
=b
H[n]
φ
i(r) =
ε
i
φ
i(r)(1.8)
with veff[n(r)] :=vext(r) + Zn(r0)
|r−r0|d3r0+
δ
Exc[n(r)]
δ
n(r)(1.9)
Here the
ε
iare merely the Lagrange parameters of the normalisation constraint, and
not one-electron energies4. Correspondingly the Kohn-Sham equations (Eq. (1.8))
are not independent one-electron equations — they are coupled via the effective
potential veff[n(r)], which depends on all (occupied)
φ
i(r), and hence Eq. (1.8) can
only be solved self-consistently.
The last term of the effective potential (Eq. (1.9)) is the exchange-correlation potential
vxc[n(r)]. Generally this potential is unknown. For special cases, however, like the
homogeneous electron gas at high (or low) density, the explicit form has been eval-
uated. Assuming now that the exchange-correlation contribution to the total energy
3This formalism is not used uniformly in the literature: Exc[n]is sometimes excluded from T[n]
and included in Eq. (1.4) directly.
4Only in special cases, when the removal of electron jdoes not influence the remaining
ε
i, then one
may interpret
ε
jas the one-electron energy of electron j(Koopmans’ theorem [28]).
6
Chapter 1. Modelling the Crystal: Theories and Methods
is varying only a little, one can use approximations for vxc based on a homogeneous
electron gas. The approximation used in this work is the well-approved local den-
sity approximation (LDA) of density functional theory. In this approximation, if
expressing the exchange-correlation contribution via an integral, the integrand
xc
depends only locally on the density n(r):
Exc[n] = Zn(r)
xc[n(r)] d3r≈Zn(r)
LDA
xc (n(r)) d3r(1.10)
The main drawback of LDA is the underestimation of the forbidden electronic band
gap of semiconductor crystals. Along with this, density functional theory describes
the ground state in a given Hilbert space only. Therefore excited states can usually
not be described properly anyway.
For a given spatial arrangement of nuclei one can now determine the ground
state electron density n(r)within the Born-Oppenheimer approximation by solv-
ing Eq. (1.8) self-consistently. Therefore the external potential vext(r)in Eq. (1.9) is
given as the electrostatic potential of the nuclei. The total energy of all electrons in
the potential of all nuclei can then be evaluated via Eq. (1.4). Adding the Coulomb
repulsion Enuc of the nuclei yields the total energy of the system consisting of elec-
trons and nuclei:
Etot[n] = E[n] + Enuc Rj (1.11)
with Enuc Rj=1
2
M
X
i,j6=i
ZiZj
Ri−Rj(1.12)
Ziis the atomic number of atom iand Riits coordinates. With this energy at hand,
one can search for the equilibrium of forces — the local minima of the energy sur-
face. This structural optimisation is usually done with the help of algorithms in-
volving atomic forces. The latter can be obtained via the Hellmann-Feynman theo-
rem [29,30]. For atom-centred basis sets the so called Pulay corrections have to be
applied [31].
The next sections will give two examples for implementations of DFT. Both involve
additional approximations to reduce the computational effort.
1.1.2 The pseudopotential approach
In this section, the pseudopotential approach to DFT — as implemented in the AIM-
PRO computer code — will be introduced briefly. This introduction avoids dis-
cussing any of the arising problems like singularities in potentials, relativistic effects
for heavier atoms and the transferability of pseudopotentials. Further details can be
found in more specific literature on the AIMPRO code [17,32] and the approach in
general [33,34,35].
In practice, the bonding between atoms in solids is predominantly governed by
their valence electrons. The core electrons of the closed inner shells basically behave
like in an isolated atom and create a screening effective potential. Thus it seems a
1.1. Density functional theory
7
promising idea to only consider the valence electrons when self-consistently calcu-
lating the electron density. The remaining core electrons as well as the electrostatic
potential of the nuclei are included in a new effective potential — the so called pseu-
dopotential vps(r), which represents the potential created by all ions. Then the overall
effective potential in Eq. (1.8) becomes:
veff[nv(r)] = vps(r) + Znv(r0)
|r−r0|d3r0+vxc[nv(r)] (1.13)
Here nv(r)is the valence electron density. The pseudopotential vps(r)is given as the
superposition of all single ionic pseudopotentials:
vps(r) =
M
X
j=1
vps
j(r−Rj)(1.14)
Here the Rjare the positions of the respective nuclei. To obtain the ionic pseudopo-
tential vps
j(r)one starts with the effective all-electron potential veff
j(r)of a neutral
atom. Stripping off the valence electron contributions yields:
vps
j(r) = veff
j(r)−Znv
j(r0)
|r−r0|d3r0−vxc[nv
j(r)] (1.15)
There are several recipes how to construct pseudopotentials in detail. In this work
the norm-conserving pseudopotentials of Bachelet, Hamann and Schl¨uter [35] are
used. Assuming a frozen core, the same pseudopotential can be used for every spatial
distribution of atoms and different charge states.
The advantages of the pseudopotential approach are obvious: Comparing all-
electron energies of similar systems might lead to relatively large errors, since two
large numbers are subtracted. The problem is resolved if only the valence electrons
contribute to the difference in energy. Another crucial advantage is the reduction
of computational complexity. The number of electrons, and with this the number of
orthogonal wavefunctions, to be included explicitly in the self-consistent calculation
of the total energy is reduced dramatically if only valence electrons are considered.
Hence the modelling of larger systems becomes possible5.
The choice of basis set: For a practical implementation of this approach, one has
to decide for a basis in terms of which the Kohn-Sham orbitals are expanded. In
AIMPRO the choice is a basis set of Cartesian Gaussian orbitals centred at R
ν
:
ψν
(r) = (x1−R
ν
1)n1
ν
(x2−R
ν
2)n2
ν
(x3−R
ν
3)n3
ν
e−a
ν
(r−R
ν
)2(1.16)
With specially selected and real ni
ν
it is possible to describe atom-centred s-, p- and
d-like functions. Expanding the Kohn-Sham orbitals
φ
i(r)into Cartesian Gaussian
orbitals yields:
φ
i(r) = X
ν
C
ν
i
ψν
(r)(1.17)
5At the time of writing this work, AIMPRO on parallel supercomputers was capable of conveniently
treating models consisting of a few hundred atoms.
8
Chapter 1. Modelling the Crystal: Theories and Methods
Inserting the expansions of the Kohn-Sham orbitals in Eq. (1.8), followed by left-side
multiplication with
ψν
(r)and integration over all space, reduces Eq. (1.8) to a set of
algebraic secular equations:
X
ν
C
ν
i(H
µν
−
ε
iS
µν
)=0 (1.18)
Here the C
ν
iare the expansion coefficients and H
µν
and S
µν
the Hamiltonian and
overlap matrix respectively:
H
µν
=Z
ψ
∗
µ
(r)−∇2
2+veff[nv(r)]
ψν
(r)d3r(1.19)
S
µν
=Z
ψ
∗
µ
(r)
ψν
(r)d3r(1.20)
The strong point of Gaussian basis functions is their localisation. Therefore consid-
erably fewer basis functions are necessary to approximatively describe a localised
wavefunction. Dislocations (line defects) usually yield wavefunctions localised in at
least two dimensions. Since most models used in this work include a great amount
of vacuum, the use of a localised basis is almost inevitable.
1.1.3 The density-functional based tight-binding approach
We will now introduce the Density-functional based tight-binding method (DFTB)
— the second approach used in this work to approximately solve the Kohn-Sham
equations (1.8). Just like in the last section, the description will be introductory
and lack many details. For further insight the original work of Seifert et al. [36]
and Porezag et al. [18,37] or the review in [19] are recommended. A compact but
nevertheless detailed overview including a sketch of the historical development can
be found in [38].
To understand the basic idea of the DFTB method, one best starts with the explicit
form of the total energy in DFT after introducing the Kohn-Sham orbitals. This is
obtained easily by inserting Eq. (1.7) and (1.5) in Eq. (1.4):
E[n] =
occ
X
i
niZ
φ
∗
i(r)−∇2
2+1
2Zn(r0)
|r−r0|d3r0+vext(r)
φ
i(r)d3r+Exc[n](1.21)
Here we additionally introduced normalised occupation numbers nifollowing
Janak [39] :
n(r) =
occ
X
i
ni|
φ
i(r)|2;N=
occ
X
i
ni(1.22)
These occupation numbers nicontain the information how many electrons occupy
the respective Kohn-Sham orbital. In a spin-restricted formalism as used here, the
maximum occupation per orbital is 2.
1.1. Density functional theory
9
The idea is now to simplify Eq. (1.21) by applying a Taylor-expansion around a
reference density n0(r)and neglecting higher order contributions. Introducing
n(r) = n(r)−n0(r)we obtain (Foulkes and Haydock [40]):
E[n] =
occ
X
i
niZ
φ
∗
i(r)b
H[n0]
φ
i(r)d3r−1
2ZZ n0(r)n0(r0)
|r−r0|d3r0d3r
+Exc[n0]−Zvxc[n0(r)]n0(r)d3r
+1
2ZZ 1
|r−r0|+
δ
2Exc[n]
δ
n2n0
n(r)
n(r0)d3r0d3r+O(
n3)(1.23)
b
H[n]was defined in Eq. (1.8). The first and second lines give the zero-th order terms
in
nand the third line gives the second and third order. All first order contributions
cancel.
Choosing a localised (atom-centred) set of basis functions, the second to fifth terms
in Eq. (1.23) can be expressed as a sum of two-centre integrals6. Subsuming those
terms as E2 cent and neglecting three-centre and higher-order contributions yields
the total energy in DFTB:
E[n]≈
occ
X
i
niZ
φ
∗
i(r)b
H[n0]
φ
i(r)d3r+E2 cent =EDFTB (1.24)
DFT calculations show, that the density of an atom surrounded by other atoms in a
molecule or a solid is slightly compressed compared to the density of a free atom.
Hence in DFTB the starting density n0(r)is expressed as a superposition of weakly
confined neutral atoms, so called pseudo-atoms7. In all DFTB implementations used
in this work, the respective pseudo-atomic wavefunctions are represented by Slater-
type orbitals
ϕν
(r).
Expanding the Kohn-Sham orbitals of Eq. (1.24) into the atom-centred Slater-type
orbitals
ϕν
(r)yields (compare Eq. (1.17)):
φ
i(r) = X
ν
C
ν
i
ϕν
(r)(1.25)
This resembles an LCAO basis (linear combination of atomic orbitals ). Further we
approximate the two-centre contributions by the sum of short-ranged repulsive pair
potentials vij
rep:
E2 cent ≈1
2X
i6=j
vij
rep Ri−Rj=Erep ({Ri})(1.26)
6For the exchange-correlation terms this only holds approximately.
7These pseudo-atom densities result from self-consistent DFT calculations with an additional weak
parabolic constriction potential.
Just as a note for those familiar with the internal details of the DFTB method: In this work the diamond
C–C interaction includes the superposition of the potentials of the pseudo-atoms. These potentials are
compressed by an additional term (r/r0)2with a compression radius of r0=1.42 ˚
A. In the case of
silicon carbide, however, it is the superposition of the electron densities of the pseudo-atoms which is
used. Here the C compression radii are rwav
0=2.7 ˚
A and rden
0=7.0 ˚
A for the wavefunction and the
density respectively. For Si the chosen radii are rwav
0=3.3 ˚
A and rden
0=6.7 ˚
A.
10
Chapter 1. Modelling the Crystal: Theories and Methods
Inserting Eq. (1.25) and (1.26) in (1.24) gives the DFTB energy in terms of the expan-
sion coefficients C
ν
i. Subsequent variation with respect to the expansion coefficients
and subject to normalisation yields the DFTB secular equations (compare Eq. (1.18)):
X
ν
C
ν
iH0
µν
−
ε
iS
µν
=0 (1.27)
If R
ν
denotes the atom site the orbital is centred at, then the Hamiltonian and the
overlap matrix are given as:
H
µν
=
ε
atom
µ
:
µ
=
ν
Z
ϕ
∗
µ
(r)b
H[n0]
ϕν
(r)d3r:R
ν
6=R
µ
0 : else
(1.28)
S
µν
=Z
ϕ
∗
µ
(r)
ϕν
(r)d3r(1.29)
Notice, that the diagonal elements of H
µν
are taken to be atomic orbital energies of a
free atom, to ensure the right energies for dissociated atoms. Once the pseudo-atom
orbitals have been calculated self-consistently in DFT, the non-diagonal elements of
the Hamiltonian H
µν
and the overlap matrix S
µν
can be tabulated as a function of
distance between the two centres. Thus both matrices can be calculated in advance.
Once these tables have been generated, Eq. (1.27) can be solved straight away and
non self-consistently for any given coordinates of the nuclei. The DFTB energy then
becomes:
EDFTB =
occ
X
i
ni
ε
i+Erep ({Ri})(1.30)
Apart from the repulsive energy, the total energy is now determined. Assuming
approximate transferability of the repulsive potentials vij
rep(|Ri−Rj|), they are ob-
tained in comparison with self-consistent DFT calculations for selected reference
systems (small molecules or infinite crystals): Varying one distance |Ri−Rj|the re-
spective repulsive potential is tabulated just like the matrix elements H
µν
and S
µν
.
In practice, the repulsive pair potentials also contain the respective ion-ion repulsion
as given in Eq. (1.12). Further, the compensating effect of the repulsive potential al-
lows one to use a minimal basis of Slater-type orbitals, reducing matrix size and thus
speeding up all calculations.
As has been just shown, all DFT integrals can be calculated in advance. In the cal-
culation of the eigenvalues
ε
iand the total energy this saves a great deal of compu-
tational effort and speeds up the calculation dramatically. Such a speed up allows
a structural relaxation of systems containing more atoms than all-electron DFT or
even pseudopotential methods8. This advantage will prove crucial when it comes
to comparing DFT-based atomistic calculations with the results of elasticity theory
in Chapter 4— a task, where models containing a relatively large number of atoms
are needed.
8At the time of writing this work, DFTB on workstations was capable of conveniently relaxing
models consisting of more than 700 atoms — more than in the DFT-pseudopotential approach on
parallel supercomputers.
1.1. Density functional theory
11
SCC-DFTB: For some systems with strong ionic bonding character the approxi-
mations of standard DFTB might fail. Especially the second order term in Eq. (1.23)
becomes too large to be transferable and simply tabulated within the repulsive po-
tential. The term has then to be treated self-consistently. This has been successfully
achieved in the self-consistent-charge extension of DFTB (SCC-DFTB), where the
charge transfer through
n(r)is approximated by spherical atom-centred charge
fluctuations. For further details see [19].
In this work however, the self-consistent-charge extension is not used. Test calcula-
tions have shown, that even in silicon carbide the non-SCC treatment is sufficient to
describe at least the atomic structure and energy of the dislocations investigated. As
all subsequent electronic structure calculations are performed within AIMPRO, the
DFTB calculations can be restricted to the non-SCC formalism. Given the size of the
involved models (∼500 atoms), this reduces the computational effort considerably.
12
Chapter 1. Modelling the Crystal: Theories and Methods
1.2 Linear elasticity theory
While the preceding sections presented methods which allow an atomistic descrip-
tion of a semiconductor crystal as a system consisting of atoms and electrons, we
will now turn towards continuum theory: Elasticity theory basically describes the
relation between strain and stress fields, displacements and elastic strain energy in
a continuous and elastic medium. This section starts with an brief introduction of
the fundamental equations and definitions of elasticity theory. After explaining the
contracted matrix notation and discussing the isotropic case, the Voigt average for
anisotropic crystals is presented, giving merely the results and skipping any details.
We basically follow the notation of Hirth and Lothe [41], who give a good review
of those aspects of elasticity theory which are essential in dislocation theory. More
complete texts on the whole subject are those of Love [42], Sokolnikoff [43], Nye [44]
and Landau and Lifshitz [45].
1.2.1 Fundamental concepts and equations
Elasticity theory is a continuum theory: It describes an elastic solid by means of a
continuous medium. Let us now assume a homogeneous medium, in other words,
2
31
σ
23
σ
32
1
σ
22
σ
12
σ
21
σ
11
σ
σ13
x3
x
x
σ33
Figure 1.1: Stress distribution on an in-
finitesimal volume element.
the mass density %(r)is constant in the case
of zero strain. Let r= (x1,x2,x3)be a vec-
tor in orthogonal Cartesian coordinates and
σ
ij(r)be the stresses in the medium. As
shown in Fig. 1.1 for an infinitesimal vol-
ume element,
σ
ij(r)is the force in xidirec-
tion per unit area on a plane normal to the xj
direction. In the following the dependence
of variables on rwill not be given explic-
itly. In mechanical equilibrium there is no net
torque on the element and if we also rule out
internal torques, then
σ
ij is symmetric:
σ
ij =
σ
ji (1.31)
Further, there is also no net force on the element. We obtain the equilibrium equations
of classical elasticity written with the Einstein convention:
∂
σ
ij
∂xj
+fi=0∀i(1.32)
Here fis the body force per unit volume. Such forces can arise, for example, from a
charge distribution in the presence of an electric field.
A body which is acted upon by stress, deforms. Let now ube the displacement field
after deformation, then we can define the strains
ε
ij and the stiff rotations
ω
ij:
ε
ij =1
2∂ui
∂xj
+∂uj
∂xi(1.33)
1.2. Linear elasticity theory
13
ω
ij =∂ui
∂xj−∂uj
∂xi
(1.34)
Both Eq. (1.33) and (1.34) are restricted to first order terms in the derivatives, assum-
ing only small displacements and stresses. The factor 1/2 in Eq. (1.33) is convention-
ally used for transformation purposes. This differs from the definition of the shear
strain
γ
in engineering. Shear strain is given, when i6=j:
γ
ij =2
ε
ij i6=j(1.35)
In linear theory (small distortions ∂ui/∂xj)Hooke’s law applies — we can assume
a linear dependence of the stress on the deformation. Since the stiff rotations
ω
ij
cannot give rise to stresses, we obtain:
σ
ij =cijkl
ε
kl (1.36)
The cijkl are called elastic constants. Considering now the reversible deformation of a
volume element under differential strain, one can determine the strain energy density
by integration:
es=1
2cijkl
ε
ij
ε
kl (1.37)
In principle, with Eq. (1.36) and (1.37), we are now able to calculate the elastic strain
energy for a given strain. However, it should be noted, that here we did not discrim-
inate between a fixed external and an embedded coordinate system, which deforms
with the medium: Only the terms common to both descriptions are retained. For
small displacements and stresses this is sufficient, for strongly distorted parts of a
crystal however, the results of linear theory have to be questioned.
1.2.2 The contracted matrix notation
Let us turn back to Eq. (1.36) and the elastic constants. The symmetry of
ε
ij imposes
several symmetries on the tensor of the cijkl indicating there are only 21 independent
elastic constants among the 81 elements of the tensor. Hence in matrix notation we
can write the tensor as a symmetric (9 ×9) matrix representation {cijkl }with 21
independent entries. We obtain for Eq. (1.36):
σ
11
σ
22
σ
33
σ
23
σ
31
σ
12
σ
32
σ
13
σ
21
=
c11 c12 c13 c14 c15 c16 c14 c15 c16
c22 c23 c24 c25 c26 c24 c25 c26
c33 c34 c35 c36 c34 c35 c36
c44 c45 c46 c44 c45 c46
c55 c56 c45 c55 c56
c66 c46 c56 c66
c44 c45 c46
c55 c56
c66
·
ε
11
ε
22
ε
33
ε
23
ε
31
ε
12
ε
32
ε
13
ε
21
(1.38)
14
Chapter 1. Modelling the Crystal: Theories and Methods
In Eq. (1.38) we abbreviated the first two and the last two indices of the cijkl into
single indices running from 1 to 9. Thus, the indices mand nof the cmn correspond
to the index pairs ij and kl of the cijkl as follows:
ij or kl : 11 22 33 23 31 12 32 13 21
mor n: 1 2 3 4 5 6 7 8 9 (1.39)
Since the indices 7, 8, 9 correspond to 4, 5, 6 respectively, we can conveniently write
Eq. (1.38) in a contracted (6 ×6) matrix notation as:
σ
11
σ
22
σ
33
σ
23
σ
31
σ
12
=
c11 c12 c13 c14 c15 c16
c22 c23 c24 c25 c26
c33 c34 c35 c36
c44 c45 c46
c55 c56
c66
·
ε
11
ε
22
ε
33
γ
23
γ
31
γ
12
(1.40)
Notice that here we have to use the shear strains
γ
ij as given in Eq. (1.35) instead of
the
ε
ij. This allows a symmetric matrix.
The results so far all presumed an anisotropic medium with no symmetry. In crys-
tals however, symmetry can usually help to reduce the number of independent elas-
tic constants dramatically. For cubic symmetry the cijkl have to be invariant to 90◦
rotations. A comparison of tensor entries yields:
{cnm}cubic =
c11 c12 c12 000
c11 c12 000
c11 000
c44 0 0
c44 0
c44
(1.41)
Similarly we obtain for hexagonal symmetry with the third axis perpendicular to
the basal plane [44]:
{cnm}hexagonal =
c11 c12 c13 0 0 0
c11 c13 0 0 0
c33 0 0 0
c44 0 0
c44 0
1
2(c11 −c12)
(1.42)
1.2.3 The link to isotropic elasticity theory
After the two examples of cubic and hexagonal symmetry, we will now discuss the
case of an isotropic medium. In such a medium all possible directions are equiva-
lent, so that the {cijkl}are invariant to arbitrary rotations. A short calculation yields
as a reduction from cubic symmetry in Eq. (1.41) to the isotropic case:
c44 =1
2(c11 −c12)(1.43)
1.2. Linear elasticity theory
15
With this only two of the three constants are independent — the shear modulus
µ
and
the Lam´e constant
λ
:
µ
=c44 =1
2(c11 −c12);
λ
=c12 (1.44)
With these, Eq. (1.40) writes simply as:
σ
11 = (
λ
+2
µ
)
ε
11 +
λε
22 +
λε
33
σ
23 =
µ γ
23
σ
22 =
λε
11 + (
λ
+2
µ
)
ε
22 +
λε
33
σ
31 =
µ γ
31
σ
33 =
λε
11 +
λε
22 + (
λ
+2
µ
)
ε
33
σ
12 =
µ γ
12
(1.45)
Further we can introduce two alternative elastic constants. The ratio of simple ten-
sile stress to strain, the Young’s modulus E, and the ratio of transverse contraction to
elongation in simple tension, the Poisson’s ratio
ν
, are commonly used elastic con-
stants:
E=
µ
(3
λ
+2
µ
)
µ
+
λ
;
ν
=
λ
2(
µ
+
λ
)(1.46)
Finally, we can also express the bulk modulus in terms of the elastic constants. The
bulk modulus gives the ratio between the applied pressure p=−1
3(
σ
11 +
σ
22 +
σ
33)
and the resulting compression −e=−
ε
11 −
ε
22 −
ε
33 of a crystal:
B=−p
e=1
3(3
λ
+2
µ
)=1
3(c11 +2c12)(1.47)
1.2.4 The Voigt average of elastic constants
As shown in the last section, assuming an isotropic medium simplifies elasticity
theory dramatically. Consequently, later in Chapter 2this assumption allows the
analytical solution of many dislocation related problems. Crystals usually do not
show exactly isotropic behaviour, but fortunately, in many cases, they are approxi-
mately isotropic. Thus, we need a way of sensibly averaging the anisotropic elastic
constants to approximate isotropic constants. There are two basic approaches to
this: One can either average over the elastic constants themselves, or over their in-
verse, the elastic compliances. The former, the so called Voigt average [46], models
a polycrystal in which every grain is in the same state of strain, whereas the latter,
the so called Reuss average [47], models a situation where every grain has the same
stress. Experience has shown, that in most cases, especially when describing local
strain around a dislocation, the Voigt average is most appropriate [41]9. The Voigt
average can be determined by the use of invariants to arbitrary rotations: The trace
of {cijkl }and the strain energy density of homogeneous expansion. One obtains as
the average isotropic elastic constants:
µ
=1
30 3cijij −ciijj
λ
=1
15 2ciijj −cijij(1.48)
9The Reuss average has proven valuable in the description of long-range internal stress fields.
16
Chapter 1. Modelling the Crystal: Theories and Methods
Eq. (1.48) yields respectively for a crystal with cubic (Eq. (1.41)) and for a crystal
with hexagonal (Eq. (1.42)) symmetry:
µ
cubic =1
5(3c44 −c12 +c11)
λ
cubic =1
5(−2c44 +4c12 +c11)
(1.49)
µ
hexagonal =1
30 (7c11 −5c12 +2c33 +12 c44 −4c13)
λ
hexagonal =1
15 (c11 +c33 +5c12 +8c13 −4c44)
(1.50)
Chapter 2
An Introduction to Dislocation
Theory
It is the main purpose of this chapter to introduce the concepts and definitions con-
cerning dislocations. The reader not familiar with dislocation theory will be pro-
vided with the basic ideas and the necessary vocabulary. This includes selected re-
sults from linear elasticity theory whenever this supports the general understanding
or will be useful for later chapters.
The introduction given here can of course not replace a thorough discussion as given
for example by Hirth and Lothe [41].
2.1 The Burgers vector
s
`
d
Figure 2.1: The Burgers integration circuit
If stress is applied to a crystal, it deforms.
Let u(r)be the displacement field de-
scribing the displacement of a small vol-
ume element from its original position af-
ter deformation. This implies a contin-
uum description of the crystal as in Sec-
tion 1.2. In an ideal crystal the closed in-
tegral circuit over u(r)will always yield
zero — even after arbitrary elastic defor-
mation. Real crystals, however, are far from being ideal: Structural line defects with
a line direction `(r)may exist, where the integral circuit around the defect results in
a non-zero vector:
b=I∂u
∂sds(2.1)
This defines the so-called Burgers vector bof the line defect. Line defects with a
Burgers vector b6=0 are called dislocations. They form either during the growth
process of the crystal, or through subsequent plastic deformation.
17
18
Chapter 2. An Introduction to Dislocation Theory
b
3
3
3
3b
`
`
Figure 2.2: Edge and screw dislocation in a simple cubic lattice, only a small region of the
lattice is drawn. Left: Edge dislocation. Right: Screw dislocation. In addition to the line
direction `and the Burgers vector b, a possible Burgers circuit for each dislocation is given.
The closure fault of this circuit defines the Burgers vector b.
If we now abandon the continuous description of the crystal and take it as a periodic
arrangement of discrete lattice sites1, then it becomes obvious that bcannot be arbi-
trary but has to be somehow restricted by the structure of the lattice: If the Burgers
integration circuit only encircles the dislocation and no other defects, and if the inte-
gration path runs through perfect (but strained) lattice only, then the Burgers vector
bhas to be a linear combination of the lattice translational vectors. Dislocations of
this type are called perfect dislocations as opposed to partial dislocations, which are
accompanied by a secondary structural lattice defect, e.g. a stacking fault. Partial
dislocations will be discussed later in Chapter 4.
2.2 Edge and screw dislocations
All considerations in this section will be clarified and illustrated with the example
of a simple cubic lattice. However, the conclusions drawn are not restricted to this
lattice type. Assuming a simple cubic lattice, the local line direction `(r)and a min-
imum Burgers vector bcan be either parallel or perpendicular, resulting in two ele-
mentary dislocation types with minimal Burgers vector: The edge dislocation and the
screw dislocation. Both are schematically depicted in Fig. 2.2 and can be characterised
as follows:
The edge dislocation: Topologically speaking we obtain the edge dislocation by
inserting a half-plane of lattice sites into the crystal. The dislocation line is the line
where the half-plane terminates. As a result the Burgers vector is perpendicular to
the line direction:
bedge ·`(r) = 0 (2.2)
1With either one atom or a “basis” of several atoms at each lattice site. The lattice itself is defined
by its translational symmetry given by translational vectors: A translation along one of these vectors
maps the whole lattice onto itself. The crystal is then described as the periodic arrangement of the
afore mentioned “basis” of atoms following the translational lattice vectors. A simple introduction
into these basics of solid state physics can be found in the book of Kittel [48].
2.3. Straight dislocations in linear elasticity theory
19
The screw dislocation: This dislocation type can be constructed by shearing one
part of the crystal with respect to the other part at a half-plane only. The dislocation
line is given as the line where the half-plane terminates. The Burgers vector is then
parallel to the line direction:
bscrew ×`(r) = 0 (2.3)
Superpositions of both types resulting in mixed type dislocations are possible.
Fig. 2.2 also shows the Burgers circuit in the discrete lattice. Note that the circuit
— now of course in discrete steps from lattice site to lattice site — must be a circuit
which would be closed in an ideal undislocated lattice, e.g. a parallelogram. The
closure fault defines the Burgers vector b. Usually bis given in units of the lattice,
in internal and strained coordinates, thus bdoes not depend on the exact Burgers
circuit path through more or less strained lattice: Both dislocations in Fig. 2.2 have
a Burgers vector with the length of one minimal lattice translation, even though the
length in external coordinates may vary due to varying strain.
2.3 Straight dislocations in linear elasticity theory
Linear elasticity theory as described in Section 1.2 proves to be a very useful tool to
describe the long-range elastic strain effects of dislocations. In the following we will
discuss the elastic strain energy of an isolated straight dislocation and the interaction
forces between two straight dislocations. Curved dislocations are excluded since
they play no important role in this work.
2.3.1 The elastic strain energy of a straight dislocation
The strain energy of an infinite straight dislocation in an otherwise perfect crys-
tal can be calculated analytically using linear elasticity theory. As shown in Ap-
pendix Aone writes the energy per unit length of a dislocation, contained in a cylin-
der of radius Rand length L, as
E(R)
L=k(
β
)|b|2
4
π
ln R
Rc+Ec
L,R≥Rc, (2.4)
where
β
is the angle between band the line direction `of the dislocation, Rcis
the core radius and Ec/Lthe core energy per unit length. The energy factor k(
β
)
depends on
β
and the elastic properties of the material. Assuming an isotropic
medium, k(
β
)can be evaluated as
k(
β
) =
µ
cos2
β
+sin2
β
1−
ν
!(2.5)
with
µ
being the shear modulus and
ν
the Poisson’s ratio as defined in Eq. (1.44)
and (1.46).
20
Chapter 2. An Introduction to Dislocation Theory
ln( / )R
E
c
c
0
E
R
Rc
R
Figure 2.3: The elastic strain energy of a disloca-
tion integrated in a surrounding cylinder. The x-
axis is scaled logarithmically.
The logarithmic term in Eq. (2.4)
diverges for both the limits R→
0 and R→
. Continuum the-
ory supposes that the atomic dis-
placements vary slowly over the di-
mensions of a unit cell and this
breaks down at the core resulting
in the divergence as R→0. Also,
for Rof the order of magnitude of
interatomic distances, continuum
theory cannot describe the discrete
system correctly. As a result, below
a certain radius Eq. (2.4) does not
give a good description of the elastic energy: The core radius Rcis defined by the
condition that (2.4) ceases to be applicable for radii R<Rc. In other words, the dis-
location core is the minimum region which cannot be described by elasticity theory
and therefore, discrete (atomistic) models are necessary to evaluate the core energy
Ec. If the core energy is known, the elastic strain energy for R≥Rccan be plotted
as shown in Fig. 2.3.
As already mentioned, Eq. (2.4) also diverges for R→
. Consequently, in an
infinite crystal we cannot evaluate a finite total elastic energy of a dislocation, but
only its core energy, the energy factor k(
β
)and core radius which together describe
the variation of the strain energy with R.
2.3.2 The elastic interaction between two straight dislocations
Let us consider two straight and parallel dislocations A and B of arbitrary Burgers
vectors bAand bBin an infinite and isotropic crystal. Then a somewhat involved
calculation yields the elastic interaction energy per unit length:
EAB(R)
L=−
µ
2
π
(bA·`) (bB·`)+(bA×`)·(bB×`)
1−
ν
ln R
R0
−
µ
2
π
(1−
ν
)R2[(bA×`)·R] [(bB×`)·R](2.6)
Here Ris the distance vector between the two dislocations and Rits length. `gives
the line direction (|`|=1). Eq. (2.6), which was first developed by Nabarro [49], al-
lows the calculation of the interaction energy except for a constant shift ∝ln(R0/˚
A).
Similar to the core energy in Eq. (2.4), this shift cannot be determined in linear elas-
ticity theory. However, this shift does not influence the elastic force between the
two dislocations. By differentiation we obtain the radial component of the interac-
tion force per unit length:
FAB(R)
L=−∂
∂REAB(R)
L=
µ
2
π
R(bA·`) (bB·`)+(bA×`)·(bB×`)
1−
ν
(2.7)
In analogy to this, the angular component can be derived by differentiation
−(1/R)∂/∂
ϑ
perpendicular to R. Eq. (2.7) will be useful in the calculation of equi-
librium stacking fault widths in Section 4.4.1.
2.4. Dislocation motion
21
glide
Figure 2.4: Dislocation glide in a simple cubic lattice: Lattice sites are shown as empty circles
and the glide plane lies horizontally. The small arrows indicate the glide direction if one
proceeds from the structure shown on the left to the one shown on the right.
climb
Figure 2.5: Dislocation climb in a simple cubic lattice: Lattice sites are shown as empty
circles and the glide plane lies horizontally. The small arrows indicate the climb direction if
one proceeds from the structure shown on the left to the one shown on the right.
2.4 Dislocation motion
When talking of dislocation motion, we have to differentiate between two basic
types: Dislocation glide and dislocation climb:
Dislocation glide: A dislocation in an otherwise perfect and infinite crystal will
be in equilibrium and it will remain on its position. However, if either external
forces act upon the crystal or if other defects are present, then this will generate a
glide plane
b
`
Figure 2.6: The glide plane of a dislocation
strain field which might result in a net
force on the dislocation line and it will
possibly move (see for example Eq. (2.7)).
For dislocations with an edge component
(b×`6=0) this motion is restricted to the
plane defined by the Burgers vector band
the line direction `(Fig. 2.6). In this plane
the dislocation is able to move rather easily by breaking and forming new bonds
as schematically depicted in Fig 2.4. This process is called dislocation glide and
the plane accordingly glide plane. Pure screw dislocations, of course, do not have a
featured glide plane in that sense.
Since during glide the number of lattice sites of the whole system remains constant,
we speak of dislocation glide as conservative motion.
22
Chapter 2. An Introduction to Dislocation Theory
Dislocation climb: The second sort of dislocation motion is called dislocation
climb. As shown in Fig. 2.5, during climb the dislocation moves from one glide
plane to another. In the case of a pure edge dislocation with minimal Burgers vector,
the direction of motion points exactly along the inserted half-plane: The half-plane
either shrinks (upward climb, Fig. 2.5), or it expands (downward climb). Hence, as
opposed to glide motion, for climb motion an absorption or creation of lattice sites
is necessary. In a real crystal this can be achieved through the release or trapping of
either interstitial atoms or lattice vacancies. Thus, climb does not require a driving
elastic force acting upon the dislocation line.
Since dislocation climb involves a net change in the number of lattice sites, we speak
of non-conservative motion.
2.5 The dissociation of dislocations
Looking at Fig. 2.2 it is obvious that a larger Burgers vector leads to larger lattice
distortion in the region at and near the dislocation. Hence it seems plausible, that in
terms of elastic strain energy only dislocations with the one or two smallest Burgers
vectors are stable: A dislocation with a larger Burgers vector bmight dissociate into
dislocations A and B with smaller vectors bAand bB. For topological reasons in this
process the overall Burgers vector has to be conserved:
b=bA+bB(2.8)
Fig. 2.7 shows the dissociation of an edge dislocation with a Burgers vector of two
times a minimal lattice translation in a simple cubic lattice. A comparison of energies
using Eq. (2.4) gives a simple criterion for dissociation:
The generalised Frank criterion for dissociation: A perfect dislocation with a Burg-
ers vector bwill dissociate into two perfect dislocations A and B if
k(
β
)|b|2>k(
β
A)|bA|2+k(
β
B)|bB|2. (2.9)
For the original criterion F. C. Frank assumed an energy proportional to |b|2only
and therefore neglected the variation with
β
Hirth and Lothe [41].
A B
Figure 2.7: Dislocation dissociation in a simple cubic lattice: Lattice sites are shown as empty
circles and the glide plane lies horizontally. The arrows indicate the direction of motion for
the two resulting dislocations.
2.6. The influence of lattice periodicity: Kinks and jogs
23
2.6 The influence of lattice periodicity: Kinks and jogs
Whilst the last sections dealt with straight perfect dislocations, we will now turn
towards those deviations from the straight line, that represent a lateral offset. In the
description of those so-called kinks and jogs the influence of lattice periodicity on the
energy of a dislocation plays a major role.
2.6.1 The periodic displacement potential of a crystal
Plastically shearing a crystal on a rational plane2involves the stretching, breaking
and forming of new bonds accordingly to the periodicity in this plane along the
direction of shear. This means, microscopically the shear translation xwill depend
periodically on the applied shear stress
σ
and not linearly. The classical approach to
this problem was that of Frenkel [50], who assumed a sinusoidal periodicity of the
displacement potential energy, yielding a sinusoidal expression for the shear stress:
σ
=
σ
0sin 2
π
x
T(2.10)
Here Tis the length of the minimal lattice translation vector in the plane. It is ob-
vious, that a dislocation line moving in the same plane will experience a similar
displacement potential that reflects the lattice periodicity. Starting from this idea,
one can develop a dislocation model that includes lattice periodicity. The first and
phenomenological approach is known as the Frenkel-Kontorova model [51,52,53],
where a one-dimensional array of spring-connected balls lying on a periodic sub-
strate serves as an analogue. However, this approach merely gives a qualitative
description.
The Peierls-Nabarro model [54,55] gives a more formal solution for the displacement
potential. In this model we speak of a dislocation resting in a Peierls valley. In mo-
tion it has to periodically overcome the barriers (Peierls hills) between the valleys.
An approximation for dislocation core energies results from this model. However,
since in this work lattice periodicity will be treated atomistically, it is unnecessary to
introduce the formalism of Peierls and Nabarro. A good overview is given in [41].
2.6.2 Dislocation kinks
Let us consider a dislocation, where segments have been displaced within the
glide plane, keeping the same line direction `. Stable low energy configurations
will always be those, where each of the segments lies in a Peierls valley of the
glide plane, as depicted in Fig. 2.8. The connections between the segments —
those parts, where the dislocation line crosses a Peierls hill — are called dislo-
cation kinks. A pair consisting of a right kink (RK) and a left kink (LK) might
be formed on a straight dislocation by thermal fluctuations in the crystal: Af-
ter the pair the dislocation runs in the same Peierls valley again. In fact it is
2A rational plane is a plane defined by two non parallel lattice translational vectors.
24
Chapter 2. An Introduction to Dislocation Theory
RK
LK LK
Figure 2.8: A kinked dislocation in
the glide plane. Straight solid lines
in the plane are Peierls valleys and
dashed lines are hills. LK and RK
denote left kinks and right kinks re-
spectively.
assumed that for small forces3on the dislocation line, glide involves the ther-
mal creation and migration of kinks. For their migration along the dislocation
one has to imagine a similar variation of the Peierls potential — now however
along the dislocation line and not perpendicular to it. Thus, glide velocity is con-
trolled by the kink formation energy and the kink migration barrier. In semiconduc-
tors, where for atomistic processes like kink formation and migration quantum
mechanical effects play an important role, the Peierls-Nabarro model and elastic-
ity theory are not sufficient to calculate the related energies. Details on how this
L
`
LK aRK
Figure 2.9: A dislocation kink pair.
problem can be solved using DFT-based ap-
proaches will be given later in Section 4.5. How-
ever, here we can still calculate the elastic inter-
action energy of kinks: If we assume an infinite
dislocation with a kink pair where the kink seg-
ments of the dislocation are perpendicular to the
line direction `, then the elastic kink–kink inter-
action energy is given as [41]:
ELK,RK(L) = −
µ
a2
8
π
L(1−
ν
)n|b·`|2(1+
ν
) + |b×`|2(1−2
ν
)o(2.11)
Fig. 2.9 shows the corresponding situation. Lis the kink–kink separation. The kink
height ais given by the distance between the Peierls valleys, or in other words, by
the lattice translational symmetry in the glide plane. The overall formation energy
of the kink pair now can be written as:
Epair(L) = Ef(LK) + Ef(RK) + ELK,RK(L)(2.12)
Here Efdenotes the formation energy of the respective single kink. Eq. (2.12) will
play an important role in the calculation of the kink formation energies in Sec-
tion 4.5. By differentiation of Eq. (2.11) we obtain the attractive force between kinks:
FLK,RK(L) = −d
dLELK,RK(L)
=−
µ
a2
8
π
L2(1−
ν
)n|b·`|2(1+
ν
) + |b×`|2(1−2
ν
)o(2.13)
Note that the elastic interaction force between the two kinks falls off more rapidly
with their separation L(∝1/L2) than the elastic force between two parallel straight
dislocations (∝1/R) in Eq. (2.7), where Rdenotes the separation of the dislocations.
3Small forces in this context are forces that are not large enough to lift the dislocation as a whole
over the Peierls hill.
2.6. The influence of lattice periodicity: Kinks and jogs
25
`
J
Figure 2.10: A dislocation jog. The
upper of the two glide planes is only
drawn half, so the underlying plane
with the dislocation line can be seen.
J denotes the position of the jog.
2.6.3 Dislocation jogs
Similarly to displacing segments of the dislocation within the glide plane, one can
imagine the same normal to it. The connections between the displaced segments are
called dislocation jogs. They are an analogue to dislocation kinks in the preceding
section. Fig. 2.10 shows a single jog between two adjacent glide planes. Such a jog
is also called unit jog, as opposed to a superjog, which spans across several glide
planes. As kinks correspond to glide motion, jogs correspond to climb motion. This
implies that jogs will neither form, nor migrate along the dislocation line by mere
thermal activation. They are sessile and can only move along the dislocation line by
trapping or releasing interstitial atoms or vacancies.
Analogous to the kink–kink interaction one can evaluate the elastic interaction en-
ergy of a pair consisting of an up jog (UJ) and a down jog (DJ) in a geometry similar
to that of the kink pair in the last section [41]:
EUJ,DJ(L) = −
µ
a2
8
π
L(1−
ν
)n|b·`|2(1+
ν
) + |b×`|2o(2.14)
Here for a unit jog adepends on the glide plane separation. For pure screw dislo-
cations (b×`=0) with Eq. (2.14) we obtain the same result as for the kink–kink
interaction energy in Eq. (2.11). This reflects the fact that screw dislocations do not
have a featured glide plane: For a screw dislocation jog with respect to one glide
plane one always finds another glide plane where this jog resides in. With respect
to the latter plane the jog has to be considered a kink.
Chapter 3
Dislocations in Tetrahedrally
Bonded Semiconductors
To prepare for the analysis of the examples of dislocations in various tetrahedrally
bonded semiconductors, this chapter will give an overview of the low energy types
of dislocations in the respective crystal structures. After discussing perfect and par-
tial dislocations belonging to the crystal’s main slip system, further types are intro-
duced briefly.
Following Eq. (2.4), low energy dislocation structures are those with a minimum
Burgers vector. Further, in Section 2.1 we learnt, that the Burgers vector of a perfect
dislocation is always a linear combination of the lattice translational vectors. The
same holds for the dislocation line direction. Thus, if the lattice structure is known,
one can already specify possible low energy types of dislocations.
The prevalent bonding configuration in semiconductor crystals is that of each atom
being bonded to four immediate neighbour atoms. The electronic configuration of
each atom is then approximately given as an sp3hybrid. Ideally, the four neighbours
are located at the vertices of a tetrahedron around the central atom, forming bond
angles of 109.5◦. With this bonding several periodic structures are possible. The
two most common crystal structures are those of cubic symmetry and a hexagonal
structure known as 2H. Both can be described as a lattice with a two-atom basis, or
equivalently, as two intersecting lattices [48]. In Fig. 3.1 the structures are depicted
for compound semiconductors, where the basis consists of two atoms of different
species. In both cases the lattice is that of a closed-packed structure: A face-centred
cubic (fcc) lattice and a hexagonal closed-packed (hcp) lattice1. The characteristic
parameters of the respective structure are its lattice constants. The cubic cell is char-
acterised by just one lattice constant a0. Ideally, one lattice constant would be suf-
ficient to describe the hexagonal cell as well. In real crystals however, variations in
bond lengths and angles lead to deviations from the ideal ratio 1.633 between cell
height and edge length of the basal hexagon. Thus the structure is defined by two
lattice constants a0and c0(see. Fig. 3.1).
1In a hard sphere model of the lattice, where each sphere is in direct contact with its neighbours,
both fcc and hcp have the maximum possible space-filling (74%). For comparison: The space-filling of
body-centred cubic (bcc) lattices is 68%.
27
28
Chapter 3. Dislocations in Tetrahedrally Bonded Semiconductors
(111)
c
(0001)
c0
a0
a0
a3
a2
a1
a1
a2
a3
Figure 3.1: Unit cells of tetrahedrally bonded semiconductors. Left: The cubic structure, also
known as the zincblende structure. It can be constructed as a face-centred cubic lattice with
a two-atom basis. The lattice constant a0is given as the edge of the cubic cell. The three
perpendicular basis vectors are labelled a1to a3.Right: The hexagonal 2Hstructure, also
known as the wurtzite structure. It can be constructed as a hexagonal closed-packed lattice
with a two-atom basis. The two lattice constants are given as one edge of the basal hexagon
(a0) and the height of the unit cell (c0). The basis vectors are labelled a1to a3and c. The
ailie in one plane and form angles of 120◦.a3is conventionally introduced for reasons of
symmetry in the description.
In both unit cells a plane with the highest density of lattice sites is drawn: (111) and (0001)
for cubic and hexagonal symmetry respectively. The minimum lattice translations within
these planes are shown as dashed lines.
3.1 Crystal slip systems and perfect dislocations
In the process of plastic deformation one part of the crystal might be sheared macro-
scopically with respect to the other part. This so-called crystal slip usually occurs on
specific crystallographic planes only. This can be explained microscopically: Crystal
slip involves the formation and propagation of dislocations [41]. Those dislocations
which have minimum Burgers vectors are easiest to form. Since a Burgers vector
is always given as a lattice translation, dislocations are preferentially formed on
planes containing the minimum lattice translations. These planes are the preferred
slip planes. A slip plane contains both the Burgers vector and the line direction
of the generated dislocations. Hence the plane of crystal slip is identical with the
glide plane of the dislocations involved in the process. Let us now investigate the
preferred slip systems of the two crystal lattices introduced above.
3.1.1 The main slip systems
The {111}h110islip system in fcc lattices: In the cubic structure the planes of
highest lattice site density are the {111}planes. Fig. 3.1 (left) shows one specific
plane of that family. These planes contain the minimum lattice translations and are
thus preferred slip planes. The slip system is defined by the plane and the direction
of slip, conventionally written as {111}h110i. For each plane we have three possible
3.1. Crystal slip systems and perfect dislocations
29
directions. Specifically in the (111) plane the minimum lattice translations (Burgers
vectors) are 1
2[1¯
10],1
2[¯
101]and 1
2[01¯
1]. They are drawn as dashed lines in Fig. 3.1
(left).
Since the line direction, at least locally, is also given by lattice translations, the afore
mentioned directions yield two basic dislocation types with minimum Burgers vec-
tor:
•The screw dislocation with line direction and Burgers vector parallel, e.g.
`= [1¯
10]and bs=1
2[1¯
10].
•The 60◦dislocation, where line direction and Burgers vector form an angle of
60◦, e.g. `= [1¯
10]and b60 =1
2[0¯
11].
The {0001}h11¯
20islip system in hcp lattices: Whilst in the fcc system there is a
whole family of planes with the highest lattice site density, in the hcp structure there
is only one, the (0001) or basal plane, as shown in Fig. 3.1 (right). The slip system is of
the {0001}h11¯
20itype and the minimum lattice translations are given as 1
3[11¯
20],
1
3[1¯
210]and 1
3[¯
2110]. They are drawn as dashed lines in Fig. 3.1 (right). The two
basic dislocation types with minimum Burgers vector are:
•The screw dislocation with line direction and Burgers vector parallel, e.g.
`= [11¯
20]and bs=1
3[11¯
20].
•The 60◦dislocation, where line direction and Burgers vector form an angle of
60◦, e.g. `= [11¯
20]and b60 =1
3[2¯
1¯
10].
3.1.2 Glide and shuffle structures
In a lattice with a one-atom basis the Burgers vector and the line direction uniquely
define the dislocation type. However as mentioned before, the crystal structure of
tetrahedrally bonded semiconductors is that of a lattice with a two-atom basis. A
direct consequence is, that dislocations with a featured glide plane exist in two ge-
ometrically different types: As illustrated in Section 2.2, the edge component of a
dislocation involves an extra half-plane of lattice sites on one side of the glide plane.
With a two-atom basis, for one specific glide plane the extra half-plane can termi-
nate on either of the two atoms, resulting in two geometrically different types — the
glide and the shuffle dislocation. Fig 3.2 shows the atomistic situation for the perfect
60◦dislocation in cubic material: If the half-plane terminates between two closely-
separated (111) planes we obtain the glide structure, if it terminates between widely
separated planes we obtain the shuffle structure. The transition between shuffle and
glide is a climb process and thus involves the trapping or release of interstitial atoms
or vacancies (compare with Fig. 2.5 in Section 2.4). As we will see in later chapters,
the glide structure is usually the energetically more stable structure.
Further, in compound semiconductors AB with two atom species Aand B, we have
to distinguish the atom species of the terminating line of atoms of the extra half-
plane. Conventionally the respective structures are called
α
and
β
dislocations.
Hence depending on the combination of glide and shuffle with
α
and
β
there are
4 different types of 60◦dislocations.
30
Chapter 3. Dislocations in Tetrahedrally Bonded Semiconductors
[112] [112]
[111]
Ab
c
B
climb Ab
c
B
Figure 3.2: The glide and the shuffle structure of the perfect 60◦dislocation in cubic material.
Only two (1¯
10)layers along the dislocation line are shown for each structure. The extra half-
plane is enclosed by a shaded box. Depending on where the half-plane terminates, we obtain
the glide or the shuffle structure. The transition between both is given by a climb process
as in Fig. 2.5.Left: The 60◦glide dislocation. Here the half-plane terminates between two
closely-spaced (111) planes Aand b.Right: The 60◦shuffle dislocation. Here the half-plane
terminates between two widely-spaced (111) planes band B.
B
A
C
B
A A
B
A
B
A
[112] [1100]
[111]
[0001]
Figure 3.3: The Stacking sequence in the cubic and hexagonal lattice. In both structures the
respective unit cell from Fig. 3.1 is drawn. Left: The cubic structure. The view is along [1¯
10]
and the (111)slip planes lie horizontally. Right: The hexagonal structure. The view is along
[11¯
20]and the (0001)basal slip planes lie horizontally.
B
A
C
B
B’
[1100][112]
[111]
[0001]
A’
B’
C
A
B
Figure 3.4: Intrinsic stacking faults in the cubic and hexagonal lattice. In both structures the
stacking fault plane is shaded. Left: An intrinsic stacking fault in the cubic structure. Right:
An intrinsic stacking fault in the hexagonal structure.
3.2. Crystal stacking and partial dislocations
31
3.2 Crystal stacking and partial dislocations
Looking only at one of the two sub-lattices, the preferred slip planes mentioned
in the last section are also the planes stacked with the widest separation. The
cubic and the hexagonal structure can not only be distinguished by their distinct
unit cells, but alternatively also by their respective stacking sequence of these
planes. As can be seen in Fig. 3.3, the stacking sequence is ···ABC|ABC ···in
cubic2and ···AB|AB|AB ··· in hexagonal material3. Faults in this stacking se-
quence are very common planar defects. If the normal sequence is maintained
on both sides of the fault, then we speak of an intrinsic stacking fault (ISF), other-
wise of an extrinsic fault. In this work only intrinsic faults are considered. Fig. 3.4
shows an ISF in cubic and in hexagonal material. In the ISF plane one type of
atoms is displaced with respect to the bulk plane in normal stacking. To illus-
trate this, we compare plane Cof the cubic crystal structure (Fig. 3.3 (left)) with the
[110]
[112]
bulk
ISF
oo’
Figure 3.5: The offset between bulk
and faulted region (ISF) for the sub-
lattice of one atom species.
respective plane B0in the cubic stacking fault
(Fig. 3.4 (left)). Fig. 3.5 shows the atom posi-
tions of the displaced species in the ISF plane
and in the corresponding bulk plane. The off-
set is o=1
6[1¯
21]or equivalently 1
6[11¯
2]and
1
6[¯
211]. The situation is exactly the same in
the hexagonal structure. Actually one only has
to relabel the crystallographic directions [1¯
10]
and [¯
1¯
12]in Fig. 3.5 with [11¯
20]and [1¯
100]re-
spectively, since the stacking fault lies in the
basal plane. In hexagonal coordinates we ob-
tain an offset of o=1
3[10¯
10]or equivalent.
Let us now assume a dislocation, which is not surrounded by ideal lattice only, but
which borders an intrinsic stacking fault in its glide plane. This type of dislocation
is called Shockley partial dislocation [41]. The minimum Burgers vector of such a dis-
location is given as the offset vector obetween the faulted and unfaulted region of
the glide plane.
Fig. 3.6 (left) shows the resulting principal Burgers vectors. The two partials are:
•The 30◦Shockley partial. E.g. in cubic material: `= [1¯
10]and b30 =1
6[1¯
21]
and in hexagonal material: `= [11¯
20]and b30 =1
3[10¯
10].
•The 90◦Shockley partial. E.g. in cubic material: `= [1¯
10]and b90 =1
6[¯
1¯
12]
and in hexagonal material: `= [11¯
20]and b90 =1
3[1¯
100].
2Therefore the cubic structure is also alternatively referred to as the 3Cstructure, since the same
three planes are periodically repeated in the [111]direction. It should be noted, that we attach a
collective label (A,B,C) to actually a pair of planes — one plane per sub-lattice. To be precise one
would have to label the sub-lattices separately as done in Fig. 3.2. This yields for the cubic structure
· · · aAbBcC|aAbBcC ···.
3Analogously to the cubic structure, the hexagonal structure presented here is referred to as 2H.
Also other hexagonal structures are possible, for example 4Hand 6H, with a stacking sequence of
· · · ABAC|ABAC · · · and ···ABCACB|ABCACB···respectively. 4Hand 6Hare very common poly-
types of silicon carbide crystals.
32
Chapter 3. Dislocations in Tetrahedrally Bonded Semiconductors
ISF
`
b60
b60
−
b
b
b
b30
90
b90
30
90
Figure 3.6: The dissociation of the perfect 60◦dislocation. Left: Burgers vector of the perfect
60◦dislocation and of the 30◦and 90◦Shockley partials. The vectors of the partial dislo-
cations are given as the offset vectors between faulted and unfaulted region of the glide
plane (compare Fig. 3.5). We obtain −b90 or b90 depending on which side of the dislocation
the faulted region is located. The perfect 60◦dislocation can be decomposed into the two
Shockley partials: b60 =b30 +b90.Right: Schematic sketch of the corresponding dissociation
reaction in the glide plane.
As shown in Fig. 3.6 (left), the Burgers vector of the perfect 60◦dislocation is given
as the sum of the two Shockley partials: b60 =b30 +b90. The Burgers vectors of
the Shockley partials are smaller than that of the perfect dislocation: The difference
is large enough, to ensure that the Frank criterion (2.9) is usually satisfied. Hence
a dissociation is very likely if the energy of the stacking fault is not too large. The
dissociation reaction is given as:
b60 −→ b30 +ISF +b90 (3.1)
The geometrical situation is shown schematically in Fig. 3.6 (right): The two partials
are separated by an intrinsic stacking fault between them. Their equilibrium dis-
tance is determined through the energy of the stacking fault and the partial – partial
interaction force, usually given as the elastic interaction as in Eq.(2.7). Fig. 3.7 gives
an impression of the microscopic arrangement of the two partials and the stacking
fault in cubic a crystal. The dissociation distance shown is rather small: In real semi-
conductor crystals the two partials are very often dissociated with widths of 10 or
more lattice constants, depending on the stacking fault energy.
Analogously the screw dislocation can dissociate into two 30◦partials enclosing an
intrinsic stacking fault:
bs−→ b30 +ISF +b−30 (3.2)
Here b−30 is given as 1
6[2¯
1¯
1]for cubic and 1
3[¯
12¯
10]for hexagonal material, assuming
`,bsand b30 as introduced above. In a compound the two 30◦partials are of opposite
termination (
α
and
β
type) to maintain stoichiometry.
It has to be noted, that in this chapter all atomic dislocation structures were deduced
from displacing the lattice according to the respective combination of Burgers vector
and line direction. They do not represent real structures! The atomic structures close
to the dislocation line — the so called core structures — might reconstruct and vary
with material. Examples will be given in the following chapters.
3.3. Further classes of dislocations
33
90°
30°
[111]
[112]
Figure 3.7: The dissociation of the perfect 60◦glide dislocation seen atomistically. Only two
(1¯
10)layers along the dislocation line are shown. The intrinsic stacking fault separating the
90◦and 30◦Shockley partials is enclosed by a shaded box. The very core region of each
partial is located at the respective end of the stacking fault.
3.3 Further classes of dislocations
All dislocations discussed so far can be generated through plastic deformation of
the crystal and belong to a low energy slip system. In real crystals however, other
classes of dislocations might be present. The most common are misfit dislocations,
threading dislocations and grain boundary dislocations:
Misfit dislocations: Semiconductor crystals are very often grown on a substrate
with a considerable lattice mismatch. Misfit dislocations compensate this mismatch
and are basically the termination lines of additional lattice planes. They are very
often similar or equivalent in structure to dislocations of the crystal’s main slip sys-
tem.
Threading dislocations: During crystal growth, unevenness of the substrate sur-
face or the collision of growth islands might lead to the nucleation of dislocations
with line direction parallel to the direction of growth. These dislocations thread
the epilayer and are very common in hexagonal semiconductors like hexagonal
GaN [56]. The two basic types are the threading screw (ts) and the threading edge dislo-
cation (te), both with a line direction of `= [0001]and Burgers vectors bts = [0001]
and bte =1
3[1¯
210]respectively.
Grain boundary dislocations: These dislocations occur at the boundary between
two tilted crystal grains in a polycrystal. They compensate for the tilt angle between
the two grains. They are very often similar or equivalent in structure to dislocations
of the crystal’s main slip system or to threading dislocations.
Chapter 4
Dislocations in Diamond
After a short introduction to diamond as a material and to the experimental evidence
for dislocations in diamond, this chapter demonstrates, how the methods described
in Chapter 1can be combined to describe the different aspects of dislocations in
semiconductors. With the 30◦glide partial dislocation as a first example, the chosen
model geometry (a supercell-cluster hybrid) and its convergence with size are tested
in comparison with elasticity theory. Subsequently the low energy core structures of
the remaining predominant perfect and partial dislocations of the {111}h110islip
system in diamond are determined and their core energies and the long range elastic
energy contributions are calculated. This is then followed by a discussion of possible
dissociation reactions and dislocation glide of the partials is considered. The latter
requires to model kink formation and migration processes on the atomistic level.
In the last part of this chapter it is shown how the calculated core geometries can
be used as input coordinates for HRTEM simulations, while the calculation of the
electronic structures allows the modelling of EEL spectra. Thus a direct link to ex-
periments can be established.
4.1 Introduction and background
Diamond exhibits some interesting physical properties, like extreme hardness, high
heat conductivity, high refractive index and a wide indirect electronic band gap
(5.49 eV)[57], which make it an interesting material not only as a gemstone but also
for technological applications. The development of several techniques to produce
synthetic diamond at rather low costs allows its technological application: Taking
advantage of its high hardness, synthetic diamond powder is preferentially used for
the abrasive coating of cutting tools.
Conventional growth techniques for single crystal semiconductors usually involve
the melting of the material (e.g. the Czochralski method) — this unfortunately fails
for diamond due to its high melting point at the high pressure needed to avoid
graphitisation. However, chemical vapour deposition (CVD) growth of diamond
produces thin films, which might be used for semiconductor devices [58]. The
wide band gap would make diamond an especially attractive semiconductor in op-
toelectronics or for high-temperature high-power devices. p-type doping of dia-
35
36
Chapter 4. Dislocations in Diamond
mond can be achieved by boron implantation, but n-type doping yet remains an
unsolved problem, hindering the widespread application of diamond as a semicon-
ductor1. Nevertheless, CVD diamond films are currently being used in high-power
microwave devices, as heat spreaders in power transistors and as optical infrared-
transmissive windows — to name just a few applications. Recent developments
look promising for high voltage devices. ABB and De Beers Industrial Diamonds
(UK) report single-crystal CVD diamond with carrier lifetimes >2
µ
s, carrier mo-
bilities around 4000 cm2/Vs and 4 kV diodes with a breakdown voltage >4 MV/cm
[58,60].
4.1.1 Experimental evidence for dislocations in diamond
In general, natural gem-quality type Ia diamonds, where nitrogen is present in
an aggregated form, (see Dyer et al. [61] for the classification scheme) are almost
dislocation-free [57]. Chemically pure type IIa, however, has been reported to con-
tain a dislocation density of up to 107cm−2and boron containing type IIb also
proves to be dislocation-rich [62,63,64]. Further, CVD-grown polycrystalline di-
amond reveals high densities of dislocations — sometimes up to 1012 cm−2. Some
originate at the substrate-interface and propagate through the thin film [65], but
others lie at or near grain boundaries [66]. As in many other fcc semiconductors,
the slip system in diamond is of the {111}h110itype. This means that perfect
dislocations lying on {111}planes possess h110iBurgers vectors and line direc-
tions [41,67,68,69] (see also Fig. 4.1). Details on this slip system have been given in
Chapter 3.
Using weak-beam electron microscopy, Pirouz et al. [69] found that dislocations in
type IIa diamond are dissociated into glide partials separated by an intrinsic stack-
ing fault ribbon of width 25 – 42 ˚
A. Further, many extended dislocation nodes and
dipoles consisting of 60◦dislocations were observed. This is similar to the properties
of deformed Si and Ge.
Further it is known, that dislocations in diamond influence the electrical and optical
properties of the material: In undoped CVD diamond and natural type-II diamond
dislocations have been correlated with the so-called A-band cathodoluminescence
at 2.8 – 2.9 eV [62,63,70,71]. This luminescence is polarised with the electrical
field parallel to the dislocation line [62,63] and is associated with under-coordinated
carbon atoms [71].
4.1.2 HPHT treatment — a threat to the international gem trade
More recently, it has also been observed that colour changes produced in natural
brown diamonds by high-pressure, high-temperature annealing (HPHT) are linked
to plastic deformations and dislocations [72]. Even complete decolouring of natu-
ral brown diamond by HPHT treatment has proven possible and the technique is
applied in gem industry (General Electric and Lazare Kaplan): The majority of di-
amonds is brown and useless as gem stones. If treated however, they can be sold
1However, n-type doping seems to improve over recent years. Nowadays phosphorous n-type
doping at least seems to be 100% reproducible [59] with increasing carrier lifetimes.
4.1. Introduction and background
37
Figure 4.1: A weak-beam image of a typical dislocation distribution in diamond (diffraction
vector g= [¯
202]indicated). To give some examples of the dislocations shown: AA’ and
BB’ are glide dislocations; GG’ and HH’ are dipoles consisting of 60◦dislocations and M, N
and O are faulted dipoles (dissociated dipoles consisting of 4 partials). For detailed infor-
mation on this image see Ref. [69]. (Reproduced from Ref. [69] with kind permission from
the authors.)
as a cheaper alternative. Since the detection of HPHT treatment is far from trivial,
dubious companies now try to sell treated diamonds as natural untreated stones. If
they succeed, their profit is enormous, as is the loss for the major diamond selling
companies2. Understandably, research to find ways of easy detection and thus elim-
inate this commercial threat to the gem trade is part of the major companies’ gem
defensive work [74,75].
4.1.3 Earlier theoretical work
In contrast to silicon, relatively little theoretical work has been carried out on dis-
locations in diamond. Early studies by Nandedkar and Narayan [76] involved the
usage of classical empirical potentials while recent ab-initio studies were restricted to
the 90◦partial glide dislocation [77,78,79]. Also work related to hydrogen and soli-
tons at dislocations [80,81,82] and to graphitisation at core structures [83] consid-
ered the 90◦partial only. Hence a systematic investigation of the core structures, energies,
mobility and electrical activity by means of DFT-based calculations was still missing.
2HPHT costs are around 800 US $ per stone, yielding colourless diamonds worth an average value
of 15,000 US $. However, not all treated diamonds appear to be colourless, IIa-type stones might show
pink colouring and Ia green or even fancy yellow. The untreated natural equivalent of the latter can be
sold for up to 100,000 US $ per carat [73]. Thus, treated stones also threaten the top end of the market.
38
Chapter 4. Dislocations in Diamond
4.2 The atomic scale modelling of dislocations
As discussed in Section 2.3, continuum elasticity theory is unable to address prop-
erties of the core where the strain diverges. Instead, the core energy and structure
of a dislocation requires an atomistic treatment. As we will see later in this section,
the modelling of the core often involves a considerable number of atoms — mak-
ing a fully self-consistent treatment by means of the DFT-pseudopotential approach
(AIMPRO, Section 1.1.2) computationally very expensive. Hence here the method
of choice is the DFTB method described in Section 1.1.3, which is capable of conve-
niently performing structural relaxations of more than 700 atoms on a workstation
or even over 1500 on state-of-the-art supercomputers. To verify the correct repre-
sentation of dislocation core structures by DFTB, a range of small dislocated clusters
and supercells containing around 200 atoms have been modelled with both meth-
ods, DFTB and AIMPRO. Even though these small clusters and supercells were not
large enough to model a dislocation embedded in otherwise perfect bulk material,
they are sufficient to compare the representation of typical bonding situations in
dislocation cores between different methods. In this comparison DFTB has proven
capable to model core structures comparably to AIMPRO — both approaches yield
very similar lowest energy structures and deviations in bond lengths and angles are
well below 2 %
As shown in Chapter 2, the determination of energies and forces in continuum elas-
ticity theory always involves the elastic constants of the respective material. To now
sensibly compare atomistic results with elasticity theory, one has to apply the elastic
constants corresponding to the respective atomistic method and not the experimen-
tal ones. Hence the knowledge of the elastic constants within the DFTB method is
crucial. The three independent elastic constants of diamond in a conventional cubic
unit cell were found by suitably deforming the cell and calculating the respective
total energies. The integration over the Brillouin zone was accomplished using a
Monkhorst-Pack-optimised set of 3 ×3×3k-points [84]. The resulting elastic con-
stants are given in Table 4.1. These are about 40 GPa larger than the experimental
ones giving errors ranging from 3.7 % for c11 to 38 % for c12. The calculated bulk
modulus exceeds the experimental one by 10 %3.
3Just for comparison, AIMPRO gives B=466 GPa, which is 5 % too large.
Table 4.1: Some elastic properties of diamond: Comparison between calculated and experi-
mental data. The first three columns give the independent elastic constants cij of diamond.
The resulting shear modulus
µ
, the Lam´e constant
λ
, the Poisson’s ratio
ν
and the bulk mod-
ulus Bare calculated as Voigt averages following Eq. (1.49),(1.46), and (1.47). All values are
given in GPa, except for
ν
, which is dimensionless.
c11 c12 c44
µ
Voigt
λ
Voigt
ν
Voigt B=1
3(c11 +2c12)
Exp.a1076 125 577 536 84 0.068 442
DFTBb1116 172 608 554 118 0.088 487
aExperimental data by McSkimin and Bond [85] (ultrasonic waves) and Grimsditch and Ramdas
[86] (Brillouin scattering). Within the accuracy given here, both report the same values.
bValues obtained using the DFTB method.
4.2. The atomic scale modelling of dislocations
39
Some remarks on precision and errors: The large relative deviation for c12 seems
considerable, and in fact it is this error which gives rise to the rather large error in
the bulk modulus and the Poisson’s ratio. One has to keep in mind though, that the
quantities essential in dislocation related calculations are not Bor
ν
, but
µ
, 1 ±
ν
and 1 −2
ν
(see all expressions for energies and forces in Chapter 2). The respective
deviations are at maximum only 3.4 %. Consequently, the observed large errors for
the bulk modulus and the Poisson’s ratio have no adverse effect on the precision of
the calculations performed in this work.
4.2.1 The supercell-cluster hybrid as the model of choice
Having shown the capability of the DFTB method to describe the significant elastic
properties of diamond with acceptable accuracy, one now has to decide about the
way how to model dislocations with this method. Generally there are two main
approaches used to model defects in semiconductors atomistically: The cluster and
the supercell approach. For the case of dislocations this means that either an atom
cluster containing a single dislocation is considered or a supercell containing a dis-
location multipole. The long range elastic effects are treated differently in each case
but neither approach treats these effects rigorously, as will be discussed below.
`
B
A
Figure 4.2: Schematic sketch of a dis-
location in a cluster. A and B denote
the points of dislocation–surface in-
tersection.
The cluster approach: Here a single straight
dislocation is inserted into an atomic cluster. As
a result, the dislocation line intersects with the
surface of the cluster. Fig. 4.2 depicts a clus-
ter, whose surfaces are cut as crystallographic
planes. For small clusters it is advisable to im-
pose the displacements expected from elasticity
theory for a dislocation in an infinite crystal at
the surface of the finite cluster. In other words
one then has to embed the cluster in a continu-
ous medium, which represents the surrounding
crystal environment. However, if the cluster is
large enough, the difference with a freely relaxed surface is negligible [87].
In DFT-based methods where the electronic energy is explicitly included, the cluster
surfaces are usually terminated with hydrogen atoms to avoid electronic gap states
associated with dangling bonds or surface reconstructions, which might induce ad-
ditional strain in the cluster core.
The main advantage of the cluster model is the possibility to study a single and iso-
lated dislocation, but its principal disadvantage stems from the part of the disloca-
tion line which intersects the surface. The dislocation core structure will be distorted
near the surface, where it loses its periodicity. Thus, the cluster has to be consider-
ably extended in the direction of the dislocation line. Experience from modelling
dislocations in Si shows that clusters of around 700 atoms with approximately 6
repeat distances along the dislocation line are necessary, as applied in [2].
40
Chapter 4. Dislocations in Diamond
Figure 4.3: Schematic sketch of a dislocation
dipole in a supercell.
The supercell approach: Here a dislo-
cation dipole or multipole is placed in a
supercell. The periodic boundary con-
ditions require an overall Burgers vec-
tor of btot =0. Thus, the most sim-
ple configuration is a dipole with two
dislocations of opposite Burgers vector.
Fig. 4.3 depicts an example supercell
containing a dipole, indicating its peri-
odically repeated images in one dimen-
sion as an example. In the case of a
dipole, the volume per single dislocation is only half that of the cell. Since DFT
calculations are nowadays limited to a few hundred atoms, usually the two disloca-
tions are separated by a relatively small distance. A result of this is a considerable
dislocation–dislocation interaction within and across the cell boundaries to neigh-
bouring cells and in compound semiconductors also a considerable charge transfer
might occur. The elastic interaction can be described by a procedure similar to an
Ewald sum, but the resulting expressions are not easy to handle [88]. Blase et al. [79]
perform this summation by taking advantage of an analytical solution found for tilt
boundaries. However, the same authors conclude that the core reconstruction of a
dislocation depends on the stress state. This implies that the stress of the selected
supercell geometry influences the core structure and hence possibly the core energy.
Seeing the advantages and disadvantages of both the cluster and the supercell, it is
a rather straightforward idea to combine the advantages of both in a supercell-cluster
hybrid approach — minimising the problems associated with either of the preceding.
Figure 4.4: Schematic sketch of a dis-
location in a supercell-cluster hybrid
model.
The supercell-cluster hybrid approach:
Here the dislocation is placed in a model
which is periodic along the dislocation line,
however, it is non-periodic with a hydrogen-
terminated surface perpendicular to the line
direction [9,89]. This allows to maintain the
‘natural’ dislocation periodicity as a line defect
and at the same time avoid the interactions
between dislocations in different cells. The
single dislocation in the hybrid model is sur-
rounded by twice the amount of bulk crystal
compared to the pure supercell containing a dipole as in Fig. 4.3 — assuming both
models are of the same volume and neglecting surface effects. In other words, in
the hybrid model it is by far easier to give a good representation of the surrounding
bulk crystal. The latter in combination with the capability of modelling a single and
isolated dislocation (avoiding dislocation–dislocation interaction) makes the hybrid
model the ideal choice to describe line defects4.
4The hybrid approach — just as the pure cluster approach — is however not suitable for DFT
methods based on plane wave basis sets, since these methods have considerable problems dealing with
the large amount of vacuum surrounding the model: The then necessary high energy-cutoff makes the
cluster and hybrid models computationally very demanding. Thus whenever plane wave methods are
applied to describe dislocations, the pure supercell is still a very popular model, regardless of all its
4.2. The atomic scale modelling of dislocations
41
Table 4.2: The calculated intrinsic stacking fault energy
γ
in diamond and selected experi-
mental data.
Exp.aExp.bExp.cDFTBd
γ
(mJ/m2) 285 ±40 279 ±41 295 ±30 293
aData from weak-beam electron microscopy [90] .
bEstimated from dissociated glide dislocations in weak-beam electron microscopy [69].
cEstimated from extended nodes in weak-beam electron microscopy [69].
dCalculated with the DFTB method in a supercell-cluster hybrid.
The supercell-cluster hybrids used in this work are usually of double-period length
having an approximate radius of three or more lattice constants. As we will see
in the next section, this is sufficient to describe the energetics and structure of the
dislocations examined here. To check if these models are further capable of describ-
ing the intrinsic stacking fault that accompanies partial dislocations, we calculate its
energy in a supercell-cluster hybrid. Here the stacking fault energy is given as the
difference in total energy between a supercell-cluster hybrid containing the stack-
ing fault and a bulk-like hybrid with the same number of C and H atoms. The two
structures are geometrically optimised using a conjugate gradient algorithm until
all forces are well below 5 ×10−3eV/ ˚
A. As shown in Table 4.2, the method gives
results well within the experimental errors.
4.2.2 The elastic energy as a size-convergence criterion
Whenever modelling defects in semiconductors on an atomistic level, one has to test
whether the chosen model geometry is suitable and if its size — or in other words the
number of atoms considered — is sufficient to describe the defect as embedded in an
otherwise perfect crystal lattice. In particular for dislocations, which are extended
structural defects of the lattice with a considerable strain field, a sufficient model
size is crucial if the surface of the model is relaxed freely5. This convergence in size
can be checked in comparison with elasticity theory. As we know, the elastic energy
contained in a cylinder centred on the dislocation core shows logarithmic behaviour
in the continuum limit. Per unit length we obtain (see Section 2.3.1, Eq. (2.4)):
E(R)
L=k(
β
)|b|2
4
π
ln R
Rc+Ec
L,R≥Rc(4.1)
Unfortunately we cannot directly compare this expression with the total energy ob-
tained by DFT-based calculations, since this total energy contains all contributions
from the interaction of the atomic nuclei and electrons. Elasticity theory, however,
only gives the difference of energy with unstrained bulk material — the elastic en-
ergy. Hence the DFT total energy can only be compared with the elastic energy if the
total energy of the same group of atoms in perfect material is subtracted. The DFTB
method allows an easy determination of the total energy projected onto a single
deficiencies mentioned above.
5As mentioned earlier, a different approach is to embed the model in a bulk crystal by imposing
the displacements expected from continuum elasticity theory at the model’s surface.
42
Chapter 4. Dislocations in Diamond
[110]
[112]
Figure 4.5: The core re-
construction of the 30◦
glide partial projected
into the (111) glide plane.
The stacking fault area is
shaded. Left: The unre-
constructed core structure.
Right: The reconstructed
core structure. The recon-
struction bonds are about
18 % stretched compared
to bulk diamond.
atom i. The formation energy6projected onto atom iis then defined as the differ-
ence between the total energy Ei
tot and the total energy of the same type of atom in
perfect material Ebulk
tot :
Ei
f=Ei
tot −Ebulk
tot (4.2)
To compare now the spatial distribution Ei
fof the formation energy in a dislocated
supercell-cluster hybrid with the elastic energy as given in Eq. (4.1), one has to sum
over all atoms in a cylinder C(R,L)of radius Rand length Lfollowing the cor-
responding geometry. This procedure yields the radial formation energy per unit
length7:
Ef(R)
L=1
LX
C(R,L)
Ei
f−
γ
R
δ
ISF (4.3)
The second term on the right hand side is the stacking fault energy, which has to
be subtracted for models containing a Shockley partial dislocation. For such models
δ
ISF is defined to be 1 and 0 else.
The 30◦glide partial in diamond will now serve as an example to illustrate the con-
sistence between elastic energy and the DFT-based radial formation energy. In prin-
ciple, the low energy core structure of the 30◦glide partial in diamond is similar to
that shown in Fig. 3.7. However, the unreconstructed single-period core structure
is only metastable. As can be seen in Fig. 4.5, the three-fold coordinated core atoms
6If we consider a compound semiconductor crystal with two or more atomic species, then the
chemical potentials of each species might have to be taken into account, depending on the conditions
under which the crystal was grown.
7In principle, all DFT-based methods allow the definition of a spatially resolved density of total
energy
ε
tot(r)This is obvious for the electronic energy. But also the long range terms (Ewald sum
of the charges) can in principle be divided onto the respective charge positions (
δ
-function-like for
the nuclei [91]). Starting from this, we can define a formation energy density by subtracting the total
energy density of the respective bulk material:
ε
f(r) =
ε
tot(r)−
ε
bulk
tot
The sum in Eq. (4.3) then becomes an integral. Alternatively, it is also possible to project this energy
density onto atom positions using Voronoi polyhedra [92].
4.2. The atomic scale modelling of dislocations
43
R/R )
c
ln(
553 GPa (anisotropic)
E/L (eV/Å)
Ec
Rc= 3.1 Å
A
B
CA
BC
0.5 10−0.5−1
1.0
0.5
Figure 4.6: The radial formation energy of the 30◦glide partial for hybrid models of different
size. Left: Ef(R)/Las obtained with Eq. (4.3) for hybrid model A(182 carbon atoms), B(462
carbon atoms) and C(870 carbon atoms). The gradient of the dashed line corresponds to the
elastic energy in Eq. (4.1) with k(
β
) = 553 GPa, a result obtained in anisotropic elasticity the-
ory. The core radius Rcand energy Eccan be obtained directly from the graph. See text for a
detailed description. Right: Section through the three corresponding hybrid models ((1¯
10)
plane). The hydrogen termination at the surface is not shown. The inner white area marks
the core region (Rc=3.1 ˚
A). The subsequently surrounding areas give the dimensions of
the three respective models.
can form bonded pairs along the dislocation line, leaving each core atom now four-
fold coordinated. This reconstructed core is found approximately 0.6 eV per formed
atom-pair lower in energy than the unreconstructed core.
Just as the stacking fault energy in the last section, Ebulk
tot is determined in a bulk-like
model of the same shape and approximate size as the dislocated model. Fig. 4.6
shows Ef(R)/Lfor hybrid models of three different sizes (A182, B462 and C870
carbon atoms) plotted against ln(R). All three models are of double-period length
along the dislocation line ([1¯
10]direction). All three graphs show approximately
linear behaviour from a core radius Rc=3.1 ˚
A onwards. The corresponding core
energy is Ec/L=0.88 eV/ ˚
A. Of course the core radius is only vaguely defined and
it is somewhat arbitrary what one calls “approximately linear”. The gradient of
the linear part should be given as the pre-logarithmic factor in Eq. (4.1). Isotropic
elasticity theory yields k(
β
)=567 GPa and anisotropic theory as described in Ref. [93]
and [41] gives k(
β
)=553 GPa using the DFTB elastic constants from Table 4.1. The
corresponding gradient is drawn in Fig. 4.6 and the correspondence seems to be
reasonably good. The oscillations in all curves are an effect of the discrete sum and
they should disappear in the limit R→
.
From a certain radius onwards, the energy in each of the graphs increases rather
rapidly and non-linearly. This can be explained by the radius getting too close to
the surface of the respective model, where the strained bulk lattice is not repre-
sented sufficiently since surface effects dominate. For even larger radii the integra-
tion cylinder is not completely contained in the model and strain contributions from
the model’s edges lead to a sudden break-down of the formation energy. Hence one
can directly conclude from Fig. 4.6, that the maximum radius where the models give
a good representation of the surrounding strained bulk lattice — at least in terms of
elasticity — is about 4.2, 8.4 and 12.4 ˚
A for model A,Band Crespectively. In other
44
Chapter 4. Dislocations in Diamond
words, in principle the smallest model with 182 carbon atoms is able to describe the
dislocation as embedded in an otherwise perfect crystal. However, from curve A
alone this would be hard to see: The nearly correct gradient between Rcand 4.2 ˚
A
could be a random coincidence. As a consequence and to be on the safe side for dis-
locations with larger Rc, for all further calculations supercell-cluster hybrid models
similar in size to B(462 carbon atoms) will be used.
Some remarks on the effects of the k-point sampling: The structural calculations
were performed at the
-point only, but models with double-period length and a
1×1×2 or a 1 ×1×4k-point scheme gave similar energies and structures. We
found less than 4 % difference in bond length for reconstruction bonds, less than
1 % for bulk-like bonds and up to 8 % difference in formation energy depending on
the k-point sampling. Especially the latter may seem considerably large, however
for dislocations of the same type but with a different core structure, the deviation
seems to be an approximately constant shift. Also, compared to the large differences
in core energies for the different dislocations modelled in this work, the k-point
scheme related differences are negligible. Generally however, k-point sampling is
an important issue and its effects always have to be tested.
4.3 Core structures and energies
Hitherto the 30◦glide partial served as an example to illustrate the procedure used
in this work to model the core structure and energy of dislocations. This procedure
will now be applied to find the low energy core structures of the predominant per-
fect and partial dislocations of the {111}h110islip system in diamond. Therefore
the structures and their corresponding core energies and the long range elastic en-
ergy contributions are calculated.
4.3.1 Core structures
The minimum Burgers vectors band line directions `for the {111}h110islip system
in cubic material have been given in Chapter 3. With those known, one can construct
atomistic models of the corresponding dislocations by displacing the atoms of an
appropriate supercell-cluster hybrid according to band `. If required, an additional
half-plane of atoms or a stacking fault has to be introduced. The so obtained disloca-
tion core structures however, are nothing more but a mere first guess. As shown in
the last section, it can be energetically favourable if the core reconstructs by forming
new bonds and possibly breaking existing ones. Since core reconstructions can be
rather complicated, it is not guaranteed to find the overall lowest energy structures
by simply relaxing one starting structure by means of a conjugate gradient algo-
rithm. The reconstruction found then might just be one of many local minima of the
corresponding energy-surface. Thus, in this work usually several different starting
structures for a given combination of band `are structurally optimised. The low
energy core structures obtained are shown in Figs. 4.7 and 4.8. In the following, each
structure will be described and discussed briefly.
4.3. Core structures and energies
45
(a) shuffle screw (b) 60° glide (c) 60° shuffle
(d) 30° glide (e) 90° glide (SP) (f) 90° glide (DP)
[111]
[112]
[111][110] [110]
[112]
Figure 4.7: Dislocation core structures of the {111}h110islip system in diamond. The re-
laxed core structures of (a) the [1¯
10]shuffle screw dislocation, (b),(c) the undissociated 60◦
dislocation and (d) – (f) the glide set of Shockley partials are shown. For each structure,
the upper figure shows the view along the dislocation line projected into the (1¯
10)plane,
and the lower figure shows the (111)glide plane. In (a) two (111)planes are shown, one
in dark and one in light grey. The region of the intrinsic stacking fault accompanying the
partials is shaded in the respective structures. In case of the 90◦partial SP and DP denote the
single-period and double-period core reconstruction.
46
Chapter 4. Dislocations in Diamond
(b) 30° shuffle (I) (c) 30° shuffle (I’)
(d) 90° shuffle (V) (e) 90° shuffle (I) (f) 90° shuffle (I’)
(a) 30° shuffle (V)
[111]
[112]
[111][110]
[112]
[110]
B
A
Figure 4.8: Dislocation core structures of the {111}h110islip system in diamond. The re-
laxed core structures of (a) – (c) the 30◦shuffle partial and (d) – (f) the 90◦shuffle partial
are shown. For each structure, the upper figure shows the view along the dislocation line
projected into the (1¯
10)plane, and the lower figure shows the (111)glide plane. For both
the 30◦and 90◦partial several stable core structures are possible. We distinguish between
the vacancy structure (V), the interstitial structure (I) and an alternative interstitial structure
(I’). The region of the intrinsic stacking fault accompanying the partials is shaded. In (f) the
glide plane atoms are depicted in dark grey, and the underlying plane in light grey.
4.3. Core structures and energies
47
The b=1
2[1¯
10]screw dislocation (Fig. 4.7 (a)) : This dislocation shows the struc-
ture one would expect from locally shearing the crystal between two {111}glide
planes. Some of the core bonds approximately parallel to the dislocation line are
considerably weakened giving 17 % elongation due to the high stress and large core
displacements.
For the structure shown in Fig. 4.7 (a) the core line (drawn as a small circle in
the [1¯
10]projection) lies between two sets of two widely spaced planes (drawn as
dashed lines)8. Consequently the structure given here can be considered a shuffle
structure. However the pure screw is a special case, since it has no featured glide
plane. Dislocation motion from shuffle to glide and vice versa preserves the num-
ber of lattice sites — no interstitial or vacancy emission or trapping is necessary.
Also mixed shuffle-glide cores are possible. Hence the structure given here is just
an example and more details on the screw dislocation will be given in Section 4.3.3.
The 60◦dislocation (Fig. 4.7 (b,c)) : This dislocation is obtained by inserting (or
removing) a {111}half-plane of atoms. Fig. 4.7 shows the half-plane for the shuffle
and the glide cases framed with lines.
The relaxed core structure of the glide dislocation shows bond reconstruction of
the terminating atoms of the half-plane with neighbouring atoms. Reconstruction
bonds along the dislocation line impose a double-period structure.
The shuffle dislocation does not reconstruct. The terminating atoms possess dan-
gling bonds normal to the glide plane which give rise to electronic gap states, as
will be shown later. Further the local stress “below” the dislocation line stretches
one bond extremely by 80 % (dashed line), hence breaking it.
The glide set of Shockley partials (Fig. 4.7 (d–f)) : In the glide set of partial dislo-
cations each dislocation is accompanied by a stacking fault.
The 30◦glide partial reconstructs forming a line of bonded atom pairs. The recon-
struction bonds are 18 % stretched compared to bulk diamond. The structure is
double-periodic and has been described in the last section already.
For the 90◦glide partial there are two principal structures: A single-period (SP) and
a double-period (DP) reconstruction. The SP structure results simply from forming
bonds in the glide plane connecting the stacking fault region with the bulk region.
These bonds are 13 % stretched. The DP structure can be obtained from the SP
structure by introducing alternating kinks. Here the deviations from the bulk bond
length is slightly less. The DP structure was first proposed for silicon by Bennetto
et al. [94]. All atoms in both the SP and the DP structure are fully four-fold coordi-
nated. This is in contrast to the ethene-like cores proposed recently [83]. Since the
latter structures are higher in energy, here they are not discussed further9.
8In the case of a pure screw dislocation no inserted half-plane is present. However comparing the
lattice displacement associated with a screw as given by elasticity theory with the atom coordinates of
the relaxed atomistic model, one can still determine the localisation of the core line.
9However, they could still exist at raised temperatures and then might assist graphite formation.
48
Chapter 4. Dislocations in Diamond
The shuffle set of 30◦partials (Fig. 4.8 (a–c)) : To form the shuffle set of 30◦partials
one can either add or remove a row of atoms along the dislocation line. In contrast
with the 60◦dislocation, adding and removing a line of atoms leads to different
structures as the stacking fault is displaced with respect to the termination of the
inserted half-plane.
Upon the removal of one row of atoms from Fig. 4.7 (d), one obtains a vacancy struc-
ture (V) shown in (a). Unlike that found for silicon10, no strong bond reconstruction
occurs at the core but the distance between atoms Aand Bis reduced from 2.52 ˚
A to
2.08 ˚
A.
(b) and (c) are distinct interstitial structures formed by the addition of a row of atoms
to the core (I). In (b) the additional row of atoms shows almost planar sp2bonding
geometry while (c) shows strong bond angle distortions, but is at least metastable.
The stress induced by adding the row of atoms pushes the surrounding lattice apart
and breaks bonds.
The shuffle set of 90◦partials (Fig. 4.8 (d–f)) : The 90◦shuffle set is also formed
by climb of the glide partial. In a similar way to the 30◦shuffle it possesses both
vacancy and interstitial forms.
The vacancy structure (d) appears to be symmetric in the glide plane and a line of
bonded dimers is formed, leaving the dislocation with a double-period11. The bonds
of the dimers to neighbouring atoms in the glide plane, however, are weak and 22 %
stretched, and one row of dangling bonds remains unreconstructed.
We find two different metastable interstitial-like core structures. In the first one (e)
all atoms are fully four-fold coordinated, with appreciable bond length and angle
distortion, while the second (f) contains many dangling bonds. The main elements
in the latter structure are four membered [01¯
1]zig-zag chains forming a double-
period core.
4.3.2 Core energies
Having all the structures of the last section relaxed to their minimum in total energy,
it is now possible to obtain their core radii and energies via the procedure described
in Section 4.2.2. The results are given in Table 4.3, retaining the sequence of struc-
tures as in Figs. 4.7 and 4.8. The energy factors k(
β
)have been evaluated in a linear
fitting procedure for ln(R/Rc)>0. For comparison, the energy factors resulting
from linear isotropic as well as anisotropic elasticity theory are also given — both
those based on the elastic constants as obtained by the DFTB method and those
10 This structure, or a small segment of it, has been suggested as a candidate for the K1/K2 EPR
signal detected in plastically deformed Si [95]. More recently in Si, the single vacancy and a row of
vacancies at the 30◦partial have been considered theoretically by Lehto and ¨
Oberg [96] and by Justo
et al. [97]. These authors found a strong rebonding at least for the single vacancy which enabled all
dangling bonds to be eliminated.
11 An interesting aspect of this structure is its symmetry: If the corresponding glide structure
(Fig. 4.7 (e)) moves as a whole via applied stress, then the dislocation will proceed via a saddle point
with the same approximate symmetry as the vacancy structure (d) itself. Thus, the vacancies seem to
have pinned the dislocation into what is normally a saddle point structure.
4.3. Core structures and energies
49
Table 4.3: The calculated energy factors k(
β
), core radii Rcand core energies Ecof the dis-
locations of the {111}h110islip system in diamond. The corresponding structures can be
found in Fig. 4.7 and 4.8 in the same sequence. To facilitate comparison between different
dislocations with different core radii, the core energy E0
ccorresponding to a radius of 6 ˚
A
is introduced. In some cases — mainly the interstitial structures of the 30◦and 90◦partials
— the fluctuations in Ef(R)were rather large, leading to errors of around 20 % in the fitting
process. Affected values are marked with an asterisk.
shuffle screw 60◦glide 60◦shuffle
k(
β
)a(GPa) 582 570 591
k(
β
)isotropicb(GPa) 554 (536) 593 (566) 593 (566)
k(
β
)anisotropicc(GPa) 536 (524) 587 (561) 587 (561)
Rc(˚
A) 4.3 4.1 4.2
Ec/L(eV/ ˚
A) 3.32 2.13 2.60
E0
c(R=6˚
A)/L(eV/ ˚
A) 3.94 2.55 3.18
30◦glide 90◦glide (SP) 90◦glide (DP)
k(
β
)a(GPa) 573 601 599
k(
β
)isotropicb(GPa) 567 (546) 607 (576) 607 (576)
k(
β
)anisotropicc(GPa) 553 (536) 603 (574) 603 (574)
Rc(˚
A) 3.1 3.0 3.3
Ec/L(eV/ ˚
A) 0.88 1.32 1.18
E0
c(R=6˚
A)/L(eV/ ˚
A) 1.23 1.74 1.57
30◦shuffle (V) 30◦shuffle (I) 30◦shuffle (I’)
k(
β
)a(GPa) 637 605 762∗
k(
β
)isotropicb(GPa) 567 (546) 567 (546) 567 (546)
k(
β
)anisotropicc(GPa) 553 (536) 553 (536) 553 (536)
Rc(˚
A) 6.0 5.0 5.5
Ec/L(eV/ ˚
A) 2.55 3.31 4.32
E0
c(R=6˚
A)/L(eV/ ˚
A) 2.55 3.43 4.39
90◦shuffle (V) 90◦shuffle (I) 90◦shuffle (I’)
k(
β
)a(GPa) 642∗602∗640∗
k(
β
)isotropicb(GPa) 607 (576) 607 (576) 607 (576)
k(
β
)anisotropicc(GPa) 603 (574) 603 (574) 603 (574)
Rc(˚
A) 4.4∗4.1∗4.5∗
Ec/L(eV/ ˚
A) 1.89 3.69∗2.67∗
E0
c(R=6˚
A)/L(eV/ ˚
A) 2.09 3.94∗2.86∗
aFit to Ef(R)vs. ln(R/Rc)plot for R≥Rcfollowing Eq. (4.1).
bEvaluated using Eq. (2.5). The first number is given by the elastic constants obtained in the DFTB
method, the number in brackets by the experimental constants (for both see Table 4.1).
cEvaluated using linear anisotropic elasticity theory as suggested in [93] and [41].
50
Chapter 4. Dislocations in Diamond
based on the experimental constants. Some of the interstitial-type shuffle partial
dislocations show a rather uneven linear part leading to larger errors in the deter-
mination of the energy factors, the core radii and sometimes even the core energies.
These are marked with an asterisk. Models of larger diameter might help to increase
the accuracy.
As mentioned earlier, the determination of the core radii is a bit arbitrary. Hence
one should not draw strong conclusions from Rc. However, some general trends
are obvious: All the undissociated structures have similar core radii of about 4 ˚
A.
Especially in the case of the Shockley partials, the shuffle set appears to have con-
siderably larger core radii compared to the glide set. This should be kept in mind
since the reliability of the calculations is lower the larger the core radius gets.
A direct comparison of core energies only makes sense between dislocations of the
same angle
β
between Burgers vector and line direction. Further, since the core
radius Rcusually varies with different dislocations, one should not compare Ec=
E(Rc)but E0
c=E(R0)for an arbitrary radius R0≥Rccommon to all dislocations.
Here the choice is R0=6˚
A.
Since the screw dislocation will be discussed later separately, this section focuses on
the 60◦dislocations and their dissociation products, the 30◦and 90◦partials. The
next paragraphs give the general trends.
Glide and shuffle dislocations: As a first result the undissociated 60◦disloca-
tion appears to be most stable in the glide structure: The shuffle structure is
∼630 meV/ ˚
A higher in energy. Due to the variety of partial shuffle structures,
one cannot give a characteristic energy difference, however here the lowest energy
structures are also always of the glide type12.
Single- and double-period reconstructions: One of the few problems concerning
dislocation core structures in diamond which has been treated from a theoretical
point of view before is the relative stability of the SP and DP structures of the 90◦
glide partial: Nunes et al. [78] find the DP core to be 235 meV/ ˚
A lower in energy
in DFT calculations. Similarly, the low-stress quadrupole calculations of Blase et al.
[79] yield 169 – 198 meV/ ˚
A. This is all in good agreement with the ∼170 meV/ ˚
A
difference found in this work by means of the DFTB method and supercell-cluster
hybrid models13. This difference in energy between the SP and the DP structure is
rather small, suggesting that possibly both structures co-exist.
Vacancy and interstitial shuffle structures: Within the shuffle set of 30◦partials
as well as within the shuffle set of 90◦partials, the vacancy structures are clearly
12 The core energy of the 60◦perfect glide dislocation is slightly lower than that of a pair of 90◦and
30◦glide partial dislocations by 0.25 eV/ ˚
A suggesting that dissociation would be prohibited. How-
ever, the contribution of the long range stress field, to be discussed in Section 4.4, reverses this.
13 Comparing the different results obtained by different methods and models, one can discover a
trend here: The smaller the stress induced by the dislocation–dislocation interaction within the partic-
ular model geometry, the smaller is the energy difference between SP and DP. In Ref. [79] the stress is
already minimised by optimising a quadrupole in a supercell, giving a smaller difference than found
in [78]. And finally, supercell-cluster hybrid models with no dislocation–dislocation interaction give
the smallest difference (this work). However, one should keep in mind, that the approximations of
DFT used in the three approaches are rather different, so this trend could still be an artificial effect.
4.3. Core structures and energies
51
Table 4.4: A comparison of the DFTB results for the core radius and energy of the 90◦partial
dislocation (SP) with independent classical potential and LDA-pseudopotential calculations.
r0is the radius for which in Eq. (4.1)Ecbecomes zero. The core energy for LDAPP had to be
extrapolated to Rc=3˚
A to facilitate comparison.
TPaLDAPPbDFTBc
Rc(˚
A) 3.0 — 3.0
r0(˚
A) 0.004 0.4 0.38
E0
c(Rc=3˚
A)(eV/ ˚
A) 4.69 1.26 1.32
aWork by Nandedkar and Narayan [76] using Tersoff potentials in a cylindrical model.
bWork by Blase et al. [79] using LDA-pseudopotentials.
cCalculated with the DFTB method (see Table 4.3).
favoured by approximately 900 and 800 meV/ ˚
A respectively. This is a direct re-
sult of the high strain induced by the additional row of atoms in the case of the
interstitial-type partials.
Comparison with independent calculations: For a comparison of the core radii
and core energies with independent calculations we are restricted to the single-
period reconstruction of the 90◦partial, which has been discussed in the literature.
Results from different methods are shown in Table 4.4: Tersoff potential (TP)[76],
LDA-pseudopotentials (LDAPP)[79] and DFTB. In Ref. [79] (LDAPP) a different def-
inition of the core radius is applied — it is taken as the radius r0where in Eq. (4.1)
Ecbecomes zero. For details see also Appendix A. Since Rcas used in this work
cannot be determined from their published data, Table 4.4 gives the extrapolated
alternative core radius r0for the TP and DFTB results instead. Further, E0
c(Rc)had
to be extrapolated for LDAPP. As can be seen, the core radius and energy obtained
in DFTB are in very good agreement with the LDAPP results14.
The core radius Rcof the TP calculation is exactly the same as found in this work.
For the corresponding core energy however, Tersoff potentials yield an extremely
large value, exceeding DFTB and LDAPP by more than 300 %. This reflects the
difficulties in constructing an atomic potential to describe the energy of a heavily
distorted dislocation core.
14 However, the LDAPP calculations in Ref. [79] are computationally by far more expensive. And
also, being plane wave pseudopotential calculations, the used model was a pure supercell. Therefore the
effort to minimise and subtract the artificial dislocation–dislocation interaction inherent to the super-
cell was apparently rather large. This demonstrates that the appropriate choice of a good combination
of method and model is often more important than using the most exact approximation for the energy.
52
Chapter 4. Dislocations in Diamond
4.3.3 The screw dislocation — a special case
Hitherto the detailed discussion of the pure screw dislocation has been avoided and
only the very simple structure of the shuffle screw was briefly introduced. The pure
screw is a special case, since it has no featured glide plane and there is no such
thing as dislocation climb. But still, depending on the location of the dislocation
core line — between widely or between closely separated {111}planes — one can
distinguish between “shuffle” and “glide” structures in a similar way as in the case
of the perfect 60◦dislocation. This core line is defined by the radial displacement
field being centred at it. When constructing a pure screw dislocation, first this origin
has to be chosen, and then the surrounding atoms are displaced with respect to their
positions in a perfect undislocated lattice15.
For a [1¯
10]screw dislocation there are two sets of {111}planes containing its line
direction. Hence in terms of shuffle and glide, there are three unique possible types
as illustrated in Fig. 4.9 (Left): Shuffle, glide and mixed. The first two have shuffle
or glide character respectively with respect to both the horizontal and the inclined
set of {111}planes, whereas the mixed type has shuffle character with respect to
one and glide character with respect to the other set of {111}planes.
Core structures: The relaxed structure of the shuffle screw dislocation can be
found in Fig. 4.7 (a). The bond angles and lengths near the core are stretched but
all bonds are bulk-like and all atoms remain four-fold coordinated. This is a result
of the origin of the displacement field not intersecting with any bonds. This is very
different for the glide screw and for the mixed screw, where the origin lies exactly
on a row of bonds (see Fig. 4.9 (Left)). As a result, the two neighbouring atoms con-
nected by one of those bonds are displaced with respect to each other by half the
15 In cylindrical coordinates with r= (r,
ϑ
,z)centred on the dislocation line, the displacement field
is given as u(r) = (0, 0, |bs|
ϑ
/(2
π
) ). Varying
ϑ
from 0 to 2
π
in a Burgers circuit then yields the correct
Burgers vector, in cylindrical coordinates bs= (0, 0, |bs|).
180 meV/Å
E(eV/Å)∆
1
0
[111]
[112]
860 meV/Å
560 meV/Å
M
G
GM
SS
Figure 4.9: The three unique types of the 1
2[1¯
10]screw dislocation: Shuffle, glide and mixed
(labelled S,Gand Mrespectively). Left: The core positions projected into the (1¯
10)plane.
The two sets of {111}planes containing the line direction are indicated. Right: Relative core
energies of the three different types and barriers between them. The x-axis gives an arbitrary
coordinate for the transformation G→S→M.
4.3. Core structures and energies
53
[111]
[110]
[112] [112]
A
B
BA
Figure 4.10: The 1
2[1¯
10]
glide screw dislocation
(Gin Fig. 4.9). Atoms
Aand B(and equiva-
lent atoms along [1¯
10])
are only three-fold coor-
dinated. Left: The view
projected into the (1¯
10)
plane. Right: The view
projected into the (111)
plane.
Burgers vector along [1¯
10].
In case of the glide screw this leads to a planar sp2bonding situation for both atoms.
Subsequent structural optimisation pushes the two atoms apart, resulting in an ap-
preciable C–C bond length. Fig. 4.10 shows the relaxed structure, which is basically
that of two 30◦partial glide dislocations in the same Peierls valley. However there
is no bond reconstruction along the dislocation line.
For the mixed-type screw dislocation the two atoms closest to the origin of the dis-
placement are widely separated in the [1¯
10]projection. Thus, after displacement
their distance is large enough to break the connecting bond. Subsequent relax-
ation does not lead to a bond reconstruction. The mixed structure was suggested
by Koizumi et al. [98], and for III-V compound semiconductors the same authors
claim it to be more stable than the shuffle structure.
Energetics and stability: To model the transformation from G→S→M, the
atoms of a bulk hybrid model were displaced according to the process described
earlier in this section. Varying the origin of the displacement field between Gand
Sor Sand Mallows one to scan for barriers between start and final position. The
structure is then relaxed for each point of the scan grid16.
Fig. 4.9 (Right) shows the relative core energies of the glide, the shuffle and the
mixed type. The glide screw appears to be the lowest energy structure. The shuffle
screw has a ∼860 meV/ ˚
A higher core energy, is however at least metastable with
a barrier of ∼180 meV/ ˚
A. Finally the mixed type exceeds the shuffle type by ∼
560 meV/ ˚
A. The latter structure is unstable and represents a saddle point between
two adjacent sites of the shuffle core.
16 To prevent the origin of the displacement field moving towards a stable or metastable position
during relaxation, the two atoms adjacent to Gor respectively Mare constrained parallel to the (1¯
10)
plane. The curve in Fig. 4.9 (Right) represents about 20 grid points for G→Sand 10 grid points for S
→M. Details on how to model barriers for dislocation motion will be given in Section 4.5.
54
Chapter 4. Dislocations in Diamond
4.4 The dissociation of dislocations in diamond
In Section 3.2 the two basic dissociation reactions of the perfect 1
2[1¯
10]screw and
the 60◦dislocation have been introduced. Furthermore, shuffle and glide set dis-
locations with an edge component can be transformed into each other by adding
(or removing) one row of atoms along the dislocation line (climb). A perfect glide
dislocation can always dissociate into two glide partials. Hence if a perfect shuffle
dislocation dissociates, the extra row of atoms involved can go with either partial.
Therefore, the following conservative (not involving net absorption or emission of
vacancies or interstitials) dissociation reactions are allowed:
1
2[1¯
10]screw −→ 30◦glide +ISF +30◦glide
1
2[1¯
10]screw −→ 30◦shuffle +ISF +30◦shuffle
60◦glide −→ 30◦glide +ISF +90◦glide
60◦glide −→ 30◦shuffle +ISF +90◦shuffle
60◦shuffle −→ 30◦glide +ISF +90◦shuffle
60◦shuffle −→ 30◦shuffle +ISF +90◦glide
With the core energies of the participating partial dislocations at hand (Table 4.3),
one can distinguish which combinations of partial dislocations are energetically
favoured and which are not. Based on this it is possible to rule out certain disso-
ciation reactions. The following reactions are energetically favoured17:
1
2[1¯
10]screw −→ 30◦glide +ISF +30◦glide
60◦glide −→ 30◦glide +ISF +90◦glide (DP/SP)
60◦shuffle −→ 30◦glide +ISF +90◦shuffle (V)
At this point it still remains unclear if dissociation occurs or not. For all reactions the
Frank criterion (Eq. 2.9) holds. This criterion assumes a dissociation with almost in-
finite separation, so that contributions from dislocation–dislocation interaction can
be neglected. The dissociation into partials, however, involves the creation of an in-
trinsic stacking fault between the two partials. Hence an almost infinite separation
is prohibited by the then almost infinite stacking fault energy. At a finite partial–
partial separation, however, all energy contributions have to be taken into account,
including the partial–partial interaction. As a consequence, the Frank criterion is
not sufficient to decide in favour of dissociation into partials.
In the next two sections this problem will be addressed by elasticity theory and
atomistic modelling.
17 The dissociation of the screw into two 30◦shuffle partials is 2.6 eV/ ˚
A higher in energy than the
dissociation into two glide partials (twice the energy difference between 30◦shuffle (V) and 30◦glide).
Similarly, the 60◦glide dislocation will dissociate into two glide partials. Comparing the sum of core
energies of a 30◦glide and the lowest energy 90◦shuffle with that of the lowest energy 30◦shuffle
and 90◦glide favours the first by ∼800 meV/ ˚
A. Here elasticity theory yields at least qualitatively a
similar result: Assuming the vacancy (both lowest energy shuffle partials are vacancy structures) to
be a centre of isotropic contraction or expansion, then the vacancy only interacts with the hydrostatic
component of the internal stress, which for a straight dislocation is produced by its edge component
only [41]. Thus, a vacancy will be preferentially drawn to the partial with the Burgers vector with the
largest edge component — in this case the 90◦partial dislocation.
4.4. The dissociation of dislocations in diamond
55
(eV/Å)E/L
ISF width (Å)
E /L
ISF
E /L
AB
de
0 30 402010
0
−0.5
Figure 4.11: The competing en-
ergy contributions for dissocia-
tion into partials: The stacking
fault energy EISF/L=
γ
Rand
the elastic partial–partial inter-
action EAB/L∝ln(R/R0)as
given in Eq. (2.6) for a 90◦and a
30◦partial. The sum of both has
a minimum at the equilibrium
partial separation deas given in
Eq. (4.6). In elasticity theory EAB
is known except for a constant
offset.
4.4.1 The equilibrium separation of partials
When it comes to the dissociation of perfect dislocations into Shockley partials, then
the two competing energy contributions in the elastic limit are the stacking fault
energy and the elastic partial–partial interaction energy. The first grows proportion-
ally with the partial–partial separation R(energy per unit length: EISF/L=
γ
R) and
the interaction energy lessens logarithmically with R(Eq. (2.6)). Fig. 4.11 shows a
preliminary energy balance: Unfortunately the offset ∝ln(R0)of the elastic interac-
tion energy is unknown and consequently at this stage a complete energy balance is
still impossible. In equilibrium however, the force FISF/L=−
γ
resulting from the
stacking fault and the partial–partial repulsion FAB/L, as given in Eq. (2.7), cancel
each other and one easily obtains the equilibrium partial separation:
de=
µ
2
πγ
(bA·`) (bB·`)+(bA×`)·(bB×`)
1−
ν
(4.4)
Thus the dissociation of a 1
2[1¯
10]screw or a perfect 60◦dislocation leads to the fol-
lowing equilibrium stacking fault widths:
de(screw) =
µ
8
πγ
3−1
1−
ν
|b30|2(4.5)
de(60◦) =
µ
4
πγ
1
1−
ν
|b30||b90|(4.6)
Table 4.5: The calculated stacking fault widths for the dissociated 1
2[1¯
10]screw and 60◦dis-
location in diamond given by Eq. (4.5,4.6).
Experimental dataaDFTBb
de(screw)(˚
A) 30.5 30.4
de(60◦)(˚
A) 33.9 35.0
aCalculated using the experimental elastic constants and the arithmetic mean of the experimental
stacking fault energies (see Tables 4.1 and 4.2).
bCalculated with the DFTB elastic constants and stacking fault energy (see Tables 4.1 and 4.2).
56
Chapter 4. Dislocations in Diamond
glide screw 60° glide
(2)
(4)
(0)
(4)
(2)
(0)
Figure 4.12: The first stages
of dissociation of the glide
screw and the 60◦glide
dislocation. The view is
projected into the (1¯
10)
plane. The top structure la-
belled (0) shows the respec-
tive undissociated perfect
dislocation. The stacking
fault in the second (2) and
fourth (4) dissociation step
is shaded. Left: The disso-
ciation of the 1
2[1¯
10]glide
screw into two 30◦Shock-
ley partials. Right: The dis-
sociation of the 60◦glide
dislocation into a 90◦(SP)
and a 30◦Shockley partial.
Table 4.5 shows the resulting widths obtained with the DFTB stacking fault energy
and elastic constants as well as with the averaged experimental values. The widths
vary between 30 and 35 ˚
A. This is an order of magnitude larger than the core radii,
giving confidence of being in the elastic limit, where the stacking fault energy and
the elastic partial–partial interaction are the dominating contributions. Consistently,
dissociation in the {111}h110islip system of diamond has been reported in weak-
beam electron microscopy [69]. The dissociation widths vary from 25 to 42 ˚
A. The
large variation in widths is a reminder that pinning of dislocations by point defects
may be important in preventing the equilibrium stacking fault width being attained.
Also, the energy minimum at the equilibrium distance is rather flat, as can be seen
in Fig. 4.11.
Even though there is experimental proof for dissociation, from a theory point of
view it can still not be conclusively demonstrated whether dissociation is energeti-
cally favoured overall, since the offset in the interaction energy is unknown in elas-
ticity theory: The possibility persists that even though deis a local minimum in
energy, the undissociated dislocation is even lower. Hence atomistic calculations
are called for to model the first steps of dissociation.
4.4.2 Modelling the first stages of dissociation atomistically
Dissociation of a perfect dislocation into two partials bordering a stacking fault
means creating a defect whose core is considerably extended within the glide plane.
Thus, describing such a defect requires larger hybrid models than describing just
one isolated perfect or partial dislocation. The models used here for the dissocia-
tion of the 1
2[1¯
10]glide screw and the 60◦glide dislocation contain about 600 carbon
atoms. Without the core regions of the partials getting too close to the hybrid’s
surface, this size allows dissociation up to the fourth step — which corresponds
to a stacking fault width of 8.75 ˚
A. Fig. 4.12 shows the relaxed geometries for the
4.4. The dissociation of dislocations in diamond
57
(eV/Å)E/L
ISF width (Å)
(1)
(2)
(3)
(4)
(4)
(3)
(2)
(1)
60°
screw
1.06 eV/Å
680 meV/Å
180 meV/Å
(0)
0 10 20 30
0
−0.4
−0.8
0.4
Figure 4.13: The dissociation energy of the glide screw and the 60◦glide dislocation. The
relative atomistic DFTB energies of the first four dissociation steps and of the undissociated
dislocation are labelled (1) to (4) and (0) respectively. Zero energy is set to the undissociated
dislocations. The solid lines represent the sum of the stacking fault energy and the elastic
interaction energy as given in continuum theory (Eq. (2.6)). Shading below ∼6˚
A indicates
the region where the two core radii of the respective partial dislocations overlap — the
region where continuum elasticity theory fails.
respective undissociated dislocation and the second as well as the fourth dissoci-
ation step. The corresponding discrete relative energies are presented in Fig. 4.13.
Now, with the atomistic results at hand, the unknown energy offset ∝ln(R0/˚
A)
for the elastic interaction energy EAB/Lin Eq. (2.6) can be found easily by adjusting
EAB/L+EISF/Lto the atomistic DFTB energy of the fourth dissociation step. The
procedure yields R0=1.7 ˚
A and R0=4.4 ˚
A for the screw and the 60◦dislocation re-
spectively. In Fig. 4.13 the resulting continuous energies are drawn as solid lines for
both dislocation types. One can observe that the atomistic calculation and contin-
uum theory still agree for the third stage of dissociation. For smaller stacking fault
widths in the region of overlapping dislocation core radii, however, the deviation
becomes obvious and finally for R→0 the continuum result diverges.
Both for the screw and for the 60◦dislocation the final dissociation energy at equilib-
rium separation between the partials is clearly below the energy of the undissociated
dislocation (−1.06 eV/ ˚
A and −0.680 eV/ ˚
A respectively). Hence also from a theory
point of view, dissociation into partial dislocations is strongly favoured18. A striking
difference between the dissociation of the two types of dislocations is the presence
of an energy barrier of ∼180 meV/ ˚
A to initiate dissociation of the 60◦dislocation,
whereas for the screw there is no such barrier. This barrier should have a consider-
able effect, since it is approximately 5 ˚
A wide and cannot be overcome in a single
step. This might explain the presence of undissociated 60◦dislocations as observed
in weak-beam electron microscopy (compare Fig. 4.1 and Ref. [69]).
18 In the case of the dissociation of the 60◦glide dislocation only the single-period 90◦partial is
considered here. The energy of the respective systems containing a double-period 90◦instead would
be approximately 170 meV/ ˚
A lower.
58
Chapter 4. Dislocations in Diamond
(0) (1) (2) (3)
Figure 4.14: Dislocation glide of
partials by kink formation and
migration. The Peierls valleys
are drawn as dashed lines. (0)
shows the straight dislocation
line in one valley, in (1) a kink
pair has formed, whose single
kinks then migrate along the dis-
location line (2). Finally, in (3),
the shown segment of the dis-
location has moved to the next
Peierls valley.
4.5 Kinked Shockley partials and dislocation glide
All energies given in the last section in the context of dissociation into partial dislo-
cations are energies in the Peierls valleys only. Thus, each dissociation step requires
to overcome the intermediate Peierls hill. Unless at very high temperature or under
very high stress, the partial dislocations will not move over such hill as a whole, but
piecewise. This motion of the glide partials will be modelled in this section.
4.5.1 Dislocation glide by kink formation and migration
Glide motion arises from stress acting on the dislocation core. This stress can result
from external forces on the crystal or, as discussed in the last section, from the in-
teraction with neighbouring dislocations. It leads to the formation of kinks at the
dislocation line. When the stress is insufficient to overcome the Peierls barrier, kinks
must be generated by a thermal process and motion occurs by their migration along
the dislocation line. Formation and migration then controls the dislocation velocity
in the absence of pinning centres or other obstacles. Fig. 4.14 schematically shows
the process. Since kinks on a dislocation line can only be created in pairs, their
density in thermodynamic equilibrium is controlled by the double-kink formation
energy 2Ef, whereas their rate of motion depends on the kink migration barrier Wm.
In the case when strong obstacles like point defects and impurities are absent the
activation energy for the glide motion of short dislocation segments Qis given as
the sum of 2Efand Wm19. One obtains for the glide velocity [41]:
vdisl ∝e−Q/(kT)with Q=2Ef+Wm(4.7)
For long dislocation segments however, the activation energy is controlled by the
sum of the single-kink formation energy Efand the kink migration barrier. The
exact expressions and their rather lengthy derivations shall not be given here, since
in this work they are only of marginal interest. For more details see for example
Ref. [41].
19 If the dislocation is pinned by obstacles and subsequently released, then it is the corresponding
pinning energy, which controls the velocity [99,100].
4.5. Kinked Shockley partials and dislocation glide
59
In the next two sections the two crucial parameters Efand Wmof dislocation glide
will be modelled atomistically for the two glide partials involved in the dissociation
processes discussed earlier. An investigation of reconstruction phase shifts in the
context of dislocation kinks is excluded here and only the lowest energy structures
are introduced 20.
4.5.2 The 90◦glide partial
[110]
LK RK
Figure 4.15: A double-kink in the 90◦
partial (SP). The stacking fault is
shaded.
As discussed earlier in Section 2.6.2, kinks in
a dislocation interact with each other. There-
fore, to minimise the effects of the interaction
of the kink-pair itself with its periodic images
along the dislocation line, in this section the
supercell-cluster hybrid model is now aban-
doned in favour of pure cluster models. For
the 90◦glide partial (SP) a C420H214 model
is used, which extends 6 lattice translations
along the dislocation line. This geometry con-
veniently allows to introduce a double-kink
with a kink–kink separation of two lattice
translations (√2a0), where each kink is sepa-
rated √2a0from the surface. Fig. 4.15 shows
a section of the (111) glide plane containing the relaxed and reconstructed double-
kink. All atoms are four-fold coordinated and within the (111) glide plane, in other
words neglecting the different stacking on either side of the dislocation line, the
structures of the left kink (LK) and the right kink (RK) are identical. The formation
energy of the kink pair with separation L=√2a0is obtained as the difference in
total DFTB energy between the model containing the kinked dislocation and a clus-
ter of the same stoichiometry and geometry containing the unkinked dislocation
segment. The calculation yields the formation energy Epair(L=√2a0) = 1.00 eV.
Assuming the formation energy of a single left kink and a single right kink to be
the same (Ef(LK)≈Ef(RK)≈Ef) and further including the energy of the faulted
region generated alongside with the kinks, Eq. (2.12) reads:
2Ef=Epair(L)−ELK,RK(L)−aL
γ
(4.8)
Here ais the kink height and ELK,RK(L)the elastic kink–kink interaction energy of
the 90◦partial (Eq. (2.11)):
ELK,RK(L) = −
µ
a2
8
π
L
1−2
ν
1−
ν
|b90|2(4.9)
With the kink height a=p3/8a0given as the Peierls valley distance, one obtains a
single-kink formation energy of Ef=520 meV — for the chosen kink–kink separa-
tion, elastic interaction and the contribution of the stacking fault nearly cancel each
other.
20The discussion of the structures of kinks and phase shifting defects as given in Refs. [101,102] for
silicon can be directly applied to diamond as well. The lowest energy structures are basically the same.
60
Chapter 4. Dislocations in Diamond
= 2.48 eV
m
W
Kink separation
Ef
2
KK’
S
atom 1
= 1.04 eV
Energy (eV)
1
2
0
atom 2
K’
KS
1
2
3
4
K’
S
K
Energy (eV)
0
Figure 4.16: Kink migration at the 90◦(SP) glide partial. Upper panel: The three main stages
of one elementary kink migration step. The relaxed structures of the starting kink K, the
saddle point Sand the migrated kink K’ are shown projected into the glide plane. The
faulted region is shaded and arrows indicate the motion of the two atoms involved.
Lower left panel: The energy surface of the corresponding process leading from Kvia Sto K’.
The two parameters are defined via the two atoms involved (see text and Fig. 4.17).
Lower right panel: Schematic representation of the energy of the glide process. As shown in
Fig. 4.14, a kink pair is formed and subsequent migration of the two kinks enlarges their sep-
aration. K,Sand K’ are labelled for an arbitrary migration step. The dashed line connecting
the minima represents the formation energy of the kink pair. The energy contribution of the
expanding stacking fault is not included in the graph.
To model kink migration along the dislocation line is more complicated. The ele-
mentary migration step has to be parameterised so that the minimum energy path
leading from the starting to the final structure can be found and the barrier deter-
mined. Fortunately, all kink migration steps discussed in this work involve mainly
the motion of two atoms, which break bonds with neighbouring atoms and form
new bonds in the migration process. All other atoms remain at their respective lat-
tice sites. For a right kink at the 90◦(SP) partial this is illustrated in Fig. 4.16 (upper
panel). The parametrisation of the motion (or diffusion) of one of the involved atoms
is explained in Fig. 4.17. Varying the two parameters independently, yields a two-
dimensional energy surface as given in Fig. 4.16 (lower left panel)21. This procedure
was applied to a cluster containing a dislocation segment with a single right kink, as
well to one with a single left kink. The migration barrier appears to be Wm=2.48 eV
for both.
21 The energy surface shown was obtained at 10 ×10 points in the two-dimensional parameter space
by relaxing the whole structure subject to constraining the two primary atoms to lie each in a plane
perpendicular to the connecting line between their respective initial and final positions. In the vicinity
of S, the parameter mesh was refined by a factor of 10.
4.5. Kinked Shockley partials and dislocation glide
61
B
(2’ )
(3’ )
(2)
(3)
(1)
(1’ )
A
Figure 4.17: The parametrisation of a diffusing atom. A
and Bare the positions of the moving atom in the relaxed
starting and relaxed final structure respectively. The move-
ment of each atom is modelled as follows: Starting at A,
the atom is displaced towards B. The resulting position is
(1). The whole model is relaxed constraining the moving
atom to a plane perpendicular to the connecting line be-
tween Aand B. The resulting position is (1’). This process
is repeated until Bis reached. The structural configura-
tions corresponding to (1’),(2’),(3’), . ..then lie on a low
energy path from Ato B. And the movement of one atom
can be parameterised by its distance from Aprojected onto
the connecting line between Aand B.
With the kink migration barrier as well as the formation energy now at hand, the
activation energy in the kink migration process evaluates to Q90 ≈3.5 eV for short
dislocation segments following Eq. (4.7). The energy corresponding to the whole
glide process of the 90◦(SP) glide partial is schematically shown in Fig. 4.16 (lower
right panel) for the first four kink migration steps. There the dashed line connecting
the minima represents the formation energy of the kink pair and the solid line the
energy of the minimum energy path for glide motion. The first few minima are
considerably lower due to the attractive kink–kink interaction. Since the elementary
processes are the same for kink migration along the 90◦(DP) glide partial, a similar
barrier and formation energy can be expected.
4.5.3 The 30◦glide partial
After having gone in detail through the simple case of kink formation and migration
along the 90◦(SP) glide partial, let us now discuss the the equivalent processes at the
30◦glide partial. With the latter partial adopting a double-period core reconstruc-
tion, the situation is far more diverse. Depending on the kink position relative to
the 30◦partial reconstruction bonds two different structures for the left kink as well
as for the right kink are possible. These different structures will be labelled LK1,
LK2 and RK1 and RK2 for the two left and right kinks respectively. Fig. 4.18 shows
sections of the (111) glide plane containing relaxed and reconstructed double-kinks.
Like in the case of the 90◦partial, for each kink all atoms are four-fold coordinated.
To obtain the kink formation energies, clusters similar to those for the 90◦partial are
used. However, unlike at the 90◦partial, the structures of left and right kinks are
very different, and therefore a similar formation energy cannot be expected. Since
in a cluster the formation energy of a single-kink cannot be determined, it is only
possible to obtain the sum of two formation energies similar to Eq. (4.8):
Ef(LKX) + Ef(RKY) = Epair(X,Y,L)−ELK,RK(L)−aL
γ
(4.10)
Xand Ycan be 1 or 2 (compare Fig. 4.18). For the 30◦partial the elastic kink–kink
interaction energy evaluates as (Eq. (2.11)):
ELK,RK(L) = −
µ
a2
32
π
L
4+
ν
1−
ν
|b30|2(4.11)
62
Chapter 4. Dislocations in Diamond
LK1 RK1 LK2 RK2
Figure 4.18: Double-kinks in the 30◦partial. The stacking fault is shaded. Left: Left kink and
right kink of type 1 separated by three lattice translations. Right: Left kink and right kink of
type 2 separated by three lattice translations.
After calculating the formation energies of all possible four combinations
(LKX,RKY) comparison yields:
•Ef(LK1) + Ef(RK1) = 3.00 eV
•Ef(LK2) = Ef(LK1) + 2.18 eV
•Ef(RK2) = Ef(RK1) + 0.78 eV
LK1 and RK1 are the lowest energy kinks22. Thus in the kink migration pro-
cess it is an LK1-RK1 double-kink that will be preferentially formed at the initial
stage. However, subsequent migration inevitably involves the two high energy
kinks as intermediate structures. As a consequence, four migration barriers have
to be determined. The corresponding migration steps LK2 →LK1 →LK2’ and
RK2 →RK1 →RK2’ are shown in Fig. 4.19. As one can see, LK2 →LK1 is just the
reverse process of LK1 →LK2’ since all structures involved are symmetric within
the glide plane if the different stacking on either side of the dislocation is ignored.
The same holds for RK2 →RK1 and RK1 →RK2’. As expected, the two barriers
for left kink migration are found identical within 1 % error and so are the two right
kink barriers:
•Wm(LK) = 3.49 eV
•Wm(RK) = 2.66 eV
Fig. 4.19 (lower panel) shows the resulting energy of the glide process at the 30◦
partial for a moving right and left kink separately. In principle, all which has been
said for the 90◦partial applies here as well. However having two different barriers
for left and right kink migration respectively results in a modified expression for the
partial velocity. Eq. (4.7) is replaced by [102]:
vdisl ∝e−Ef(LK1)+Ef(RK1)
kT e−Wm(LK)
kT +e−Wm(RK)
kT (4.12)
22 There is no apparent reason why LK2 is so much higher in energy. Nunes et al. [102] encounter
a similar phenomenon for the right kink in silicon, and attribute it to the medium-range behaviour of
the associated strain field.
4.5. Kinked Shockley partials and dislocation glide
63
Kink separationKink separation
LK2’ LK2
LK1 RK1
RK2 RK2’
2
3
4
5
6
3.00 eV 3.00 eV
0.78 eV
2.18 eV
left kinks right kinks
Energy (eV)
2.66 eV3.49 eV
1
0
LK2 LK1 LK2’
RK2 RK1 RK2’
Figure 4.19: Kink migration at the 30◦glide partial. Upper panel: The elementary kink mi-
gration steps of the left kink LK2 →LK1 →LK2’. The relaxed structures of a starting kink
LK2, and the subsequent kinks LK1 and LK2’ are shown projected into the glide plane. The
saddle point structures are not given. The faulted region is shaded and arrows indicate the
motion of the two involved atoms. Middle panel: The elementary kink migration steps of the
right kink RK2 →RK1 →RK2’. Lower panel: Schematic representation of the energy of the
glide process. As shown in Fig. 4.14, a kink pair is formed and subsequent migration of the
two kinks enlarges their separation. Since for the 30◦partial the migration barriers differ
for left and right kinks, the corresponding processes are shown separately. The dashed line
connecting the minima represents the formation energy of the respective lowest energy kink
pair. The energy contribution of the expanding stacking fault is not included.
Due to its considerably lower migration barrier, the right kink is more mobile than
the left kink and dominates the partial velocity. However, unlike in the case of the
90◦partial, where the migration barriers clearly control the velocity, here the rather
large kink formation energy plays an almost equal role.
As can be seen easily now, the 90◦partial appears to be the by far more mobile partial
dislocation, and hence it can be expected to move faster under applied stress and
raised temperatures. Overall, the activation energies found are rather large and
under low pressure glide motion will only be observed at high temperatures.
64
Chapter 4. Dislocations in Diamond
Comparison with independent theoretical work Similar atomistic calculations of
kink formation energies and mobilities have been reported in Ref. [102] for silicon
only, and thus cannot be compared directly. The structures found for silicon are very
similar to those found for diamond in this work. However, formation energies and
barriers appear to be considerably lower in silicon than they are in diamond, which
reflects diamond being of course the by far harder material. Also, for silicon the left
kinks appear to be the more mobile species at the 30◦partial. It should be noted,
that all modelling in Ref. [102] utilises supercells, however, the resulting kink–kink
interaction terms are apparently ignored.
For the 90◦partial in diamond theoretical results for the activation barrier Q90 vary
between 2.9 eV for a soliton mechanism as described in Ref. [80] and 3.3 eV in
Ref. [77] — both lower than the 3.5 eV found here.
In this work, as well as in all the theoretical work cited above, the effects of point
defects as obstacles to dislocation motion are ignored. However, point defects might
have a major influence on glide motion [99,100]. This has been discussed for silicon
in Refs. [2,3].
4.6. Electron microscopy — a first link to experiments
65
4.6 Electron microscopy — a first link to experiments
So far only the theoretical aspects and in particular the modelling of dislocations
have been discussed. Results were mainly compared with other independent cal-
culations. This is of course legitimate, but the capabilities of the models and ap-
proaches used and the quality of the results can only be judged in comparison with
experiments. Therefore this section will establish a first link to experimental find-
ings beyond the mere proof of the presence of dislocations in diamond.
As already mentioned in the previous sections, 60◦and screw dislocations in dia-
mond and their dissociation have been observed early on. A typical weak-beam
TEM image was given in Fig. 4.1 (1983, [69]). These early studies cannot yield any
information concerning the atomic core structure of dislocations. However, electron
microscopy has improved since and high resolution imaging nowadays allows to vi-
sualise the atomic structure of dislocations in diamond to some extent (Appendix B).
Fig. 4.20 depicts a high-resolution transmission electron microscopy (HRTEM) im-
age of a dissociated 60◦dislocation in diamond. This, as all further HRTEM imaging
in this section and also the corresponding image simulations and part of the analysis
were performed by Bert Willems and Oleg Lebedev (EMAT, University of Antwerp).
Assuming both dislocations in Fig. 4.20 are minimum Burgers vector dislocations,
the projected Burgers vector allows an easy identification of the left one as a 90◦and
the right as a 30◦Shockley partial. However, the contrast pattern does not identify
single atoms. Also an exact localisation of the two dislocation cores is impossible.
The latter is possibly a result of the dislocation not being straight through the whole
thickness of the layer, but kinked back and forth. This then gives a rather diffuse
image of the core region. The intrinsic stacking fault in between the two partials can
be identified clearly though. The stacking fault is at least 35 ˚
A wide, but might be
considerably larger. Given the rather flat energy minimum for the predicted equi-
librium width of 35 ˚
A in Fig. 4.13, such variation seems reasonable — especially
10 Å
30°
90°
[112]
[111]
Figure 4.20: Cross sectional HRTEM image of a dissociated 60◦dislocation in natural brown
type IIa diamond. The two Burgers circuits around the 90◦(left) and 30◦(right) partial are
drawn projected into the (1¯
10)plane, yielding a projected Burgers vector of 1
6[¯
1¯
12]and
1
12 [¯
1¯
12]respectively. The glide plane is marked by two small arrows. (HRTEM performed
by Bert Willems, EMAT, University of Antwerp.)
66
Chapter 4. Dislocations in Diamond
since preceding sections predict considerable barriers between two adjacent Peierls
valleys.
Figure 4.21: Simulated HRTEM image of
the 30◦glide partial with a defocus
f=
−40 nm and a sample thickness of 4 nm.
Atom positions are indicated as circles.
To relate the observed contrast patterns to
the atom positions calculated for various
dislocation core structures in this work,
TEM image simulation is a useful tool.
Some details on the underlying calculations
are given in Appendix B.2. Fig. 4.21 shows
a simulated HRTEM contrast pattern for a
given defocus and sample thickness. The
dislocation is the 30◦glide partial as shown
in Fig. 4.7 (d). The input atom coordi-
nates for the simulations are the relaxed
atom positions of the corresponding hybrid
model. Simulated images like these can
now be compared with experimental im-
ages to identify atomic core structures.
Fig. 4.22 gives the simulated images for the 60◦dislocation and all low energy Shock-
(a) 60° glide (b) 60° shuffle (c) 30° glide
(d) 90° glide (SP) (e) 90° glide (DP) (f) 90° shuffle (V)
Figure 4.22: Simulated HRTEM images of low energy dislocation core structures with a de-
focus
f=−40 nm and a sample thickness of 4 nm. The input atom coordinates for the
simulations are the relaxed atom positions of the respective hybrid model. Fig. 4.7 and 4.8
show the corresponding atomic structures in the same orientation. Each image is approxi-
mately centred on the dislocation core line and the (111) glide plane lies horizontally.
4.6. Electron microscopy — a first link to experiments
67
experiment
simulation
Figure 4.23: Comparison of the experimental and the simulated HRTEM image of an undis-
sociated 60◦dislocation. Left: Experimental image. The Burgers circuit shown yields a pro-
jected Burgers vector of 1
4[¯
1¯
12]identifying the dislocation. The overlaying mesh of lines ap-
proximately connects the points of highest intensity in the contrast pattern. Right: The mesh
of the experimental image superimposed with the corresponding mesh of the simulated im-
age of a 60◦glide dislocation (Fig. 4.22 (a)). The latter is mirrored to facilitate comparison.
The small offset between the two meshes is intentional, to allow both to be seen clearly. As
mentioned in the text, the main deviation between the simulated image and the experimen-
tal one arises from the artificial bending of the (111)plane in the hybrid models applied in
this work. (HRTEM performed by Bert Willems, EMAT, University of Antwerp.)
ley partials23. The glide and shuffle type of the 60◦dislocation appear to be rather
distinct in their HRTEM image. Also the vacancy shuffle structure of the 90◦partial
can be easily distinguished from the two glide structures. However, the difference
between the single and double-period reconstruction of the 90◦glide partial is minis-
cule — at least they can hardly be distinguished by eye. An effect common to all
simulated images is the bending of the (111) plane. The larger the edge component
of the dislocation, the larger the effect. This bending is a result of the limited size of
the hybrid models: The inserted material in the upper half (with respect to [111]) of
the model leads to an artificial expansion, which would not occur if the dislocation
was embedded in an infinite crystal. This further complicates the comparison with
experimental HRTEM images24.
In Fig. 4.23 the experimental and simulated cross sectional HRTEM images of a
60◦dislocation are compared. Having the largest edge component of all investi-
gated dislocations, the hybrid model of the undissociated 60◦dislocation shows the
23 Comparing Fig. 4.22 with Fig. 4.21 shows us how easily the human eye and the subsequent neu-
ral data-processing are deceived: Without the atom positions being shown, one might easily assume
the bright spots to be atoms or atom pairs and the large dark areas to be the empty [1¯
10]channels.
However, the real situation for the given defocus and specimen thickness is exactly the opposite.
This demonstrates the importance of image simulations when it comes to interpreting experimental
HRTEM images.
24To reduce this effect by enlarging the diameter of the hybrid models would result in a large in-
crease in computational cost. The alternative — to abandon the free relaxation of the hybrid’s surface
— is a non-trivial problem: A rigid surface, even with the surface atom positions calculated in elas-
ticity theory, would not allow core structure specific volume expansion. With the core that strongly
confined, the resulting core energies turn out to be unreasonably large. Hence these two most simple
approaches to improve the geometrical boundary conditions are not really feasible.
68
Chapter 4. Dislocations in Diamond
90° glide (SP) 90° glide (DP) 30° glide
90° 30°
Figure 4.24: Comparison of the experimental and the simulated HRTEM image of a dissoci-
ated 60◦dislocation. Top: The experimental image. The two regions containing the 90◦and
30◦Shockley partial (left and right respectively) are framed in white. Apparently the two
partials do not lie on the same glide plane — the two glide planes are indicated by small
arrows. Bottom: The mesh derived from the simulated images of the two types of 90◦glide
partials and the 30◦glide partial is shown superimposed on the respective region of the
experimental image. (HRTEM performed by Bert Willems, EMAT, University of Antwerp.)
strongest artificial bending of the (111)plane, leading to considerable deviation.
However, keeping this in mind, at least locally the agreement is reasonable. Simi-
larly, in Fig. 4.24 a 90◦and 30◦partial are shown. Here the agreement is much better,
especially for the 30◦glide partial. Unfortunately the contrast pattern does not allow
a rigorous decision over the periodicity of the core reconstruction of the 90◦partial.
However comparison with the simulated single-period structure shows slightly less
deviation. Thus in this specific case the SP reconstruction seems more likely. One
should keep in mind however, that the diffuse contrast of the 90◦partial might re-
sult from the partial not being straight, but heavily kinked. Then the question of
periodicity becomes obsolete anyway, since the DP structure can be simply seen as
a periodically kinked SP structure (Fig. 4.7).
Conclusions: The cited experimental work proves the existence of some of the
predicted dislocation types in the investigated specimens of natural HPHT-treated
brown type IIa diamond.
At least within the rather crude approach to compare simulated and observed
HRTEM images, the observed core structures are in reasonable agreement with
those calculated in the earlier sections.
Also, an undissociated 60◦dislocation could be identified, which is in agreement
with the predicted barrier to its dissociation as shown in Fig. 4.13.
4.7. Electronic structure calculations and electron energy-loss
69
4.7 Electronic structure calculations and electron energy-loss
So far all calculated quantities have been exclusively related to dislocation struc-
tures, their interaction and their core energies. Except for test and reference cal-
culations, in this context DFTB and continuum elasticity theory were the methods
predominantly applied. However, as most other types of defects, dislocations also
effect the electronic structure of the crystal in their vicinity. In this section, these ef-
fects will be investigated within the DFT-pseudopotential approach as implemented
in AIMPRO (see Section 1.1.2).
In DFT the electronic structure of the ground state is given as the spatial distribu-
tion of the electron density, and the corresponding Kohn-Sham eigenvalues (the
ε
i
in Eq. (1.8)). In a periodic crystal the latter is usually best described as a band struc-
ture with k-dependent (quasi-)continuous eigenvalues. As mentioned before, pure
diamond has a relatively large (5.49 eV) and indirect band gap (the valence band
maximum (VBM) and the conduction band minimum (CBM) are associated with
different k-points)25. It should be noted, that DFT in LDA usually underestimates the
band gap: With the models used in this work AIMPRO yields a band gap 1.48 eV too
small for diamond.
Dislocations are line defects, periodic only along their line, but not in the two per-
pendicular dimensions. Hence the electronic structure in a region of the crystal con-
taining a dislocation is that of a one-dimensional periodic arrangement, parame-
terised by an only one-dimensional k-space. The band structure of the surrounding
bulk crystal in that region is therefore projected onto the corresponding direction
in its three-dimensional k-space26. Therefore, when comparing a dislocated region
with a bulk region of the crystal, we have to compare the respective projected band
structures27.
There are three distinct major origins of dislocation related changes in the projected
band structure:
Dangling bonds: Some of the dislocation structures discussed in previous sec-
tions appeared to have dangling bonds — resulting from atoms in an electronic
sp3-hybrid configuration, bonded to only three other neighbour atoms, instead of
four. Such dangling bonds might result in states deep within the band gap. These
states are usually strongly localised in the vicinity of the dangling bonds.
25 In semiconductors and insulators the eigenvalues of the occupied Kohn-Sham eigenstates are sep-
arated from those of the unoccupied states by a forbidden energy gap, wherein no states are present. In
terms of band structures we speak of the valence band (occupied states) the conduction band (unoccupied
states) and the band gap in between the two.
The concept of band structures as well as its application to cubic or hexagonal crystals shall not be
revised further. The subject is covered in most textbooks on quantum mechanics and a brief introduc-
tion is given in [48].
26 This is similar to the situation at surfaces, where the three-dimensional k-space is reduced to a
two-dimensional space.
27 Of course strictly speaking a band structure is not defined for a finite region of the crystal, but
only for an infinite crystal. However, for large enough regions, and assuming the spatial coordinates
to be “good” quantum numbers at least on the large scale, one can approximately speak of a “local”
band structure and local defect-induced changes to it.
70
Chapter 4. Dislocations in Diamond
Strained bonds and different bonding configurations: In the very core region of
most dislocations, bond angles and lengths deviate considerably from those found
in the bulk crystal. This is usually the case for reconstruction bonds, but also oth-
erwise “bulk-like” bonds at the core are often considerably stretched or compressed
due to the strain field. Also non-sp3hybridisations might occur.
All these lead to changes in the density of states below the VBM and above the CBM.
However, if the strain is large enough, then localised gap states, usually close to the
band edges, are possible.
The medium range strain field: All dislocations are surrounded by a strain field.
The strain results in a local reduction of the band gap in the vicinity of the dislocation
core. This effect leads to carrier confinement and might yield localised transitions
near the dislocation, even if no states in the gap are present [103].
Within the approach used in this work, the band gap reduction due to the strain
field can be observed to some extent, however it cannot be quantified and thus will
not be discussed further.
4.7.1 The computational approach
To obtain a good representation of the Kohn-Sham eigenstate spectra associated
with the dislocation core structures discussed in preceding sections, the DFTB
method in its minimal basis approach is not sufficient: The valence band and defect
states close to it are represented reasonably well, conduction band states however
deviate considerably from more accurate DFT-pseudopotential calculations (AIM-
PRO)28. The conduction bands play an important role in electron energy-loss spec-
troscopy (EELS), which will be discussed later in this section. Hence the use of
DFT-pseudopotential calculations is essential at this point.
To obtain the projected band structure corresponding to one of the various low en-
ergy dislocation core structures in Section 4.3, the outer shells of atoms are removed
from the relaxed structure and the new surfaces are terminated with hydrogen.
The models thus obtained are still of the supercell-cluster hybrid type, however
smaller in diameter to allow treatment within the computationally more expensive
approach: The number of atoms per model now ranges between 60 and 160 atoms
depending on the core radius and the periodicity of the structure29. These structures
are then relaxed with the AIMPRO code. In all calculations the electronic wavefunc-
tions are expanded in a set of s,p, and datom-centred Gaussian orbitals with 28
variational degrees of freedom per atom and the Brillouin zone integration during
self-consistency is performed using a set of between 2 and 4 Monkhorst-Pack re-
duced k-points along the axis in k-space [84], which corresponds to the dislocation’s
periodicity.
28 But still one has to keep in mind that the description of the conduction band states suffers from
DFT, at least in the form applied in this work, being a ground-state theory.
29 This might sound rather small, and for the calculation of core energies it would be too small
indeed (compare Fig. 4.6), however for all investigated cases the actual core structures were still rep-
resented well after relaxation of the smaller hybrid models.
4.7. Electronic structure calculations and electron energy-loss
71
To then obtain the electronic band structure of the relaxed structure, the Kohn-Sham
eigenstates are calculated at approximately 25 different k-points along the same axis.
Electron energy-loss spectroscopy and its simulation: The experimental tech-
nique yielding the most direct information on the electronic band structure is elec-
tron energy-loss spectroscopy. One has to distinguish between low-loss EELS, where
the signal results from the scattering of high-energy primary electrons with sec-
ondary valence electrons which undergo a transition to empty conduction or gap
states, and core-excitation EELS, which results from electronic transitions between
deep atomic core states and empty gap or conduction band states. For the former,
the signal obtained is proportional to Im(−1/
ε
(E)) where
ε
=
ε
1+i
ε
2as the dielec-
tric function and Eis the energy loss. Core-excitation EEL spectra are related to an
angular momentum projected local density of states at the respective atom. In the
case of the s-like diamond Kedge this will be the p-projected local density of states.
More details on both the experimental as well as the modelling aspects of EELS can
be found in Appendix C.
Some remarks on the effects of boundary conditions: In this work EEL spectra
are calculated exclusively in supercell-cluster hybrid models. However, modelling
in pure supercells might yield different results since the boundary conditions in-
fluence the electronic structure. This has been tested for dislocations in gallium
nitride [11], and differences in the EEL spectra due to the different boundary condi-
tions seem to occur only at high energies well above the band edge.
The theoretical results presented in the following have been obtained in collabo-
ration with Caspar J. Fall (University of Exeter, UK), who implemented the EELS
functionality into the AIMPRO code and subsequently performed the EELS simula-
tions.
4.7.2 Calculated band structures and EEL spectra
The electronic structure calculations turn out to be computationally rather expen-
sive, since a considerable number of k-points is required for the simulated EEL
spectra: Depending on periodicity and model size, for the EELS calculations in this
section 50–150 k-points have been used. On this account not all core structures as
given in Fig. 4.7 and 4.8 are investigated, but only a selection of structures, which
will either exist predominantly or show some interesting features.
Fig. 4.25 shows the calculated band structures, low-loss EEL spectra and K-edge core
excitation EEL spectra for the perfect 60◦glide and shuffle dislocations. The results
for the lowest energy Shockley partials of the {111}h110islip system, namely the
30◦and 90◦(DP) glide partials and the vacancy structure of the 90◦shuffle partial,
are given in Fig. 4.26. The length of the k-axis reciprocally depends on the period-
icity length along [1¯
10]of the core reconstruction. Therefore, being the only single-
period structure, the 60◦shuffle dislocation appears with twice the length in k-space.
Both low-loss and core EELS at the dislocation core are shown in comparison with
spectra calculated in bulk supercells. As shown in Appendix C, the calculated bulk
spectra agree rather well with experiments.
72
Chapter 4. Dislocations in Diamond
bulk
bulk
60° glide60° shuffle
low−loss EELS
low−loss EELS
[111]
[112]
[110]
[111]
[112]
[110]
121086420 (eV)
−2
0
2
4
6
8
−4
(eV)
1050
−2
0
2
4
6
8
−4
121086420 (eV)
(eV)
0 5 10
(eV)
(eV)
(a.u.)(a.u.)
(a.u.) (a.u.)
core EELS
core EELS
Figure 4.25: The projected band structures and simulated EEL spectra of the perfect 60◦dis-
location. In the band structures (left column) only the bands close to the gap are drawn,
the rest of the valence and conduction band region is shaded. The origin of the Brillouin
zone is at the far left of each band structure. The conduction band is shifted upwards by
1.48 eV to match the experimental gap. States within the gap are shifted proportionally to
their distance in energy from the valence band top. For the low-loss EEL spectra (middle
column) the results for different beam orientations are shown. The line of plus signs gives
the bulk spectrum. The K-edge core EEL spectra (right column) are displayed with a Gaus-
sian broadening of 0.8 eV and 8–10 inequivalent atoms neighbouring the dislocation core
have been taken into account. The dashed line gives the bulk spectrum. Upper panel: The
60◦glide dislocation. The band structure is folded once (reducing the Brillouin zone), since
the structure is period-doubled (See Fig. 4.7 (b)). Lower panel: The 60◦shuffle dislocation.
In the band structure the Fermi level, up to which the bands are occupied, is shown as a
dashed line. The dislocation is assumed to be uncharged.
Glide dislocations: All glide dislocations show no evidence of deep gap states.
This directly stems from the reconstructed nature of the cores which allow all atoms
to be four-fold coordinated with bond angles and lengths not too far from bulk mate-
rial. Consequently, only small changes in the corresponding simulated EEL spectra
are observed: Only for beam directions perpendicular to the dislocation line ([¯
1¯
12]
and [111]) is the low-loss spectrum slightly enhanced in the 6–8 eV region.
4.7. Electronic structure calculations and electron energy-loss
73
bulk
bulk
bulk
90° shuffle (V) 90° glide (DP) 30° glide
low−loss EELS
low−loss EELS
low−loss EELS
[111]
[112]
[110]
[111]
[112]
[110]
[111]
[112]
[110]
1050 (eV)
(a.u.)
1050 (eV)
(a.u.)
1050 (eV)
(a.u.)
−2
0
2
4
6
8
−4
(eV)
−2
0
2
4
6
8
−4
(eV)
−2
0
2
4
6
8
−4
(eV)
121086420 (eV)
(a.u.)
121086420 (eV)
(a.u.)
121086420 (eV)
(a.u.)
core EELS
core EELS
core EELS
Figure 4.26: The projected band structures and simulated EEL spectra of the low energy
partial dislocations. All band structures are folded once (reducing the Brillouin zone), since
the structures are period-doubled (See Fig. 4.8). For more details see also caption of Fig. 4.25.
Upper panel: The 30◦glide partial dislocation. Middle panel: The 90◦(DP) glide dislocation.
Lower panel: The 90◦(V) shuffle dislocation. In the band structure the Fermi level, up to
which the bands are occupied, is shown as a dashed line. The dislocation is assumed to be
uncharged.
74
Chapter 4. Dislocations in Diamond
Shuffle dislocations: As can be seen in Fig. 4.7 (c) and 4.8 (d), the two investi-
gated shuffle dislocations each possess a row of three-fold coordinated atoms. The
resulting dangling bonds protrude along [¯
1¯
1¯
1]into the wide [1¯
10]core channel. In
both cases this results in a band at mid-gap position with similar dispersion. For an
uncharged dislocation this band will be only half occupied. At the 90◦shuffle (V)
structure the band is folded, leading to two gap states per k-point. By that it clearly
shows its single-period character — the double periodicity stems from the recon-
struction bonds along the dislocation line only, which has hardly any influence on
the dangling bonds30.
The presence of electronic bands around mid-gap is consistent with the observed
energy of band-Aemission at 2.8–2.9 eV, which has been correlated with threefold
coordinated atoms in extended defects [71]. The 60◦shuffle core additionally con-
tains a stretched bond that contributes to an additional gap state that is higher in
energy.
For the subsequent EELS calculations the material is assumed to be p-type, leav-
ing all gap states empty. Under these conditions the 60◦shuffle dislocation shows
strongly enhanced low-loss EEL absorption in the 3–12 eV range for beam orienta-
tions perpendicular to the dislocation line. Similarly, for the same beam orientation,
the vacancy structure of the 90◦shuffle partial is found with enhanced absorption
in the 6–9 eV range. Additional strong absorption at the latter occurs in the 0–2 eV
range for electron beams parallel to the dislocation.
In core-excitation EELS on these dislocations the empty gap states create supple-
mentary absorption peaks below the conduction band minimum. This might ac-
count for previous experimental findings [104].
4.7.3 Experimental EELS
Experimentally, EEL spectra can be taken with high spatial resolution, which allows
to distinguish between spectra taken at dislocation core regions and those taken in
dislocation-free areas of the crystal. This section will give an example of EELS being
applied to dislocations in diamond.
All spectra shown are the result of experimental work performed by A. Guti´errez-
Sosa and U. Bangert at UMIST in Manchester (UK), A. E. Mora and J. W. Steeds
at the University of Bristol (UK) and J. E. Butler at the Naval Research Laboratory,
Washington DC (USA).
The investigated CVD-grown polycrystalline diamond sample was boron doped
with an estimated boron concentration of 1021 cm−3. Electron energy-loss spec-
troscopy was conducted in a scanning transmission electron microscope (STEM),
operated at 100 kV. More details on sample preparation and the EELS experiments
are presented in Ref. [6].
30 To give some more details: The gap states of the 90◦shuffle (V) partial are formed from p-like
orbitals of the threefold coordinated carbon atoms. As mentioned, because of the double-period re-
construction, the band structure is folded and there are two gap states for each wave vector. In the
energetically lower gap state at k= (0, 0, 0), the p-like orbitals are repeated along the core with a
phase change of
π
between successive atoms, while in the energetically higher gap state they are all in
phase.
4.7. Electronic structure calculations and electron energy-loss
75
40 nm
2 4 6 8 10 12 14
low−loss EELS
(eV)
dislocation
bulk
Figure 4.27: Experimental low-loss EELS and TEM on dislocations in CVD diamond. Left:
Experimental low-loss EEL spectra for bulk (plus signs) and oblique partial dislocations
(dashed line) with an electron beam along h111i. Each graph shown results from several tens
of EEL spectra taken. Right: TEM image of CVD diamond. The arrow marks the dislocation
whose EEL spectrum is shown on the left, presumably a 30◦glide partial of a dissociated
screw.
(All experiments including sample-growth and preparation were performed by A.
Guti´errez-Sosa and U. Bangert at UMIST in Manchester (UK), A. E. Mora and J. W. Steeds at
the University of Bristol (UK) and J. E. Butler at the Naval Research Laboratory, Washington
DC (USA))
Plan view TEM of the investigated sample showed grain clusters up to a diame-
ter of 0.2 mm with common h110ior h112igrowth directions. Each such cluster
contained up to ten individual grains and the dislocations were predominantly ori-
entated along h110idirections. Fig. 4.27 shows the spectra obtained on and off such
a dislocation. Unfortunately the grain size was too small to allow any determina-
tion of the Burgers vector. However, considering their origin and formation they
are most likely to be dissociated screw dislocations [66]. In other words their exper-
imental spectra have to be compared with that of the 30◦glide partial in Fig. 4.26.
In the experiment, the dislocation was oriented neither parallel nor perpendicular
to the beam, but oblique. Hence its spectrum does not directly correspond to either
of the calculated spectra. However, the calculated spectrum with a beam orienta-
tion of [111]perpendicular to the dislocation line can be taken as an approximate
representation of the experimental situation: Both the experimental spectrum as the
calculated spectrum clearly show an increase in the 6–8 eV region compared to the
respective bulk spectrum. Also the relative increase around 7 eV is common to both
spectra.
Further spectra taken with a beam orientation along the dislocation line appear to
be less intense and rather featureless — a trend which can also be observed in all
theoretical spectra.
76
Chapter 4. Dislocations in Diamond
4.7.4 Electronic structure calculations — conclusions
The calculations have shown that, at least among the low energy dislocations of the
{111}h110islip system in diamond, and assuming the dislocations to be undeco-
rated by point defects, only the shuffle set of dislocations appears to induce deep
electronic states into the band gap. This suggests that the origin of the band-Alu-
minescence as observed in natural type-II diamond is either due to point defects
bound to dislocation lines, such as impurities or jogs, or to shuffle segments.
As a consequence of the absence of deep gap levels associated with the glide set
of dislocations, only the calculated EELS spectra of shuffle dislocations show major
differences in comparison with the respective bulk spectrum. Still, it was possible to
correlate the features of experimentally observed low-loss EEL spectra in polycrys-
talline CVD diamond with those of 30◦glide dislocations. This supports the earlier
experimental presumption of dissociated screw dislocations being the predominant
species near grain boundaries. In the quoted experimental work on polycrystalline
CVD, no indication of shuffle segments could be found.
Earlier core EELS performed by Bruley and Batson [104] on natural type-II diamond
shows a supplementary absorption below the conduction band, which is associ-
ated with dislocations. This agrees well with the spectra calculated for both the 60◦
shuffle dislocation and the 90◦shuffle (V) partial, suggesting shuffle segments to be
present in their sample.
4.8. Summary and conclusions (diamond)
77
4.8 Summary and conclusions (diamond)
In this chapter, the low energy dislocations of the {111}h110islip system in dia-
mond have been modelled in an approach combining DFT-based atomistic calcula-
tions with linear isotropic (and to some extent anisotropic) elasticity theory. In this
approach elasticity theory has proven invaluable to describe the long range elastic
strain effects associated with dislocations and also served as a good convergence
criterion for the size of the atomistic models used.
The atomistic modelling — in particular the DFTB method — allowed a convenient
determination of dislocation core energies and of the energy offset in the dissociation
of dislocations. In elasticity theory the latter properties are both unknown. Wher-
ever there was overlap between the elasticity theory description of dislocations and
their atomistic modelling, the agreement was very good: The atomistic calculations
were converged to within the elastic limit.
The predicted low energy core structures and dissociation distances match well with
those observed experimentally in (HR)TEM. In particular to compare the calculated
structures with high-resolution micrographs, the application of image simulations
based on the calculated coordinates was crucial.
Based on the predicted core structures it was then possible to calculate the corre-
sponding electronic structures in a pseudopotential approach (AIMPRO). The sub-
sequent simulation of both core and low-loss EEL spectra allowed a direct link to
recent experiments to be established.
For the two probably predominant partial dislocations — the 30◦and the 90◦glide
partials — the kinetics of thermally activated glide motion have also been modelled,
assuming a kink formation and migration mechanism.
Comparison with earlier theoretical work: There are relatively few independent
DFT-based calculations on dislocations in diamond and they are limited to a small
subset of the problems treated in the work presented here. Where comparison is
possible the difference between this work and those earlier calculations is small —
around 6% for glide activation energies and below 5% for dislocation core energies
and their relative differences. This, along with the good agreement with experimen-
tal work gives confidence, that the chosen approach of combining two different ap-
proximations to DFT with elasticity theory for the long range effects, is well adapted
for the problem. However, being a first step towards a multiscale approach, it is by
far less computationally expensive than pure DFT-pseudopotential calculations and
covers are larger variety of length scales and associated properties.
Further, the chosen supercell-cluster hybrid model has proven very successful to
model core structures and energies as well as the electronic properties of straight
perfect and partial dislocations.
In the following section the important results are listed explicitly, followed by pos-
sible implications.
78
Chapter 4. Dislocations in Diamond
4.8.1 Selected results
Comparing the shuffle structure of the undissociated 60◦dislocation with the glide
structure, one finds the latter to be 630 meV/ ˚
A lower in line energy. However, since
a shuffle →glide transition involves absorption or emission of vacancies or inter-
stitials, one would expect shuffle segments to be present if formed by plastic defor-
mation. These might then undergo the transition to glide structures in a thermal
annealing process.
There is a barrier of around 180 meV/ ˚
A to the dissociation of the perfect 60◦dis-
location into a 30◦and a 90◦Shockley partial. Due to its width this is a consider-
able barrier which is not easily overcome. This explains the experimental observa-
tion of undissociated 60◦dislocations even though an overall energy-gain of around
680 meV/ ˚
A strongly favours dissociation.
The situation is different for the 1
2[1¯
10]screw dislocation, here the three proposed
structures (glide, shuffle and mixed) can be transformed into each other by the mere
breaking and forming of bonds. The shuffle screw is found to be metastable, but
around 860 meV/ ˚
A higher in energy than the glide structure which resembles two
30◦glide partials in the same Peierls valley. The dissociation of those two partials
yields a further energy of approximately 1 eV/ ˚
A without any considerable barrier.
Thus spontaneous dissociation into two 30◦glide partials is strongly suggested. In-
deed no undissociated 1
2[1¯
10]screw has been observed experimentally.
The equilibrium distance between two dissociated Shockley partials, or in other
words the equilibrium stacking fault width, can be determined easily by means of
elasticity theory only. Using the elastic constants and stacking fault energies ob-
tained within the DFTB method, the equilibrium distance was evaluated to around
30 and 35 ˚
A for the screw and 60◦dislocation respectively. This corresponds well
with the experimentally observed widths between 25 and 42 ˚
A. The large variation
in observed widths might be explained by the energy minimum being very flat com-
pared to the activation energies required to overcome the barrier between adjacent
Peierls valleys.
A comparison of core energies ruled out many of the possible dissociation reac-
tions, leaving only the 30◦and 90◦glide partials and the vacancy structure of the
90◦shuffle partial likely to exist. Consistent with earlier calculations, in this work
the double-period core reconstruction of the 90◦glide partial is favoured over its
single-period reconstruction.
The glide motion of the two glide partials has been considered as a process of kink
formation and subsequent migration. Under these assumptions the 90◦glide dislo-
cation proves to be the by far more mobile species, with a thermal activation energy
of 3.5 eV (the sum of the double-kink formation energy and the migration barrier)
for short dislocation segments. At the 30◦glide partial the migration barriers are on
average 0.5 eV larger and the double-kink formation energy exceeds that of the 90◦
partial even by 2 eV. The resulting average thermal activation energy of the 30◦glide
partial is found to be Q30 ≈6.1 eV. If a dissociated dislocation moves as a whole,
then its speed is controlled by the slower partial. Hence the thermal activation en-
ergy for both the dissociated screw and for the dissociated 60◦glide dislocation will
be given as that of the 30◦glide partial.
4.8. Summary and conclusions (diamond)
79
Finally turning towards the electronic properties, only the perfect and partial shuffle
dislocations appear to induce deep electronic states/bands in the band gap. The ma-
jority of these states arises from dangling bonds at the dislocation line and is located
around mid-gap. As shown, the gap states should lead to a considerable increase in
low-loss EELS absorption and might also well explain the supplementary core EELS
absorption below the conduction band edge, which is experimentally observed in
natural type-II diamond. This would then further suggest shuffle segments to be
present in that material, which could well be responsible for the well-known band-
Acathodoluminescence around 2.8–2.9 eV.
In polycrystalline CVD diamond, low-loss EEL spectra obtained near grain bound-
aries on presumably dissociated screw dislocations seem to correspond well with
the calculated spectra of 30◦glide partials. However, for a good comparison and to
draw strong conclusions, further experimental EEL spectra of different partial and
perfect dislocations would be required.
4.8.2 The decolouring of brown diamonds by HPHT treatment
The brown colouring, which is found in most natural diamonds, renders them use-
less as gem stones. Even though it is assumed to be related to plastic deformation
and extended defects, its exact origin remains unknown. As mentioned in the in-
troduction to this chapter, in recent years colour improvement via HPHT annealing
became feasible, threatening the gem market. Hence a deeper understanding of the
origin and annealing of the colouring is of particular interest and there is a strong
interest to find definite methods of identifying HPHT treated stones.
Several indications point towards a possible scenario involving a shuffle →glide
transition in the annealing and decolouring process:
1. The absence of deep electronic gap states for all perfect and partial glide dis-
locations excludes them from being responsible for the brown colouring if un-
decorated. All low energy shuffle structures however, give rise to deep bands
in the upper two thirds of the electronic gap, which might account for the
colour.
2. The shuffle dislocation is found to be higher in line energy, suggesting a shuffle
→glide transition in the annealing process.
3. Calculated vibrational spectra show modes (around 1500 cm−1) above the Ra-
man frequency for the shuffle dislocation only [105]. These modes might ac-
count for a Raman line found in brown diamond, which anneals out above
2000◦C [106], further supporting the transition mentioned above. However,
the Raman signal appears to disappear before the actual decolouring.
4. In the case of dissociated shuffle dislocations, vacancy structures are more sta-
ble than those involving interstitials. Thus, a shuffle →glide transition at par-
tials would always involve vacancy emission (or interstitial trapping). This
is in agreement with the observed formation of vacancy–nitrogen complexes
subsequent to annealing [72]. This process might well be fed by the vacancies
released during a shuffle →glide transition.
80
Chapter 4. Dislocations in Diamond
Summarising all findings, it still remains unclear whether a shuffle →glide transi-
tion occurs before the actual annealing of the colour, or if the 1500 cm−1Raman signal
becomes just too small to be detected. It can also not be excluded that the observed
line has a different origin and is not related to dislocations at all.
4.8.3 Outlook
Further (HR)TEM studies and EELS experiments on HPHT treated decoloured dia-
mond as well as on untreated brown diamond are necessary. Then, in combination
with the methods presented here, it might be possible to decide in favour or against
an annealing mechanism involving dislocations undergoing a shuffle →glide tran-
sition.
Further, the electronic structure of kinks and jogs might be of interest, as well as
dislocations with line directions different from h110i.
Methodological: In the long run, the way of combining different methods to
model the effects of dislocations on different length scales could be improved. A
true embedding approach is imaginable, where the region treated atomistically is
embedded in a larger region which is described by elasticity theory. The crucial
point would then be the interface/coupling between the different regions, which is
far from trivial.
Chapter 5
Dislocations in Silicon Carbide
In this chapter selected dislocation-related problems in silicon carbide (SiC) will be
addressed, which are of particular interest and technologically relevant. Threading
dislocations as mentioned in Section 3.3, which are formed during growth, are very
common defects in SiC, but excluded from this work. Hence the only dislocations
considered are the low energy dislocations in the basal plane1.
After giving a short introduction into SiC as a material and explaining the differ-
ent polytypes, the glide set of basal dislocations is analysed, discussing the main
core reconstructions and energies. This is followed by an investigation of the glide
motion of 30◦and 90◦partial dislocations, assuming a kink formation and migra-
tion process as already introduced for the case of diamond. Finally, the electronic
structure of the discussed defects is examined.
Since all the basic concepts of the modelling of dislocations were discussed in de-
tail with diamond as an example, this chapter will mainly present results and their
implications. Whenever there is a considerable difference or problems specific to
modelling dislocations in SiC, then those will be pointed out.
5.1 Introduction and background
Silicon carbide is a compound semiconductor consisting of the two components sili-
con and carbon. It possesses a wide electronic band gap of 2.39 – 3.33 eV (depending
on the polytype [107]) and has extreme properties similar to diamond. Its high ther-
mal conductivity, extreme hardness and resistance to thermal shock, oxidation and
corrosion, are useful for engineering applications in extreme environments [108].
As a semiconductor, the rather high breakdown electric field strength make SiC a
promising material for high-power, high-temperature and high-frequency devices.
Single crystal SiC CVD-growth is technologically reasonably controlled and appre-
ciable n-type conductivity can be achieved by doping with nitrogen. For p-type
usually aluminium or boron are used as dopants [109,110,111].
1The term “basal plane” is used here for the hexagonal polytypes and the cubic polytype alike. In
the latter it simply refers to an arbitrary {111}plane.
81
82
Chapter 5. Dislocations in Silicon Carbide
C
A
B
C
A
B
C
A
B
A
C
A
A
B
C
A
B
C
3 −SiCC4 −SiCH6 −SiCH
Figure 5.1: The Stacking sequences of 3C-, 4H- and 6H-SiC. Carbon atoms are represented
in dark grey, Silicon atoms in yellow (light grey in black and white prints). Only the planes
of one sub-lattice are labelled. The arrow indicates the h111idirection for the cubic polytype
(3C) and h0001ifor the hexagonal polytypes (4Hand 6H).
In device development, a strong emphasis lies on high-power diodes: Bipolar
6H-SiC diodes have been reported with reverse breakdown voltages as high as
4.5 kV [111] and more recently 4H-SiC PiN diodes have shown reverse breakdown
voltages >5 kV with capacities to transmit very high powers >100 MW. How-
ever, these devices show a considerable degradation under forward-biased opera-
tion [112,113,114,115]. More details on this problem are given in Section 5.1.2.
Further, high current operation of devices is limited by the material quality. In par-
ticular micropipes, which thread the layer and are commonly associated with dis-
locations, have a disastrous effect: Even a single micropipe can lead to an early
electrical breakdown of the device [116]. However as mentioned above, micropipes
and threading dislocations will not be discussed further in this work.
5.1.1 The different polytypes of SiC
Unlike most other semiconductors, which only occur in one or two different crys-
tal structures each, silicon carbide is known to exist in over 170 different poly-
types [117] with cubic, hexagonal and rhombohedral symmetry. Three very com-
mon polytypes are 3C(cubic) with the stacking sequence ···ABC|ABC ···and 4H
and 6H(hexagonal) with the respective stacking sequences ···ABAC|ABAC ···
and ···ABCACB|ABCACB···as shown in Fig. 5.1. More details on the concept
of crystal stacking can be found in Section 3.2.
Table 5.1 gives the experimental lattice constants and the electronic band gap for
different polytypes. In all cases the average bond lengths resulting from the lattice
constants are almost identical. The band gaps differ however. Especially that of
cubic SiC is considerably smaller than those of the hexagonal polytypes.
The energy differences between the different polytypes are rather small, and con-
sequently stacking faults are low energy defects. Just to give an example, Fig. 5.2
shows an extended stacking fault node in a plastically deformed 4H-SiC single
crystal. The stacking fault and three equivalent partial dislocations with Burgers
vectors b=1/3[1¯
100],b=1/3[01¯
10]and b=1/3[10¯
10]can be identified using
5.1. Introduction and background
83
Figure 5.2: TEM weak-beam
contrast experiments carried
out on an extended disloca-
tion node in a 4H-SiC single
crystal. The crystal was plas-
tically deformed at 1300◦C.
Arrows indicate the diffrac-
tion vector: (a) g= [¯
12¯
10],
(b) g= [2¯
1¯
10], (c) g=
[¯
1¯
120], (d) g= [¯
101¯
1]. The
beam direction for (a) – (c)
was B= [0001]and B=
[¯
1012]for (d). Using the g·
b=0 invisibility criterion
in (a) – (c) allows us to iden-
tify the dislocations labelled
A, B and C as partials with
a respective Burgers vector
of 1/3[1¯
100], 1/3[01¯
10]and
1/3[10¯
10]. Similarly, in (d)
the stacking fault node be-
tween the partials can be
seen. (Reproduced from
Ref. [118] with kind permis-
sion from the authors.)
the g·b=0 invisibility criterion [118] (see also Appendix B). Experimental results
agree on stacking fault energies of around 15 and 3 mJ/m2for 4H- and 6H-SiC re-
spectively [119,120]. These numbers accord well with recent theoretical calculations
by Miao et al. [121]. Further, the calculated energy of an intrinsic fault in 3C-SiC was
found to be negative (−3.4 mJ/m2[122] and −0.14 mJ/m2[121]). Since a stacking
fault in cubic material is nothing else but a hexagonal inclusion, this negative stack-
ing fault energy reflects 3Cbeing a less stable polytype than for example 4Hor 6H.
Compared to diamond with a stacking fault energy around 300 mJ/m2(Table 4.2),
all values reported for SiC are absolutely miniscule. Hence extended stacking faults
are relatively easy to form. This results in devastating effects on SiC devices, which
will be discussed in the next section.
Table 5.1: The experimental lattice constants (a0,c0/p) and the excitonic band gap Egof
different SiC polytypes. The unit cells for 3Cand 2Hare given in Fig. 3.1, those of 4Hand
6Hare similar to that of 2H, but include not two, but four and six bilayers along [0001]
respectively. c0is divided by the number pof bilayers to facilitate comparison. All values
are taken from Ref. [107]. Those of Ref. [123] are almost identical.
3C2H4H6H
a0(˚
A) 4.360 3.076 3.073 3.081
c0/p(˚
A) — 2.524 2.513 2.520
Eg(T=4 K) (eV) 2.39 3.33 3.26 3.02
84
Chapter 5. Dislocations in Silicon Carbide
Figure 5.3: A time sequence of
plan-view electroluminescence
images of partial dislocation
motion in 4H-SiC at 30◦C and
under a current density of
7 A/cm2. The reference grid
size is 100 ×100
µ
m. The
image shows only the spectral
range of 700 ±20 nm, revealing
the partial dislocations as bright
lines and threading dislocations
as bright spots. (Reproduced
from Ref. [115] with kind per-
mission from the authors.)
5.1.2 The degradation of SiC PiN diodes under forward-bias
Despite all afore mentioned advantages of SiC for high-power applications, a major
problem in SiC power-device technology remains unsolved: Recent experiments
have shown, that SiC PiN diodes2degrade during forward-biased operation of the
device even at moderate current densities [112,113,114,115], sometimes below
1 A/cm2[124]. This degradation means a considerable drop in the forward voltage,
rendering the device useless after few days of constant operation.
The voltage drop is accompanied by the formation, propagation and growth of
stacking faults of triangular and sometimes rhombic shape [114]. Very recently the
stacking fault edges, which lie along h11¯
20idirections, were identified by means
of transmission electron microscopy and X-ray topography to be Shockley partials
with Burgers vectors of 1
3h1¯
100i-type [124,125]. Fig. 5.4 shows high-resolution TEM
images of presumably 30◦Shockley partials bordering a stacking fault in 4H- and
6H-SiC diodes.
Since neither the mechanical stresses nor the temperature are high enough to over-
come the barrier to dislocation glide motion, it is believed that a recombination-
enhanced dislocation glide mechanism (REDG) is responsible for the observed ef-
fect3. The REDG mechanism requires non-radiative electron-hole recombination
sites along the dislocation line. The released recombination energy then has to be
redirected to assist the formation and migration of kinks at the dislocation and thus
substantially lower the thermal glide activation energy [128].
Galeckas et al. [115] used optical emission microscopy (OEM) in spectrally selective
2A PiN diode is a diode without a direct p-n-junction, but p- and n-doped material are separated
by an undoped region (intrinsic). When unbiased, it has a wide forbidden zone. The p- and n-type
regions are usually heavily doped. Under forward-bias this allows enough carriers from the doped
zones to diffuse into the forbidden zone to enable sufficient current flow.
3Weeks et al. [126] first proposed a theory of recombination-enhanced defect reactions. If a de-
fect, which is involved in a defect reaction (e.g., annealing reactions, defect diffusion), induces a deep
level in the electronic band gap, then the reaction might be enhanced by an electron–hole recombina-
tion associated with that deep level. The theory was later refined by Sumi [127] and first applied to
dislocation glide motion by Maeda and Takeuchi [128].
5.1. Introduction and background
85
4H 6H
Figure 5.4: HRTEM images of partial dislocations in 4H- and 6H-SiC diodes. The basal
plane lies horizontally and the stacking sequences are given at the left and right edges. Bulk
stacking is indicated by white triangles and the fault by a black triangle. Both images are
centred on the partial dislocation core, which is surrounded by a Burgers circuit (indicated
by white dots). Left: 30◦partial dislocation in 4H-SiC. Right: 30◦partial dislocation in 6H-
SiC. (Reproduced from Ref. [124] with kind permission from the authors.)
plan-view to demonstrate stacking fault expansion under low forward current. As
can be seen in Fig. 5.3, the moving Shockley partials appear as bright lines, indicat-
ing the presence of radiative recombination centres on the dislocation line. Based
on spectral analysis, a radiative recombination around 2.8 eV at the stacking fault
and an also radiative recombination around 1.8 eV on the moving Shockley par-
tials is identified. The findings for the stacking fault are in good agreement with
recent theoretical work [121,129]. Further the activation energy for partial glide
motion under forward-bias is found to be 0.27 eV [115], substantially lower than the
estimated value of 2.5 eV obtained from the temperature-dependence of the yield
stress [130] or the brittle-to-ductile transition temperature [115,131]. This supports
a REDG mechanism as mentioned above and preliminary suggests a non-radiative
recombination of approximately 2.2 eV at the Shockley partials [115].
If no solution is found to prevent the formation and propagation of stacking faults,
then this might prove fatal for the development of a SiC-based high-power switch-
ing technology. In fact the Swiss-Swedish company ABB (Asea Brown Boweri) re-
cently retreated from the SiC semiconductor market for exactly this reason.
5.1.3 Earlier theoretical work
As mentioned in the previous section, stacking faults play an important role in de-
vice degradation in SiC. Their energy and electronic structure have been investi-
gated by means of DFT-based calculations recently [121,122,129]. However, only
very little theoretical research has been carried out on the basal plane dislocations,
which border the intrinsic faults and whose motion leads to stacking fault growth.
In the literature the only DFT-based work on these dislocations is that of Sitch et al.
[132], who modelled thermally activated dislocation motion in SiC. However, their
work was restricted to the 90◦glide partial in 3C-SiC only. Hence for a better under-
standing of the recombination-enhanced glide motion in SiC devices, further inves-
tigations are necessary.
86
Chapter 5. Dislocations in Silicon Carbide
5.2 Modelling bulk SiC — the elastic constants
The dislocation structures investigated in this section will be very similar to those
discussed in the case of diamond in the preceding chapter. For this reason the details
of modelling and convergence tests will not be presented again at this point. It is
important however, to check the accuracy of the applied method with respect to
the elastic properties of SiC, since this can and will vary with material and method,
basis sets or parameters4. Therefore in the following the anisotropic and isotropic
elastic constants of SiC will be modelled within the DFTB method.
As discussed in preceding sections, SiC exists in many polytypes. However, their
difference in stacking does not change the elastic properties much. Especially among
the hexagonal types we can expect the elastic constants to be almost identical.
Hence, for reasons of simplicity, the calculations here are restricted to 3Cand 2H,
the latter being a representative of the hexagonal polytypes. Even though experi-
mentally 2Hplays almost no role, it is chosen here since it has the smallest unit cell
of all hexagonal polytypes, allowing a rather inexpensive calculation.
Just as in the case of diamond, the independent elastic constants of 3C- and 2H-SiC
were obtained by suitably deforming a conventional unit cell and calculating the
respective total energies. The integration over the Brillouin zone was accomplished
using a Monkhorst-Pack-optimised set of 3 ×3×3k-points [84]. Tables 5.2 and
5.3 show the results for the cubic and the hexagonal polytype alongside with an
incomplete selection of experimental and previous theoretical data.
3C-SiC: Unfortunately, not much experimental data is available on 3C-SiC. In fact
there seems to be no confirmed direct measurement of the full set of elastic constants.
As already noticed by Lambrecht et al. [135], Lee and Joannopoulos [134] refer to
“experimental data” but do not give any reference or mention how it was obtained.
In later publications — including the reference work of Landolt and B¨ornstein [107]
— these values were often cited as experimental values, even though the work of
Lee and Joannopoulos [134] is purely theoretical and the experimental work cannot
be traced back from that reference. Hence their origin remains a mystery and they
are not quoted here5.
Instead in Table 5.2 the empirical tight-binding results are given alongside with cal-
culations using Tersoff potentials (T), full-potential linear-muffin-tin-orbital (LMTO)
calculations and valence force fields (VFF). The Tersoff results are remarkably close
to the much less approximate LMTO calculations. Overall however, the VFF ap-
proach shows the best agreement with experiments — especially if one compares
µ
and
ν
. The empirical tight-binding calculations (ETB) as well as the density-
functional based tight-binding approach of this work (DFTB) deviate especially in
the off-diagonal constant c12 — a trend already observed in the case of diamond
(Table 4.1). The experimental values given for c11,c12 and c44 have been derived by
Lambrecht et al. [135] from sound velocities measured by Feldman et al. [138].
4Within the DFTB method especially the choice of the DFT reference systems can have a major
influence on quality of the calculations in this respect (see Section 1.1.3 and in particular page 10).
5Even the original authors — who quote those results without a reference — could not help in that
matter [142].
5.2. Modelling bulk SiC — the elastic constants
87
Table 5.2: Elastic properties of 3C-SiC: A comparison between calculated and experimental
data. The first three rows give the independent elastic constants ci j of 3C-SiC. Those values
for the shear modulus
µ
and the Poisson’s ratio
ν
marked with an asterisk are calculated
as Voigt averages following Eq. (1.49) and (1.46). All values are given in GPa, except for
ν
,
which is dimensionless.
DFTBaTbETBcLMTOdVFFeExp.fExp.g
(3C) (3C) (3C) (3C) (3C) (3C) (3C)
c11 487 420 363 420 428 390
c12 218 120 209 126 165 142
c44 232 260 149 287 246 256
µ
193∗216∗120∗231∗200 192 203∗
ν
0.241∗0.130∗0.300∗0.116∗0.187 0.168 0.153∗
aValues obtained using the DFTB method.
bTheoretical results of Tersoff [133] using semiempirical interatomic potentials.
cTheoretical results of Lee and Joannopoulos [134] using an empirical tight-binding scheme.
dTheoretical results of Lambrecht et al. [135] using a full-potential LMTO approach.
eTheoretical work of Mirgorodsky et al. [136] in a valence force field approach.
fExperimental data by Carnahan [137].
gData derived from sound velocities measured by Feldman et al. [138]. For details see Ref. [135].
Table 5.3: Elastic properties of hexagonal SiC: A comparison between calculated and exper-
imental data. The first five rows give the independent elastic constants ci j of the respective
polytype. Those values for the shear modulus
µ
and the Poisson’s ratio
ν
marked with an
asterisk are calculated as Voigt averages following Eq. (1.50) and (1.46). All values are given
in GPa, except for
ν
, which is dimensionless.
DFTBaVFFbExp.cExp.dExp.dExp.e
(2H) (2H) (6H) (4H) (6H) (4H/6H)
c11 563 520 502 501
c12 193 145 95 111
c13 140 89 52
c33 672 585 565 605 565 553
c44 162 170 169 163
µ
190∗201 194∗
ν
0.242∗0.174 0.207 0.212 0.212 0.161∗
aValues obtained using the DFTB method.
bTheoretical work of Mirgorodsky et al. [136] in a valence force field approach.
cExperimental data by Arlt and Schodder [139], average of two measurements.
dExperimental data by Karmann et al. [140] by measuring piezoelectric properties.
eExperimental data by Kamitani et al. [141] (Brillouin scattering).
88
Chapter 5. Dislocations in Silicon Carbide
Hexagonal SiC: Unlike the case of the cubic polytype, for 4H- and 6H-SiC plenty
of experimental data is available. However, the experimental determination of c13
seems to be a problem, and the only value given here is derived from measurements
on only one particular sample, as explained in Ref. [141]. In the same work, no ma-
jor differences between the two hexagonal polytypes 4Hand 6Hcould be found.
The difference between 4Hand 6Hin Ref. [140] might be explained by the different
temperature of the sample: The value for 6Hwas determined at room temperature,
whereas that for 4Hwas measured at 20 K. If this reflects a general trend, then the
elastic constants of SiC depend considerably on the temperature, and lower temper-
atures lead to larger values. This can then explain, why many calculations — usu-
ally zero-temperature calculations — give larger elastic constants than determined
in experiments.
As mentioned in Section 4.2, the relatively large deviations in
ν
do not influence the
accuracy of dislocation modelling much, since the important quantities are
µ
, 1 ±
ν
and 1 −2
ν
. As can be seen in Tables 5.2 and 5.3 the deviation of the DFTB result
from the experimental value is negligible in the case of
µ
. However, the error in
ν
cannot be quantified precisely since the experimental results are not coherent: The
experimental values themselves differ from each other by as much as 40 %.
5.3 Straight Shockley partials in the basal plane
As pointed out in Section 5.1.2, the technologically relevant dislocations in the basal
plane of SiC are the 30◦and 90◦Shockley (glide) partial dislocations bordering stack-
ing faults. Hence the investigations presented here will be entirely restricted to those
partials.
5.3.1 Core structures
Since the basal plane in all hexagonal polytypes and the {111}planes in 3C-SiC are
identical except for the surrounding crystal stacking, the core structures of disloca-
tions whose line direction and Burgers vectors lie in these planes can be expected to
be identical within the accuracy of the methods employed in this work. So at least in
terms of structures and line energies it would be sufficient to limit all investigations
to one polytype only. However, to be on the safe side, all core structures will be mod-
elled in both 3C-SiC and 2H-SiC. The latter polytype is chosen as a representative for
hexagonal stacking for reasons of simplicity, even though experimentally 2Hplays
no significant role. These two polytypes allow us, to look at the (basal) glide plane
embedded in either a locally cubic or hexagonal stacking sequence respectively6.
The structural optimisation was performed with the DFTB method in supercell-
cluster hybrid models of similar dimensions to those described in Section 4.2.2. All
structures and also energies presented in the following were obtained without the
6Possible glide planes in 4Hand 6Hare locally embedded in cubic stacking of the nearest and
next-nearest planes on one side and hexagonal stacking on the other. Further in 6H, glide planes with
locally cubic stacking on both sides are possible (Fig. 5.1).
5.3. Straight Shockley partials in the basal plane
89
self-consistent-charge extension of DFTB7. Further all calculations were performed
at the
-point only, which — as demonstrated in Section 4.2.2 — is sufficient. Details
on the structural optimisation process can be found in Section 4.3.1.
As already mentioned in Section 3.1.2, within compound semiconductors each dis-
location core with an edge component can exist in two variants. Depending on the
atom type of the terminating line of atoms, we will speak of a silicon core or a car-
bon core. This combined with the three low energy glide partial cores — 30◦and
single- and double-period reconstruction of the 90◦partial — allows six low energy
core structures. Fig. 5.5 shows the optimised structures in the 3Cpolytype. The
following paragraphs give a brief description:
The 30◦glide partial (Fig. 5.5 (a),(d)): Very similar to the case in diamond, the
30◦glide partial reconstructs forming a line of bonded atom pairs. As given in Ta-
ble 5.4, for the silicon core the Si–Si reconstruction bond length is almost exactly
like in bulk silicon. The equivalent C–C reconstruction bonds in the the carbon core
however, are stretched by 17 % compared to bulk diamond.
The 90◦glide partial (SP) (Fig. 5.5 (b),(e)): In the single-period structure of the 90◦
glide partial, reconstruction bonds are formed which connect the faulted with the
unfaulted region of the glide plane. In the silicon-terminated core these bonds are
of comparable length to bulk silicon. Just as at the 30◦partial, at the carbon core the
reconstruction bonds are very stretched (Table 5.4).
The 90◦glide partial (DP) (Fig. 5.5 (c),(f)): This double-period structure can be ob-
tained from the single-period structure by introducing alternating kinks. For both
the silicon and the carbon core this results in a shortening of the reconstruction
bonds, leaving the Si–Si bonds slightly compressed compared to bulk silicon. The
C–C bonds, however, still appear about 10 % stretched compared to bulk diamond.
The undissociated dislocations are not shown. Their structures are similar to the
equivalent structures in diamond. Although as for the partials presented here, the
carbon- and silicon-terminated dislocation cores differ from each other in recon-
struction bond lengths and angles.
7Even though in contrast to diamond, silicon carbide is a rather polar compound semiconductor,
the self-consistent-charge variant of the DFTB method (SCC-DFTB, see Section 1.1.3) gave no substan-
tial difference in structure or relative energies compared to the standard variant. This might be due to
the rather bulk-like bonding situation for all glide dislocation cores.
Table 5.4: Relative reconstruction bond lengths of the Shockley partials in 3C-SiC. All bond
lengths are given relative to the respective bulk bond length in silicon or diamond, as ob-
tained within the DFTB method.
30◦glide 90◦(SP) glide 90◦(DP) glide
Si–Si (bulk Si) +0.7 % −0.4 % −1.4 %
C–C (bulk diamond) +17.0 % +14.6 % +9.7 %
90
Chapter 5. Dislocations in Silicon Carbide
(d) 30° C glide (e) 90° C glide (SP) (f) 90° C glide (DP)
(a) 30° Si glide (b) 90° Si glide (SP) (c) 90° Si glide (DP)
[111][110]
[112]
[111][110]
[112]
Figure 5.5: The relaxed core structures of the Shockley partials in the {111}plane of 3C-SiC.
(a) – (c) show the silicon-terminated partials. (d) – (f) show the carbon-terminated partials.
For each structure, the upper figure shows the view along the dislocation line projected
into the (1¯
10)plane, and the lower figure shows the (111)glide plane. The region of the
intrinsic stacking fault accompanying the partials is shaded. In case of the 90◦partial SP and
DP denote the single-period and double-period core reconstruction.
5.3. Straight Shockley partials in the basal plane
91
[0001]
[1100]
[1120]
r
h
bh
cc
Figure 5.6: The relaxed
core structure of a 30◦
carbon Shockley partial in
2H-SiC. Left: The view
along the dislocation line
projected into the (11¯
20)
plane. Right: The view pro-
jected into the (0001)basal
plane. The region of the
intrinsic stacking fault ac-
companying the partial is
shaded.
Fig. 5.5 shows the partial dislocations in the cubic stacking sequence only. All struc-
tures however, have also been optimised in the hexagonal 2Hstacking. Fig. 5.6
gives the core structure of the carbon-terminated 30◦partial in 2H-SiC as an exam-
ple. Compared to its equivalent structure in 3C-SiC (Fig. 5.5 (d)), where the stacking
fault was given as the hexagonal region of the glide plane (left half), here the sit-
uation is reversed and the cubic region (right half) defines the stacking fault. The
actual bond angles and lengths within the glide plane appear to be identical within
the accuracy of the methods applied. Table 5.5 gives the differences in selected bond
lengths for all investigated partial core structures. The largest deviation is found in
the case of the carbon-terminated 90◦double-period partial. But even here the dif-
ference of approximately 0.4 % is negligible, if one keeps in mind that the respective
bond is 10 % stretched compared to bulk diamond.
As a consequence for all partials, the core structures seem to be almost entirely in-
dependent of the surrounding crystal stacking8. The same can be expected for the
core energies and will be tested in the next section.
83Cand 2Hgive an either purely cubic or purely hexagonal stacking environment. Since the
differences between the corresponding core structures in both stackings are miniscule, one can expect
them to be just as small in other stacking sequences, which are (at least locally) of mixed cubic and
hexagonal nature, like 4Hand 6H.
Table 5.5: Differences in core bond lengths of the Shockley partial dislocations in 2H-SiC
compared to 3C-SiC. As shown in Fig. 5.6,rdenotes the reconstruction bond. cand hare
the bonds connecting the central line of atoms with the cubic and hexagonal region of the
glide plane respectively. bconnects the central line of atoms with the bulk region in the
adjacent plane. r, which is given for the double-period 90◦partials, gives the average of the
two slightly different reconstruction bonds in each structure. The numbers given represent
the change in bond length from 3Cto 2Hfor Si-core and C-core structures.
30◦glide 90◦(SP) glide 90◦(DP) glide
r c h b r r
Si +0.08 % +0.06 % +0.09 % +0.10 % −0.17 % −0.10 %
C+0.08 % +0.08 % +0.08 % +0.17 % −0.25 % −0.39 %
92
Chapter 5. Dislocations in Silicon Carbide
Table 5.6: The calculated energy factors k(
β
)for 30◦and 90◦partials in cubic (C) and hexag-
onal SiC (H). Both isotropic and anisotropic elasticity theory are applied.
30◦90◦
C H C H
k(
β
)isotropica(GPa) 208 (212) 205 (203) 254 (240) 251 (231)
k(
β
)anisotropicb(GPa) 195 (191) 191 (191) 249 (231) 246 (227)
aEvaluated using Eq. (2.5). The first number is given by the elastic constants obtained in the DFTB
method, the number in brackets by the experimental constants based on Feldman et al. [138] (3C) and
Kamitani et al. [141] (4H/6H) (for both see Table 5.2 and 5.3).
bEvaluated using linear anisotropic elasticity theory as suggested in [93] and [41].
5.3.2 Core energies
Before discussing the core energies it is useful to investigate the influence of the
polytype on the energy factors for the two partials, as these play an important role in
obtaining the core energies following Eq. (2.4). With the elastic constants calculated
in Section 5.2, the energy factors can be obtained easily using Eq. (2.5) or — a bit
more elaborately — by means of anisotropic theory as described in [41]. Table 5.6
gives the energy factors obtained for the 30◦and 90◦partial dislocations in both
3C- and 2H-SiC, based on the DFTB elastic constants. For comparison the values
based on the experimental constants for 3Cand 4H/6Hare shown in brackets. The
difference between the energy factors in cubic and hexagonal stacking appears to be
negligible in the case of the DFTB energy factors. Only the isotropic values based on
experiments differ by up to 5 %. One has to keep in mind though, that the observed
elastic constants for cubic and hexagonal stacking were obtained in very different
experiments9. Overall the absolute values obtained in the anisotropic calculation
are slightly lower than the isotropic values. The same effect was already observed
for dislocations in diamond (see Table 4.3).
In Section 4.2.2 the spatial distribution of the formation energy was used to obtain
the core radius Rcand the core energy Ecof a dislocation in comparison with the
elastic energy as given in continuum theory (Eq. (2.4)). Now the same procedure
will be applied to the Shockley partials in 3C- and 2H-SiC. However, since SiC is a
compound semiconductor, one has to be careful when speaking of formation ener-
gies: In this section the radial formation energies and also the resulting core energies
are all given relative to a stoichiometric crystal10.
9Taking
µ
and
ν
of Carnahan [137] as the cubic isotropic constants instead of Feldman et al. [138],
reduces the difference between the cubic and hexagonal case to almost zero.
10 As mentioned in Section 4.2.2, within the DFTB method for each atom in a modelled structure
an atomic total energy can be defined easily. The “formation energy” of a particular atom in that
structure is then obtained by subtracting the total energy of the same type of atom in a perfect —
and hence stoichiometric — SiC crystal (Eq. (4.2)). In terms of chemical potentials, here those chemical
potentials, which correspond to a stoichiometric crystal growth for silicon and carbon, are assumed.
5.3. Straight Shockley partials in the basal plane
93
(eV/Å)E/L
ln( /Å)R
90° (DP) in 3C−SiC
C core
Si core
580 meV/Å
0
1
1 1.5 2
Figure 5.7: The radial formation en-
ergy of the double-period reconstruc-
tion of the 90◦glide partial in 3C-SiC.
Ef(R)/Lis given by Eq. (4.3). The gra-
dient of the dashed line is obtained as
a fit to the linear part of the energy,
and corresponds to the elastic energy in
Eq. (2.4). The fit yields k(
β
) = 244 GPa
for the C-terminated core and k(
β
) =
250 GPa for the Si-terminated core. A
circle marks the approximate point for
each core, where for R→0 the energy
deviates considerably from the straight
line. This point defines the respective
core radius Rcand the core energy Ec.
As an example, Fig. 5.7 shows the resulting radial formation energy per unit length
for the double-period 90◦partial dislocations in 3C-SiC (Fig. 5.5 (c),(f)). The C core
structure is found about 580 meV/ ˚
A higher in line energy than the Si core.
The fitted energy factors, core radii and core energies for all investigated Shockley
partials are given in Table 5.7. For all partials the energy factors obtained in the
fitting procedure agree with those found in elasticity theory (given in Table 5.6).
On average the agreement is better with anisotropic elasticity theory. Compared to
diamond, all core energies are very small.
Some particular results will be discussed in the following paragraphs.
Table 5.7: The calculated energy factors k(
β
), core radii Rcand core energies Ecof the basal
plane Shockley partials in SiC. The corresponding structures can be found in Fig. 5.5 for
3C-SiC in the same sequence. To facilitate comparison between different dislocations with
different core radii, the core energy E0
ccorresponding to a radius of 6 ˚
A is introduced.
30◦Si 90◦Si (SP) 90◦Si (DP)
3C2H3C2H3C2H
k(
β
)a(GPa) 203 194 249 251 250 242
Rc(˚
A) 5.3 5.5 3.9 3.4 6.0 5.2
Ec/L(eV/ ˚
A) 0.44 0.45 0.51 0.46 0.59 0.54
E0
c(R=6˚
A)/L(eV/ ˚
A) 0.48 0.47 0.68 0.69 0.59 0.60
30◦C 90◦C (SP) 90◦C (DP)
3C2H3C2H3C2H
k(
β
)a(GPa) 194 197 242 237 244 244
Rc(˚
A) 4.5 4.7 5.0 4.1 4.7 4.1
Ec/L(eV/ ˚
A) 0.77 0.78 1.20 1.13 1.08 1.02
E0
c(R=6˚
A)/L(eV/ ˚
A) 0.86 0.85 1.27 1.27 1.17 1.17
aFit to Ef(R)vs. ln(R/Rc)plot for R≥Rcfollowing Eq. (2.4).
94
Chapter 5. Dislocations in Silicon Carbide
Silicon- and carbon-terminated partials: Compared with the Si core structures,
the corresponding C core structures have a higher line energy: 380 meV/ ˚
A in case
of the 30◦partials and around 580 meV/ ˚
A in case of the 90◦partials. To some extent
this higher line energy might be attributed to the reconstruction bonds: In the Si core
the reconstruction bonds are of about the same length as they would be in perfect
bulk silicon, giving a rather low energy configuration. In the C core however, the
bonds are stretched by 10 % or more compared to diamond (Table 5.4), which is
energetically less favourable.
Single- and double-period reconstructions: In both polytypes the Si core double-
period reconstruction of the 90◦partial appears around 90 meV/ ˚
A lower in energy
than the single-period structure. Similarly, the difference for the carbon-terminated
structures is 100 meV/ ˚
A. With the difference that small, both reconstructions can be
expected to co-exist.
Effects of the polytype: Since the polytype seems not to have any notable effect
on the structure of the partials in the glide plane, one would expect the same for
the energies. And in fact there appears to be no considerable difference in the line
energies of equivalent partials — the maximum differences in E0
c(R=6˚
A)/Lare of
the order of 10 meV/ ˚
A, which is negligible. Larger deviations occur only for the core
radii and the resulting energies. But as mentioned in Section 4.2.2, the definition of
these core radii is rather vague anyway, and no strong conclusions can be drawn
from it.
With the Shockley partials showing the same energies in the 3Cand the 2Hpolytype,
the same can be expected for 4Hand 6Has well. Hence in terms of structure and
energy the stacking sequence seems to have no impact on the partial dislocations in
the basal plane of SiC.
5.4. Dislocation glide motion
95
5.4 Dislocation glide motion
Recombination-enhanced glide motion of partial dislocations presumably causes
the degradation of bipolar SiC devices as described in Section 5.1.2. To gain more
insight into this mechanism, it is an important prerequisite to understand the un-
derlying atomistic processes of dislocation glide. Therefore, in this section the glide
motion of the involved partial dislocations is modelled as a thermally activated pro-
cess involving the formation and migration of kinks — details on this process, which
assumes the absence of strong obstacles to dislocation motion, were given in Sec-
tion 4.5.1. The resulting kink formation energies and migration barriers will then
allow us to estimate the thermal activation energies for glide motion in SiC.
The cluster models and methods used in this section are exactly the same as de-
scribed in Section 4.5 for dislocation glide in diamond. For this reason most of the
details will not be repeated here. The calculations, which are all performed using
the DFTB method, are restricted to the cubic polytype only. But as we have learnt
in the preceding sections, changing the polytype has almost no impact on the dis-
location structures and resulting energies in the basal plane. Hence all calculations
described below will be representative for all polytypes alike.
5.4.1 The glide motion of 90◦partial dislocations
Fig. 5.8 (far left panels) shows relaxed kink structures at the 90◦silicon and the 90◦
carbon single-period partial projected into the basal plane. All atoms are four-fold
coordinated. The difference in the reconstruction bond lengths of the two different
partials (Si–Si or C–C bonds respectively) leads to slightly different bond angles, but
qualitatively both structures are very similar and close to that found for diamond
(Fig. 4.15).
The kink formation energy Efis obtained by comparing a cluster containing a
straight dislocation segment with one of the same stoichiometry containing a
double-kink. The elastic kink–kink interaction energy and the stacking fault energy
are subtracted following Eq. (4.8) and (4.9). Table 5.8 gives the resulting single-kink
formation energies — for the silicon and the carbon partial both around 0.5 eV.
The elementary kink migration step of the right kink is depicted in Fig. 5.8, where
the kink K, the saddle point Sand the migrated kink K’ are shown for both partials.
In the process only two atoms move considerably and break and form new bonds.
The motion of these two atoms — one silicon and one carbon atom — is used to
parameterise the migration step (see Fig. 4.17). Fig. 5.9 (left) gives the correspond-
ing two-dimensional energy surfaces11. The energy surfaces differ considerably for
the two 90◦partials. For the carbon partial the saddle point is well pronounced
and the barrier can only be overcome by varying both parameters. The silicon and
the carbon atom seem to be equally involved in the process. At the silicon par-
11 As in Section 4.5 the energy surface is obtained at 10 ×10 points in the two-dimensional param-
eter space by relaxing the whole structure subject to constraining the two primary atoms to lie in a
plane perpendicular to the connecting line between the initial and final position. In the vicinity of the
saddle point S, the parameter mesh is refined by a factor of 10. The model used is a dislocated cluster
containing a single kink.
96
Chapter 5. Dislocations in Silicon Carbide
90° C
90° Si
K’
KS
K’
KS
Figure 5.8: Kink migration at the 90◦(SP) glide partial. The relaxed structures of the starting
kink K, the saddle point Sand the migrated kink K’ are shown projected into the glide
plane. The faulted region is shaded and arrows indicate the motion of the two involved
atoms. Upper panel: The silicon-terminated core. Lower panel: The carbon-terminated core.
tial, however, the motion of the carbon atom clearly dominates the process12. The
whole procedure was also applied to the two left kinks. But since the correspond-
ing structures are approximately symmetric in the glide plane, no difference in the
migration barriers could be found. As can be seen in Table 5.8, the barrier at the
silicon-terminated partial is about 1 eV larger compared to the carbon-terminated
partial. With diamond having a larger cohesive energy than silicon, this might seem
contra-intuitive at first glance. However one has to keep in mind, that the C–C re-
construction bonds at the partial are considerably stretched compared to diamond,
which lowers the energy required to break the bond.
From the kink formation energies and the migration barriers, the activation energy
of the kink migration process can be evaluated following Eq. (4.7). Table 5.8 gives
the results. With the kink formation energies being almost the same at both par-
tials, the migration barrier is the deciding quantity: In a glide process controlled by
the formation and migration of kinks, when no strong obstacles to dislocation mo-
tion are present, the carbon partial is clearly the more mobile 90◦partial dislocation.
The energy corresponding to the whole glide process of the 90◦partial is schemati-
cally shown in Fig. 5.9 (right) for the first four kink migration steps, assuming short
dislocation segments. There the dashed line connecting the minima represents the
formation energy of the kink pair and the solid line the energy of the minimum en-
12 To some extent this might be explained by the Si atom only moving by roughly 0.6 ˚
A with re-
spect to the surrounding crystal, whereas the C atom moves by 1.2 ˚
A. This of course scales the two
parameters accordingly. Also the different bonding situation for the two atoms will play a role: The
movement of the Si atom involves the breaking and reforming of Si–Si bonds, which are rather “soft”,
whereas the movement of the C atom breaks and reforms “hard” Si–C bonds.
5.4. Dislocation glide motion
97
Energy (eV)
1
2
0
Energy (eV)
1
2
0
carbon
silicon
carbon
silicon
Kink separation
KK’
S
Ef
2 = 0.91 eV
Kink separation
S
Ef
2 = 1.03 eV
= 3.06 eV
m
W
K’
K
= 1.83 eV
m
W
90° C
90° Si 90° Si
90° C
S
S
0
1
2
3
2
1
0
Energy (eV) Energy (eV)
K’
KK’
K
Figure 5.9: The energies and barriers of kink formation and migration at the 90◦(SP) glide
partial. Left diagrams: The energy surface of the process leading from the starting kink K
via the saddle point Sto the migrated kink K’. The two parameters are defined via the
two involved atoms (see text and Fig. 4.17). Right diagrams: Schematic representation of
the energy of the glide process. As shown in Fig. 4.14, a kink pair is formed and subsequent
migration of the two kinks enlarges their separation. K,Sand K’ are labelled for an arbitrary
migration step. The dashed line connecting the minima represents the formation energy of
the kink pair. The energy contribution of the expanding stacking fault is not included in the
graph. Upper panel: The silicon-terminated core. Lower panel: The carbon-terminated core.
ergy path for glide motion. The first few minima are considerably lower due to the
attractive kink–kink interaction.
For reasons of simplicity dislocation glide was once again modelled for the single-
period reconstruction of the 90◦partial only. However since the elementary pro-
cesses are very similar for the double-period structure, similar kink formation ener-
gies and migration barriers can be expected.
Table 5.8: Kink formation energies Efand migration barriers Wmfor the 90◦Shockley par-
tials. The resulting thermal glide activation energy Q90 =2Ef+Wmassumes short dis-
location segments. The number in brackets gives the respective value for long segments
(Ef+Wm).
90◦(SP) glide (all energies in eV)
EfWmQ90
Si 0.515 3.06 4.09 (3.58)
C 0.455 1.83 2.74 (2.29)
98
Chapter 5. Dislocations in Silicon Carbide
30° C LK
30° Si LK
LK2 LK1 LK2’
LK2 LK1 LK2’
Figure 5.10: The elementary kink migration steps of left kinks at the 30◦glide partial:
LK2 →LK1 →LK20. The relaxed structures of the high-energy kink LK2, the low en-
ergy kink LK1 and the migrated high-energy kink LK2’ are shown projected into the glide
plane. The faulted region is shaded and arrows indicate the motion of the two involved
atoms. Upper panel: The silicon-terminated core. Lower panel: The carbon-terminated core.
30° C RK
30° Si RK
RK2 RK1 RK2’
RK2 RK1 RK2’
Figure 5.11: The elementary kink migration steps of right kinks at the 30◦glide partial:
RK2 →RK1 →RK20. The relaxed structures of the high-energy kink RK2, the low en-
ergy kink RK1 and the migrated high-energy kink RK2’ are shown projected into the glide
plane. The faulted region is shaded and arrows indicate the motion of the two involved
atoms. Upper panel: The silicon-terminated core. Lower panel: The carbon-terminated core.
5.4. Dislocation glide motion
99
5.4.2 The glide motion of 30◦partial dislocations
Since the growing stacking faults observed in degrading bipolar SiC devices are bor-
dered by dislocations on four sides (Fig. 5.3), there will be both 90◦and 30◦partials
present. Further, as already demonstrated in Section 4.5.3 for diamond, the kink
structures and the mechanisms for kink migration at the 30◦partial are different
from those at the 90◦partial, leading to substantially different glide activation bar-
riers. Therefore in this section kink formation and migration at both 30◦partials are
examined in detail.
Due to the double-periodicity of the reconstructed 30◦dislocation shown in
Fig. 5.5 (a) and (d), two variants of left kinks (LK1 and LK2) and two variants of
right kinks (RK1 and RK2) are possible. The left and middle panels of Fig. 5.10
depicts the different relaxed and reconstructed kink structures for the left kinks, dis-
tinguishing between the Si-terminated and the C-terminated partial. Fig. 5.11 shows
the same for the right kinks. As can be seen, in the case of LK1 and RK1 the kink
is located on a reconstruction bond, whereas LK2 and RK2 are located between two
reconstruction bonds. As in the case of diamond, all atoms in all structures are four-
fold coordinated. In principle the structures at the silicon core and the carbon core
are very similar, with only differences due to the different character of the Si–Si and
C–C bonds. The bond lengths and as a consequence also the bond angles differ
between the two partials. The high-energy left kinks (LK2) appear with an interest-
ing feature: The kink forms a reconstruction bond, which is of the opposite type to
the reconstruction bonds of the respective partial — at the silicon-terminated par-
tial with Si–Si reconstruction bonds the kink reconstructs with an “alien” C–C bond,
and vice versa (Fig. 5.10 (left)).
The kink formation energies are obtained by introducing kink pairs in dislocated
cluster models. The energy of the generated faulted area and the kink–kink inter-
action energy have to be subtracted according to Eq. (4.10) and (4.11). Since the
structures of left and right kinks are very different, one cannot expect similar for-
mation energies. Hence the modelling of kink pairs does not allow a calculation of
explicit single-kink energies, but only the sums of two kink formation energies. For
Table 5.9: Kink formation energies Efand migration barriers Wmfor the 30◦Shockley par-
tials. One has to differentiate between left and right kink (LK and RK).
Ef(LK)gives the
energy difference between the low energy (LK1) and the high-energy left kinks (LK2) and
Ef(RK)accordingly for the right kinks. To facilitate comparison with the 90◦partials, for
the 30◦partials an average activation energy Q30 = (2Ef(LK1) + Wm(LK) + 2Ef(RK1) +
Wm(RK))/2 is introduced. This glide activation energy assumes short dislocation segments.
The number in brackets gives the respective value for long segments.
30◦glide (all energies in eV)
Ef(LK1) + Ef(RK1)
Ef(LK)Wm(LK)
Ef(RK)Wm(RK)Q30
Si 1.62 2.87 3.79 0.28 2.87 4.95 (4.14)
C 2.21 2.11 3.00 0.33 1.78 4.60 (3.50)
100
Chapter 5. Dislocations in Silicon Carbide
0.28 eV
2.87 eV
1.62 eV
2.87 eV
3.79 eV
1.62 eV
4
5
6
RK1
RK2 RK2’
Kink separation
LK1
LK2
LK2’
Kink separation
Energy (eV)
left kinks right kinks
0.33 eV
1.78 eV
2.21 eV
4
5
6
2.11 eV
3.00 eV
2.21 eV RK1
RK2
RK2’
Kink separation
LK1
LK2
LK2’
Kink separation
Energy (eV)
right kinksleft kinks
30° Si
30° C
1
2
0
1
2
0
Figure 5.12: The energies and barriers of kink formation and migration at the 30◦glide
partial. The graphs schematically represent the energy of the glide process: As shown in
Fig. 4.14, a kink pair is formed and subsequent migration of the two kinks enlarges their
separation. Since for the 30◦partial the migration barriers differ for left and right kinks,
the corresponding processes are shown separately. The dashed line connecting the minima
represents the formation energy of the kink pair. The energy contribution of the expanding
stacking fault is not included in the graph. Upper panel: The silicon-terminated core. Lower
panel: The carbon-terminated core.
details see Section 4.5.3. Table 5.9 gives the resulting formation energies of the low
energy kink pair Ef(LK1) + Ef(RK1), as well as the extra energy
Efneeded if ei-
ther the low energy left kink LK1, or the low energy right kink RK1, is replaced by
the respective high-energy kink LK2 or RK2. With around 0.3 eV, this extra energy
is particularly low for the right kinks RK2 at both partials and larger than 2 eV for
the left kink LK2. The two right kinks appear with similar bond angle distortions,
whereas among the two left kinks it is clearly LK2 which shows the strongest dis-
tortions (>20◦at the two atoms closest to the kink). This might explain the large
energy difference between LK1 and LK2.
As in the case of the 90◦partial, the elementary migration steps are modelled in dis-
located clusters containing a single kink. Fig. 5.10 and 5.11 show the kink migration
steps LK2 →LK1 →LK20and RK2 →RK1 →RK20at the two different partials.
The resulting energy surfaces for each single step look very similar to those at the
90◦partial shown in Fig. 5.9 and again at the silicon partial it is the motion of the
carbon atom, which dominates the process. Due to symmetry in the glide plane
LK2 →LK1 and LK1 →LK20(or the equivalent right kink processes) are approx-
imately twin processes and yield the same migration barriers within 1 %. Table 5.9
5.4. Dislocation glide motion
101
gives the resulting migration barriers Wmfor both left and right kinks and Fig. 5.12
schematically shows the resulting energy of the migration processes of left and right
kink at each partial. Overall the right kinks are more mobile.
As explained in Section 4.5.3, Eq. (4.12) describes the velocity of thermally activated
glide motion with two distinct migration barriers for left and right kinks. However
to allow an easy comparison, as a crude approximation an average activation barrier
Q30 = (2Ef(LK1) + Wm(LK) + 2Ef(RK1) + Wm(RK))/2 can be defined for short
segments. For long segments the same expression holds, however with half the
formation energies. Both values are given in Table 5.9. Among the 30◦partials the
carbon-core is slightly more mobile than the silicon-core partial, but the difference
is less pronounced than for the 90◦partial.
5.4.3 Dislocation glide motion — summary
The structures and basic processes involved in the formation and migration of kinks
at basal plane partial dislocations in SiC are very similar to those found in the {111}
plane of diamond. Also the general trends with respect to barrier heights and for-
mation energies are the same. Overall, the 90◦partial is the thermally more mobile
species. Further, the purely thermal mobility of carbon partials in the absence of
strong obstacles appears to be higher than that of silicon partials, and the velocities
of undecorated neutral partials are largely controlled by their kink migration energy.
Comparison with experiments: Experimental results in 6H-SiC give activation en-
ergies in the range of Q=2.1 to 4.8 eV [143,144]. This is reasonably close to the
activation energies found in this work, which range from 2.3 eV for the carbon-
terminated 90◦partial to 4.1 eV for the silicon-terminated 30◦partial — assuming the
glide process to involve the motion of long dislocation segments.
The carbon partials being the more mobile species is in contradiction to the conclu-
sions of Pirouz and Yang [145]. However, LACBED (large angle convergent beam
electron diffraction) experiments seem to indicate, that the Si partials in 3C-SiC are
smooth whereas the C partials are zig-zagged [146], suggesting they are pinned by
obstacles. In this case the glide process would not mainly be controlled by the for-
mation and migration of kinks, but by the energy needed for the pinning and un-
pinning at the obstacles.
Comparison with independent theoretical work: In comparison with earlier DFT-
based results for the 90◦(SP) partials in SiC by Sitch et al. [132], the kink migration
barrier found here is about 0.4 eV higher at both the silicon and the carbon partial.
This might result from the use of much larger cluster models in this work, which
give a more accurate representation of the stress response of the surrounding bulk
material. In addition, in Ref. [132] the migration barrier was deduced from the bar-
rier to double-kink formation, rather than kink expansion as modelled here. The
formation energy of the smallest double-kink at the C partial is 0.4 eV lower than
found in the present work. A more striking difference is found for the Si partial.
In this work the kink formation energy appears to be similar to that at the C core,
whereas in Ref. [132] it is negligible. Nevertheless, both calculations conclude that
the thermal activation barrier for the carbon partial is lower than that of the silicon
partial.
102
Chapter 5. Dislocations in Silicon Carbide
5.5 Electronic structure calculations
In this section the electronic structure of straight and kinked Shockley partial dislo-
cations in the basal plane of SiC will be investigated.
The band structures of straight Shockley partials: In Section 5.3 it was shown that
the effects of the stacking sequence of SiC on the atomic structure and line energy
of the Shockley partials is negligible. Hence one could jump to the conclusion that
the same holds as well for the electronic structure of the partials. However, as can
be seen in Table 5.1, the electronic band gap varies from 2.39 to 3.33 eV, depending
on the polytype [107,123]. Also, the band structures themselves vary with the poly-
type. If the partial dislocations induce defect states in the band gap, then a wider
band gap might allow states to appear in the gap, which in case of a smaller band
gap would disappear into either the valence or the conduction bands. As a con-
sequence, the electronic structure of the partials has to be investigated at least for
the two extremes, which are once again 3C-SiC with a band gap of 2.39 eV and 2H-
SiC with a band gap of 3.33 eV. Therefore, in this section the projected band struc-
tures of the discussed Shockley partials in 3C- and 2H-SiC are modelled within the
DFT-pseudopotential approach (AIMPRO, see Section 1.1.2). As already described
in Section 4.7, the band structure calculations are performed in smaller hybrid mod-
els than those for the calculation of the core energies — for SiC the model size is
on average 110 atoms. The structures are relaxed with the AIMPRO code, using a
set of 2 k-points along the the axis in k-space, which corresponds to the dislocation
line direction. To then obtain the projected electronic band structure of the relaxed
structure, the Kohn-Sham eigenstates are calculated at 21 different k-points along
the same axis.
Fig. 5.13 shows the projected band structures of bulk 3C- and 2H-SiC as well as 2H-
SiC containing an intrinsic stacking fault. The calculated width of the gap of bulk
2 −SiC bulkH2 −SiC ISFH3 −SiC bulkC
(eV)
−2
0
2
4
(eV)
−2
0
2
4
(eV)
−2
0
2
4
Figure 5.13: The band structures of bulk and faulted SiC projected onto the k-space axis
corresponding to the basal plane dislocation line direction ([1¯
10]in 3Cand [11¯
20]in 2H
with respective periodicity lengths of a0/√2 and a0). Only those bands which are close to
the gap are drawn, the rest of the valence and conduction band regions are shaded. The
origin of the Brillouin zone is at the far left of each band structure.
5.5. Electronic structure calculations
103
(d) 30° C (e) 90° C (SP)
(a) 30° Si (b) 90° Si (SP)
3 −SiCC3 −SiCC
(f) 90° C (DP)
(c) 90° Si (DP)
(eV)
−2
0
2
4
(eV)
−2
0
2
4
(eV)
−2
0
2
4
(eV)
−2
0
2
4
(eV)
−2
0
2
4
(eV)
−2
0
2
4
Figure 5.14: The band structures of glide partials in 3C-SiC projected onto the k-space axis
corresponding to the dislocation line direction [1¯
10](with a periodicity length of a0/√2
for (b) and (e) and √2a0for all others). (a) – (c) show the band structures of the silicon-
terminated partials and (d) – (f) show those of the carbon-terminated partials. Fig. 5.5 gives
the corresponding core structures in the same sequence. Only those bands which are close
to the gap are drawn, the rest of the valence and conduction band regions are shaded. The
origin of the Brillouin zone is at the far left of each band structure. Being double-period
reconstructions, (a), (c), (d) and (f) appear with a Brillouin zone of half width.
SiC deviates from experimental results. Two competing effects have to be consid-
ered: Commonly in LDA the band gap is under-estimated. On the other hand in the
chosen supercell-cluster hybrid models the wavefunction is confined with respect
to two dimensions, leading to a widening of the gap. Both effects compensate to
some extent, yielding a gap just 18 % too small for both the 3Cand 2Hpolytype.
As predicted by K¨ackell et al. [122], the intrinsic stacking fault in 3C-SiC does not
introduce any deep electronic gap states and therefore its band structure is not dis-
played. In 2H-SiC however, an intrinsic fault represents a layer of cubic inclusion in
the lattice, so one would expect empty and localised gap states near the conduction
104
Chapter 5. Dislocations in Silicon Carbide
(a) 30° Si (b) 90° Si (SP)
(d) 30° C (e) 90° C (SP)
2 −SiCH2 −SiCH
(c) 90° Si (DP)
(f) 90° C (DP)
(eV)
−2
0
2
4
(eV)
−2
0
2
4
(eV)
−2
0
2
4
(eV)
−2
0
2
4
(eV)
−2
0
2
4
(eV)
−2
0
2
4
Figure 5.15: The band structures of glide partials in 2H-SiC projected onto the k-space axis
corresponding to the dislocation line direction [11¯
20](with a periodicity length of a0for (b)
and (e) and 2 a0for all others). (a) – (c) show the band structures of the silicon-terminated
partials and (d) – (f) show those of the carbon-terminated partials. Only those bands which
are close to the gap are drawn, the rest of the valence and conduction band regions are
shaded. The origin of the Brillouin zone is at the far left of each band structure. Being
double-period reconstructions, (a), (c), (d) and (f) appear with a Brillouin zone of half width.
band minimum (CBM). Indeed localised bands are observed, which reach as far as
approximately 0.5–0.6 eV below the CBM (see Fig. 5.14, right panel). In terms of
energetic position and dispersion, these bands remarkably resemble the lowest con-
duction bands of the 3Cpolytype. Similar bands have been predicted by Miao et al.
[121] for intrinsic faults in 4H(∼0.3 eV below the CBM) and by Iwata et al. [129]
in 4Hand 6H(∼0.2 eV below the CBM). Of course the supercell-cluster hybrids
used here, which are optimised to model dislocations, are not ideal to describe bulk
or faulted SiC. These models do not allow the calculation of a “full” band struc-
ture along selected high-symmetry directions in the Brillouin zone. Hence a direct
comparison with Ref. [121] and [129] is impossible.
5.5. Electronic structure calculations
105
The projected band structures of the Shockley partials in 3C-SiC are displayed in
Fig. 5.14. As all investigated dislocations are glide partials, no dangling bonds are
present. For elementary semiconductors like diamond this is often equivalent to the
absence of deep defect states in the gap. In a compound semiconductor like SiC,
however, the reconstruction bonds of the glide partials are of a different nature (Si–
Si or C–C) compared to those in bulk material (Si–C). Therefore the electronic states
associated with those reconstruction bonds might have an energetic position very
different from states in bulk material. As can be seen in Fig. 5.14, only the silicon-
terminated partials (a) – (c) appear to introduce deep defect bands into the gap —
reaching as far as 0.4 eV above the valence band maximum (VBM). Analysis shows
that the corresponding wavefunctions are strongly localised at the Si–Si reconstruc-
tion bonds. At the 30◦carbon partial an empty band splits off from the conduction
bands at the edge of the Brillouin zone. Apart from that no further considerable
changes in the band structure can be observed13.
The equivalent band structures in 2H-SiC are shown in Fig. 5.15. Near the valence
band they are all almost identical with the corresponding ones in 3C. The only
considerable difference between the two polytypes is that in the case of 3Cthe lowest
unoccupied bands are of conduction band character, whereas in the case of 2Hthey
are localised bands of the intrinsic stacking fault.
The electronic structure of kinks: After having investigated the electronic struc-
ture of the straight Shockley partials, the question now arises if any additional elec-
tronic levels are introduced in the band gap due to the presence of kinks. To avoid
kink–kink interaction, pure cluster models containing a single kink only are used14.
The structures are relaxed using AIMPRO and their resulting electronic Kohn-Sham
eigenvalue spectra are given in Fig. 5.16 and 5.17. The obtained gap widths vary
due to the differences in cluster size and geometry for the different types of partials
and kinks. As a general observation for both 30◦partials, the high energy kinks
(LK2 and RK2) give rise to a slightly more pronounced local gap narrowing than
their low energy counterparts (LK1 and RK1). This can probably be interpreted as a
consequence of the larger strain field associated with LK2 and RK2.
In the case of the silicon-terminated 30◦and 90◦partials the highest occupied lev-
els are localised gap levels of the Si–Si reconstruction bonds. These characteristic
levels correspond to the Si–Si related band above the VBM predicted in the band
structures of Si-terminated partials (Fig. 5.14 and 5.15). Also most of the kink struc-
tures in carbon-terminated partials do not alter the original electronic structure of
the partials much. Overall, only three kink structures show more outstanding elec-
tronic characteristics:
30◦Si LK2 : Here the highest occupied level is pushed about 0.4 eV into the band
gap. This effect is probably related to the local strain at the kink structure.
13 The slight variations in the width of the gap for the different partials result from differences in
model geometry and size.
14 The cluster stoichiometries are: Si94C89H128 (straight 30◦Si), Si94C89H124 (left kinks in 30◦Si),
Si110C103H142 (right kinks in 30◦Si) and Si125C125H150 (straight and kinked 90◦partials). The 30◦C
clusters are constructed by simply swapping Si and C. The chosen stacking sequence is 3C.
106
Chapter 5. Dislocations in Silicon Carbide
30° Si 30° Si LK1 30° Si LK2 30° Si RK1 30° Si RK2
30° C 30° C LK1 30° C LK2 30° C RK1 30° C RK2
(eV)
1
2
3
0
−1
−2
(eV)
1
2
3
0
−1
−2
Figure 5.16: The Kohn-Sham eigenvalue spectra of kinked 30◦partials in SiC. The highest
occupied and the lowest unoccupied states are indicated by two filled and two empty circles
respectively. As some of these states are localised gap levels (see text), they do not neces-
sarily represent the valence band maximum or the conduction band minimum. The spectra
are unscaled and energy-zero is chosen arbitrarily through a shift constant for all structures.
Upper panel: The Si-terminated partial. Lower panel: The C-terminated partial. The far left
spectra give the respective unkinked partial.
5.5. Electronic structure calculations
107
90° Si K 90° C K90° Si 90° C
(eV)
1
2
3
0
−1
−2
Figure 5.17: The Kohn-Sham eigenvalue spectra of kinked 90◦partials in SiC. The highest
occupied and the lowest unoccupied states are indicated by two filled and two empty circles
respectively. As some of these states are localised gap levels (see text), they do not neces-
sarily represent the valence band maximum or the conduction band minimum. The spectra
are unscaled and energy-zero is chosen arbitrarily through a shift constant for all structures.
Left pair: The Si-terminated partial. Right pair: The C-terminated partial. The left spectrum
in each pair gives the respective unkinked partial.
30◦C LK2 : A feature common to all silicon-terminated partials can also be found
in the carbon-terminated LK2 structure: As shown in Fig. 5.10, this is the only kink
in a C-terminated partial which appears with an “alien” Si–Si reconstruction bond.
It is this bond which in the neutral charge state gives rise to the occupied gap level
close to the VBM — very similar to the Si partials.
30◦C RK1 : As can be seen in Fig. 5.11, the kink reconstructs with a considerable
bond angle distortion. This distortion might give rise to the empty level observed
in the upper half of the gap.
108
Chapter 5. Dislocations in Silicon Carbide
5.6 Summary and conclusions (SiC)
In this chapter the approach and methods, which were presented in detail in Chap-
ter 4, and tested with diamond as an example, have been applied to model 30◦and
90◦Shockley partials in the basal plane of silicon carbide. Their low energy core re-
constructions, energies and electronic structures were investigated in both the cubic
(3C) and one hexagonal polytype (2H). Furthermore, dislocation glide was mod-
elled as a mechanism of kink formation and migration. The obtained thermal acti-
vation energies were in reasonable agreement with experimental results and show
the same trends as independent theoretical work.
Recombination-enhanced dislocation glide might cause the experimentally ob-
served degradation of bipolar devices. Therefore, the electronic structures of all
involved kinks have been calculated as they could play a role in this process.
In the following section the important results are listed explicitly, followed by a
discussion of possible implications.
5.6.1 Selected results
Generally speaking, the core reconstructions of partial dislocations in the basal plane
of SiC are very similar to those found in the preceding chapter for diamond. Unlike
in diamond though, in SiC one has to distinguish between two types of core re-
constructions for each dislocation: Silicon-terminated and carbon-terminated cores
with respectively Si–Si and C–C reconstruction bonds. For all investigated struc-
tures, the Si–Si core bonds are of comparable length to bonds in bulk Si. The C–C
bonds of the carbon-terminated cores, however, are all stretched by approximately
15 % compared to the diamond equilibrium bond length. As a consequence the
carbon-terminated cores possess higher core energies.
Furthermore, the results strongly indicate that for all partials both energy and local
structure are almost entirely independent of the surrounding crystal stacking. The
same can be expected for 4H, 6Hand all further hexagonal stacking sequences.
Comparing the single-period and the double-period reconstructions of the 90◦par-
tials, in SiC the latter is only about 90–100 meV/ ˚
A lower in energy. Thus both will
probably co-exist at room temperature, and most certainly at device operating tem-
peratures.
The glide motion of the four different Shockley partials — 30◦and 90◦, both Si-
and C-terminated — has been considered as a process of kink formation and subse-
quent migration. Overall, the 90◦partial was found to be the thermally more mobile
species. Further, at least in the absence of strong obstacles, the thermal mobility of
carbon-terminated partials appears to be higher than that of silicon partials. The
glide activation energies found — ranging from 2.3 eV to 4.1 eV for long dislocation
segments — agree well with experimental results, but the carbon partials being the
more mobile species is in contradiction to the conclusions of Pirouz and Yang [145].
However, LACBED (large angle convergent beam electron diffraction) experiments
seem to indicate, that the Si partials in 3C-SiC are smooth whereas the C partials are
zig-zagged [146], suggesting they are pinned by obstacles. Then the glide process
5.6. Summary and conclusions (SiC)
109
would not be mainly controlled by the formation and migration of kinks, but by the
energy needed for the pinning and un-pinning at the obstacles.
Since the band gap varies with the polytype, the electronic band structures of the
partials were calculated both in 3C- and 2H-SiC. In both polytypes only the Si par-
tials appear with a localised band, which is as deep as 0.4 eV above the VBM. These
bands originate from the Si–Si reconstruction bonds. In contrast to this, the C par-
tials lack any deep states in the gap.
Independent of the type of partial, for the 2Hpolytype additional empty bands be-
low the CBM are found. These bands are associated with the intrinsic stacking fault
accompanying the partial dislocation and reach approximately 0.5–0.6 eV into the
gap. Due to the smaller gap, for 4H- and 6H-SiC one would expect the stacking
fault bands to be closer to the conduction band. And indeed theoretical investiga-
tions by Miao et al. [121] for 4Hand Iwata et al. [129] for 6Hpredict similar bands
at 0.3 and 0.2 eV below the CBM respectively. Except for the presence and positions
of the stacking fault related bands, the polytype seems to have no further influence
on the electronic structure of the basal plane partials.
Out of the 10 elementary kinks, only three locally introduce additional states in the
gap: At the 30◦Si partial the high-energy left kink (LK2) appears with an occupied
level 0.4 eV above the VBM. At the C partial the same kink (LK2) reconstructs with
a Si–Si bond. Similar to the Si–Si core bonds in all Si partials, this leads to a localised
level above the VBM. Finally, the low energy right kink (RK1) at the 30◦C partial
gives rise to an empty level in the upper half of the band gap. In contrast to the 30◦
partials, all kinks at the 90◦partials were found to be electrically inert.
5.6.2 Recombination-enhanced dislocation glide
As presented in the introduction of this chapter in detail, rapid degradation un-
der forward-biased operation turned out to be the major problem of bipolar de-
vices based on hexagonal SiC. In this irreversible process stacking faults, bounded
by Shockley partials, propagate and expand, rendering the device useless. Nei-
ther the temperature nor the stress in these devices are sufficient to overcome the
barriers to dislocation glide. Therefore it is commonly believed that the bounding
Shockley partials move in a process of recombination-enhanced dislocation glide
(REDG), where the glide activation energy is substantially lowered by energy from
non-radiative recombination at the dislocation core. So far the origin of this non-
radiative recombination could not be resolved experimentally. However, there are
several important experimental findings, which might help to understand the pro-
cess:
1. In 4H-SiC, electroluminescence experiments by Galeckas et al. [115] reveal
a 2.8 eV radiative recombination localised at the stacking faults involved in
the degradation process. This recombination has been attributed to electronic
bands induced by the fault at 0.2–0.6 eV below the conduction band mini-
mum, which were predicted by Miao et al. [121], Iwata et al. [129] and also in
this work.
110
Chapter 5. Dislocations in Silicon Carbide
2. Under forward-bias a further ∼1.8 eV radiative recombination is found at the
Shockley partials bordering the growing stacking faults [115].
3. Skowronski et al. [147] observed 30◦partials in 4H-SiC moving under
forward-bias. In their experiments, light emission of considerable intensity
as well as glide motion appeared to be restricted to one species of 30◦partial
only. The authors of Ref. [147] propose it is the silicon partial which is mobile
under forward-bias and emits light.
4. The activation energy to partial glide motion under forward-bias is found to
be 0.27 eV [115]. Comparing this with similar experimental estimates without
bias, the authors of Ref. [115] suggest a non-radiative recombination of 2.2 eV
at the Shockley partials to be responsible for the REDG mechanism. However,
no further experimental indication of such recombination exists.
As found in this work, the main difference between Si and C partials in terms of
electronic structure appears to be the presence of states 0.4 eV above the VBM in the
case of the Si partials. This, in combination with the afore mentioned experimen-
tal findings, points towards a non-radiative recombination process between the gap
states of the Si-terminated partials and those induced by the stacking fault. Assum-
ing the latter states to be around 0.3 eV below the CBM [121] together with a gap of
3.3 eV, and the states of the Si partials to be 0.4 eV above the VBM (this work), gives
an estimate of 2.6 eV for the non-radiative recombination. Considering the band gap
error in DFT, this is in reasonable agreement with the 2.2 eV suggested in Ref. [115].
It further explains why only one type of partial — the Si partial — is highly mobile
under forward-bias.
All conclusions drawn here are still rather speculative. Further experiments might
yield results to support or contradict the REDG model presented above.
Also, the origin of the 1.8 eV emission remains unclear. It cannot be explained by
the electronic gap states found for some of the kink structures either, as those do not
appear on every type of partial.
5.6.3 Outlook — an alternative model
First test calculations for the 30◦Si partial indicate, that positively charging the dis-
location line empties the Si–Si bonding states, leaving the reconstruction bonds bro-
ken. Since under forward-bias the active region of the device contains many carriers
(electrons and holes), holes might be trapped at the Si–Si bonding states, break-
ing the bond, and thus lowering the glide activation barrier. As the C–C bonds do
not induce localised states above the VBM, this mechanism would only work for
the Si-terminated partials. Hence it explains part of the experimental observations.
However, the effect of the broken bonds on the glide activation energies has to be
tested and quantified for all involved partials.
Chapter 6
Summary and Outlook
In this work the atomic structures, energies, electronic properties and the mobility of
dislocations in diamond and silicon carbide were studied. The approach presented
combines DFT-based atomistic calculations with linear isotropic (and to some ex-
tent anisotropic) elasticity theory. The latter has proven invaluable to describe the
long range elastic strain effects associated with dislocations and further served as
a good convergence criterion with respect to the size of the atomistic models used.
A convenient determination of the various energy offsets — unaccessible in elastic-
ity theory — became possible through atomistic modelling within the DFT-based
tight-binding method (DFTB). More accurate electronic structure calculations were
also performed in a localised basis pseudopotential approach (AIMPRO). Wherever
there was overlap between the elasticity description of dislocations and their atom-
istic modelling, the agreement was very good and the elastic limit of the atomistic
calculations well fulfilled.
The first three chapters of this work gave a short introduction of the methods used
and dislocation basics in general. The relevant results of elasticity theory were pre-
sented and explained.
In Chapter 4the low energy dislocations of the {111}h110islip system in diamond
were examined. The predicted low energy core structures and dissociation distances
match well with those observed experimentally in (HR)TEM. In particular to com-
pare the calculated structures with high-resolution micrographs, the application of
image simulations based on the calculated coordinates was crucial. For the 60◦dis-
location a wide barrier to dissociation into Shockley partials was found, which can
explain the experimental observation of undissociated 60◦dislocations in HRTEM
even though an overall energy-gain strongly favours dissociation.
A comparison of core energies ruled out many of the possible dissociation reac-
tions for the 1
2[1¯
10]screw and the 60◦dislocation, leaving only the 30◦and 90◦glide
partials and the vacancy structure of the 90◦shuffle partial likely to exist. Con-
sistent with earlier calculations, in this work the double-period core reconstruction
of the 90◦glide partial is favoured over its single-period reconstruction by around
170 meV.
Based on the predicted core structures it was then possible to calculate the corre-
sponding electronic structures. Only the perfect and partial shuffle dislocations ap-
111
112
Chapter 6. Summary and Outlook
pear to induce deep levels in the band gap — predominantly caused by dangling
bonds. As shown in simulations, these gap states should lead to an increase in low-
loss EELS absorption and supplementary core EELS absorption below the conduc-
tion band edge, which is observed experimentally in type-II diamond. This would
then suggest shuffle segments are present in that material, which could well be re-
sponsible for the observed band-Acathodoluminescence. Furthermore experimen-
tal EEL spectra taken on a 30◦partial agree with those predicted for the respective
glide partial.
For the two probably predominant partial dislocations — the 30◦and the 90◦glide
partials — the kinetics of thermally activated glide motion have been modelled as-
suming a kink formation and migration mechanism. The 90◦glide dislocation was
found to be the by far more mobile species, with a thermal activation energy of
3.5 eV for short dislocation segments as opposed to ∼6.1 eV for the 30◦partial.
Comparing the shuffle structures of the 60◦dislocation with the glide structures, the
latter appeared ∼0.6 eV/ ˚
A lower in line energy. However, since a shuffle →glide
transition involves absorption or emission of vacancies or interstitials, one would
expect shuffle segments to be present if formed by plastic deformation. These might
then undergo the transition to glide structures in a thermal annealing process. Fur-
thermore among the shuffle partials it is the vacancy structures which were found
to be energetically most stable. This, combined with results from Raman and pho-
toluminescence experiments, supports a scenario for the observed high-pressure,
high-temperature decolouring of brown diamond, which involves a shuffle →glide
transition via the emission of vacancies. Further experiments are necessary to draw
final conclusions in this matter.
In Chapter 5the 30◦and 90◦Shockley partials in silicon carbide were investigated.
These partials border stacking faults in the basal plane whose growth is believed to
be the cause of the degradation of bipolar devices under forward bias. Core struc-
tures and energies were found to be independent of the crystal stacking. Since the
band gap varies with the polytype however, the electronic band structures of the
partials were calculated both in 3C- and 2H-SiC. Only the Si partials appear with a
localised band as deep as 0.4 eV above the valence band maximum. The C partials
lack any deep states in the gap. For the 2Hpolytype, additional empty bands below
the conduction band minimum are found independent of the partial. These bands
are associated with the intrinsic stacking fault accompanying the partial dislocation
and reach approximately 0.5–0.6 eV into the gap.
The structures and basic processes involved in the formation and migration of kinks
in basal plane partial dislocations in SiC were found to be very similar to those in
the {111}plane of diamond. Also the general trends for thermal activation ener-
gies to glide motion are the same. Overall, the 90◦partial appeared to be the more
mobile species and the purely thermal mobility of carbon partials in the absence of
strong obstacles appears to be higher than that of silicon partials. However, some
experiments indicate that pinning by obstacles might play a role as well for some
partials. Nevertheless the activation energies of between 2.3 and 4.1 eV found for
long dislocation segments agree well with experimental observations in 6H-SiC.
The glide activation energies, in combination with the electronic structure calcu-
lations, can be related to a recombination-enhanced dislocation glide mechanism,
113
where non-radiative electron-hole recombination occurs at sites along the disloca-
tion line. The released recombination energy then has to be redirected to assist the
formation and migration of kinks at the dislocation and thus substantially lowers
the thermal glide activation energy. The results of this work — combined with pre-
vious experimental evidence — strongly point towards a non-radiative recombina-
tion between the gap levels induced by the stacking fault and those of the bordering
silicon-terminated Shockley partials. This recombination would enhance the partial
mobility considerably, and thus expedite stacking fault growth in forward-biased
bipolar SiC devices — leading to their irreversible degradation.
Methodological outlook
In the long run, the way of combining different methods to model the effects of dis-
locations on different length scales could be improved. A true embedding approach
is imaginable, where the region treated atomistically is embedded in a larger region
which is described by elasticity theory. The crucial point would then be the inter-
face/coupling between the different regions, which is far from trivial. As a success-
ful implementation would increase the accuracy of the core energies and electronic
structures, and at the same time it would reduce the computational effort as the
volume treated atomistically could be minimised.
Appendix A
Straight Dislocations in Elasticity
Theory
Throughout this work many fundamental expressions for dislocation energies and
forces are used, which are based on linear elasticity theory. In Chapter 2these ex-
pressions are introduced without giving any derivation, as deducing them would
clearly lead beyond the scope of this thesis and has been presented elsewhere. Still,
to give the reader a feeling for the basic approach, this appendix will give some sim-
ple examples. The presentation is brief, but chosen to be close to that of Hirth and
Lothe [41].
A.1 Screw dislocations in isotropic media
In this section the elastic strain energy of a straight screw dislocation in an infinite
and otherwise defect-free isotropic material will be deduced. To construct the dis-
location let us start with a cylinder of unstrained material orientated along the zor
x3axis. On this cylinder one then applies a shear displacement u(r)along the x3di-
rection across the (x1,x3)plane as demonstrated in Fig. A.1. The cut/shear surface
is defined by x1>0, x2=0. There is no displacement in the x1and x2direction
(u1(r) = u2(r) = 0)and the displacement at the cut surface is discontinuous:
lim
η
→0u3(x1,−
η
,x3)−u3(x1,+
η
,x3) = |b|,
η
>0, x1>0 (A.1)
A Burgers circuit varying
ϑ
from 0 to 2
π
gives the Burgers vector b. In an isotropic
medium by symmetry the displacement must vary uniformly with
ϑ
and one ob-
tains:
u3(x1,x2,x3) = |b|
2
π
arctan x2
x1=uz(r,
ϑ
,z) = |b|
ϑ
2
π
(A.2)
The right hand side gives the expression in cylindrical coordinates (r,
ϑ
,z)as indi-
cated in Fig. A.1. Following Eq. (1.33) the resulting non-zero strains are:
ε
13 =
ε
31 =−|b|
4
π
x2
x2
1+x2
2
,
ε
23 =
ε
32 =|b|
4
π
x1
x2
1+x2
2
(A.3)
115
116
Appendix A. Straight Dislocations in Elasticity Theory
x2
r
e
ϑ
e
z
e
x1
`
x3
b
ϑ
Figure A.1: A screw dislocation ori-
ented along the z-axis with a cylin-
der of the surrounding material
drawn. Both the Cartesian coor-
dinate system (x1,x2,x3)and a lo-
cal set of unit vectors (er,e
ϑ
,ez)of
the cylindrical coordinate system are
given. `gives the line direction and
a Burgers circuit varying
ϑ
from 0 to
2
π
gives the Burgers vector b.
Or in cylindrical coordinates simply:
εϑ
z=
ε
z
ϑ
=|b|
4
π
1
r(A.4)
Eq. (1.35) and the generalised Hooke’s law for isotropic media Eq. (1.45) finally yield
the non-zero stress components in cylindrical coordinates:
σϑ
z=
σ
z
ϑ
=
µ
|b|
2
π
1
r(A.5)
This stress field is also known as the self-stress of the screw dislocation. It trivially
satisfies the equilibrium equations of classical elasticity Eq. (1.32). With the strains
and stresses known one can now calculate the strain energy density. With Eq. (1.37)
and (1.36) we obtain:
es(r) = 1
2cijkl
ε
ij
ε
kl =1
2
σ
ij
ε
kl =1
2(
σϑ
z
ε
z
ϑ
+
σ
z
ϑεϑ
z)=
µ
|b|2
8
π
2
1
r2(A.6)
The volume integral over the infinite medium diverges. However, one can calculate
the strain energy per unit length contained between two co-axial cylinders centred
on the dislocation line:
E(r0,R)
L=ZR
r0
es(r)2
π
rdr=
µ
|b|2
4
π
ln R
r0(A.7)
Here Lis the length of the integration volume along zand r0and Rthe radius of the
inner and outer cylinder respectively. The expression diverges for both the limits
R→
and r0→0. The former divergence is physical: The strain energy per unit
length of a dislocation in an infinite volume is not defined and the absolute strain
energy of a dislocation depends on the size of the crystal. In the case of a single
dislocation in an otherwise defect free crystal one can approximately choose Ras
the average distance of the dislocation line from the surface. Due to the logarithmic
dependence in Eq. (A.7) the choice of Ris not critical.
A.2. General straight dislocations in isotropic media
117
The divergence for r0→0 shows a general problem of linear continuum elasticity
theory as presented here: The underlying assumption that the displacements vary
slowly over the dimensions of the volume investigated breaks down in the core
region of a dislocation. Furthermore, for interatomic length scales continuum theory
is unable to describe the discrete system of atoms correctly. Therefore we define the
so called core radius Rcaround the dislocation line. Rcis chosen as the smallest
possible radius, where for R≥Rcelasticity theory is able to describe the strains
and energies associated with the dislocation “correctly”. Of course this definition is
rather vague.
The so called core energy Ecstored inside the cylinder of radius Rccannot be ob-
tained in elasticity theory. Here atomistic calculations are essential and in many
cases quantum mechanical effects contribute considerably. With this core energy
introduced, Eq. (A.7) becomes:
E(R)
L=
µ
|b|2
4
π
ln R
Rc+Ec
L,R≥Rc(A.8)
This expression gives the energy per unit length contained in a cylinder of radius R
around the dislocation.
An alternative to introducing a core energy is to simply adjust r0so that Eq. (A.7)
gives the right energy. In the literature this adjusted radius r0is often called the core
radius as well. However, r0is not a physical radius but just a different representation
of the core energy — it should not be confused with the core radius Rc. In this work
r0does not play any major role.
The screw dislocation in a finite volume: In a finite volume one has to consider
surface effects. The displacement field of a dislocated rod of finite length and with
free ends appears with a twist — the so called Eshelby twist [148]. This twist can be
observed experimentally in long thin whiskers containing a screw dislocation [149,
150,151]. It also occurs when a dislocation with a screw component is modelled
atomistically in a cluster where the surfaces are allowed to relax freely. Taking the
free-surface terms into consideration, the energy of a dislocation in such a cluster of
radius Ris given as [41]:
E
L=
µ
|b|2
4
π
ln R
Rc−1+Ec
L(A.9)
A.2 General straight dislocations in isotropic media
Just as with the screw dislocation in the last section, the edge dislocation induces
planar strain into the surrounding material — this time however defined by u3=0
and ∂ui/∂x3=0. Starting from this, one can derive the non-zero stress components
via an Airy stress function as described in Ref. [41]:
σ
rr =
σϑϑ
=−
µ
|b|sin
ϑ
2
π
(1−
ν
)r,
σ
r
ϑ
=
σϑ
r=
µ
|b|cos
ϑ
2
π
(1−
ν
)r,
σ
zz =
ν
(
σ
rr +
σϑϑ
)(A.10)
118
Appendix A. Straight Dislocations in Elasticity Theory
Similarly to Eq. (A.6), the strain energy density can be determined from the self-
stresses and integration between two co-axial cylinders of radius r0and Ryields:
E(r0,R)
L=
µ
|b|2
4
π
(1−
ν
)ln R
r0(A.11)
This expression is very similar to Eq. (A.7) and hence all which has been said in
the context of the screw dislocation applies here as well. An expression similar to
Eq. (A.8) can be derived.
Since only linear theory is used, the results for the screw and the edge dislocation
can be superimposed to describe a mixed dislocation. Let us assume a dislocation
where the Burgers vector is inclined to the line direction at an angle
β
. Then the
screw component of that vector is projected as |b|cos
β
and the edge component as
|b|sin
β
. Superposition then gives the energy of the general straight dislocation:
E(R)
L=
µ
cos2
β
+sin2
β
1−
ν
!|b|2
4
π
ln R
Rc+Ec
L,R≥Rc(A.12)
The term which only depends on the elastic constants and the angle
β
between the
line direction and the burgers vector is in this work called energy factor k(
β
):
k(
β
) =
µ
cos2
β
+sin2
β
1−
ν
!(A.13)
A.3 Straight dislocations in anisotropic media
For good reason the two preceding sections were restricted to isotropic media
only. As the calculation of self-stresses in anisotropic media is by far more compli-
cated. Very often it is sufficient to describe the anisotropic crystal as approximately
isotropic, with the elastic constants given by the Voigt or the Reuss average (see Sec-
tion 1.2.4). But this might not always be accurate enough. Fully anisotropic theory
yields an expression for the self-stress energy similar to Eq (A.12):
E(R)
L=K|b|2
4
π
ln R
Rc+Ec
L,R≥Rc(A.14)
Here Kis the energy factor derived from the displacement field u(r). However
the determination of the displacement field through Eq. (1.32) and (1.36) generally
requires the solution of a sixth-order polynomial equation [41]. This can be done
numerically and was demonstrated for the determination of dislocation energies by
Teutonico [93,152].
In this work the same method is used as implemented by Heggie and Nyl´en [87]
and the anisotropic energy factors are calculated for all dislocations discussed.
Appendix B
Transmission Electron Microscopy
There are several experimental techniques which allow the visualisation of disloca-
tions and other structural defects in semiconductors. Among these the most com-
mon are surface etching in combination with conventional microscopy, X-ray to-
pography, and conventional or high-resolution electron microscopy. There exists a
variety of further methods, like EBIC (electron beam induced current microscopy),
STEM (scanning transmission electron microscopy), infrared microscopy and more.
However, as this is not meant to be a complete guide to imaging techniques, they
are excluded from this overview. Primarily this appendix presents the two trans-
mission electron microscopy (TEM) methods, which in Chapters 4and 5provide
valuable information and allow to verify some of the theoretical results.
When “shining” an electron beam onto a thin crystal specimen as used in TEM,
there are two important interactions of the electrons with the crystal’s atoms: Elas-
tic scattering and inelastic scattering. Electrons which are not scattered are simply
transmitted through the crystal. Both the elastically scattered and the transmitted
electrons are used in TEM to visualise structural defects. Inelastic electron scatter-
ing is utilised in electron energy-loss spectroscopy as described in Appendix C. In
the now following sections the basic concepts of TEM will be explained. A more
detailed description can be found in the books of Hirsch et al. [153] and Williams
and Carter [154].
B.1 Conventional transmission electron microscopy
In conventional transmission electron microscopy a monochromatic and parallel
electron beam shines on a thin sample of usually ≤100 nm thickness. The basic
setup is shown in Fig. B.1. When passing through the sample, some electrons are
transmitted through without interacting, while others are scattered. For TEM one
tries to minimise inelastic scattering/absorption as it does not contain much local
structural information. However, it usually cannot be avoided completely and in-
elastically scattered electrons lead to features in TEM micrographs known as Kakuchi
lines, whose width is inversely proportional to the atomic spacing in the crystal. The
theory of inelastic scattering is rather complicated and shall not be discussed here.
In the following only elastic scattering will be considered.
119
120
Appendix B. Transmission Electron Microscopy
electron source
condenser lens system
sample
objective lens
objective aperture
intermediate lens
projector lens system
phosphorous screen
CCD camera
(a) bright field imaging
(b) dark field imaging
objective aperture
objective lens
sample
objective aperture
objective lens
sample
diffracted
diffracted
Figure B.1: Schematic sketch of a transmission electron microscope. Left: A schematic sim-
plified representation of a TEM setup with condenser, objective, intermediate and projector
lenses. Usually between the objective aperture and the intermediate lens a further so called
diffraction lens is inserted. Right: The two basic imaging modes. In bright field imaging the
diffracted beam is masked by the objective aperture and the transmitted beam is used. In
the case of dark field imaging the situation is just the other way round.
The origin of the TEM image contrast: Elastic scattering in a crystal occurs with
highest intensity whenever the Bragg condition for diffraction is met (see Fig. B.2):
n
λ
=2dhkl sin
θ
,n=1, 2, 3, . . . (B.1)
Here
λ
is the wavelength of the electron beam, dhkl the spacing of a set of (hkl)crys-
tallographic planes and
θ
gives the angle between the beam and that set of planes.
The diffraction intensity is strongest for planes with low indices hkl. Since in a crys-
tal there are several such planes, generally several diffracted beams or reflexes are
observed. It is common to label the different Bragg reflexes with the reciprocal vec-
tor ghkl of their corresponding plane — in this context also called the diffraction
vector.
In the vicinity of a structural defect, such as a dislocation, the lattice is strained and
the lattice planes are bent locally. This results in local changes of the Bragg condition
and thus in a change in intensity for the diffracted and transmitted beams.
B.1. Conventional transmission electron microscopy
121
2θ
d
g
diffracted
Figure B.2: Bragg diffraction in the
TEM sample. gdenotes the corre-
sponding diffraction vector.
If all diffracted beams are masked by the ob-
jective aperture and hence only the transmitted
beam is used for the imaging process, then one
speaks of a bright field image (see Fig. B.1 (a)).
Bright field images are useful to give a first im-
pression of the microstructure.
For dark field imaging one of the diffracted beams
is used (Fig. B.1 (b)). To obtain the best contrast,
the specimen is tilted in the beam in a way that
the so called two-beam condition is met: The Bragg
condition is nearly fulfilled (low excitation er-
ror) for only one specific set of crystallographic
low-index planes, resulting in only two beams
of considerable intensity — the transmitted and
the diffracted beam of the chosen plane. Usually
the incident beam is tilted such that the selected
diffracted beam lies along the optical axis of the
microscope. This so called centred dark field method is essential in order to obtain
good quality images. Overall the dark field method is lower in intensity and yields
a better contrast than bright field microscopy.
When it comes to imaging very fine dislocation structures, then often the resolution
of dark field imaging as described above is not sufficient. However, it can be further
improved by so called weak-beam imaging, where the sample is tilted away from the
ideal two-beam condition, such that the exact Bragg condition is only met in a small
region close to the dislocation (large Bragg excitation error for bulk material). If this
region is narrow enough, then for example dislocation dissociation into partials can
be resolved in plan-view with resolutions around 1 nm. A typical weak-beam image
is given in Fig. 4.1.
b
b
g010
g100
Figure B.3: Illustration of the invisibil-
ity criterion for an edge dislocation:
Planes with the reciprocal vector g100
are not bent.
The invisibility criterion: Plan-view imag-
ing can not only be used to visualise dislo-
cations, but also to determine their Burgers
vector bby making use of selected diffraction
vectors under dark field two-beam or under
weak-beam conditions.
Fig. B.3 schematically shows a displaced sim-
ple cubic lattice around an edge dislocation as
obtained in (anisotropic) elasticity theory. As
one can see, the set of lattice planes with the
reciprocal vector g100 is not distorted / bent at
all. Hence for these planes the Bragg diffrac-
tion condition in the vicinity of the disloca-
tion core remains the same as in bulk mate-
rial. Consequently the dislocation remains in-
visible in dark field imaging if the corresponding Bragg reflex g100 is selected exclusively.
g100 is specified by being perpendicular to the Burgers vector band further lying in
122
Appendix B. Transmission Electron Microscopy
the glide plane, which is defined by band the dislocation line direction `. From this
one obtains the invisibility criterion for an edge dislocation:
g·b=0 and g·(b×`)=0 (B.2)
As can be seen in Fig. B.3, the glide plane itself is usually only weakly bent (recipro-
cal vector g010). Therefore a g010 Bragg reflex would give only very weak contrast.
Hence g·b=0 alone is usually sufficient as a less rigorous invisibility criterion. For
the pure screw dislocation the invisibility criterion is also given by Eq. (B.2), with
the right hand term fulfilled automatically as bs×`=0.
Imaging a dislocation with different diffraction vectors and comparing the contrast
allows an exact determination of the direction of the Burgers vector via the invisi-
bility criterion. Fig 5.2 gives a typical example.
B.2 High-resolution transmission electron microscopy
Conventional TEM in plan-view can be a useful tool to observe dislocations and
to determine their line directions and Burgers vectors. Its resolution is too limited
though, to provide information about the actual atomic structure at the dislocation
core. This can be achieved by high-resolution electron microscopy (HRTEM). To
obtain the required resolution of the order of interatomic distances (∼0.2 nm), high-
resolution microscopes need a better mechanical and electrical stability and higher
accelerating voltages (>300 keV) than a conventional TEM. In contrast to the latter,
the HRTEM image is a result of a multitude of differently scattered beams and their
phase difference (see next paragraph). In order to obtain a good quality image, the
sample has to be very thin (∼10 nm) and the optical axis has to lie precisely along
a crystallographic low-index direction. For a good contrast, the atoms along the
chosen low-index direction have to lie exactly on top of each other. Therefore line
defects like dislocations can only be imaged in cross-section, with the dislocation
line as parallel to the electron beam as possible.
The next paragraph will give the rough idea of HRTEM imaging, for more details
see for example Ref. [155].
The origin of the HRTEM image contrast: Basically HRTEM works as a phase
contrast microscope: After passing through the sample, the diffraction pattern of
the electron beam is given as the Fourier transform of the periodic potential experi-
enced. The following objective lens brings all diffracted beams and the transmitted
beam back together again, resulting in an interference pattern which is described
by the Fourier back-transformation. The created image thus gives a magnified rep-
resentation of the periodic potential which the electrons experienced when passing
through the sample. This image is magnified further by the subsequent electron-
optical system. Overall, magnifications of the order of magnitude of 106can be
achieved. As there is no such thing as a perfect phase contrast electron microscope,
the obtained image depends on several factors. The most important are the align-
ment of the beam, its coherence, the specimen thickness, the defocus of the objective
lens and the chromatic and spherical aberration.
B.3. Alternative techniques
123
HRTEM simulation: A defect destroys the periodicity of the lattice. Since intuition
fails when it comes to predict the effects of that broken periodicity on the Fourier
transform, one cannot interpret the observed contrast patterns of defects easily. In-
terpretation is much easier and safer in comparison with simulated HRTEM images
based on atomistic defect models.
Image simulation can be performed using Bloch waves, or in a multi-slice approach
as described by Moodie [156]: The atomistic model is divided into slices perpen-
dicular to the incident beam. For each slice the crystal potential is projected onto
a plane. The propagation of the beam through the sample is then described as a
succession of scattering events of the incident wavefront at the planes representing
the slices with intermediate propagation through vacuum between the slices. The
rather complicated scattering theory involved is described in detail in Ref. [157]. The
subsequent objective lens is modelled by a Fourier transform which has to take the
defocus into account as well as possibly several other specifications of the HRTEM
setup. In this procedure, the defocus caused by the imperfect objective lens can be
described by a so called phase contrast transfer function. This function is specific
for each microscope and has to be determined experimentally by imaging the same
area of a sample with varying defocus.
The so obtained images can then be compared with experimental images as demon-
strated in Section 4.6. All simulated images shown have been generated with the
multi-slice method using the commercial CRYSTAL KITTM and MACTEMPASTM soft-
ware packages.
B.3 Alternative techniques
Although this appendix mainly covers transmission electron microscopy, two alter-
native or supplementary techniques — X-ray topography and surface etching — are
now explained briefly.
X-ray topography: Unlike in the case of light or electron beams, there are no ef-
ficient lenses for X-rays. Hence the construction of an imaging apparatus similar
sample
aperture
film
diffracted
primary beam
X−ray
mechanical coupling
scan
Figure B.4: Schematic X-ray topography setup. The diffracted beam is masked by an aper-
ture. Sample and film are mechanically coupled during the scan.
124
Appendix B. Transmission Electron Microscopy
to a conventional or transmission electron microscope is impossible. Nevertheless
X-rays can be used in a scanning technique as illustrated in Fig. B.4: An X-ray beam
with a quasi one-dimensional cross section shines on a thin sample. This sample is
oriented that the Bragg condition for diffraction is (nearly) met for one set of low-
index lattice planes. As described earlier in Appendix B.1 for electron beams, the
strain field around the dislocation locally deforms the lattice and thus changes the
Bragg condition in its vicinity. As a result, the intensity of the primary and of the
diffracted beam vary according to the strain field. To image the strain field, the sam-
ple is scanned with one of the beams — usually the diffracted beam — masked by
an aperture.
X-ray topography usually gives a resolution of around 5
µ
m and allows a limited
Burgers vector analysis via the Bragg condition.
original surface
etched surface
Figure B.5: Sketch of an etched sur-
face. Dislocations are drawn as lines.
Chemical surface etching: By means of con-
ventional optical microscopy the limited resolu-
tion usually prohibits to directly image disloca-
tions intersecting with an untreated surface of
the crystal. The basic idea is to visualise the lo-
cation of a dislocation by marking it with surface
features of sufficient size, which can then be de-
tected in conventional microscopy. This can be
done by surface etching: Some solutions pref-
erentially dissolve the strained material around
structural defects intersecting the surface. Ex-
posing a polished surface to such a solution for
an appropriate amount of time results in the formation of so called etch pits where the
defect intersects the surface. This is schematically shown in Fig. B.5. It is possible to
determine the type of the defect by the shape of the etch pit, revealing dislocations,
stacking faults and other defects.
The advantages of chemical etching are its high sensitivity and low cost. On the
other hand however, since the etching mechanisms are usually not understood in
detail, the interpretation of the obtained images is rather prone to errors. Also the
detection limit remains generally unclear and Burgers vectors or core structures can-
not be identified1.
1Only in some cases, when screw and edge dislocations result in etch pits of distinct shape or
depth, then the type of a dislocation may be identified as screw, edge or mixed.
Appendix C
Electron Energy-Loss Spectroscopy
Some of the dislocation core structures discussed in this work appear with under-
coordinated atoms or with reconstruction bonds of very different nature compared
to bulk material. In many cases this results in electronic states deep in the band gap.
This change in electronic structure causes various effects — for example optical ac-
tivity or recombination-enhanced dislocation glide as described in Chapter 5. All
these effects can be measured and quantified experimentally, however the most di-
rect spatially resolved observation and “measurement” of electronic gap states can
be achieved by electron energy-loss spectroscopy (EELS).
As explained in Appendix B, electron microscopy makes use of transmitted and
elastically scattered (diffracted) electrons to generate an image of the investigated
structure. In contrast EELS utilises inelastic scattering of electrons to probe the elec-
tronic structure. Implemented in an electron microscope, the fine electron beam
allows high spatial resolutions of the order of atomic distances1. Combining EELS
with TEM has the further advantage that obtained spectra can be directly correlated
with defect structures observed in TEM. Microanalytical EELS experiments were
first proposed and demonstrated by Hillier and Baker [158] more than half a cen-
tury ago. However, only several decades later did experimental techniques become
advanced and stable enough to allow an easy application.
The following sections will give a brief introduction into the basics of EELS and
EELS simulation. More details can be found in the book of Egerton [159] and in
Refs. [160,161].
C.1 The basic principles
As primary high-energy electrons pass through a thin sample, some are scattered
inelastically and lose a fraction of their energy. Fig. C.1 (left) illustrates how this
energy-loss is measured in EELS. The energy is lost as electronic transitions between
the valence band and conduction band states of the solid are induced, resulting in
so called low-loss spectra. The interaction can be that of single valence electron exci-
1Typically EEL spectrometer are combined with scanning electron microscopes (STEM) and the
technique is often referred to as high-resolution EELS (HREELS).
125
126
Appendix C. Electron Energy-Loss Spectroscopy
entrance aperture / TEM viewing screen
multipole optics
E
∆E − E
detector
(b) EELS excitations(a) EELS setup
magnetic prism
E
VB
CB
core
low−energy
excitation
core
excitation
Figure C.1: A simplified EELS setup and the principle EELS excitations. Left: A simplified
sketch of a spectrometer as it can be installed underneath an electron microscope as shown in
Fig. B.1.Right: Representation of the principle EELS excitations / transitions. core denotes
a deep atomic core state and VB and CB a valence and conduction band states respectively.
tation events or can also encompass collective modes of the solid, called plasmons.
If the transitions occur between tightly bound atomic core states and the conduction
band one speaks of core-excitation EELS or energy-loss near edge structure (ELNES)
as shown in Fig. C.1 (right). Typical spectral energy ranges are 0–50 eV for low-loss
and 100-300 eV for core-excitation EELS. Very small losses of the order of 1–100 meV,
which arise from the excitation of phonons, often cannot be resolved by the energy
filter of a (S)TEM EEL spectrometer and then simply contribute to the zero-loss peak
of (approximately) elastically scattered electrons.
Inelastic scattering and the dielectric function: The interaction of high-energy
electrons passing through a solid can be described as an interaction with the field of
polarisation which is caused by the electric field of the solid’s electrons [162,163].
The quantity actually observed in EELS experiments is the fraction of electrons
which are scattered into a solid angle d
having lost an energy between
Eand
E+dE. To link this quantity with the interaction with the field of polarisation, we
introduce the differential cross section d
σ
2
d
dE. Assuming an isotropic medium, this
cross section can be related to the macroscopic dielectric function
ε
(q,
ω
)[164,165]:
d
σ
2
d
dE∝1
q2Im−1
ε
(q,
ω
)(C.1)
Here qgives the momentum transfer and
ω
=E/¯h. The last term, the imaginary
part of the negative reciprocal dielectric function, is also known as the loss function.
In the single particle approximation the imaginary part of the dielectric function
C.2. The simulation of EELS
127
ε
(q,
ω
) =
ε
1(q,
ω
) + i
ε
2(q,
ω
)can be written as [166,167]:
ε
2(q,
ω
) = 4
π
e2
q2X
n,j|h
n|eiq·r|
ji|2
δ
(En−Ej−¯h
ω
)(C.2)
Where
is the unit cell volume, and |
niare single particle states, with respective
energies En. The real part of the dielectric function can be obtained easily via a
Kramers-Kronig transformation.
C.2 The simulation of EELS
Eq. (C.2) basically represents a joint density of states, weighted by matrix elements
h
n|eiq·r|
ji. Electronic structure calculations can be used to calculate these matrix
elements. In this work EEL spectra are shown which were obtained based on pseu-
dopotential calculations (AIMPRO). The next sections will give some details on how
the necessary matrix elements can be approximated in an approach for low-loss and
core-excitation EELS respectively.
Low-loss EELS: As mentioned above, low-loss EELS involves excitations from oc-
cupied valence band states to empty conduction band states. In the long-wavelength
dipole approximation (q→0), the matrix elements in Eq. (C.2) can be simplified fur-
ther using eiq·r≈1+iq·r:
ε
`
2(E) = 4
π
e2
X
c,vZBZ |h
c
k|r`|
v
ki|2
δ
(Ec
k−Ev
k−E)d3k(C.3)
Here cand vdenote conduction and valence bands respectively and kgives the
Brillouin zone vector — excitations with
k6=0 are excluded in this simplified
approach. As the dielectric function depends on the polarisation of the exciting
wave through q, it has to be calculated for different orientations `of rof the electron
beam. The integral over kcovers the whole Brillouin zone which is sampled with
1000–5300 Monkhorst-Pack-optimised [84]k-points for bulk cells and 100–150 k-
points for larger models containing dislocations. As LDA underestimates the band
gap, a scissors operator is applied to the band structure, shifting the conduction
bands upwards to match the experimental gap. Defect states in the gap are shifted
proportionally to their distance from the valence band maximum. A Lorentzian
broadening scheme for the
δ
-function in Eq. (C.3) allows an analytical Kramers-
Kronig transformation to obtain the real part of the dielectric function.
Core-excitation EELS: Core-excitation spectra result from transitions from atomic
core states to empty conduction band states. However, in pseudopotential calcu-
lations as presented in this work, the core states are not treated explicitly but are
only included through the pseudopotential. Therefore here Eq. (C.3) is not applica-
ble to compute core-excitation spectra. Still, core-excitations can be modelled at a
very simple level: Final state effects (i.e. electron-hole interactions) are completely
128
Appendix C. Electron Energy-Loss Spectroscopy
core EELSlow−loss EELS
theory
theory
CVD
CVD
IIb
(a.u.)
0 10 20 30 40 (eV)
(a.u.)
0 10 20 30 40 (eV)
Figure C.2: Theoretical and experimental EELS on diamond. Left: Low-loss EEL spectrum
(solid line) compared with experimental data on CVD diamond (dashed line) [6]. Right:
Theoretical K-edge EEL spectrum of bulk diamond (solid line), compared with experimental
data on CVD diamond (dashed line) [6] and on natural type-IIb diamond (plus signs) from
Bruley and Batson [104]. The experimental spectra have been arbitrarily scaled and the
energy zero is set at the conduction band minimum, where the spectra have been aligned.
neglected, the core bands are assumed to have no dispersion and the initial state is
assumed to be localised on a single atom. Under these assumptions, core-excitation
spectra are related to an angular momentum projected local density of states on that
atom. For an s-like initial core state — as in the diamond K-edge spectrum — only
transitions into p-like states contribute. The spectrum at one atom is then propor-
tional to the p-projected density of conduction band states localised at R.
IR(E) = X
cZBZ |h
c
k|pRi|2
δ
(Ec
k−Ei,R−E)d3k(C.4)
The angular part of the projecting function pRis the standard Legendre polynomial
while the radial part is chosen to be constantly 1 (in the respective units) for radii
smaller 1 a.u., and zero otherwise. Varying the radius has no considerable influ-
ence on the computed EEL spectra [6]. As discussed in Ref. [13], the energy Ei,Rof
the core level at Rcan be reconstructed from the differences in the potential at the
atomic core between bulk material and the atom of interest. Consequently Eq. (C.4)
allows us to approximately obtain the spectral contributions from one atom, with-
out the need to reconstruct the core-wavefunction of the final state. Summation over
all atoms within a chosen volume finally reproduces the situation in an electron mi-
croscope, where the electron beam probes a small region containing several atoms.
Fig. C.2 shows the resulting calculated low-loss and core-excitation spectra of bulk
diamond in comparison with experimental data. Qualitatively the theoretical and
experimental spectra agree reasonably well and the peak positions are reproduced.
Above 20 eV the differences between theory and experiment become significant.
These may be due to surface effects in the performed experiments or the crude ap-
proximations made in the theoretical approach.
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Page backreferences: To facilitate locating the context of citations, numbers at the very end
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Acknowledgements
First of all I wish to thank my thesis advisor in Paderborn, Thomas Frauenheim,
for his support and for allowing me to expand my research activities according to
my own interest. During my extended stays at the University of Exeter, UK, it was
Robert Jones, whose enthusiasm and guidance kept me on the track which eventu-
ally lead to this thesis. Bob, together with Malcolm Heggie (University of Sussex,
UK), provided me with inspiring ideas and their experience in dislocation research
was invaluable for my work. It was Malcolm, who opened my eyes to many aspects
of elasticity theory.
Financially my research has been supported by the Deutsche Forschungsgemeinschaft,
the Engineering and Physical Sciences Research Council (UK) and the European Union
through various funding schemes.
I would like to express my gratitude to all those colleagues and collaborators who
directly contributed to the contents of this thesis: The diamond HRTEM work pre-
sented in Section 4.6 and to be published in Ref. [7], was performed by Bert Willems
(EMAT, Antwerp) on natural diamonds supplied by the DTC Research Centre / De
Beers. Section 4.7 — the discussion of electron energy-loss spectra in CVD diamond,
published in Ref. [6] — is the result of a collaboration with Aurora Guti´errez-Sosa
and Ursel Bangert (both UMIST, Manchester), who performed the experiments, and
Caspar Fall (Exeter) who implemented the EELS functionality into the AIMPRO code
and performed the simulations based on my dislocation models. With regard to the
degradation of bipolar SiC devices discussed in Chapter 5, I am grateful to Pirouz
Pirouz (Case Western Reserve University) and Marek Skowronski (Carnegie Mellon
University) for their interest in my work and their cooperation. Furthermore both
gave permission to use some of their microscopic images (Fig. 5.2,5.3 and 5.4). The
last-minute calculation of most of the SiC dislocation band structures and Kohn-
Sham spectra in Section 5.5 was only possible thanks to the effort of Sven ¨
Oberg at
the Department of Mathematics, Lule˚a University of Technology, Sweden.
Furthermore I am grateful to Gotthard Seifert (Technische Universit¨at Dresden). He
was always open for discussion and gave advice when it came to the basics of den-
sity functional theory or the DFTB method. Still, my first attempts of using this
method would have been futile without the start-up help of Gerd Jungnickel and
Joachim Elsner, who gave me an introduction to the method and its practical appli-
cation. Also, Gerd’s DFTB implementation DYLAX considerably reduced the efforts
to calculate elastic constants. In a similar context I thank Patrick Briddon (Newcas-
tle) for maintaining and constantly improving the AIMPRO code, which has proven
ever so useful when more accurate electronic structure calculations were required.
139
Many thanks to Ben Hourahine and Jonathan Goss for their critical reading of the
manuscript. They prevented many a Teutonic wording construct and were a great
help in identifying those passages of the text which needed rephrasing or clarifica-
tion. In addition, as I am not an expert in experimental techniques, Appendices B
and Cwere checked for major flaws in the description of TEM and EELS techniques
by Bert Willems (EMAT, Antwerp) and Annette Kolodzie (Cavendish Laboratory,
Cambridge).
I am also indebted to all those people involved in the administration of the various
computer systems I used at Paderborn and Exeter. Here in particular Michael Stern-
berg has to be named, whose efforts ensured a very smooth operation of the Pader-
born UNIX and Linux systems. Michael also kindly provided the L
A
T
EX template and
hence had a major influence on the graphical layout of this thesis. Further, in the
context of Paderborn computing, Zolt´an Hajnal, Christof K¨ohler and Peter K¨onig
have to be mentioned. Zolt´an also played the key-role in a thrilling fetch-the-key-
from-the-river-Pader expedition. At Exeter, where the major part of this manuscript
was compiled, it was John Rowe who kept the system up and running.
I would like to thank Uwe Gerstmann, Michael Sternberg and Thomas K¨ohler for
many a late-evening discussion, covering not only physics but many fields of life,
and Marc Amkreutz for taking up the challenge of sharing a Paderborn office with
me, and while I was away in Exeter with my substitute, a giant fluffy chicken.
I further thank all those unnamed, who in one way or another contributed to the
inspiring atmosphere I enjoyed at the Physics departments of Paderborn and Ex-
eter and during my rather short stay in the Theoretical Chemistry Group at the
University of Sussex. Non-scientific life at those places included many sources of
motivation, like walks in Dartmoor and at the Cornish coast, the Paderborn Physics
pizza-and-movie nights and several pub and club crawls in Exeter and Brighton.
My thanks to all who played their part within.
Last but not least, thanks to my parents for their sympathy and understanding even
when I regularly missed their birthdays due to my extended stays abroad.
140
