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The Atomic Structure of
Diamond Surfaces and Interfaces
Michael Sternberg
The Atomic Structure of
Diamond Surfaces and Interfaces
Dissertation
zur Erlangung des akademischen Grades
Doktor der Naturwissenschaften (Dr. rer. nat.)
vorgelegt dem
Fachbereich Physik der Universit¨
at Paderborn
Michael Sternberg
Paderborn, 2001
Vom Fachbereich Physik der Universit¨
at Paderborn als Dissertation genehmigt.
Tag der Einreichung: 27. September 2001
Tag der m¨
undlichen Pr¨
ufung: 14. November 2001
Promotionskommission
Vorsitzender Prof. Dr. rer. nat. Gerhard Wortmann
Erstgutachter Prof. Dr. rer. nat. Thomas Frauenheim
Mitgutachter Dr. habil. Hans-Gerd Busmann
Beisitzer PD Dr. Siegmund Greulich-Weber
Archiv
Elektronische Dissertationen und Habilitationen der Universit¨
at Paderborn
http://www.ub.upb.de/volltext/ediss
Version: 14. November 2001
Michael Sternberg, The Atomic Structure of Diamond Surfaces and Interfaces.
PhD Thesis (in english), Department of Physics, University of Paderborn, Germany (2001).
137 pages, 41 figures, 24 tables.
Abstract
This thesis investigates several theoretical issues on the growth and structure of diamond
films produced by chemical vapour deposition. The work is divided into two major parts,
the first being methodological in character, and the second devoted to applications.
After an overview on electronic structure theory certain aspects of the density-functional-
based tight-binding method (DFTB) are examined, primarily its connection to the underly-
ing density functionals. A review of the approximations taken in various implementations
of this scheme over the years is given as well. In the present work this method is extended to
a linear scaling (O(N)) formulation. The first part concludes with test results and a general
assessment.
In part two, the standard DFTB method is applied to several problems of interest in the
context of diamond materials. First, an overview of the properties of diamond bulk and
surfaces is given. Energy and geometry data set the stage for the subsequent calculations
and serve as performance benchmarks. The remaining chapters address questions related to
the relatively new class of ultrananocrystalline diamond (UNCD) thin films grown primar-
ily from C2species. A complete mechanism for the growth of a diamond (110) monolayer
is established by investigating total energies and adsorption barriers on an initially clean
surface, followed by chain growth and coalescence, and finishing with the filling of surface
vacancies. The mechanism is qualitatively different from conventional methyl-based dia-
mond growth. A stabilisation of the diamond phase over graphite due to C2was observed
in the absence of hydrogen.
The internal structure of UNCD films is investigated by considering high-angle high-
energy (100) twist grain boundaries. They differ significantly from those in microcrystalline
diamond, where low-angle and tilt grain boundaries, as well as stacking faults and twins
prevail. First, the structure of pure grain boundaries is established, followed by an investi-
gation of the effects of impurities. The grain boundaries are confirmed to be very narrow,
essentially spanning only two atomic monolayers. The atomic structure is characterised by
threefold coordinated atoms which amount to about 50% of all interface atoms and intro-
duce electronic levels into the diamond band gap. The electrical conductivity observed in
the films is attributed to carbon π-states in the grain boundary regions. It is found that nitro-
gen impurities are energetically easier to incorporate into the grain boundaries than into the
grain bulk and that nitrogen promotes threefold coordination of carbon atoms in the grain
boundary. A shift in the Fermi energy towards the conduction band at larger nitrogen con-
centrations was noticed. These mechanisms support experimental evidence on enhanced
electrical conductivity due to nitrogen. In contrast, hydrogen saturates dangling bonds and
so lowers the film conductivity.
Keywords
diamond, surfaces, grain boundaries, growth, conductivity, density functional theory, tight-
binding, linear scaling, O(N)
PACS
71.15.Dx Computational methodology
61.43.Bn Structural modeling: serial-addition models, computer simulation
61.72.Mm Grain and twin boundaries
68.35.-p Solid surfaces and solid-solid interfaces: Structure and energetics
81.15.Aa Theory and models of film growth
Michael Sternberg, Die Atomare Struktur von Diamantoberfl¨achen und Grenzfl¨achen.
Dissertation (in englischer Sprache), Fachbereich Physik, Universit¨
at Paderborn (2001).
137 Seiten, 41 Abbildungen, 24 Tabellen.
Kurzfassung
Diese Dissertation untersucht verschiedene theoretische Fragen zu Wachstum und Struk-
tur von Diamantschichten, welche mittels chemischer Dampfphasenabscheidung hergestellt
werden. Die Arbeit ist in zwei Hauptteile gegliedert, von denen der erste methodischen
Charakter hat und der zweite Anwendungen gewidmet ist.
Nach einem ¨
Uberblick zur Theorie der Elektronenstruktur werden einige Aspekte der
Dichtefunktional-basierten Tight-Binding Methode (DFTB) beleuchtet, insbesondere deren
Beziehung zu den zugrundeliegenden Dichtefunktionalen. Ferner wird ein R¨
uckblick zu
den angewandten N¨
aherungen in den verschiedenen Implementierungen des Verfahrens
im Verlauf der letzten Jahre gegeben. In der vorliegenden Arbeit wird die Methode um eine
linear skalierende (O(N)) Formulierung erweitert. Der Teil schließt mit Testergebnissen und
einer allgemeinen Einsch¨
atzung ab.
Im zweiten Teil wird die Standard-DFTB-Variante auf verschiedene Probleme im Um-
feld von Diamantmaterialien angewandt. Zun¨
achst wird ein ¨
Uberblick zu Eigenschaften
des Diamantfestk¨
orpers und seinen Oberfl¨
achen gegeben. Daten zu Energie and Geome-
trie ebnen den Weg zu den anschließenden Rechnungen und dienen zur Absch¨
atzung ihrer
Zuverl¨
assigkeit. Die verbleibenden Kapitel behandeln Fragen zur relativ neuen Klasse von
ultrananokristallinen Diamantschichten (UNCD), welche haupts¨
achlich mittels C2-Spezies
gewachsen werden. Anhand von Berechnungen zu Gesamtenergien und Barrieren an einer
urspr¨
unglich reinen Oberfl¨
ache, gefolgt von Kettenwachstum und -verschmelzung sowie
abgeschlossen durch die Auff¨
ullung von Oberfl¨
achenleerstellen wird ein vollst¨
andiger Me-
chanismus zum Wachstum einer (110) Monolage aufgestellt. Der Mechanismus unterschei-
det sich qualitativ von konventionellem, methylbasiertem Wachstum. Eine Stabilisierung
der Diamantphase gegen¨
uber Graphit unter Einfluß von C2und ohne Wasserstoff wurde
beobachtet.
Schließlich wird die innere Struktur der UNCD-Schichten anhand von vergleichsweise
hochenergetischen (100) Großwinkeldrehkorngrenzen untersucht. Die Korngrenzen unter-
scheiden sich von jenen in mikrokristallinem Diamant, wo Kleinwinkelkorngrenzen, Kipp-
korngrenzen sowie Zwillingsbildung vorherrschen. Zun¨
achst wird die Struktur von rei-
nen Korngrenzen aufgestellt, gefolgt von Betrachtungen zum Einfluß von Defekten. Die
Korngrenzen werden als extrem d¨
unn best¨
atigt, da sie lediglich zwei atomare Monolagen
umfassen. Die atomare Struktur ist gekennzeichnet durch dreifach koordinierte Atome,
welche etwa 50% aller Grenzfl¨
achenatome ausmachen und elektronische Zust¨
ande in der
Bandl¨
ucke hervorrufen. Die an den Schichten beobachtete elektrische Leitf¨
ahigkeit wird
Kohlenstoff-π-Zust¨
anden innerhalb der Korngrenzen zugeschrieben. Es wird festgestellt,
daß Stickstoffverunreinigungen in die Korngrenzen energetisch g¨
unstiger als in das Innere
der Kristalle einzuf¨
ugen sind sowie außerdem die Dreifachkoordinierung von Kohlenstoff-
atomen f¨
ordern. Eine Verschiebung des Ferminiveaus zum Leitungsband hin wurde bei
h¨
oheren Stickstoffkonzentrationen konstatiert. Diese Mechanismen st¨
utzen experimentelle
Befunde einer erh¨
ohten elektrischen Leitf¨
ahigkeit infolge von Stickstoff. Im Gegensatz dazu
s¨
attigt Wasserstoff freie Bindungen ab und senkt daher die Leitf¨
ahigkeit.
Schlagw¨orter
Diamant, Oberfl¨
achen, Korngrenzen, Wachstum, Leitf¨
ahigkeit, Dichtefunktional-Theorie,
Tight-Binding, Lineare Skalierung, O(N)
Contents
Contents v
List of Figures ................................... viii
List of Tables .................................... x
Symbols and abbreviations ............................ xi
Introduction 1
1 Theoretical Foundations 5
1.1 Wave mechanics and density functional theory ............. 5
1.2 The density as basic variable ........................ 7
1.3 The Hohenberg-Kohn variational principle ............... 7
1.4 The Kohn-Sham equations ......................... 9
1.5 Solutions to the Kohn-Sham equation .................. 11
1.5.1 The Kohn-Sham total energy ................... 12
1.5.2 Charges and forces ......................... 13
2 The DFTB Method 15
2.1 Historical sketch ............................... 15
2.2 Variational approach and stationary principle .............. 18
2.3 Approximations in DFTB .......................... 20
2.3.1 Pseudo-atomic starting density .................. 21
2.3.2 Tight-Binding integrals and the two-centre approximation . . 22
2.3.3 Repulsive potential ......................... 23
2.3.4 Second-order corrections ..................... 25
2.4 The DFTB secular equation ........................ 28
3 Order-N Method and Implementation 31
3.1 Introduction ................................. 31
3.2 Energy functional and charges ...................... 32
v
vi Contents
3.2.1 Energy functional .......................... 32
3.2.2 Localisation of support functions ................. 34
3.2.3 Hamilton and overlap matrices .................. 35
3.2.4 Atomic charges and SCC energy contributions ......... 38
3.3 Electronic minimisation .......................... 39
3.3.1 The Lagrange multiplier η..................... 41
3.3.2 The adaptive secant method .................... 42
3.4 The calculation of forces .......................... 45
3.5 Accuracy and performance ........................ 46
3.5.1 Energy and charges ......................... 47
3.5.2 Forces ................................ 48
3.5.3 Scaling ................................ 49
3.5.4 Assessment ............................. 50
4 Properties of Diamond 51
4.1 History and economics ........................... 51
4.2 Bulk structure and properties ....................... 52
4.3 Diamond surface structure ......................... 55
4.3.1 General properties ......................... 55
4.3.2 The individual surfaces ...................... 58
4.4 Reference calculations on diamond bulk and surfaces ......... 60
4.4.1 Atomic and diatomic energies .................. 60
4.4.2 Bulk systems ............................ 61
4.4.3 Surfaces ............................... 63
4.4.4 Summary .............................. 68
5 Growth of (110) Diamond Using Pure Dicarbon 71
5.1 Introduction ................................. 71
5.2 Simulation setup .............................. 73
5.3 Adsorption and energetics of small carbon clusters ........... 73
5.3.1 Initial C2deposition ........................ 73
5.3.2 Addition of C2........................... 76
5.3.3 C4clusters .............................. 77
5.3.4 C6and C8clusters ......................... 78
5.3.5 Adsorption barriers ........................ 79
5.3.6 Surface defect formation ...................... 79
5.3.7 Surface vacancy filling ....................... 80
Contents vii
5.3.8 Graphitisation and rebonding ................... 80
5.4 Surface diffusion of C2........................... 83
5.5 Molecular dynamics depositions ..................... 84
6 Structure and Impurities in UNCD Grain Boundaries 85
6.1 Introduction ................................. 85
6.1.1 Grain boundary primer ...................... 87
6.1.2 Grain boundary supercell setup ................. 90
6.2 Twist (100) grain boundaries without impurities ............ 91
6.2.1 Structure and bonding ....................... 91
6.2.2 Energetics .............................. 96
6.2.3 Energy levels ............................ 97
6.3 Nitrogen substitutional impurities .................... 99
6.3.1 Structure and bonding ....................... 99
6.3.2 Energetics .............................. 102
6.3.3 Energy levels ............................ 102
6.4 Silicon substitutional impurities ...................... 106
6.5 Hydrogen addition ............................. 106
6.6 Summary ................................... 109
7 Summary and Conclusions 111
A Atomic Units Reference 115
B Approximations for Exchange and Correlation Energies 118
Bibliography 121
Acknowledgements 135
Colophon 137
viii List of Figures
List of Figures
2.1 Example for generating the repulsive potential ............. 25
2.2 Interpolation functions for the γab term in SCC-DFTB ......... 27
3.1 Spatial relations between localisation regions .............. 36
3.2 Line minimisation update step λmin vs. iteration ............ 41
3.3 Flowchart for the O(N)minimisation of the total energy ....... 42
3.4 Calculated energy, charges, and η-parameter during iteration . . . . 43
3.5 Examples for the application of the adaptive secant method ..... 45
3.6 Accuracy of O(N)force calculation .................... 48
3.7 Scaling behaviour for CPU time and memory requirements ...... 50
4.1 The crystal structure of diamond and lonsdaleite ............ 53
4.2 Geometry of diamond surfaces ...................... 56
4.3 {111}−(1×1)surface net (hexagonal) ................. 58
4.4 {111}−(2×1)surface net ........................ 58
4.5 {110}−(1×1)surface net (rectangular) ................ 59
4.6 {100}−(1×1)surface net (square) ................... 59
4.7 {100}−(2×1)surface net ........................ 59
5.1 Cross-section SEM images of as-grown UNCD films .......... 72
5.2 Overview of relaxed structures from subsequent C2depositions . . . 74
5.3 Initial steps for deposition of a C2molecule ............... 74
5.4 Continued deposition of C2next to an existing adsorbate ....... 77
5.5 Continued deposition of C2on top of an existing adsorbate ...... 78
5.6 Continued deposition of C2with high insertion energy ........ 79
5.7 Final stages of chain growth and coalescence .............. 81
5.8 Graphitisation and induced rebonding after additional deposition . . 82
6.1 Crystallographic relation for twist grain boundaries .......... 88
6.2 Setup of a grain boundary simulation cell ................ 89
6.3 Side view of the periodic cell for a Σ13 grain boundary ........ 91
6.4 Comparison of Σ13, Σ5(2×2), and Σ29 grain boundaries ....... 92
6.5 Radial distribution function J(r)for Σ13 ................. 94
6.6 Overview of individual interface planes ................. 95
6.7 Electronic density of states for diamond and twist grain boundaries . 97
List of Figures ix
6.8 Local density of states for a diamond twist grain boundary ...... 98
6.9 Schematic band structure for a diamond Σ13 grain boundary ..... 98
6.10 Substitution sites selected in a Σ13 grain boundary ........... 100
6.11 Relaxed local structure around a substitutional nitrogen atom . . . . 101
6.12 Localisation of states near substitutional N in diamond (P1 centre) . . 103
6.13 Density of states for a Σ13 grain boundary with nitrogen impurities . 105
6.14 Side and top view of a grain boundary with a silicon impurity . . . . 107
6.15 Relaxed local structure around a substitutional silicon atom . . . . . 107
6.16 Relaxed structure of a Σ13 grain boundary with hydrogen ...... 108
6.17 Molecular dynamics path of hydrogen atoms in a grain boundary . . 109
xList of Tables
List of Tables
2.1 Schemes for pseudo-atom calculation in DFTB ............. 22
2.2 Integral types in the DFTB Hamiltonian ................. 23
2.3 Chemical hardness for some elements .................. 27
3.1 Comparison of O(N)total energies and charges ............ 47
3.2 Scaling of O(N)CPU time and memory requirements ......... 49
4.1 Elementary properties of low-index diamond surfaces ......... 55
4.2 Pseudo-atom parameters for SCC-DFTB ................. 60
4.3 SCC-DFTB reference calculations for diatomic molecules ....... 61
4.4 Reference energies of bulk diamond ................... 62
4.5 Reference energies of graphite ....................... 62
4.6 Reference energies of diamond surfaces ................. 64
4.7 Calculated geometry of diamond {111}surfaces ............ 65
4.8 Calculated geometry of diamond {110}surfaces ............ 67
4.9 Calculated geometry of diamond {100}surfaces ............ 68
4.10 Summary of stable reconstructions on diamond surfaces ....... 69
5.1 Energy barriers and adsorption energies of C2during growth stages 75
5.2 Diffusion barriers and change in total energy along diffusion path . . 83
6.1 Structure and energetics of twist grain boundaries ........... 90
6.2 Coordination and bondlengths for atoms in grain boundary planes . 93
6.3 Coordination for nitrogen and carbon atoms in grain boundary planes 100
A.1 List of the CODATA recommended values of fundamental constants 116
A.2 Values in SI units of some atomic units .................. 116
A.3 Conversion between units of energy and values of energy equivalents 117
A.4 Energy equivalents for electromagnetic radiation ............ 117
Symbols and abbreviations xi
Symbols and abbreviations
This list is limited to the most important symbols. Others are explained in the text
when first used. Appendix Acontains symbols and reference data for fundamental
constants and atomic units (which are used throughout).
DFT density functional theory
KS Kohn-Sham
LCAO linear combination of atomic orbitals
TB tight binding
DFTB density functional based tight binding
SCC self-consistent charge
O(N)Order-N(scaling linearly with system size N)
LR localisation region (range) for one-electron wave functions Φ(r)(q.v.)
LRSK localisation region for Slater-Koster integrals
OV overlap region for Φi(r)with other Φj(r)
CVD chemical vapour deposition
UNCD ultrananocrystalline diamond
GB grain boundary
b
HHamilton operator
n(r)electron density
occ number of occupied one-electron states (in summation bounds)
Ψ({xi})many-particle wave function of spatial and spin variables xi
Φ(r),|Φione-electron wave function (molecular orbital)
ϕ(r),|ϕibasis function (usually atomic)
i,jlabel for molecular orbitals
µ,ν, . . . label for basis functions, regardless of atomic site
a,b, . . . label for atoms
Rapositions of atoms
ra,raspatial vector relative to an atomic position ra=r−Ra, and its length
ra=|ra|
µ[a](set of) basis functions located on specified atom a
a[µ]atom corresponding to a specified basis function or set thereof.
H,Hij Hamilton matrix (element) for molecular orbitals (≡εifor eigenstates)
S,Sij overlap matrix (element) for molecular orbitals (≡δi j for eigenstates)
h,hµν Hamiltonian matrix (element) between basis functions
s,sµν overlap matrix (element) between basis functions
xii Symbols and abbreviations
Crystallographic notation
The following are the standard recommended symbols to describe crystallographic
planes and directions (Miller indices), emphasising the distinction between a generic
notation for a family of planes or directions, and a specific notation for individual
ones, to be used where the distinction is important. A frequent exception, however,
employed here as well, is the generic use of (hkl)for planes to improve readability
in text.
(hkl)plane, specific. (100)6= (010)
{hkl}plane, generic. (100),(010),(¯
100), . . .
[hkl]direction, specific. [100]6= [010]
hhklidirection, generic. [100],[010], . . .
Introduction
On two occasions I have been asked [by members of parliament] ‘Pray,
Mr Babbage, if you put into the machine wrong figures, will the right
answers come out?’ [. . .] I am not able rightly to apprehend the kind of
confusion of ideas that could provoke such a question.
– Charles Babbage (1792-1871)
This thesis investigates several theoretical aspects of the growth and structure of
diamond films produced by chemical vapour deposition.
Diamond as material
Gemstone-quality diamond is found only in nature, in strongboxes, and occasion-
ally in ballrooms, where it sparkles at its best. Materials scientists have a farther
reaching interest in diamond as material of choice for current and future applica-
tions. Diamond has an attractive combination of physical properties like extreme
hardness, high refractive index, high heat conductivity, high IR-transmissivity and
interesting electronic properties like a wide bandgap and negative electron affinity.
Natural diamond is the paragon for many of these parameters.
The majority of mined diamonds is of industrial grade and is consumed largely
as grinding material. Research efforts in the 1950s led to the development of sev-
eral technologies to produce synthetic diamond. By far the largest share of pro-
duction derives from a high-pressure high-temperature (HPHT) process developed
for industrial scales contemporaneously by the General Electric Company (GE) in
the US and Allmanna Svenska Elektriska Aktiebolaget (ASEA) in Sweden, see [1,
“Matter”]. This type of material is used mostly in the form of grit or sintered pow-
der in coatings for cutting and grinding tools to improve their hardness and wear
resistance. However, efforts to produce high quality diamond encounter signifi-
cant problems, since classical techniques for growing large single crystals, e.g. the
Czochralski method, fail, because the melting point of diamond under reasonable
pressure is so high that it is not known, see [1, “Structure of carbon allotropes”].
A completely different strategy suitable for modern applications is the actual growth
of diamond by chemical vapour deposition (CVD). It was developed independently
by Eversole and subsequently Angus in the USA and by Spitsyn and Deryagin in
the USSR [2]. This method allows the direct deposition of diamond polycrystalline
1
2Introduction
films on substrates of possibly complicated shapes. By growing films to a thick-
ness in the millimetre range and subsequently etching away the substrate, one may
even fabricate free platelets, or wafers. Devices based on diamond thin films are
being developed e.g. for cold electron emitters and high-temperature applications,
while platelets find application as heat spreaders or as mechanically robust infrared-
transmissive windows.
The CVD growth process must be delicately controlled since the formation of di-
amond competes thermodynamically with that of graphite. At room temperature
(298 K) and atmospheric pressure, the free energy for graphite is lower than that of
diamond by 30 meV/atom [3]. Conventionally, methane (CH4) is used as feed gas
in a low-pressure chamber. To stabilise diamond over graphite, the methyl content
in the gas has to be kept in the 1% range, with hydrogen constituting the rest, acting
as catalyst. The resulting growth rate and film morphology of CVD films depend on
the growth conditions; typical rates are nowadays of the order of a few micrometres
per hour, resulting in polycrystalline films with a typical grain size of 1 micrometre.
It was a long-standing assertion that hydrogen in its role of etching graphitic de-
posits is essential for the growth of diamond. In the 1990s research at Argonne
National Laboratory led to a new type of diamond film grown from fullerene-
fragmented C2precursors in hydrogen-poor, argon-rich plasmas [4]. The resulting
films differ markedly in their morphology from conventional microcrystalline films.
They consist of ultrananocrystalline diamond with crystallite sizes in the range of
3–10 nanometres. This is at least an order of magnitude smaller than occasionally
observed submicron grain sizes in traditional films, and so justifies a separate clas-
sification for the new films.
The role of theory
Theoretical studies help to understand and overcome some of the problems in-
volved in the production of CVD diamond. While a great deal is known experi-
mentally about bulk properties and crystal defects, detailed information about the
atomic surface structure is derived largely from quantum-mechanical models in-
volving the calculation of total energies, atomic forces, and electronic properties.
The spectrum of theoretical methods for atomic structure calculations ranges from
high-accuracy techniques based on configuration interactions, over the large realm
of Hartree-Fock and density-functional based methods, and into empirical force-
fields. Each class of methods has its particular domain of applicability, characterised
roughly by the system size, i.e., the number of atoms which can comfortably be han-
dled at a desired accuracy level.
Surfaces and interfaces are in general more complex than highly symmetric bulk
systems, and therefore have larger periodic cell sizes. This fact curtails the ap-
plicability of high-accuracy wave-function- and density-functional methods alike.
Classical potentials, on the other hand, allow to investigate mesoscopic phenom-
ena like crack propagation or surface roughening but cannot provide a description
of the electronic properties of the material. It is the tight-binding methods of the
medium-accuracy level, such as the one employed here, which provide a quantum
mechanical description of chemical bonding sufficient to describe the electronic and
Introduction 3
atomic structure of large scale bulk, surface and interface systems. They may also
yield insight into system evolution, e.g. during growth and phase transitions.
Outline
The subject of the present work is the application of a particular density-functional
based tight-binding method (DFTB) to diamond and related systems. The work is
divided into two major parts, the first being methodological in character, and the
second devoted to applications.
The goal of part one is to consider a linear-scaling development of the tight binding
formalism in the context of density functional theory. The standard implementation
of tight-binding involves the solution of an eigenvalue problem which scales as N3
in terms of the size of the system, Nrepresenting the number of basis functions.
In line with several attempts during the last decade to find alternative solutions
which scale linearly with the system size, one goal of this thesis is to extend our
tight-binding method to such an Order-Nformulation. To this end, the background
of the density-functional theory is given in Chapter 1and that on a tight-binding
formulation derived directly from it in Chapter 2. The extension of this tight-binding
method into an O(N)formulation is then presented in Chapter 3, which concludes
the first part.
In part two, the standard DFTB method is applied to several problems of interest
in the context of diamond materials. The first chapter in the second part (4) gives
an overview of the properties of bulk diamond and especially diamond surfaces—
including hydrogenated surfaces and various reconstructions. The data on energies
and geometries not only provide the basis for the subsequent calculations but also
serve as a benchmark for the performance of the method.
The next two Chapters (5and 6) address questions related to the new class of ul-
trananocrystalline diamond (UNCD). While the conventional methyl-based growth
process is believed to be largely understood, the predominant growth species for
UNCD films is C2produced in great abundance from the fragmentation of C60
fullerenes. To shed light on this process, Chapter 5addresses the question of growth
due to C2on the (110)-face which is the fastest growing face. In this chapter, a com-
plete mechanism for the growth of a diamond (110) monolayer is established by
investigating total energies and adsorption barriers for clusters on an initially clean
surface, followed by chain growth and coalescence, and finishing with a mechanism
for filling surface vacancies. In this context, a stabilisation effect of the diamond
phase over graphite was observed without hydrogen participation.
In Chapter 6the structural properties of UNCD films are investigated in a study of
high-angle high-energy (100) twist grain boundaries. Such grain boundaries occur
in the films as a result of the more or less random orientation of the small grains.
Again, they differ significantly from those in microcrystalline diamond, where low-
angle and twist grain boundaries, as well as stacking faults and twins prevail, all
of which have been extensively studied. In this work, the structure of pure grain
boundaries is established, followed by an investigation of the effects of impurities.
4Introduction
Appendix Acontains reference material on relevant atomic units which the reader
and author alike will find convenient to be included. Some notes on the widely
used local density approximation (LDA) and generalised gradient approximations
(GGA) to the density functional theory have found a place in Appendix B.
Chapter 1
Theoretical Foundations
In this chapter, we will briefly review the density functional theory, to the extent
appropriate for the scope of the present work. The background, precise mathemat-
ical formulation, and extensions are reviewed regularly and at length in the liter-
ature [5,6,7,8]. The most recent treatise from the horse’s mouth [9] discusses the
connection between density functional theory and venerable wave mechanics, an
approach most appealing to take as a starting point.
1.1 Wave mechanics and density functional theory
The basic problem posed in electronic structure calculations is to determine the time-
independent many-electron wave function [10]
Ψ=Ψ({xi}),i=1 . . . N(1.1)
for Ninteracting electrons moving in a static potential, usually the potential of the
atomic nuclei of a solid or molecule. The parameters xiin general contain the spatial
and spin coordinates, i.e., xi≡(ri,si), but the spin variables will not play a role in
most of the present work. The wave function for stationary states is the solution of
the Schr¨
odinger equation [11]:
b
HΨ(n)({xi}) = E(n)Ψ(n)({xi}). (1.2)
It has solutions for a number of different states, labelled n. The state with the lowest
energy is the ground state. The non-relativistic and stationary Hamiltonian in the
Schr¨
odinger equation is given by:1
b
H=
N
X
i=1
−∆i
2+
N
X
j>i
1
|ri−rj|+Vext(ri)
. (1.3)
1Atomic units are used; cf. Appendix A.
5
6Chapter 1. Theoretical Foundations
In atomic systems, the external potential Vext(r)is the potential of nuclei with atomic
numbers Zaand positions Ra:
Vext(r) = −
Nat
X
a=1
Za
|r−Ra|(1.4)
This Hamiltonian embodies the adiabatic or Born-Oppenheimer approximation,
which describes the system of electrons in the static field of the nuclei, an approxi-
mation justified by the large difference in mass, and therefore, length and time scales
for the motion of nuclei and electrons. Consequently, all physical properties of the
system depend parametrically on the positions of the nuclei, i.e.,
Ψ=Ψ({xi},{Ra})(1.5)
E=E({Ra}). (1.6)
For the purpose of electronic structure calculations, this dependency will be put
aside. It will be revisited in the calculation of forces on atoms for the purpose of a
quasi-classical integration of motion.
The many-particle Schr¨
odinger equation (1.2), a second-order partial differential
equation of many variables, cannot in principle be solved exactly because of its com-
plexity due to electron-electron interaction. Straightforward approximate solutions
are sought within the Hartree- and Hartree-Fock theory [12,13,14] and their ex-
tensions, developed steadily since the early days of wave mechanics in the 1920s.
The Hartree theory expresses the many-particle wave function Ψin terms of a sim-
ple product of one-electron orbitals, while on the Hartree-Fock level this is gener-
alised to a Slater determinant of spin orbitals. The determinant form ensures that
the solution be antisymmetric with respect to particle exchange, as dictated by the
Fermi statistics. The Hartree-Fock theory thus treats particle exchange interactions
correctly while it does not, however, describe correlation effects, i.e., further many-
particle effects. Extensions to the single-determinant HF theory invoke a linear
combination of many different Slater determinants to describe configuration interac-
tion between different states. The highest accuracy solutions of this type achievable
nowadays involve up to 109determinants. As a result of the mutual interaction
of all electrons across the whole configuration space of spatial and spin variables,
the computational complexity is exponential in the number of electrons. Therefore,
the excellent accuracy of wave function methods comes at an insurmountably steep
price tag when the number of atoms exceeds about 10–20.
Density functional theory (DFT) provides a radically different approach by recasting
the basic problem in terms of the electron density [15]. The spin-free density in terms
of the wave function can be formally defined as follows:
n(r) = X
s1...sNZΨ∗(rs1,r2s2, . . . rNsN)Ψ(rs1,r2s2, . . . rNsN)dr2. . . drN. (1.7)
The concept of employing the electron density in an energy functional dates back to
the Thomas-Fermi theory [16,17], which has been reviewed by Lieb [19] and Jones
and Gunnarsson [5].
1.2. The density as basic variable 7
With the appropriate approximations devised since its inception in 1964, DFT has
paved the way for the successful modelling of ever larger systems. Walter Kohn, as
one of the fathers of DFT, has been honoured “for his development of the density-
functional theory”, by the award of the 1998 Nobel Prize in Chemistry [20], jointly
with John Pople “for his development of computational methods in quantum chem-
istry”.
1.2 The density as basic variable
At the heart of DFT stand two rigorous mathematical statements. The first is a
lemma linking the density and the potential of a Hamiltonian, and the second is a
theorem about the existence of a universal functional of the electron density describ-
ing the ground state energy of a system of Nelectrons within an external potential.
The theorem will be discussed in a moment. First, here is the lemma:
Lemma 1 (Basic Lemma of Hohenberg-Kohn) The ground state density n(r)of a
bound system of a fixed number of interacting electrons in an external potential Vext(r)
determines this potential uniquely (to within an additive constant).
The proof (for non-degenerate ground states) is rather simple and need not be re-
peated here. Suffice to say that it represents the density in terms of wave func-
tions, invokes the Rayleigh-Ritz principle2and proceeds by reductio ad absurdum.
The lemma is mathematically exact and holds for any density, including any one of
degenerate states, as shown later by Kohn [22] and others [6].
Since the density determines the potential, it establishes the full Hamiltonian of the
system, and hence, determines all physical quantities derived through b
H, including
the many-particle eigenstates Ψ(n)({xi})of the Schr¨
odinger equation.
One issue remains to be resolved in a satisfactory manner, namely, that not all well-
behaved positive functions n(r)are indeed the ground state density corresponding
to some external potential Vext(r), as has been demonstrated by example indepen-
dently by Levy and Lieb [23,24]. This so-called v-representability problem is mostly a
formal one, with little impact on the applicability of DFT for real systems.
It should be noted that the basic lemma does not hold in its generalisation to spin-
dependent density functional-theory, as was pointed out early on by von Barth and
Hedin [25] and clarified recently by Eschrig [26]. The Levy/Lieb formulation, as
shown below, does away with the requirement of a unique mapping of density to
potential.
1.3 The Hohenberg-Kohn variational principle
For the study of molecules and solids our primary interest is the total energy of a
given arrangement of atoms; ultimately, we want to find, within suitable boundary
conditions, the atomic arrangement that is lowest in total energy and hence the most
2An introductory discussion of the Rayleigh-Ritz minimal principle is given, e.g., in Ref.[21].
8Chapter 1. Theoretical Foundations
stable. Besides contributions from the internuclear interactions, this task requires
knowledge of the energy of the electronic system.
The Schr¨
odinger equation is essentially an eigenvalue problem in the space of nor-
malised wave functions. The lowest eigenvalue, representing the ground state en-
ergy, may be found from the Rayleigh-Ritz principle (cf. footnote 2on the preceding
page). In the original work by Hohenberg and Kohn [15] this principle was refor-
mulated in terms of trial densities. A refined description, known as the constrained
search method, was given later by Levy and Lieb [23,24] as follows.
For a physical system, a given density determines, by Lemma 1, its external po-
tential, and therefore, the Hamiltonian and thus, in turn, by the Schr¨
odinger equa-
tion (1.2), the many-particle wave function Ψ({ri}). Hence, the latter is strictly a
functional of n(r), albeit not a trivial one. Accordingly, Levy and Lieb define Ψ({ri})
as minimum of a universal functional of a trial density as
F[˜
n(r)] = min
e
Ψ→˜
nhe
Ψ|b
T+b
U|e
Ψi, (1.8)
where b
Tand b
Uare the operators of the kinetic energy and the Coulomb repulsion
of the interacting many-electron system, respectively. e
Ψextends over the class of
wave functions yielding the argument density ˜
n. The functional F[˜
n]is universal
in the sense that it does not refer to an external potential. By Lemma 1,Vext is
fixed for a given density. Therefore, F[˜
n]also minimises, within the space of trial
wave functions reproducing the particular argument density ˜
n(r), the total energy
functional
E[˜
n(r)] = ZVext(r)˜
n(r)dr+F[˜
n(r)] . (1.9)
The key difference between this definition and the original formulation [15] is, that
F[˜
n]is defined for all “N-representable” densities ˜
n(r), whether “V-representable”
or not.
A second minimisation step, now in the space of all densities for which the func-
tional F[˜
n]is defined, finally minimises (1.9) for all normalised densities ˜
nby the
following theorem:
Theorem 1 (Hohenberg-Kohn variational principle) For all N-representable densi-
ties the functional (1.9) has a minimum, which it assumes at the ground state density nGS.
E[˜
n(r)] ≥EGS ,EGS =min
˜
n(r)→NE[˜
n(r)] = E[nGS(r)] . (1.10)
As for Lemma 1, the theorem also holds for any one of a number of possibly degen-
erate ground state densities.
The electronic ground state energy of a system of interacting electrons is thus rigor-
ously expressed in terms of their spatial charge density instead of the many-particle
wave function.
1.4. The Kohn-Sham equations 9
1.4 The Kohn-Sham equations
The difficulties in finding the energy and ground state density of a system of inter-
acting electrons according to (1.10) are formidable. Most prominently, the functional
F[˜
n(r)] is not explicitly known.
Kohn and Sham (KS) suggested in 1965 [27] a separation of F[˜
n(r)] into appropriate
major components:
F[n(r)] = T0[n(r)] + EH[n(r)] + Exc[n(r)] . (1.11)
The components are:
•T0[n(r)] is the kinetic energy functional for a fictitious system of non-interacting
electrons producing the same density as n(r).
•EH[n(r)] is the so-called Hartree energy, arising classically from the mutual
Coulomb repulsion of all electrons,
EH[n(r)] = 1
2ZZ n(r)n(r0)
|r−r0|drdr0. (1.12)
•The last term, Exc[n(r)], called the exchange-correlation functional, is a correc-
tion term, which accounts for all many-body effects in F[n(r)]. The treatment
of this term decides upon the viability of any DFT implementation. Equa-
tion (1.11) is the definition of Exc[n(r)].
The Hohenberg-Kohn functional (1.9) now takes the form:
EKS[n(r)] = T0[n(r)] + ZVext(r)n(r)dr+1
2ZZ n(r)n(r0)
|r−r0|drdr0+
+Exc[n(r)] . (1.13)
All terms but the last of the preceding equation are known analytically. As inher-
itance from the Hohenberg-Kohn functional, the functional form of the correction
term Exc[n(r)] is still unknown and must be approximated. Practical applications
of the DFT are classified according to the approximations taken for the exchange-
correlation functional Exc. Details are discussed in Appendix B.
In order to describe the kinetic energy functional T0[n(r)], KS instituted single-
particle orbitals Φi(r)representing the density. The orbitals are normalised as
hΦi|Φii=Z|Φi(r)|2dr=1 . (1.14)
Janak [28] introduced normalised occupation numbers nias follows:
n(r) =
occ
X
i=1
ni|Φi(r)|2;N=
occ
X
i=1
ni. (1.15)
10 Chapter 1. Theoretical Foundations
In the general, spin-unrestricted case, the label ispans spin variables, and the occu-
pation numbers are 0 or 1. However, in a spin-restricted formalism, the occupation
numbers represent orbitals occupied by two electrons of equal and opposite spin
and are then integers between 0 and 2. By extension, small thermal excitations may
be modelled through fractional occupation numbers subject to a Fermi-Dirac dis-
tribution, ni=f(εi). Janak notes that the energy functional (1.13) is represented
correctly only for these limited cases, and not for arbitrary sets of ni.
The kinetic energy in the KS formulation is now written in terms of an expectation
value over the single-particle orbitals, T0[n(r)] ≈T[{Φi}], with
T[{Φi}] =
occ
X
i=1
niΦi−∆
2Φi=
occ
X
i=1
niZΦ∗
i(r)−∆
2Φi(r)dr. (1.16)
With the preceding substitution, the ground state energy of the functional (1.13),
subject to the normalisation constraint (1.14), is found through the Euler-Lagrange
formalism. With the Lagrange parameters designated εi, the variation at the ground
state energy must vanish:
δ
δΦ∗
i(r)(E[n(r)] +
occ
X
i=1
niεi1−Z|Φi(r)|2dr)=0 ; ∀i. (1.17)
Noting that δn(r)
δΦ∗
i(r)=2Φ(r)one can carry out the variation straightforwardly and
obtains:
−∆
2+Vext(r) + VH([n(r)],r) + Vxc([n(r)],r)Φi(r) = εiΦi(r);∀i. (1.18)
This set of equations, supplemented by the density definition (1.15), is called the
Kohn-Sham equations.
Equation (1.18) is an eigenvalue equation for the εiand their eigenfunctions Φi(r).
It has the form of a single-particle Hartree equation,
−∆
2+Veff([n(r)],r)Φi(r) = εiΦi(r), (1.19)
with an effective potential
Veff([n(r)],r) = Vext(r) + VH([n(r)],r) + Vxc([n(r)],r), (1.20)
yet it describes, through Vxc, in a formally exact manner the interacting electron
system including many-body effects.
The classical contribution to Veff(r)due to the Coulomb interaction is called the
Hartree potential:
VH([n(r)],r) = δEH[n(r)]
δn(r)=Zn(r0)
|r−r0|dr0, (1.21)
1.5. Solutions to the Kohn-Sham equation 11
and contributions due to the Pauli repulsion, correlation, and the interaction contri-
bution in the kinetic energy are expressed by the exchange-correlation potential
Vxc([n(r)],r) = δExc[n(r)]
δn(r). (1.22)
Both contributions are local potentials, though they depend functionally on the en-
tire density distribution.
1.5 Solutions to the Kohn-Sham equation
An analytic solution of the Kohn-Sham equation (1.18) is quite unattainable. Instead,
the unknown functions Φi(r)are sought numerically employing routine tools from
the arsenal of mathematical physics. Two classes of approaches will yield a solution:
(a) real-space grid methods and (b) basis function expansions. Either approach will
transform the differential equation for the unknown functions into a set of algebraic
ones for unknown grid values or expansion coefficients.
Grid methods have not really been successful so far because the amplitude of the so-
lution typically varies over several orders of magnitude in space and thus presents a
great deal of numerical difficulty. Though this initial obstacle was overcome through
the use of adaptive curvilinear coordinates [29,30,31], much room for improvement
remains. Currently, the field experiences a renaissance spurred by the growth in
computing power and a drive towards parallelisation because of the inherent real-
space separability of grid-based algorithms [32,33].
Still, the more established tools are basis function methods. The general expansion
of the one-electron wave functions reads:
|Φii=
N
X
ν=1
Ciν|ϕνi;∀i. (1.23)
The unknown functions Φiare thus represented by their expansion- or Fourier-
coefficients Ciν. In principle, the basis set {|ϕνi} ought to be complete within the
configuration space. In practice, one strives to use a basis as small as possible in
order to reduce the computational effort in determining its Fourier coefficients.
Which basis set to use depends on the type of problem investigated. For periodic
structures of solids, one often employs plane waves. Plane wave functions are easy
to handle mathematically, because they are tightly knit to the Bloch paradigm. For
non-periodic structures, i.e., molecules and clusters, a great number of plane waves
is required to represent the strictly localised density distributions. Thus, for finite
systems, atomiclike basis sets are more popular.
Once a basis has been chosen, the Hamilton operator must be expressed within such
basis as a matrix. Applying the expansion (1.23), the Kohn-Sham equation (1.18)
reads:
b
H|Φii=
N
X
ν=1b
HCiν|ϕνi=
N
X
ν=1
εiCiν|ϕνi=εi|Φii;∀i. (1.24)
12 Chapter 1. Theoretical Foundations
Through multiplication on the left with any one hϕµ|and utilising the linearity of
the matrix operator one obtains a generalised hermitian eigenvalue problem:
N
X
ν=1
Ciν(hµν −εisµν)=0 ; ∀i,µ, (1.25)
with the matrix elements
hµν =hϕµ|b
H|ϕνi=Rϕ∗
µ(r)−∆
2+Veff([n(r)],r)ϕν(r)dr
sµν =hϕµ|ϕνi=Rϕ∗
µ(r)ϕν(r)dr.(1.26)
The matrices hand sare termed Hamilton- and overlap-matrix, respectively. They
depend on the atomic positions and on a well-guessed density n(r). After the ma-
trices have been established, eq. (1.25) may be solved using standard mathematical
methods. The solutions comprise the eigenvalues εiand eigenvectors Ciν.
Since the orbital solution depends on the density, and in turn, by eq. (1.15), on the or-
bitals themselves, equation (1.25) needs to be solved self-consistently. One starts with
a suitable input density n0
in(r), constructs the effective potential Veff[n(r)] through
proper approximations, solves eq. (1.25) for the orbitals Φi(r), and obtains a density
n0
out(r), which is used as new input density n1
in(r). Refined procedures construct
nj
in(r)from a weighted mixture of one or more previous densities. This procedure
must be iterated until the input and output densities agree to within a certain preci-
sion.
Upon convergence related physical observables like atomic charges, dielectric func-
tions or vibrational properties may be extracted, although some of these only at
considerable additional effort. The basic electronic structure problem, however, is
thus solved.
1.5.1 The Kohn-Sham total energy
It is instructive to write the total energy in terms of the Lagrange parameters εi. To
this end, it follows from their eigenvalue equation (1.19) after multiplication with
Φ∗
i(r)and integration,
ZΦ∗
i(r)−∆
2+Veff(r)Φi(r)dr=εihΦi|Φii=εi, (1.27)
where the orbital normalisation constraints (1.14) were used. Multiplication with ni
and summation yields
occ
X
i
niZΦ∗
i(r)−∆
2+Vext(r) + VH(r) + Vxc(r)Φi(r)dr=
occ
X
i
niεi. (1.28)
Considering the expressions for the total density, eq. (1.15) and the kinetic en-
ergy, eq. (1.16), the preceeding equation already looks very similar to the Kohn-
Sham energy functional (1.13), except for some unbalanced potential contribu-
tions. To restore EKS[n(r)], the following additive corrections are required:
1.5. Solutions to the Kohn-Sham equation 13
−1
2RVH(r)n(r)dr+Exc −RVxc(r)n(r)dr. Thus, the Kohn-Sham ground state en-
ergy in terms of the eigenvalues is given by
EKS[n(r)] =
occ
X
i=1
niεi−1
2ZVH(r)n(r)dr+Exc[n(r)] −ZVxc(r)n(r)dr. (1.29)
The first term on the right-hand-side includes most of the electronic shell structure
effects that arise from the exact treatment of the kinetic energy within the KS formu-
lation.3The additive terms are collectively known as double-counting corrections.
One might wish to assign a physical meaning to the Lagrange parameters εi. A
straightforward interpretation, known as Janak’s Theorem [28], is derived from
equation (1.29) for the minimising density:
∂E
∂ni
=εi. (1.30)
For the highest occupied electronic levels this relation is often interpreted as an ap-
proximation to true ionisation energies. However, it must be recalled that the rela-
tion stems from a fictitious one-electron system. Almbladh and von Barth [35] how-
ever, have given a proof that the highest occupied eigenvalue for isolated systems
does indeed equal the exact ionisation potential.
1.5.2 Charges and forces
In the chosen basis, the density matrix operator b
n(r,r0)takes the following form:
b
n=
occ
X
i
ni|ΦiihΦi|=X
µ,ν
occ
X
i
niC∗
iµCiν|ϕµihϕν|, (1.31)
with
Pµν =
occ
X
i
Pi
µν =
occ
X
i
niC∗
iµCiν(1.32)
being the elements of the density matrix, which may be further subdivided into their
contributions Pi
µν from individual eigenstates i, as shown.
The spatial charge density is given by (cf. eq. (1.15)):
n(r) = hr|b
n|ri=X
µX
ν
Pµνϕ∗
µ(r)ϕν(r). (1.33)
Energy-resolved charge densities n(r,εi)are obtained by using the density matrix of
the individual eigenstates, Pi
µν.
3Only for this purpose were the single-particle orbitals introduced [34].
14 Chapter 1. Theoretical Foundations
To obtain the total energy within the Born-Oppenheimer approximation, the inter-
nuclear Coulomb repulsion and the classical kinetic energy for the nuclei must be
added to the KS energy considered so far:
Etot =EKS +Enuc +EK
=EKS[n(r);{Ra}] + 1
2
Nat
X
a
Nat
X
b6=a
ZaZb
|Ra−Rb|+
Nat
X
a
ma
2˙
R2
a. (1.34)
Since the KS energy depends parametrically on the positions of the atoms, cf. (1.6),
the nuclei can be seen as moving in a potential V=EKS +Enuc. The dependence
of the total energy on atomic coordinates has the meaning of forces. This leads to a
classical equation of motion according to Newton’s law:
Fa=−∂
∂Ra
(EKS +Enuc)=ma¨
Ra,a=1 . . . Nat (1.35)
In general, the forces may be obtained analytically from the eigenstates {Ciµ}at
modest computational cost; however, for complicated Hamiltonians, numerical
derivatives must be resorted to. The forces may either be used for molecular dynam-
ics runs integrating Newton’s equation of motion or for quasistatic minimisations
of the total energy as function of atomic positions. This matter has been broadly
reviewed by Payne [36]. Because the kinetic energy is irrelevant for minimisation
procedures, the term total energy is occasionally used in this context to cover just the
sum of the KS energy and the internuclear repulsion.
Chapter 2
The DFTB Method
The preceding chapter attempted to give an overview on the background and for-
malism of the density-functional theory. Evidently, a full-potential self-consistent
solution of the Kohn-Sham equations demands considerable computational re-
sources. This demand may be satisfied up to certain practical limits from the as
yet uninhibited growth in capacity of computational hardware characterising our
era [37]. In the 1980s, using the machines en vogue, the complete and precise elec-
tronic structure of molecules or crystals with at most a handful of unique atoms
could be calculated. Today, at the beginning of the new century, that number has
risen to around 100, encompassing such exciting systems as biological molecules,
fullerene cages, new superconductors or zeolites.
A great many problems exist beyond the reach of full ab initio DFT. This includes, for
instance, structural simulations of amorphous materials, or surfaces and interfaces
of crystals and their interaction with ad-species–which forms a major part of this
work. Also, the study of smaller systems on long timescales necessitates approx-
imations. These methods retain part of the quantum mechanical description, and
they are therefore referred to as semi-empirical methods.
Even larger systems and timescales call for the use of empirical potentials [38]. These
are simplified mathematical expressions fitted to represent the complex quantum
mechanical interactions between atoms. Of course, the price to pay for the use of
approximative methods is their limited transferability to systems beyond their fit-
ting database. Nonetheless, appropriate questions may be answered for problems
like the structure of enzymes in biology, statistical growth simulations in physical
chemistry, and multiple-phase systems or the study of crack propagation for mate-
rials science.
2.1 Historical sketch
The method employed in this thesis is called density-functional based tight-binding
(DFTB) and belongs to the semi-empirical category. It is similar to empirical tight-
binding [39] but replaces, as its main characteristic, the empirical fitting procedure
for matrix elements by a well-defined integration process.
15
16 Chapter 2. The DFTB Method
Before the method is presented mathematically, a brief historical outline is called for,
given that its origins date back more than 15 years by now. The definitive review
of the method, its background, and applications can be found in Ref. [40], which is
much more accessible today than are the original sources.
DFTB emerged from a fully self-consistent linear combination of atomic orbitals
(LCAO) model by Eschrig et al. [41,42]. This approach is characterised as follows:
•Expansion of (a) basis functions, (b) molecular density, and (c) molecular po-
tential into Slater-Type functions (STF).
•Split of the molecular problem into formally independent atomic Kohn-Sham
problems and their self-consistent solution.
•Calculation of molecular orbitals within a valence-basis which is implicitly or-
thogonalised to all core functions. Formally, this may be represented by a
pseudo-potential approach. The valence basis itself is non-orthogonal.
Seifert et al. [43] retained the essential points but suggested several approximations
to simplify and greatly speed up the calculation as follows:
•non-self-consistent treatment of the molecular problem.
•Superposition of the (fixed) potential of neutral atoms as molecular potential.
•Neglection of certain integrals for the calculation of Hamiltonian matrix ele-
ments.
In the early 1990s, the method was applied with great success to relatively large
atomic systems, notably, amorphous carbon [44]. To describe the total and cohesive
energy of such systems and to facilitate molecular dynamics, the following modifi-
cation was necessary:
•Introduction of an atomic short-range repulsive potential to include (screened)
core-core interactions.
In the mid-1990s, Porezag et al. [45,46] put the basis set generation under intense
scrutiny and refined the procedure as follows:
•Improvement of the initial molecular density guess by employing a confine-
ment potential in the self-consistent pseudo-atom calculation. A similar aux-
iliary potential had been proposed early on by Eschrig, but it was applied
to wave functions only, and was more geared toward solids rather than
molecules.
Finally, Porezag and Elstner worked out a self-consistent charge (SCC) extension to
the DFTB method as derived from of a second-order expansion of the Kohn-Sham
energy w.r.t. atomic charge fluctuations. This extension is characterised as follows:
•Calculation of atomic Mulliken charges
2.1. Historical sketch 17
•ab initio derivation of Hubbard-like parameters.
•Superposition of atomiclike densities instead of potentials for the calculation
of exchange-correlation contributions to the two-centre Hamiltonian matrix
elements.
The work on the theoretical foundations has not stood isolated, but was largely
prompted by demands and shortcomings that became evident in the course of im-
plementation and application of the method to real systems. Work in this class in-
cludes:
•Calculation of vibrational properties by Th. K¨
ohler et al. [47].
•The calculation of infrared- and Raman-intensities by Porezag et al. [46,48].
•The calculation of spatial charge densities to provide links to scanning tun-
nelling microscopy (STM) results on crystalline surfaces, by the author [49,50].
•Linear-scaling formulation of the secular problem, by Stephan et al. [51,52,53]
and the author [54]. The latter implementation is reviewed in Chapter 3within
the present work.
The work on the methodological improvement continues. The current and near-
future developments involve the time-dependent description of excited states by
Niehaus et al. [55,56], and a spin-dependent formulation by Ch. K¨
ohler et al. [57].
The introduction of new methodological aspects is but one part of the development
of materials modelling. Another part concerns the actual implementation of the
new methods in computer code and its documentation for users and developers.
Original work in this respect goes back to H. Eschrig and G. Seifert. P. Blaudeck
improved upon earlier programs and assembled a program version called dylcao
around 1990, supplemented by analysis tools and viewers.
This program has been extended 1995–1997 by Haugk and Elstner to implement the
self-consistent charge extensions. Significant changes to the file formats were intro-
duced at this stage. This program version was called dftb and has since evolved into
several branches with capabilities like the treatment of van der Waals interactions
and time-dependent effects in DFT.
Since about 1992, G. Jungnickel maintained a separate development line pursuing
a more monolithic design strategy, out of which arose the menu-driven programs
dylax for the DFTB modelling and statix for structure file modifications, analysis
and viewing. The main contribution was the capability for a full k-point sampling
of the Brillouin zone rather than a Γ-point only sampling.
This concludes the historical perspective. DFTB is now capable of modelling with
sufficient accuracy the structure and properties of e.g. the bulk, surfaces and de-
fects of crystalline semiconductors, as well as organic and inorganic molecules of
medium complexity with absolute accuracies in the 1 eV range and relative accura-
cies down to 30 meV.
18 Chapter 2. The DFTB Method
2.2 Variational approach and stationary principle
The previous section recounted a historical aspect of how DFTB was developed from
a simplified description of the electronic structure of molecules to a full-fledged ma-
terials modelling method. The present section will now attempt to give a review of
the mathematical derivation of the underlying formalism up to the self-consistent
charge level, as seen from a modern perspective.
In DFT, the total energy of an atomic system is the energy of its electron distribution
plus the ion-ion repulsion:
Etot =EKS[n(r)] + Enuc({Ra}), (2.1)
The latter is trivial to obtain:
Enuc({Ra}) = 1
2
M
X
a,b6=a
ZaZb
|Ra−Rb|. (2.2)
The first term is the Kohn-Sham energy of an ideally self-consistent density dis-
tribution n(r)as given by (1.29) in section 1.4. This density is obtained from an
iterative solution of the Kohn-Sham equation (1.18). During the iteration, the po-
tentials within the equation are calculated from a non-converged input density. The
solutions of the Kohn-Sham equation are the orbitals Φi(r)which, together with an
appropriate set of occupation numbers ni, yield a new output density. As a last step
for an iteration, the Kohn-Sham energy is evaluated. Usually, this involves both
the input and output densities. For instance, the potentials of the double counting
terms are functionals of the input density, but they are multiplied with the output
density, as in the double-counting term RVxc([nin],r)nout(r)dr. It is quite possible
to circumvent this heterogeneity. Brought to its conclusion, this eventually helps
to formulate a non-self-consistent energy functional, cutting short the iteration cy-
cle entirely. Harris [58] was one of the first authors to do this. He introduced a
functional E[nin]which solely uses the input density to evaluate the total-energy ex-
pression (1.29). This functional is not strictly variational, i.e., yielding energies above
the ground state for non-converged densities, but at the ground state it is (a) equi-
valent to the Kohn-Sham functional and (b) stationary, which means deviations of
the input density from the ground state density lead to energy deviations of second
and higher order only. However, the sign of the quadratic term cannot be deduced
a priori, as Harris points out. Later, Foulkes and Haydock [59, Sec. III] analysed the
second-order term in more detail.
Foulkes and Haydock [59], wrote the Kohn-Sham functional (1.13) as:
EKS[n(r)] = T0[n(r)] + V[n(r)] (2.3)
with the potentials compounded in1
V[n(r)] = ZVext(r)n(r)dr+1
2ZVH([n],r)n(r)dr+Exc[n(r)] . (2.4)
1In Ref. [59], this term is denoted F[n], in regrettable overlap with the universal functional F[n]in
the Hohenberg-Kohn formulation, eqs. (1.8) and (1.11). The latter encompasses
b
T, but not Vext.
2.2. Variational approach and stationary principle 19
The effective potential in the KS equations is just the functional derivative of V[n],
Veff([n],r) = δV[n]
δn=Vext(r) + VH([n],r) + Vxc([n],r). (2.5)
With the kinetic energy operator denoted b
T=−1
2∆for short, the non-self-consistent
Kohn-Sham equations (1.19) for a given input density nin(r)read
b
T+Veff([nin],r)Φi(r) = εiΦi(r);∀i. (2.6)
As solution, one obtains the one-electron eigenvalues εiand orbitals Φi, from which
the output density is easily obtained using eq. (1.15) (with appropriate occupation
numbers ni):
nout(r) =
occ
X
i
niZ|Φi(r)|2dr. (2.7)
In the non-scf Harris functional, the output density itself does not enter. Employing
just the eigenvalues of Eq. (2.6) the functional is defined as follows:
E[nin] =
occ
X
i
niεi−EH[nin] + Exc[nin]−ZVxc([nin],r)nin dr. (2.8)
The form is similar to the Kohn-Sham total energy (1.29) functional for the ground
state. Naturally, the question arises: What is the deviation of E[nin]from EKS[nscf]?
Foulkes and Haydock [59] concluded from a Taylor-expansion of V[n]that this devi-
ation is of second order in the charge differences, ∆n(r) = nout(r)−nin(r), namely,2
EKS[nout] = E[nin] + E(2)[nin,∆n]
=E[nin] + 1
2ZZ δ2V[n]
δn2nin
∆n(r)∆n(r0)drdr0. (2.9)
For the kernel in the second-order term E(2)[nin,∆n]one derives from eq. (2.5):
δ2V[n]
δn2=δVeff([n(r0)],r)
δn(r0)=1
|r−r0|+δVxc([n(r0)],r)
δn(r0)(2.10)
The functional E[nin]of eq. (2.8) forms the basis for the DFTB method. The method
is implemented on two levels, called the “non-self-consistent” and “self-consistent
charge” (SCC) versions. Both approaches start from a guess of the input density,
nin(r)≡n0(r). In the non-scc approach, the Kohn-Sham equation is solved by a
single iteration to obtain the eigenvalues and an improved density nout(r), which
adapts the input density to the given arrangement of atoms. The SCC extension
retains the second-order corrections E(2)[n0,∆n]by iteratively improving the charge
fluctuations ∆n, with ∆n(r)≡nout(r)−n0(r). Obviously, a good guess of the initial
density is essential for the success of either approach. Both methods are efficient
only due to a sequence of approximations, which is outlined in the next section.
2Note that this is not a Taylor-expansion of the Kohn-Sham energy itself because the functionals on
the left and right side of the equation are different.
20 Chapter 2. The DFTB Method
2.3 Approximations in DFTB
The DFTB method applies the stationary principle introduced above while making
a number of approximations. These mostly take the form of separations into atomic
contributions of global quantities like potentials, densities, and wave functions. This
separation principle allows the effective treatment of compound atomic systems by
building as much as possible upon preparatory work, which is performed before-
hand on isolated subsystems of atoms and atom pairs. The preparatory calculations
capsule the computationally demanding tasks of establishing the Hamiltonian ma-
trix elements and double-counting terms into convenient functions and tables. The
method itself has recently been reviewed in detail in Ref. [40]. The background, jus-
tification and tests of major developments were discussed at length by the respective
authors [43,45,46,60,61].
Traditionally, the method is being presented in two separate stages, i.e., in its non-
self-consistent formulation and the SCC extension. The present treatment attempts
to give a unified view instead.
The main approximations to obtain the DFTB method from the stationary principle
of DFT are in turn:
•superposition of pseudo-atomic densities as starting density
•minimal-basis, valence-only LCAO wave functions
•two-centre Hamiltonian (neglect of crystal-field and three-centre terms)
•repulsive pair potential for the double-counting and inter-nuclear energies
•for the second-order corrections:
–monopole approximation and extrapolation of δVxc[n]/δn
–Mulliken charges
The total-energy expression of the DFTB approach is gathered from Eqs. (2.8)
and (2.9) and written as follows:
Etot[n0+∆n] =
occ
X
i
nihΦi|b
H0|Φii+Erep[n0] + E(2)[n0,∆n]
=EBS +Erep +EG, (2.11)
where
b
H0=b
T+Veff([n0],r). (2.12)
The first energy term, the so-called band-structure energy EBS, is the trace of a ref-
erence Hamiltonian b
H0over the one-electron eigenstates Φiof the system. The
second term in Eq. (2.11), Erep, constitutes a repulsion energy similar to standard
tight-binding theory, which subsumes the double counting terms of the reference
Hamiltonian, as well as the nuclear repulsion. Finally, the last term, EG, describes
atomic charge fluctuations and is subject to a self-consistency treatment.
2.3. Approximations in DFTB 21
2.3.1 Pseudo-atomic starting density
In DFTB, the starting density is chosen as superposition of slightly compressed den-
sities of neutral atoms,
n0(r) = X
a
na
0(ra);ra=r−Ra. (2.13)
The densities of free atoms are too diffuse to be a good initial guess in compound
systems. Compressed densities anticipate the density modification of free atoms
when surrounded by other atoms in a solid or molecule. Furthermore, the limited
range of the atomic density works better with a number of integral approximations
discussed later. The densities are the result of a self-consistent LDA or GGA cal-
culation3of pseudo-atoms, i.e., atoms placed within a weak parabolic constriction
potential as expressed in a modified Kohn-Sham equation:
b
T+Vat
eff[na
0] + r
r0mϕpsat
ν(r) = εpsat
νϕpsat
ν(r). (2.14)
The constriction potential is characterised by its exponent mand range r0. The expo-
nent was shown to have rather small influence on the final results [46], so that m=2
is usually used. For the range parameter, a number of calculations has lead to results
with optimal transferability in covalent systems (barring 3dtransition metals [62])
using r0≈1.85 rcov, where rcov is the covalent radius of the given element.
Owing to the slightly empirical nature of using confined orbitals, there is some de-
gree of arbitrariness as to which density and potential to choose for the calculation
of the atomic orbitals and ultimately, the Hamiltonian matrix elements. Originally,
in the approximate scheme by Seifert [43], the scf-potential of the free atom was
used. Later, Porezag [46] modified the scheme to use the scf-potential of the confined
atom, making the pseudo-atom calculation self-consistent. Finally, Elstner [60] sug-
gested an intermediate approach using a weakly confined density resulting from an
scf-potential with a density specific compression radius r%
0. This radius introduces,
besides r0, another parameter into the model, which is debatable on formal grounds.
It does, however, provide an opportunity to improve upon covalent energies and
geometries.
Table 2.1 on the next page summarises the calculation steps for the three schemes.
All schemes obtain atomic orbital energies from an scf-calculation of the free atom,
in the absence of any additional potential. Overall, the first and second scheme each
require two stages of the pseudo-atom calculation, while the third one relegates the
generation of wave functions to a third stage. Thus, the atomic orbitals in the latter
scheme can be considered as one iteration step ahead of the atomic density, which
is justifiable within the Foulkes-Haydock picture discussed in section 2.2.
In all schemes, the pseudo-atomic wave functions themselves are represented by
Slater-type orbitals (STO) characterised by coefficients aij and exponents αj:
ϕν(r)≡ϕnlm(r) =
5
X
i=1
3
X
j=0
aijrl+je−αirYlm r
r. (2.15)
3cf. Appendix B.
22 Chapter 2. The DFTB Method
Table 2.1: Summary of schemes for the DFTB pseudo-atom calculation of orbital energies
εν, effective (pseudo-)atomic potentials Veff, atomic densities na
0, and basis functions ϕν.
Seifert [43] Porezag [46] Elstner [60]
step r0/rcov result r0/rcov result r0/rcov result
(m=4) (m=2) (m=2)
1. ∞Vat
eff,na
0,εν∞εν∞εν
2. ≈1.85 ϕν≈1.85 Vpsat
eff ,na
0,ϕν5.. . 14 Vpsat
eff ,na
0
3. ≈1.85 ϕν
The optimisation of the basis sets for the first- and second-row elements was dis-
cussed in depth in Ref. [46]. In the cited work the STO representation is used only
externally. Internal to the pseudo-atom calculation, the STO-basis is in fact projected
onto a set of contracted Gaussians (i.e., linear combinations of primitive Gaussians).
2.3.2 Tight-Binding integrals and the two-centre approximation
The pseudo-atomic basis set finds its application in the LCAO expansion of the
eigenfunctions:
Φi(r) = X
aX
ν[a]
Ciνϕν(ra). (2.16)
The reference Hamiltonian expressed in this basis gives matrix elements denoted
h0
µν and non-orthogonal overlap elements sµν:
h0
µν =hϕµ|b
H0|ϕνi=Zϕ∗
µ(r)b
H0ϕν(r)dr
sµν =hϕµ|ϕνi=Zϕ∗
µ(r)ϕν(r)dr.
(2.17)
These integrals are calculated immediately following the pseudo-atom calculations
and are tabulated as function of distance between the two centres. The integrals at
general difference vectors as needed in the system, ∆Rab =Ra−Rb, are transformed
using well-known projection relations [39,63]. Due to symmetry, only 10 integrals
between basis functions remain nonzero for angular momenta up to l=2. Their
sequence as tabulated is, in standard molecular orbital notation:
ddσ, ddπ, ddδ, pdσ, pdπ, ppσ, ppπ, sdσ, spσ, ssσ.
To achieve the above two-centre representation for the Hamiltonian matrix elements,
the effective Kohn-Sham potential is formally decomposed into atomiclike contribu-
tions. Because the exchange-correlation potential is non-linear, there are two ways
to do this in practice, either to sum atomic potentials or atomic charges:
Veff([n0],r)≈PcV0
c([n0
c(rc],rc)“potential superposition”
Veff Pcn0
c(rc)“density superposition” (2.18)
2.3. Approximations in DFTB 23
Table 2.2: Integral types in the DFTB Hamiltonian h0
µν according to eq. (2.17). Centres aand
bdenote orbital centres, with basis functions µ∈a,ν∈b, and cis the potential centre.
Type Classification Centres Status
(A) onsite-terms a=b=cretained
(B) crystal-field terms a=b6=cneglected
(C) two-centre terms a6=b,c=aor c=bretained
(D) three-centre terms c6=a6=b6=cneglected
The potential superposition was applied in the original schemes by Seifert and
Porezag, while the density superposition was later introduced by Elstner.
Depending on the centres involved for the basis functions and the potential, the
Hamiltonian matrix elements fall into a number of categories, which are sum-
marised in Table 2.2. As indicated, a number of these integrals are neglected in
either approach. Besides, a valence-only basis is used, which implies a core-valence
orthogonalisation between different centres. The neglect of three-centre terms pro-
vides the largest formal and practical simplification because on the one hand the
handling for three centres is more involved than for two centres and on the other
hand there are many more combinations. Conversely, the crystal-field integrals are
relatively simple and appear to provide leverage for improvement. However, the
integral neglections work only in concert because there is error-cancellation of a
considerable degree. The justification of this process as a whole is complex and was
discussed originally by Seifert et al. [43] and later reviewed in Ref. [40].
Among the retained integrals, type (A) represents onsite energiesενof single atoms,
h0
νν =εν, (2.19)
which are obtained in the first step of the atom calculation discussed in the preceed-
ing section. In the other remaining integral type (C), the potentials and densities of
two distinct atoms are to be combined:
h0
µν =hϕµ|b
T+V0
a(ra) + V0
b(rb)
Veff([n0
a+n0
b],r)|ϕνi;a[µ],b[ν]. (2.20)
The density superposition mode is coupled to a weaker density compression for
pseudo atoms. A strong density compression, as implicit in the previous pseudo-
atom scheme, would unsuitably limit the range of the effective potential in this case.
The use of weakly compressed densities in concert with the density superposition
for the Hamiltonian matrix elements has lead to improved energies, vibrational
frequencies and reaction barriers mainly for organic molecules [60]. There is evi-
dence, however, that the band structures of crystalline systems are less well repre-
sented [62,64] than in the potential superposition previously employed.
2.3.3 Repulsive potential
In the above description, the double-counting terms of the Kohn-Sham energy, eval-
uated at the input density, and the inter-nuclear repulsion are approximated as a
24 Chapter 2. The DFTB Method
sum of short-ranged repulsive pair potentials:
Erep(n0,{Ra})≈1
2X
aX
b6=a
Vab
rep (|Ra−Rb|)(2.21)
This approximation is justified by the following observations:
•With n0represented by neutral atomic fragments, there are no long-range
Coulomb interactions in the combined electrostatic Hartree and nuclear con-
tributions to the double counting energy, EH[Pana
0] + Enuc({Ra}), due to mu-
tual screening. Furthermore, since the atomic starting densities are spherically
symmetric, the Hartree integrals for atom pairs (1.12) depend on internuclear
distance only. The same is trivially the case for the nuclear repulsion. There-
fore, these contributions together are representable by short-range pair poten-
tials without loss of accuracy.
•Contributions due to exchange and correlation are not separable into pair po-
tential form per se because of the non-linearity of the xc-functional. However, a
cluster expansion [40,59] allows to extract two-body components. Its higher-
order terms involve the overlap of the densities of three centres, which is negli-
gible for the compressed starting densities used here. The remaining two-body
terms may again be represented by pair-potentials.
All monomer contributions are contained within the atomic orbital energies εν. This
ensures that the repulsive potential actually goes to zero in the dissociation limit.
The repulsive potential is obtained from self-contained ab initio calculations of the
energy of a set of reference molecules for a range of a typical bond length. Most
conveniently, the reference molecules are dimers, but also methanelike structures.
For solid state calculations, as in diamond, which is the subject of this study, crystal
reference structures may be used. For each reference structure j, the energy differ-
ence
e
Ej
rep(r) = Ej
scf(r)−Ej
BS(r)(2.22)
is calculated as function of distance r.
As a consequence of the concepts and approximations taken thus far, the set of re-
pulsive energies has the following important properties:
•e
Erep >0. For small distances, i.e., close atoms, Erep has a steep repulsive slope
indicative of strong Pauli repulsion of the electron shells.
•It decays rapidly to zero between typical first- and second-neighbour dis-
tances. This is another indication that the pair-potential representation of the
double-counting terms embodied within Erep, is valid.
•Most importantly, e
Ej
rep(r)for different molecules are close to each other. This
property is indicative of the degree of transferability of the method, i.e., its ap-
plicability to a wide range of atomic structures, extending beyond the set of
2.3. Approximations in DFTB 25
−10
−8
−6
−4
−2
0
2
4
6
8
10
1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
Energy (eV/bond)
R (Å)
CH≡CH CH2=CH2CH3−CH3
Escf
Σ εi
~
Erep
Figure 2.1: Example for generating the repulsive potential by eq. (2.22). The C– C bond
for three different coordination numbers is scaled while the C–H bond lengths and -angles
were kept fixed at their equilibrium values. SCF-data were produced using the Gaussian
program. Data courtesy of M. Elstner [60]
reference molecules. It was found that e
Erep(r)for high-coordinated crystal
phases such as fcc and bcc often does not coincide with the curve of other
structures. Such phases cannot be treated with the present method.
For practical calculations, the set of repulsive energies are represented numerically
by a polynomial or spline for each atom type combination. The polynomial has the
form
Vrep(r) = X
n
an(Rc−r)n,r<Rc. (2.23)
The cutoff radius Rcrepresents the distance beyond which the repulsive potential
has subsided. Rcis typically chosen between 1.5—2 equilibrium bond lengths, and
the polynomial coefficients anare fitted to e
Ej
rep(r). As an alternative, a spline repre-
sentation allows greater flexibility in this respect, at the price of a somewhat more
complicated handling.
2.3.4 Second-order corrections
It remains to discuss the last term in the DFTB total-energy expression, the second-
order correction E(2)[n0,∆n], or EG(where “G” stands for γ, which is properly in-
troduced below). This term becomes important in the simulation of heteroatomic
molecules and polar semiconductors where chemical bonding is influenced consid-
erably by charge transfer effects and long-range Coulomb interactions.
26 Chapter 2. The DFTB Method
In line with previous procedures, the charge fluctuations ∆nare decomposed into
atomic contributions which are expected to decay rapidly with increasing distance
from their centre. The second-order term then reads, using (2.9) and (2.10):
E(2)[n,∆n] = 1
2X
a,bZZ 1
|r−r0|+δVxc([n(r0)],r)
δn(r0)n0
∆na(r)∆nb(r0)drdr0. (2.24)
The term is expected to contribute only a small part of the total energy, so that elabo-
rate integrations can be forgone, in favour of quite crude approximations. First, one
expresses ∆naas a mere monopole contribution:
∆na(r) = X
lm
ca
lmFa
lm(ra)Ylm ra
ra≈∆qaFa
00(ra)Y00 ;ra=r−Ra. (2.25)
Thus, eq. (2.24) takes a rather simple matrix form, which shall be denoted EG:
EG=1
2
M
X
a,b
γab∆qa∆qb, with (2.26)
γab =ZZ 1
|r−r0|+δVxc[n]
δn(r0)n0
Fa
00(ra)Fb
00(r0
b)1
4πdrdr0. (2.27)
The latter, so-called γmatrix, encapsulates the dependency of E(2)on ionic posi-
tions. This matrix, except for the Vxc part, is well-known from the CNDO-formalism
by Pople, Santry and Segal [65]. Density data enter in the form of known pseudo-
atomic starting densities and orbital relaxation functions F00(r). If the latter are as-
sumed to be fixed radial functions to be weighted by ∆qa, the only geometry param-
eter in γab will be the interatomic distance R=|Ra−Rb|.
The limit R→0 represents coinciding atomic centres aand b. In this case, γaa equals
a Hubbard-like parameter Uafor the atom. In chemical terminology, the Hubbard
parameter is related to the chemical hardness ηa≈Ua/2, which is a measure of the
ionisation potential and electron affinity of the atom. Neglecting the influence of the
environment and employing Janak’s theorem [28], Uais obtained non-empirically at
the DFT level during the pseudo-atom calculation as the derivative of the HOMO of
the free atom with respect to its occupation number:
γaa =Ua=∂2Eat
∂q2
at q=q0
=∂εa
HOMO
∂nHOMO
. (2.28)
For convenience, Table 2.3 gives the values of γaa for frequently used elements.
In the limit of large interatomic distances, γab reduces to a 1/Rdependency, since
xc interactions vanish in this case. To obtain a continuous transition between the
limits of small and large interatomic distances, various interpolation formulae were
used [66]. E.g., the one suggested by Ohno/Klopman [67,68] reads:
γab(Ua,Ub,R) = 1,sR2+1
41
Ua
+1
Ub2
. (2.29)
2.3. Approximations in DFTB 27
H Li B C N O F Si
11.06 4.69 8.05 9.91 11.71 13.46 15.18 6.74
Table 2.3: Chemical hardness Uin V/eas defined by eq. (2.28) for free spin-unpolarised
atoms as calculated within LSDA. Source: Porezag [46].
0
0.2
0.4
0.6
0.8
1
1.2
1.4
012345
γ (Hartree)
R (aB)
1/R
Slater
Klopman
1e−10
1e−09
1e−08
1e−07
1e−06
1e−05
0.0001
0.001
0.01
0.1
1
12345678
dγ (Hartree)
R (aB)
1/R − Slater
1/R − Klopman
Figure 2.2: Comparison of the Klopman and Slater interpolation functions for the scc γ
interaction, eqs. (2.29) and (2.31). The Hubbard parameter was Ua=Ub=1 H. The in-
set shows the deviation of the interpolation formulae from the Coulomb behaviour on a
logarithmic scale.
While illustrative and obvious in its behaviour for R→0 and R→∞, eq. (2.29)
proved numerically unsuitable for periodic systems, since the asymptotic Coulomb-
behaviour is reached only for distances much larger than common supercell sizes,
see Fig. 2.2. A more stable behaviour is achieved by an integration of eq. (2.27) for
known charge fluctuations, neglecting the Vxc term for the moment. Assuming a
normalised Slater-type function with a range parameter τa,
na(r) = τ3
a
8πe−τa|r−Ra|, (2.30)
the Coulomb integral (1.12) of two such charge densities reads, after quite lengthy
manipulations (for integrals of this type, see Pople and Beveridge [69, App. B]):
γab(τa,τb,R) = 1
R−S(τa,τb,R), with (2.31)
S(τa,τb,R) = e−τaR τ4
bτa
2(τ2
a−τ2
b)2−τ6
b−3τ4
bτ2
a
R(τ2
a−τ2
b)3!+
+e−τbR τ4
aτb
2(τ2
b−τ2
a)2−τ6
a−3τ4
aτ2
b
R(τ2
b−τ2
a)3!;τa6=τb(2.32)
28 Chapter 2. The DFTB Method
S(τ,R) = e−τR48 +33τR+9(τR)2+ (τR)3
48R;τa=τb≡τ. (2.33)
As before, in the limit of small distances γab should equal the Hubbard parameter
for an isolated atom. This being a known quantity, one obtains, noting eq. (2.28), a
one-to-one relation between the hitherto unspecified charge fluctuation range τaof
an atom and its Hubbard parameter:
γaa =Ua=lim
R→01
R−S(τa,R)=1
R−1
R−5
16τa=5
16τa(2.34)
Since (2.31) was derived from pure Coulomb interactions, yet the Hubbard param-
eter incorporates xc contributions, this step may seem inconsistent. To reconcile
the approaches, the following should be considered: Appreciable deviations from
the point charge behaviour of 1/Roccur only close to atoms, namely, within radii
of typical bond lengths, as Fig. 2.2 attests. Adjusting the inner limit to include xc
interactions by way of equating it to the ab initio DFT value, one corrects mainly
the onsite terms of γab. The remaining influence on first-neighbour interactions is,
although not quite negligible, still acceptable within the monopole approximation
and the somewhat arbitrary choice of the interpolation function.
For periodic systems, the Ewald technique [36,70,71] is used to take care of the
long-range Madelung-like contributions to the γmatrix. The short-range deviation
terms from the 1/Rbehaviour, i.e., S(τa,τb,R), are calculated by an explicit sum
over all neighbours within a reasonable distance of a given atom.
To finally evaluate the second-order energy contributions, one needs the atomic
charge deviations ∆qa. These are obtained from Mulliken charges, which are not
undisputed (see, e.g., Ref. [66]) but widely popular because they are easy to ob-
tain from the eigenstates without the need for elaborate spatial partitioning schemes
which have their own share of problems. Given real-valued eigenvectors Ciµin an
atomic basis with overlap matrix s, the Mulliken charges qaand charge deviations
∆qaon atoms are, using eq. (1.32) on page 13:
qa=
occ
X
i
niqi
a=
occ
X
i
niX
µ[a]X
ν
CiµCiνsµν =X
µ[a]X
ν
qµν
∆qa=qa−q0
a, (2.35)
where q0
aare the charges of the respective neutral atoms.
2.4 The DFTB secular equation
With the approximations discussed above the DFTB total-energy expression derived
from eq. (2.11) reads:
Etot =
occ
X
i
niX
µX
ν
CiµCiνh0
µν +1
2
M
X
ab
γab∆qa∆qb+Erep({Ra})(2.36)
2.4. The DFTB secular equation 29
Given a set of atomic coordinates Raand the resulting matrices h,s, and γ, the
LCAO coefficients which minimise the DFTB total energy (2.36) are found by the
variation principle subject to orbital normalisation, eq. (1.14):
∂
∂Ciµ"Etot +
occ
X
i
ni˜εi 1−X
µX
ν
CiµCiνsµν!#=0 . (2.37)
The procedure is quite similar to the standard tight binding case and was given in
detail in [46]. The resulting secular equation reads:
N
X
ν=1
Ciν(hµν −˜εisµν)=0 ; ∀i,µ, (2.38)
where, as a consequence of the minimisation of EGalongside EBS the original Hamil-
tonian matrix elements are augmented as follows:
hµν =h0
µν +1
2sµν
M
X
c
(γac +γbc)(qc−q0
c);a[µ],b[ν]. (2.39)
As for the generic Kohn-Sham system (1.25), eq. (2.38) is a generalised eigenvalue
problem which is solved using standard libraries. Since the charges in the aug-
mented Hamiltonian depend upon the coefficients of the solution, the equation has
to be solved self-consistently. The iteration is driven by Broyden mixing [72] for the
charges {qa}. Typically, between 5–20 iterations are required. Upon convergence,
one obtains the eigenvectors Ciµ, the one-electron energy levels ˜εiand derived quan-
tities, like the density of states, or charge distributions.
The total energy can be conveniently expressed in terms of the eigenvalues ˜εito
avoid an explicit calculation of the matrix products in eq. (2.36). The joint minimisa-
tion of EBS and EGleads to a double-counting-like expression for EG. A discussion
and derivation of this term has been omitted from the relevant publications on DFTB
so far [40,46,60,61]. For completeness, it is given here:
occ
X
i
ni˜εi=
occ
X
i
nihΦi|b
H|Φii=
occ
X
i
niX
µX
ν
CiµCiνh0
µν +
+
occ
X
i
niX
µX
ν
CiµCiν
sµν
2X
c
(γac +γbc)∆qc;a[µ],b[ν]
occ
X
i
ni˜εi=EBS +1
2X
c
∆qc X
a
qaγac +X
b
qbγbc!(2.40)
=EBS +1
2X
c
∆qcX
b
2qbγbc . (2.41)
Use has been made of the definition of Mulliken charges (2.35) to arrive at eq. (2.40).
Since the summations over aand bare independent in eq. (2.40), the last sums are
30 Chapter 2. The DFTB Method
in fact equal. Observing the triviality 2qb= (qb−q0
b)+(qb+q0
b)one obtains from
eq. (2.41):
occ
X
i
ni˜εi−1
2X
bc
γbc∆qc(qb+q0
b) = EBS +EG(2.42)
Renaming c→a, the total energy is given finally by the following decidedly asym-
metric expression:
Etot =
occ
X
i
ni˜εi−1
2
M
X
ab
γab(qa−q0
a)(qb+q0
b) + Erep({Ra}). (2.43)
Forces on atoms can be obtained analytically [46] without resorting to repeated en-
ergy calculations for finite displacements. They are derived directly from the total
energy expression subject to the normalisation condition for orbitals:
Fk=−∂
∂Rk"Etot +
occ
X
i
ni˜εi 1−X
µX
ν
CiµCiνsµν!# (2.44)
At the variational minimum (2.37) of the electronic degrees of freedom all contri-
butions due to the eigenvectors Ciµvanish. Only the matrices h,s, and γdepend
explicitly on the atomic positions. Using (2.36) one obtains:
Fk=−
occ
X
i
niX
µX
ν
CiµCiν"∂h0
µν
∂Rk− ˜εi−1
2X
c
(γac +γbc)(qc−q0
c)!∂sµν
∂Rk#
−(qk−q0
k)X
c
∂γkc
∂Rk
(qc−q0
c)−X
c6=k
∂Vrep(|Rk−Rc|)
∂Rk
;a[µ],b[ν].
This concludes the review of the DFTB formalism. The essential result of the DFTB
calculation is available in form of the total energy and the electronic eigenvectors.
Other physical observables like charge distributions or vibrational properties may
be extracted, the latter not without considerable numerical effort.
Chapter 3
Order-N Method and
Implementation
3.1 Introduction
Customary approaches to the solution of the Schr¨
odinger equation or the Kohn-
Sham equations such as those outlined in the preceding chapters require a workload
proportional to the third power of the number of atoms involved in the simulation
— Doubling the size of the system amounts to multiplying by eight the comput-
ing time [73,74]. This applies to both first-principle and tight-binding Hamiltoni-
ans. The reason is mathematical in nature because at the heart of either approach
lies an eigenvalue problem where typically about half of all eigenvectors must be
found. Speaking somewhat simplistically, ensuring the mutual orthogonality of the
eigenvectors spatially interweaves all parts of the solution and so requires multiple
passes.
Current implementations of first principles density-functional calculations using the
local density approximation (LDA) and standard approaches can handle up to about
1200 electrons using supercomputers and parallelised algorithms. On workstation
hardware, about 400 . . . 500 electrons appear to be a practical limit. First princi-
ples LDA calculations using linear scaling algorithms have appeared in the liter-
ature [75,76], although algorithms as robust, transferable, and efficient as those
used in standard approaches have yet to be implemented. There are also algorithms
which implement self-consistent LDA directly and employ, as in the present work,
non-orthogonal basis sets and non-orthogonal orbitals [77,78]. Their applicability
has been demonstrated for systems of moderate size built of repetitive or character-
istic building blocks.
So far, linear scaling algorithms have mainly been used within a tight-binding
framework and have provided several valuable results in the last decade [73,74,
79,80]. Such methods can help tackle a number of materials science and condensed
matter physics issues requiring a qualitative quantum mechanical description for
fairly large systems, containing, e.g., 5 000 . . . 10 000 electrons. Examples include
the study of extended defects and large scale growth simulations.
All linear scaling approaches which have appeared in the literature avoid diago-
31
32 Chapter 3. Order-N Method and Implementation
nalisation of the Hamiltonian matrix, and compute an approximate value of the
ground state energy and forces without ever calculating the Hamiltonian eigenval-
ues and eigenvectors. The main classification of a method can be made whether the
approach is based on an orbital formulation, or uses a density matrix picture.
The two key points involved in orbital based methods are the following: (i) the full
physical system is divided into subsystems, and orbitals localised in the subsystems
are defined as electronic degrees of freedom [81]; these orbitals are used during an
energy functional minimisation, instead of the Hamiltonian extended eigenstates.
The subsystems are overlapping portions of the full system, called localisation re-
gions or support regions. The extent of a localisation region depends on the physical
and chemical properties of the system but not on the entire volume of the system.
(ii) Electronic orbitals are never explicitly orthonormalised; this is accomplished by
defining an appropriate energy functional whose minimisation requires neither ex-
plicit orthonormalisation of electronic orbitals, nor the inversion of an overlap ma-
trix (S) between single particle wave functions. Such a functional is in general dif-
ferent from the functional minimised by the Hamiltonian eigenstates but it has the
same absolute minimum.
One of the major drawbacks of most TB formulations is the lack of self-consistency
between the charge density and the mean-field potential of the system. This is a
particularly severe problem for polar systems involving charge transfer, such as SiC,
GaAS and GaN and for any organic compound.
In this chapter we describe the implementation of the DFTB approach in the charge
self-consistency (SCC) version. The approach is characterised by both the non-
orthogonal basis set inherent to DFTB and non-orthogonal localised orbitals, within
the linear scaling algorithm.
The linear-scaling formulation is based on the formalism outlined by Kim et al. [82].
The present work extends this formalism in three respects:
•to make use of a non-orthogonal basis,
•to treat multiple species, and
•to include charge-self-consistency effects.
The implementation of the algorithm was carried out on a serial computer. Paral-
lelisation with load balancing might be performed later along the principles laid out
by Canning et al. [83].
3.2 Energy functional and charges
3.2.1 Energy functional
The DFTB energy expression for a system containing Nel electrons and Nat ions is
(cf. 2.11):
Etot =EBS +EG+Erep. (3.1)
3.2. Energy functional and charges 33
The O(N)functional describing the band structure energy EBS in a tight-binding
formulation has been discussed in detail by Kim et al. [82] and Mauri and Galli [84].
It expresses the energy in terms of the TB Hamiltonian b
H, a set of Moverlapping
and non-normalised orbitals |Φii(instead of molecular or extended eigenstates) and
a Lagrangian multiplier ηensuring the correct filling of the localised orbitals:
EBS({Φi},η,M) = 2
M
X
ij=1hΦj|b
H−η|ΦiiQij+ηNel
=2Tr(H0Q) + ηNel. (3.2)
The factor 2 accounts for spin degeneracy. In general, the number Mof electronic
orbitals is larger than the number of occupied states Nel/2, as discussed, e.g., in
Ref. [82].
Both H0and Qare (M×M)matrices where H0is the Hamilton matrix between the
localised support orbitals (or generalised Wannier functions), augmented by the η
Lagrange parameter; Qis a truncated Taylor expansion around the identity matrix
Iof the inverse of the overlap matrix Si j =hΦi|Φji.
Q=2I−S. (3.3)
To include charge self consistency effects within the tight-binding framework we
add to the band structure contribution the term EGfrom 2.26. The calculation of
the EGterm is known as the electronic quantum Coulomb problem. Classical al-
gorithms for its solution involve the calculation of the Madelung matrix via the
Ewald technique and exhibit O(N2)scaling. However, in recent years a number
of algorithms exhibiting linear scaling have been proposed in the literature, in par-
ticular generalisations of the fast multipole method (FMM) [85]. These algorithms
could in principle be applied to make the calculation of EGshow the same scaling
with system size as the evaluation of EBS. In this work, as in most other O(N)ap-
proaches [75,86], this problem was put aside. We note that the O(N2)evaluation of
EGbecomes a limiting factor when the time for one calculation of the Madelung ma-
trix is comparable to the time for the total number of iterations required to minimise
the electronic energy functional, at each MD step. In our current implementation
this is the case for about 8000 electrons. The bulk of our calculations consists of
repeated minimisations of the electronic energy.
In the O(N)functional, the regular DFTB basis functions {ϕµ}will be used, and
the TB hopping and overlap terms hµν =hϕµ|b
H|ϕνiand sµν =hϕµ|ϕνiare cal-
culated as usual from the directional cosines of the ionic positions and tabulated
Slater-Koster integrals. However, in order to achieve linear scaling, the range of
these interactions is limited by an imposed cutoff radius RSK. Currently, the LCAO
basis is not optimised for short-ranged interactions. The cutoff radius is chosen typ-
ically such that all second and most third neighbours are taken into account. Under
these conditions, the energy deviations of the O(N)scheme from exact diagonal-
isation results are largely due to the chosen size of localisation regions. In order
to investigate the effect of the basis cutoff, we have compared the total energies as
obtained from exact diagonalisations with full and reduced range. The results are
34 Chapter 3. Order-N Method and Implementation
given in Table 3.1 on page 47. For carbon the cutoff increases the total energy per
atom by about 30 meV, while in the mixed system of SiC the effect is a mere 5 meV
per atom. These energy deviations are all of smaller magnitude than the energy
deviations of the localised solutions discussed further. We notice that the effect of
the basis cutoff is most pronounced in the homogeneous carbon systems. To reduce
this contribution to the total energy deviation of the localised solution, the process
of basis function generation and fitting of the repulsive potential will have to be
repeated.
3.2.2 Localisation of support functions
Within the real-space basis of the LCAO approach (1.23), a localisation constraint
on orbitals can easily be imposed by restricting the expansion coefficients Ciµto
be nonzero only for a set of basis functions {µ}localised within a Localisation Region
(LR). Here, a localisation region is defined by a topological criterion as a set of atoms
including Nhnearest neighbour shells, or hops, around a central atom [83]. The
neighbour shells are identified once per ionic step from a bond map constructed
using species dependent nearest neighbour cutoff radii. In principle this step can
be performed in (Nat log Nat)operations [87]. Furthermore, if atomic relaxations or
molecular dynamics simulations are performed but no significant atomic migrations
take place, the topological information built in the initial ionic step can be upheld
throughout the simulation. In these cases, the calculation of distances is required
only for atoms which are linked within these topologies.
The localisation regions are described by a pair of topology arrays. A two-
dimensional array of fixed size lists for each central atom ithe corresponding atom
numbers of its neighbours up to the Nhth shell. Typically, we choose Nh=2 or
Nh=3. This requires for diamondlike interfacial and bulk models about 20 and 50
entries per atomic localisation region, respectively. The number of atoms actually
present in each LRiis stored in an associated one-dimensional vector.
A given localisation region carries a number ns of localised orbitals. While we typ-
ically choose ns =3, for an LR centred on a Hydrogen atom ns can be reduced
to 1.
The localised orbitals are stored in terms of their LCAO coefficients. Each atom
contributes to a localised orbital with a number nb of basis functions or, equivalently,
LCAO coefficients. Since each LR carries ns orbitals and each atom within the LR
carries nb basis functions, the LCAO coefficients are stored in a matrix of size (ns ×
nb)for each atom. These matrices are mapped in a one-to-one fashion to the atomic
lists of the LR topology description. The dimensions of the atomic matrices may
vary by species.
This storage scheme for the wave function is generally referred to as “sparse” or
“packed” storage. Essentially, all nonzero elements of a sparsely occupied matrix
(here, the LCAO coefficients Ciµ), are identified and shifted to the leftmost column
of the matrix, keeping track of the original column positions in a separate list (here,
the LR array).
3.2. Energy functional and charges 35
3.2.3 Hamilton and overlap matrices
The band structure energy term in eq. (3.2) is the trace of the product of two sparse
matrices: H0and Q=2I−S. Their matrix elements are:
H0
ij =hΦi|b
H−η|Φji=X
µν
CiµCjν(hµν −ηsµν)
Sij =hΦi|Φji=X
µν
CiµCjνsµν;µ∈LRi,ν∈LRj,(3.4)
where Φiare localised orbitals. The TB hopping and overlap elements hµν and sµν
between basis functions need to be calculated once for a given ionic configuration
from the directional cosines and atom pair specific Slater-Koster tables. In order to
keep the range of the hopping and overlap terms limited in a well-defined manner,
we impose a Euclidian cutoff radius RSK around each atom. This defines, for each
atom, a Slater-Koster region, tagged SK. The cutoff radius is species dependent and
typically chosen to be twice the threshold distance used as nearest neighbour crite-
rion. This leads to about 50 neighbours within the Slater-Koster region of a given
atom.
The elements hµν and sµν are stored similar to the wave functions in atomic subma-
trices mapped to the Slater-Koster topology lists. The dimension of each submatrix
is determined here by the number of basis functions on the central atom of a given
Slater-Koster region and the number of basis functions on the subordinate atom.
Given the complications of the sparse storage of both wave functions and Slater-
Koster elements, the evaluation of the elements of the H0and Smatrices naturally
proceeds in two steps. One of the summations over atoms in eqs. (3.4) are carried
out first:
e
CH
jµ=X
ν
Cjν(hµν −ηsµν)
e
CS
jµ=X
ν
Cjνsµν;ν∈LRj. (3.5)
These summations define so-called projected orbitals |e
ΦH
jiand |e
ΦS
ji. Similar defini-
tions of projected or conjugate orbitals have been used in other orbital based O(N)
approaches [51,52,53]. The projection can also be seen as a conversion between co-
and contravariant vectors.
The projected orbitals are stored in the same manner as the original localised orbitals
using atomic submatrices tied to an appropriate topology description. The projected
orbitals differ from the original orbitals by a bigger localisation range. The increase
in localisation range is induced by gathering contributions from atoms outside the
original LR’s through the hopping and overlap integrals. The extended localisation
regions of projected orbitals are named LRSK. Each LRSKjis constructed from its
LRjparent by linking to it the neighbours of the Slater-Koster topologies SKkof each
atom kwithin LRj:
LRSKj=LRj[
k∈LRj
SKk. (3.6)
36 Chapter 3. Order-N Method and Implementation
OV
LR LR
LRSK
h, s
LRSK
jLR
(b)(a)
k
SK
j
j
ji
j
ij
k
k
ij
H , S
Figure 3.1: (a) Spatial relation between Localisation Regions (LR), the Slater-Koster (SK)
topologies of the tight-binding hopping and overlap elements and the resulting extended
localisation regions (LRSK). (b) Contribution to matrix elements Hij and Sij between sup-
port functions from two localisation regions LRiand LRjmediated by atoms kwithin the
extension LRSKjof one of the localisation regions.
Fig. 3.1 (a) illustrates this operation.
By making use of the projected orbitals the calculation for the matrix elements H0
ij
and Sij is reduced to contributions from wave function expansion coefficients Cand
e
Clocated on the same atom (an important aspect for parallelisation):
H0
ij =X
µ
Ciµe
CH
jµ
Sij =X
µ
Ciµe
CS
jµ;µ∈LRi∩LRSKj. (3.7)
Due to the limited range of the orbitals |Φiiand |e
Φji, the matrix elements of H0and
Sinvolve only summations over specific localisation regions. The localisation is
again described in terms of a topology list, which is named OV. The construction of
this topology uses a topological addition operation similar to (3.6) and is illustrated
in Fig. 3.1 (b):
OVj=LRSKj[
k∈LRSKj
LRk. (3.8)
The number of nonzero elements of the H0and Smatrices depends on the topolog-
ical hopping range of the LR’s and the Euclidian cutoff for the construction of the
SK regions. With the extents of the contributing regions listed above, a row of the
overlap matrix contains typically 300 elements. For comparison, with an orthogo-
nal nearest neighbour Hamiltonian, a row of the H0matrix contains typically about
150 elements. The matrix Sin this case is even smaller because it is constructed us-
ing only the intersections between two LR localisation regions due to the vanishing
3.2. Energy functional and charges 37
basis overlap, in contrast to the intersection between an LR and an LRSK used for
H0.
The data entity associated with an entry in the OV topology is a submatrix of di-
mension (nsi×nsj), where nsiand nsjare the number of orbitals carried by the
contributing regions LRiand LRSKj.
The matrices H0and Sare constructed from the same topologies, imposing iden-
tical cutoffs for the hopping and overlap terms. This allows to access their matrix
elements by the same topology tables. Both matrices are symmetric; taking into
account symmetry nearly halves the computational cost of their evaluation.
In order to exploit the symmetry in sparse coded matrices, tailored information
must be built explicitly into the topology descriptions. This is a non-trivial step.
A sparse topology description contains for each atom ia sequence of indices jad-
dressed by the sparse, or indirect, index j0. In other words, a topology entry at ad-
dress (j0,i)contains the non-sparse column index jfor a given matrix element, say
Sij. The transposed element Sji is accessed by the topology entry (i0,j). The indi-
rect indices i0and j0are unrelated because they are the result of different neighbour
gathering steps for the sites iand j, respectively. In order to allow a handshak-
ing between transposed matrix elements, the topology description needs to carry a
cross-reference list i0(j0,i)besides each regular entry j(j0,i).
To actually calculate only one symmetry half of a sparse coded matrix it is neces-
sary to decide whether a desired matrix element must be calculated explicitly or
is to be taken from the transposed position within the matrix. Conventionally, the
symmetry of a full matrix is employed by limiting the index loops to the upper or
lower triangle of the matrix. This is inconvenient here because of implied load im-
balances. Instead, we apply a checkerboard type criterion introduced in [83]: Matrix
elements characterised by the full indices (i,j)are calculated explicitly, if either they
fulfil (i<j)or (i+j)is even (an EXCLUSIVE-OR relation). To simplify the decision
for repeated accesses of the matrix, the topology lists are re-arranged such that the
data elements which are to be calculated explicitly are listed first in the sparse topol-
ogy list for each row of the matrix. The number of entries matching this criterion
is stored next to the total number of entries per row. The remaining elements of a
row which do not match the selection criterion are listed after the last element that
does. In the complete list of indices, the elements at the tail of the list of row iare
being pointed at by the cross-reference list from the elements at the head of row j
and vice versa. The cross-reference list only needs to extend through the first part of
each row which contains the unique matrix elements.
To circumvent storing Qseparately, the Taylor expansion of Qfor the calculation of
the band structure energy is applied through the trivial relation
EBS =2Tr(H0jiQij) = 4Tr(H0
ij)−2Tr(H0
ijSij). (3.9)
The TB elements hµν and sµν are also symmetric. The topology table SK used to
describe their symmetry and sparse storage is organised and accessed similar to the
OV description.
38 Chapter 3. Order-N Method and Implementation
3.2.4 Atomic charges and SCC energy contributions
The EGterm in the total energy expression (2.11) requires the calculation of atomic
charges. For non-orthogonal molecular orbitals, this calculation is a trifle more in-
volved than for orthogonal states.
Using the approximate inverse overlap matrix Qfrom eq. (3.3) the charge density
for a system of non-orthogonal support functions Φi(r)is given by
%(r) = 2X
ij
Φ∗
i(r)QijΦj(r). (3.10)
Similar to the energy expression, the factor 2 accounts for the spin degeneracy. By
spatial integration within the LCAO expansion the contributions due to the non-
orthogonal basis functions can be expressed in terms of their overlap integrals and
one obtains the total charge as
qtot =2X
µν X
ij
CiµCjνsµν Qi j. (3.11)
The summation over one orbital index is carried out first:
CQ
iν=X
j
CjνQij. (3.12)
This defines, for a given i, the conjugate orbital |Ψii=Pj|ΦjiQij. Formally, it lin-
early combines all localised orbitals |Φjicontributing to the row iof the overlap
matrix. However, only the atomic components within reach of the Slater-Koster
integrals sµν around atoms from LRiare needed further and hence, only the compo-
nents of |Ψiiwithin LRSKiare required and stored for the calculation of the elements
of a bond charge matrix qµν:
qµν =2X
i
CiµCQ
iνsµν;ν∈LRSKi. (3.13)
The atomic charges qkare obtained from this expression by a Mulliken analysis,
which places the elements of the full bond charge matrix in halves onto the con-
tributing atoms, cf. eq. (2.35):
qk=X
µ[k],ν
qµν +qνµ
2. (3.14)
In this notation, µ[k]picks the basis functions µlocated on atom k. Eq. (3.11) can be
written and calculated in the same form as the band structure energy:
qtot =2Tr(SijQij). (3.15)
In this form, the direct calculation of atomic Mulliken charges is not required.
3.3. Electronic minimisation 39
The energy correction due to the self-consistent charge effects (2.26) is evaluated
using an intermediate atomic vector G, which contains the Coulomb potential for
each atom:1
Gk=X
k0
γkk0∆qk0. (3.16)
If a Hubbard-like energy correction of the form EH=PkUk(∆qk)2is used, with
species dependent constants Uk, the vector Gtakes the form
Gk=2Uk∆qk. (3.17)
Both forms of charge-related energy corrections can then be calculated from the
same expression:
EG=X
k
Gk∆qk. (3.18)
The atomic vector Gkneeds to be stored for the calculation of force components, as
described later.
3.3 Electronic minimisation
The electronic contribution to the energy functional, Eel =EBS +EG, is minimised
for a given ionic configuration and a given ηparameter with respect to the orbitals
by a conjugate gradient technique [72]. The gradient of the band structure energy is:
∂EBS
∂|Φii=|ξBS
ii=4X
jb
h0|ΦjiQij −bs|ΦjihΦi|b
h0|Φji. (3.19)
The non-orthogonality of the basis functions appears in the form of the basis overlap
operator bs, mirroring the role of the hopping elements. The gradient components
are evaluated by re-using the projected orbitals |e
ΦHiand |e
ΦSidefined in eq. (3.5).
The gradient components have the same spatial extent as the corresponding wave
functions. Therefore, only the LCAO components of |ξBS
iiwithin the associated
localisation region LRiare needed.
If SCC corrections are included, the respective energy gradients are, as a result of a
cumbersome manipulation:
∂EG
∂|Φii=|ξG
ii=4X
j
(b
g|ΦjiQij −bs|ΦjihΦi|b
g|Φji),
gµν =Gk[µ]+Gk[ν]
2sµν ; (3.20)
This is the same expression as eq. (3.19), with the operator b
h0replaced by a mod-
ified overlap operator b
g, where each basis function overlap sµν is weighted with
1Throughout this chapter, the indices kand k0are used to signify running indices.
40 Chapter 3. Order-N Method and Implementation
the average components of the Coulomb potential Gon both contributing atoms.
This allows one to re-use the routines and intermediate results of the band structure
gradients for the calculation of the charge-related gradient contributions.
In the conjugate gradient procedure, a line minimisation has to be performed along
a search direction determined by the local gradient and an admixture from the pre-
vious direction. The band structure energy along the search direction |ξiiis a fourth
order polynomial in the displacement λ:
EBS({Φi+λξi}) =
4
X
m=0
amλm. (3.21)
The coefficients of the polynomial are evaluated explicitly; this requires the calcula-
tion of the matrices hξi|bh0|Φji,hξi|bh0|ξji,hξi|Φji, and hξi|ξji. These evaluations are
similar to the calculation of the total energy itself. The matrices which combine the
wave function on one side and the gradients on the other are symmetrised before-
hand using As
ij = (Aij +Aji)/2. This allows for a symmetry reduced calculation of
the trace of the product between any of the matrices above. For the symmetrisation,
the index transposition tables of the topology are used.
The SCC energy contributions EGalong the search directions form a polynomial of
8th degree. The coefficients of this polynomial are also calculated explicitly. Because
of the presence of Mulliken bond charges and their mixing, this is much more dif-
ficult than finding the coefficients of the polynomial of the band structure energy.
This complexity can be reduced significantly because each atomic charge alone is a
fourth order polynomial along the search direction:
qk({Φi+λξi}) =
4
X
m=0
bkmλm. (3.22)
The coefficients bkm are calculated with the subroutines used to evaluate the atomic
charges, plugging in the set of overlap matrices involving gradients as defined
above. In a next step, the polynomials of atoms k0are projected to the atomic G
array using eq. (3.16), which thus represents a quartic polynomial for each atom.
The full 8th order polynomial for the SCC energy is constructed following (3.18).
Both the band structure and SCC energy polynomials are added and the minimum
λmin is calculated directly. Finally, the wave function is updated as |Φi+λminξii.
We note that in the line minimisation involved in the conjugate gradient procedure,
spurious minima along the search direction may be present. Indeed, the polyno-
mial in λmay have several minima. All but one are characterised by an unphysical
charge distribution. It would be expensive to discard solutions corresponding to an
unphysical charge only after the charge has been calculated. Unphysical solutions
are recognisable from other general properties. Firstly, the total energy in the con-
jugate gradient procedure changes only by small steps. Only a solution with the
polynomial value close to the previous energy minimum is acceptable, with the ex-
ception of the early stages of an iterative minimisation started, e.g., from random
wave functions. Negative values can be discarded as well, since they indicate a dis-
crepancy with the downhill search paradigm. Finally, experience has shown that
3.3. Electronic minimisation 41
0
1
2
0 100 200 300 400
λmin
Iteration step
Diamond + H, LR2
Graphite, LR2
Graphite, LR3
SiC, LR2
SiC, LR4
Figure 3.2: Line minimisation step λmin vs. iteration for various test systems. Starting from
random wave functions, the energy gradient itself is initially steep (has a high norm), so that
sensible wave function updates arise from small λmin. After a few steps, the gradient norm
gets smaller, and the update step lies typically within 0.5. . .1.
λmin typically has values between 0.5 and 1.0, see Fig. 3.2. Much larger values are
therefore rejected.
3.3.1 The Lagrange multiplier η
During the conjugate gradient minimisation of the energy functional, ηis treated
as an external parameter. The parameter ηhas to be adjusted so as to have the
correct number of electrons (i.e., the correct orbital filling) at the end of the iterative
minimisation. Contrary to the energy functional which depends explicitly on the
value of η, the total charge depends only implicitly on η, through the localised wave
functions minimising the energy functional at that given η. We note that the location
of the energy functional minimum changes as ηis varied. Therefore the evaluation
of the function
Q=qtot(η)−Nel (3.23)
can only be performed approximately: the value of Qis determined after a certain
number of conjugate gradient steps have been performed at a fixed η. If Qis pos-
itive, ηneeds to be reduced, which corresponds to emptying the upper occupied
states. Conversely, for negative Q, i.e., a charge deficiency in the system, ηmust
be increased. A suitable iteration scheme for ηneeds to be wrapped around the
electronic conjugate gradient algorithm. Fig. 3.3 gives an overview of the extended
iteration scheme in the form of a flowchart. The ηiteration scheme is the subject of
the next section.
42 Chapter 3. Order-N Method and Implementation
qtot ({Φi})
|
ξ
ii=4Pjn(b
H−
η
)|ΦjiQij − |ΦjiHijo
E=2Tr(HijQi j) +
η
Nel
Hij =hΦi|
b
H−
η
|Φji
Sij =hΦi|Φji
Qij =2
δ
ij −Si j ≈S−1
EF+30 eV
|Φji
E(
λ
) = P4
nan
λ
n
|Φi:=|Φi+
λ
min |
ξ
i
|Φji
n
y
calculate [conjugate] gradients
(random)
choose (start at )η
n
Ey
line minimisation along
get polynomial coeff.
update wave function
choose
calculate forces/move atoms
extrapolate wave functions
charge converged?
converged?
Figure 3.3: Flowchart for the O(N)
minimisation of the total energy
w.r.t. atomic positions (outermost it-
eration), the Lagrange parameter η
(medium iteration), and the wave
function coefficients Ciµ(inner iter-
ation).
3.3.2 The adaptive secant method
There are two facts which may cause convergence difficulties when optimising the
ηparameter. Firstly, although monotonous, the relation qtot(η)is non-linear. The
slope is lower for undercharged systems than for excess charges. Secondly and more
severely, a premature readout of the charge before the wave function is stepped
close enough to the energy minimum may be inappropriate for choosing ηfor the
next cycle of conjugate gradient steps. A balance must be found between letting
the electronic conjugate gradient algorithm for the energy minimisation run for too
many steps, and accepting an approximate value of the charge, at a given η.
Another control parameter which needs to be optimised is the the amplitude ∆η
with which ηis changed. If ∆ηis chosen too small, a change in sign for Qis not
reached for many update steps; if ∆ηis chosen too large, then the energy minimum
3.3. Electronic minimisation 43
949.4
949.6
949.8
950.0
950.2
950.4
950.6
0 500 1000 1500 2000 2500 3000
qtot
Iteration step
949.92
949.94
949.96
949.98
950.00
950.02
950.04
950.06
950.08
−0.0798−0.0797−0.0796−0.0795−0.0794−0.0793−0.0792
qtot
η/H
900
1000
1100
1200
1300
1400
1500
1600
1700

0 0.2 0.4 0.6 0.8 1
qtot
η/H















0 50 100 150 200 250 300 350 400 450
EBS/H
Iteration step
700
800
900
1000
1100
1200
1300
1400
1500
1600
1700
0 50 100 150 200 250 300 350 400 450
qtot
Iteration step














0 500 1000 1500 2000 2500 3000
EBS/H
Iteration step
(d)
(e)
(f)
(a)
(b)
(c)
Figure 3.4: Calculated quantities during stepped and feedback wave function iteration
phases (see text): (a,d) energy, (b,e) charges, and (c,f) η-parameter. The sample structure
is a stable silicon-diamond interface structure (Bs
0in Ref. [88]) with 284 atoms total (174 C,
52 Si, 10 H, 48 H0). (a,b,c) starting from random input wave functions with ηstepped; (d,e,f)
feedback-controlled iteration phase, with ∆ηdepending on the charge defect Q. The arrow
in (c) indicates the iteration sequence. Lines in (f) connect pairs of subsequent iteration steps.
is shifted far away from the previous location in wave function space, and conver-
gence proximity for the wave function may be lost.
Essentially, the root of the one-dimensional relation Q(η)is searched iteratively
starting from a pair η1and η2yielding Qvalues of opposite sign. In principle this
search could be performed by a standard root finding technique. However, the prop-
erties of the relation Qdisallow a black-box approach. As mentioned above, due to
the finite number of electronic iterations nCG,Q(η)is evaluated for a given iteration
min wave function space at a point Pmwhich is in general not the energy minimum
44 Chapter 3. Order-N Method and Implementation
location Pmin(ηm), giving rise to an error ∆Qm. This error tends to systematically
drop in subsequent iterations because the repeated underlying electronic conjugate
gradient iterations will normally reduce the distance kPm−Pmin(ηm)k, even though
Pmin(ηm)is a “moving target”. We note that Q(η)can attain different values, de-
pending on the stage of the minimisation procedure (i.e., the value of the charge
density) at which the ηvalue is visited. Therefore, the calculation of the derivative
of Q(η)is subject to large errors due to the dynamic behaviour of the charge error
∆Qm. Since most black-box root solvers rely on derivative information or expect a
“smooth” behaviour of the function, they cannot be used without applying tailored
safeguard mechanisms to avoid divergence from a known good search interval for
η.
The following scheme has been established to find the value of ηcorresponding to a
minimum of the energy functional at the correct charge, i.e., fulfilling the condition
Q(η) = 0. A sequence of electronic conjugate gradient iterations are nested within
iterations for η.
The scheme starts from a random wave function and a well-guessed η. A fixed num-
ber nCG (typically, 20. . .50) of electronic conjugate gradient iterations is performed
where ηis held constant. At the end of this set of iterations the charge deviation
Qis evaluated. Subsequently, a fixed correction ∆ηis applied and the electronic
conjugate gradient cycle is restarted. This procedure is repeated until Qchanges
sign. These steps are illustrated in Fig. 3.4(a) through (c). Now an adaptive up-
date scheme is switched on, where ∆ηis calculated depending on the value of the
charge defect Q. This phase is shown in Fig. 3.4(d) through (f), which illustrates in
particular some of the problems related to the update step ∆ηmentioned above.
The dynamic charge error ∆Qmhas the effect of letting functional values Q(η)“age”
during the ongoing electronic minimisation, as can be seen in the example calcula-
tion in Fig. 3.4(f). Therefore, the secant method has been chosen and adapted as root
finding technique [72]. It limits the influence of the charge error effects by making
use of only the latest two iteration results Q(ηm)and Q(ηm−1). For comparison, the
closely related regula falsi method may keep one particular iteration result for many
cycles and is therefore not suitable. The unknown changes in the charge error ∆Q
along the iteration for ηmay send the intersection of the secant with the ηaxis far
away from the latest search interval when Q(ηm)≈ Q(ηm−1), see Fig. 3.5 (a). As a
safeguard mechanism in this case, the update step ∆ηmis chosen to be the same as
the previous one, ∆ηm−1. In other cases, the slope ∂Q/∂ηmay appear to be negative
and would point to the wrong direction for updating η, as illustrated in Fig. 3.5 (b).
Since it is known that the derivative ∂Q/∂ηis positive this case is recognised eas-
ily. As a correction, the previous slope is used again. A third mechanism is put in
place to avoid making a much smaller step in ηthan in the previous iteration, i.e.,
avoiding the case |ηm+1−ηm| |ηm−ηm−1|. This is necessary to circumvent being
committed to small steps for many iterations.
In summary, the iteration for ηis wrapped around the conjugate gradient iterations
for the wave functions. In each set of wave function iterations, nCG steps are made.
The resulting charge is evaluated and used to control a modified secant algorithm
for finding the root of the function Q(η). Safeguard mechanisms are applied in three
cases: (1) the secant is nearly parallel to the ηaxis, (2) the slope is negative and (3)
only a small step would have to be made.
3.4. The calculation of forces 45
Q Q
(a) (b)
η η
B
C
A
A
D
D
C
B
Figure 3.5: Examples for the application of the adaptive secant method to find the root of
the function Q(η). The function can be evaluated only approximately. Consecutive function
evaluations yield the points A, B and C, each subject to an error ∆Q. The abscissa of C
is obtained from the secant AB. (a) The errors cause the secant BC to point far away from
the current search interval [ηA,ηB]. (b) The secant BC has a negative slope, despite the
function being known to be monotonously increasing. Safeguard mechanisms are employed
to remain near the current search interval.
3.4 The calculation of forces
After convergence is achieved for the wave function and the ηparameter, forces
on the atoms can be obtained analytically by calculating the derivative of the total
energy with respect to the ionic positions. We show here a rigorous derivation of the
force expression. We start the calculation by taking into account each independent
variable in the total energy expression:
Fl=−∂
∂Rl
E({Ciµ,hµν,sµν,γkk0},η,Vrep)(3.24)
Thus,
Fl=−X
iµ∂E
∂Ciµ
∂Ciµ
∂Rl−X
µν ∂E
∂hµν
∂hµν
∂Rl
+∂E
∂sµν
∂sµν
∂Rl
−X
kk0∂E
∂γkk0
∂γkk0
∂Rl−∂E
∂η
∂η
∂Rl−∂Vrep
∂Rl
(3.25)
The contributions from the LCAO coefficients Ciµcancel because the total energy
was explicitly minimised in this space and the gradient components ∂E/∂Ciµvanish
in the ideally converged case. The force contribution due to the repulsive potential
Vrep is trivial to calculate. The last outright simplification arises from the fact that
the total charge is supposed to be well converged; thus the contribution due to the
Lagrange parameter ηvanishes by virtue of ∂E/∂η=Nel −qtot =0.
The remaining force components due to the band structure energy are
FBS
l=−X
µν ∂EBS
∂hµν
∂hµν
∂Rl
+∂EBS
∂sµν
∂sµν
∂Rl
46 Chapter 3. Order-N Method and Implementation
=−2X
µν (esµν
∂hµν
∂Rl−eh0
µν +ηesµν∂sµν
∂Rl). (3.26)
The arrays eh0
µν and esµν introduced here are of the same dimension and topology
as the Slater-Koster hopping and overlap matrices hµν and sµν. These arrays are
calculated from the converged wave functions:
eh0
µν =X
ij
CiµCjνH0
ij =X
i
CiµCH
iν
esµν =X
ij
CiµCjνQi j =X
i
CiµCQ
iν(3.27)
The coefficients CQ
iνwere already needed in the calculation of atomic charges; a re-
lated orbital CH
iνis required here and is calculated analogous to (3.12) from (3.5)
and (3.7). The derivatives of the TB hopping and overlap elements themselves are
obtained using a finite difference technique.
The force components from the SCC energy contribution are:
FG
l=−X
kk0
∂EG
∂γkk0
∂γkk0
∂Rl−X
µν
∂EG
∂sµν
∂sµν
∂Rl
=−∆qlX
k
∂γkl
∂Rl
∆qk−2X
µν Gk[µ]+Gk[ν]
2esµν −e
gµν∂sµν
∂Rl
(3.28)
The first term contains the components due to the changing Madelung sums and
the second term contains those due to changes in the atomic charges. There is no
contribution due to ∂EG/∂hµν. The derivatives of the Madelung terms are obtained
analytically. The array e
gµν is obtained from the overlap matrix of the b
goperator
Gij =X
µν
CiµCjν
Gk[µ]+Gk[ν]
2sµν (3.29)
and its projection on the orbital pairs analogous to eq. (3.27) defined as
e
gµν =X
ij
CiµCjνGij =X
i
CiµCG
iν(3.30)
With all forces at hand, a Verlet algorithm for MD or an ionic conjugate gradient
scheme can be performed.
3.5 Accuracy and performance
As a measure for performance of an iterative linear scaling scheme one will look at
3 key quantities: (1) the calculation time for a single iteration step, (2) the number
of iterations required for a sufficiently converged solution and (3) the error of this
3.5. Accuracy and performance 47
Table 3.1: Comparison of total energies and charges obtained from exact diagonalisation
(diag), diagonalisation with a Hamiltonian cutoff (cutdiag) and from the O(N)functional.
The calculations were performed with Localisation Regions of topological radius 2 and 3,
marked ‘LR 2’ and ‘LR 3’, respectively.
Diamond 2D Graphite 3C SiC 3C SiC (scc)
Nat 216 96 216 216
Nel 864 384 864 864
EBS (Hartree)
diag −396.339 −178.866 −337.580 −336.897
cutdiag −396.100 −178.790 −337.537 −336.853
LR 2 −394.990 −178.044 −336.881 −336.143
LR 3 −395.792 −178.586 −337.389 −336.716
E−Ecutdiag (meV/atom)
diag −30 −21 −5−5
LR 2 140 212 83 90
LR 3 39 58 19 18
η(Hartree)
LR 2 0.15725 −0.14738 0.12983 0.23491
LR 3 0.10616 −0.17270 0.05558 0.10963
solution. With these quantities at hand, an assessment of the breakeven point should
be possible, i.e., the determination of the problem size at which the linear scaling
scheme is at odds with a traditional O(N3)scheme. There are also factors related
to the initialisation procedure which influence the breakeven point. Apart from the
time needed for the setup procedure of interatomic distances and topologies repeat-
edly required when the ionic positions are changed the overall timing also depends
on the number of initial iterations needed to guide a random start wave function
near a first minimum.
Starting from a good initial guess of the Lagrangian ηand random wave functions
the initial iteration of ηand the subordinate energy minimisation require about
150.. .250 conjugate gradient iterations in total. Significantly less are expected to be
needed after updating ionic positions due to extrapolation of previous wave func-
tion updates. Based on these values, we give as an estimate for the breakeven point
of an initial energy calculation a value of 700 atoms. Important for later molecular
dynamics studies is the breakeven point for extrapolated wave functions which we
estimate to be about 300 atoms given that only 25 conjugate gradient steps will be
needed. It must be stressed that these numbers are estimates.
3.5.1 Energy and charges
The accuracy of the solutions can be assessed from Table 3.1. We have calculated
the total energies of 3 test systems: diamond, graphite and silicon carbide in the
cubic (3C) polytype. We compare the total energies obtained from diagonalisation
and the present non-orthogonal O(N)method. For the SiC model, we addition-
48 Chapter 3. Order-N Method and Implementation
−0.2
−0.1
0.0
0.1
0.2
−0.1 0.0 0.1
O(N) (a.u.)
diagonalisation (a.u.)
z
y
x
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
0 50 100 150 200
(a.u.)
(atoms, sorted)
F (diag.)
∆F
(a) (b)
Figure 3.6: Accuracy of the O(N)force calculation vs. diagonalisation for a 216-atom dia-
mond cell with an artificial hydrogen substitutional atom and all atoms randomly displaced
by up to 0.2 ˚
A. The localisation range (LR) for the O(N)calculation is 2. (a) Force com-
ponents in x,y, and zdirections (vertically separated by 0.1 a.u.). (b) Comparison of the
magnitude of exact forces Fand their deviations ∆Fin O(N), sorted by F.
ally check the accuracy of the self-consistent charge corrections. The error in total
energy for the localised solutions with respect to extended orbitals obtained from
diagonalisation is of the order of 100 meV per atom for localisation regions of radius
2 hops and expectedly much smaller, with a value of about 30 meV, for localisa-
tion regions extending up to 3 hops. The error is highest with 233 meV total in the
case of graphite. Since this system is metallic and has delocalised electronic states a
higher deviation is expected. For metallic systems, the convergence is expected to be
polynomial with the size of the localisation region. Given that the same number of
neighbour hops was applied as in the semiconducting diamond crystal, the results
are satisfactory. For both carbon systems, the influence of the cutoff Hamiltonian as
obtained from diagonalisation is about 30 meV per atom due to the long range of the
unconstrained basis orbitals, while in the mixed SiC system, the contribution due to
the Hamiltonian cutoff alone is with 5 meV per atom negligible. In other words, the
errors in diamond are larger because the DFTB-basis overlap for pure carbon are
farther reaching in units of nearest-neighbour distance and the localisation region
cutoff in terms of neighbour hops implies a larger error.
The self-consistent charge contributions to the total energy and the charge transfer
itself agree well between diagonalisation and the localised functional. In the SiC
structure the charge transfer, i.e., the overcharge of a carbon atom in electrons, is
0.630 from direct diagonalisation, while we obtain 0.657 with localisation regions of
size 2 hops and 0.637 for a 3-hop model.
3.5.2 Forces
To judge the calculation of forces, we prepared a sample model of a 216 atom dia-
mond cell, where one atom was substituted by hydrogen. This cell was subjected
3.5. Accuracy and performance 49
Table 3.2: CPU time per electronic iteration step and memory requirements for diamond
structure test systems of varying size. The last column lists the times for the calculation of
the γmatrix. Up to 2nd neighbours were included for localisation regions and Slater-Koster
cutoffs. All tests were performed on an HP PA7200 processor.
nat tBS tBS+SCC mem BS mem γtγ
s s MB MB s
216 26 65 36 0.4 40
512 78 182 91 2.0 276
1024 159 373 187 8.0 1160
2048 325 765 396 32.0 5250
to a random displacement of all atoms, and then the total energy was minimised
as before, with a localisation range of 2 hops for the wave function. The model so
constructed represents a structure far off the equilibrium, and therefore should be a
quite reasonable test case for the algorithm. The forces for this model were calcu-
lated with both localised orbitals as described in 3.4 and using traditional diagonal-
isation. Fig. 3.6(a) shows the comparison of these results in the form of a scatterplot
for each force component on all atoms. As one would expect, the forces for carbon
atoms away from the defect site are reasonably exact, such that the total deviation
∆Fper atom is in general less than 0.02 H/aB. The deviation is independent of the
force magnitude itself, see Fig. 3.6(b). For the defect ligands, the deviation value
nearly doubles, and reaches 0.12 H/aBfor the defect atom which is in the same or-
der of magnitude as the forces themselves.
Given that forces are derivative quantities, their accuracy is expected to be lower
than that for the energy. The given example presents quite a challenge in that the
electronic structure near the defect will be considerably disturbed. Obviously, the
force calculation is not quite accurate enough near such distortions. It remains to
be seen if this would limit practical applications. Throughout the rest of the struc-
ture, however, the forces should be sufficiently accurate and suitable for an ionic
minimisation procedure.
3.5.3 Scaling
The implementation performs with linear scaling, confirmed by tests run on dia-
mond structures with the number of atoms reaching up to 2048. Results for CPU
times and memory requirements are shown in Table 3.2 and Fig. 3.7. Clearly, both
time and memory requirements for the electronic minimisation scheme indeed scale
linearly. When SCC corrections are switched on, the scaling is still linear, apart from
quadratic admixtures in the calculation of the γMadelung matrix once per ionic
step as discussed earlier. Comparing the overall timing for the non-scc and scc cal-
culations, one sees from Table 3.2 that an electronic iteration step which includes the
scc corrections requires a factor of 2.3 of the CPU time needed for the band structure
term alone. The increase is largely due to the computational cost in establishing the
atomic charges and their derivatives, since they invoke the large overlap matrix be-
tween support functions. Overall, the factor is quite satisfactory. However, caution
is required in comparing this at face value with the approximately five- to tenfold
50 Chapter 3. Order-N Method and Implementation
0
100
200
300
400
500
600
700
800
0 216 512 1024 2048
t (BS)/s
t (BS+SCC)/s
mem (BS)/MB
0
20
40
60
80
100
0 216 512 1024 2048
t (gamma)/min
mem (gamma)/MB
Figure 3.7: Scaling behaviour for CPU time and memory requirements for diamond struc-
ture test systems of varying size. Data from Table 3.2.
increase in time in the diagonalisation scheme. No systematic comparison has been
made about whether the O(N)scheme including the SCC correction requires more
electronic iteration steps than without. Given the iterative nature of the procedure
such an estimate would be difficult at this point.
3.5.4 Assessment
As mentioned in the introduction, a great deal of work has been performed in
the O(N)community [73,74,79,80]. Every major electronic structure code was
sounded using linear scaling formulations. Nonetheless, the initial enthusiasm
has somewhat subsided, because most of the new mechanisms are not really ro-
bust enough to act as black-box replacements of standard programs. Most O(N)
schemes, including the present one, are prone to be numerically unstable as the
systems undergo electronic changes, particularly the opening or closing of gaps, or
topological changes due to atomic reordering. A lot of work will have to be per-
formed to overcome these weaknesses.
Two other solutions to calculate larger systems are discernible. The first relies on the
growth of computing capacity. Moore’s law [37] was valid for more than 35 years,
having predicted an exponential growth of microchip integration density with a
doubling time of just 18 months. The same applies for the clock frequency with
which the processors are driven. There is all indication that the “law” will hold a
few more years, and perhaps beyond, when the current silicon and copper technolo-
gies might be replaced. Such developments in computing power enable extensive
growth for applications with the continued use of standard algorithms. Parallelisa-
tion takes on an increasingly important role, but requires more changes and consid-
erable resources to implement, see, e.g., Ref. [89].
The other solution is the hybridisation of various methods in what is known as
multi-scale modelling, where different length scales of the same structure are cal-
culated with different methods at an appropriate degree of accuracy [90].
Chapter 4
Properties of Diamond
4.1 History and economics
Carbon is an exceptionally versatile chemical element. Its ability to form single,
double and triple bonds leads to several allotropes of vastly different properties.
For the same reason, carbon is also the basis of organic compounds.
The two most important pure forms of carbon are graphite and diamond, both of
which have been known for at least two thousand years. The Greek names are
γραϕ´ιτηςand αδ ´αµας, respectively, although there is an ongoing dispute about the
latter. Before diamond entered the Graeco-Roman world via India, adamas (“the in-
vincible”) may have referred to other hard substances known at the time, probably
corundum (Al2O3, ruby and sapphire being its gemstone varieties). The related-
ness between diamond and graphite was not recognised until the late 18th century.
Graphite was identified as carbon by Carl Wilhelm Scheele (1742–1786) in 1779,
see [1, “pencil”]. Extending combustion experiments of Lavoisier, diamond was
shown to be carbon by Smithson Tennant (1761–1815) in 1797 [91]. More recently,
other allotropes were discovered first in the laboratory, and subsequently found in
nature, namely, fullerenes, nanotubes, and amorphous as well as nanocrystalline di-
amond. A quite large variety of carbon allotropes is found in meteorites [92], many
of them diamondlike [93,94]. The first find of meteoritic “diamonds of microscopic
size” dates back over a century ago [95].
On earth, natural diamond is found primarily in two rare types of volcanic igneous
rock formations, known as kimberlite pipes and lamproite. Due to erosion of such
primary deposits in geological time, diamond is also found in secondary and ter-
tiary alluvial deposits. Contrary to public belief, diamond is mined not only in
South Africa but on all continents save Europe and Antarctica. The world’s largest
producers are currently Australia and Congo (Kinshasa), albeit only 5% of their out-
put is of gem grade. Of comparable volume, but superior quality are the mining
operations of Botswana, Russia (Siberia) and South Africa, followed by Namibia,
whose alluvial and marine deposits yield well over 90% in gem quality [96,97].
In developing countries, diamond mining and trade is a major economic factor ri-
valling the traditional role of oil. E.g., one third of the gross domestic product (GDP)
of Botswana stems from diamond mining [98, p. 392], making its economy extremely
51
52 Chapter 4. Properties of Diamond
vulnerable. During conflicts, groups controlling the relevant areas may secure their
financing through diamond trade. Three situations of this type recently occurred
within Africa, namely, in Sierra Leone, the Congo Republic, and Angola [99, p. 390].
UN trade sanctions where put in place, if only to little effect on diamond markets in
the western world due to extensive smuggling and the lack of distinctive indicators
of origin once the stones are properly cut [100,101].
For the last decade, the (official) worldwide mining output has been around 100
million carats annually, i.e., 20 metric tonnes, of which 80% are of industrial grade.
A further 400 million carats or 80 metric tonnes of synthetic diamonds are being
produced annually [97], mostly in a high-pressure high-temperature process from
graphite, at conditions of about 2000 K and 7 GPa , see [1, “synthetic diamond”]. One
of the primary applications of industrial grade diamond is in abrasives and sintered
tool coatings; this takes the lion’s share of the annual production. Curiously, the ap-
plication of diamond in tools is not a modern invention. From characteristic groove
and fracture patterns in ancient quartz beads, there is convincing indirect evidence
for the use of diamond-fitted drill bits in ancient India as far back as 250 BC [102].
The properties which are traditionally associated with diamond are its extreme
hardness (Mohs 10) and high refractive index (2.46–2.40, blue to red [2, p. 229]). Both
of these, accompanied by a high optical dispersion give diamonds its adamantine
lustre and gemstone value. However, diamond has many more interesting proper-
ties. This is the major playing field of diamond grown by chemical vapour deposi-
tion (CVD), because this technology allows to coat surfaces and to create large-area
wafers. In most bulk properties, CVD diamond is on par with natural diamond,
although hardness and particularly strength are often reduced by factors around
two [103]. In other characteristics, CVD films are potentially superior to natural di-
amond because their morphology and impurity concentration may be controlled to
some extent as needed.
The origin of CVD diamond research dates back to 1949, when Eversole started his
work on diamond synthesis, leading to growth on seed crystals in 1952, eventu-
ally patented in 1962 [104], yet after similar work has been performed in the Soviet
Union by Deryagin in 1956. Since then, great effort went into developments to sta-
bilise the growth of diamond over graphite and subsequently increase growth rate,
building upon initiatives in the groups of John Angus in the U.S., Boris Spitsyn in
the Soviet Union, and Yoishiro Sato et al. in Japan. The definitive account on the
history of diamond CVD growth from early on was given by Angus [3,105]. Recent
developments are targeted at viable commercialisation for a broad range of applica-
tions [107].
4.2 Bulk structure and properties
The key towards understanding the extreme properties of diamond lies in its crystal
structure. The crystal structure of diamond is prototypical for four-valent semicon-
ductors. Each carbon atom is surrounded tetrahedrally by four neighbours. Out
of this basic tetrapodal building block the crystal is constructed as illustrated in
Fig. 4.1(a). In terms of Bravais lattices, the crystal is a face-centred cubic (fcc) lattice
with a diatomic basis. The basis can be taken as just one corner atom plus the centre
4.2. Bulk structure and properties 53
(a) Diamond
(b) Lonsdaleite
Figure 4.1: The crystal structure of diamond and Lonsdaleite (“hexagonal diamond”), built
by repetition of tetrahedrally bonded blocks. (a) Nearest neighbours define four corners of
a cube. Cubes are stacked to form a cubic lattice. (b) Nearest neighbours define a triangular
prism. Prisms are stacked to form a hexagonal lattice.
atom of the cubic building block in Fig. 4.1(a). A rare variety of the crystal structure
with the same nearest-neighbour configurations but hexagonal stacking is named
lonsdaleite [Fig. 4.1(b)], after Dame Kathleen Lonsdale (1903–1971). She determined
the lattice constant of individual (cubic) diamonds accurate to six significant digits
(3.56665–3.56723 ˚
A) using her novel divergent-beam X-ray technique [109]. It is in-
teresting to note that the atomic number density resulting from the crystal structure
of diamond is the highest of any known substance [3].
Diamond has the highest sound velocity of any solid and hence exhibits an excep-
tionally high heat conductivity,1about 15–20 W cm−1K−1, which is more than four
times that of copper [107, p. 62]. The reason for this virtue is a combination of three
key factors. Firstly, the carbon atoms are strongly covalently bonded in a rigid lat-
tice. Secondly, the atoms are relatively light. Thirdly, the crystal is simple enough to
reduce dissipation by anharmonicity and optical phonons [110]. Similar conditions
also hold for individual graphitic basal planes and carbon nanotubes. For both,
anisotropic thermal conductivities with maxima exceeding diamond have been con-
jectured [111, and references therein]. However, as bulk material diamond is real,
proven, and unsurpassed.
1This lends diamond stones a cold touch, and hence, the quite appropriate slang sobriquet ice.
54 Chapter 4. Properties of Diamond
This still holds for CVD diamond (despite its higher density of structural inhomo-
geneities like grain boundaries) and thus accounts for one of its major application
as heat spreader in microelectronics. In this role, there are a number of futuris-
tic applications like heat sink packs on the decimetre scale, where diamond helps
to dissipate enormous amounts of heat produced by conventional integrated elec-
tronic devices [97]. One of the problems encountered in this context is that due to the
strong covalent bonding diamond has an extremely low thermal expansion coeffi-
cient of 0.8 ×10−6K−1(at 300 K), which is lower than Invar, as Angus points out [3].
Since this is widely disparate from the chip carriers, mechanical stress ensues which
may lead to device delamination.
The phonon spectrum of diamond gives rise to a single characteristic Raman
peak [112] at 1332.5 cm−1. This peak is ubiquitously used to confirm the presence
of diamond in thin films. This includes ultrananocrystalline films, where Raman
scattering results corroborated the nature of the material. In general, the integrated
peak area as compared to that of a broad graphitic band at about 1580 cm−1is often
interpreted as a measure of film quality.
Electronically, diamond spans the range from insulator to semiconductor, depend-
ing on the impurity density. The pure material has a large indirect bandgap of
5.49 eV, making it an insulator. It is easily p-type doped by boron and forms a semi-
conductor. The acceptor level lies 0.37 eV above the valence band [113, p. 56]. In
contrast, nitrogen as a most likely n-type dopant forms deep levels at about 1.7 eV
below the conduction band minimum [114]. Natural diamond is classified into types
Ia (the most frequent and most impure), Ib, IIa and IIb (both rare and virtually pure).
These classes are largely determined by the concentration and degree of agglomer-
ation of nitrogen impurities [103, p. 51], c.f. [112]. Pure diamond is one of the best
insulators. Its resistivity is above 1014 Ωcm; through heavy boron doping, this may
be decreased to well below 1 Ωcm [113], making it a semimetal. It is one of the
most pressing challenges to find a solution to produce shallow n-type doped CVD
diamond, to pave the way for high-temperature high-power device applications.
However, the recent advent of silicon carbide (SiC) is likely to surpass diamond in
this respect.
Because of the large bandgap, pure diamond is colourless and transmissive for light
in a broad range of wavelengths from infrared (IR) well into ultraviolet (UV), which
accounts for the application of suitably polished CVD diamond wafers as extremely
robust optical windows, e.g. in weapons targeting systems. Depending on the con-
centration of various dopants and defects, diamond becomes tinted. For instance, in
boron doped diamond, acceptor levels above the valence band result in absorption
of photons with relatively low energy, lending the material a blue appearance. Con-
versely, in nitrogen-doped diamond, electrons from the deep donor states may be
excited into the conduction band by absorbed blue light, giving the material a yel-
low or red tinge. The prediction of colour, however, is a trifle more complex. Besides
the dopant levels, it also depends upon selection rules and Franck-Condon-factors.
Certain diamond surfaces have a negative electron affinity (NEA) and are therefore
capable of cold electron emission which makes them highly attractive for flat-panel
displays [115].
4.3. Diamond surface structure 55
Table 4.1: Elementary properties of low-index diamond surfaces. The nomenclature is the
same as in Fig. 4.2;aDdenotes the cubic lattice constant (expt. 3.567 ˚
A).
Face Lattice Mesh size Atoms Dangling bond density
per cell per atom per area a2
D%
{111}1db hexagonal aD/√2 1 1 4/√3 100
{111}3db hexagonal aD/√2 1 3 12/√3 300
{110}rectangular aD×aD/√2 2 1 2√2 122
{100}square aD/√2 1 2 4 173
In CVD films, diamond occurs as microcrystalline or nanocrystalline phase, with
grain sizes in the micrometre and nanometre range, accordingly. The grain bound-
aries vary widely in thickness and proportion across the various film types and play
a crucial role in establishing the physical properties of the macroscopic film. Refer-
ence data on both diamond bulk and films are to be found in the excellent recent
compilations within Prelas et al. [2].
4.3 Diamond surface structure
4.3.1 General properties
Most of the surface properties of diamond depend on the crystal face exposed.
Each surface has its own characteristics with regards to the unreconstructed (i.e.,
as-cleaved) two-dimensional surface lattice, and the number of broken bonds per
unit area. These characteristics are summarised for the key low-index Miller faces
in Table 4.1, and their geometric relation is illustrated in Fig. 4.2. Evidently, there
are two different ways to cut along a {111}plane, either leaving one or three dan-
gling bonds per surface atom. The resulting faces are labelled “1db” and “3db”
faces, respectively. The last column in Table 4.1 establishes the relative strengths of
the various surfaces as predicted by the so-called bond scission model. This model
assumes that the cleavage energies are proportional to the density of bonds to be
severed when the crystal is cut along a certain plane. It is long known that natural
cleavage occurs along {111}faces, which is indeed explained by the bond scission
model. There are eight such faces forming an octahedron, a familiar shape of natural
diamonds. The nature of cleavage was largely considered to be understood, yet the
debate has recently been revived to understand the fracture process [116].
On conventional CVD diamond, the faces exposed will be those which grow slow-
est, because the faster growing faces will quickly grow themselves out of existence,
their edges stacking up to the other faces. The slowest growing faces are {111}and
{100}, leading to crystallites of cubo-octahedral morphology. Their shape is usually
classified in terms of a growth parameter α=√3v100/v111, which relates the growth
rates of the two indicated faces (see e.g. [117]). The crystallites will be cubic atα=1,
cubo-octahedral for 1 <α<3, and octahedral at α=3. The α-parameter depends
on growth conditions like the gas composition and the substrate temperature, and
can be adjusted in a controlled fashion.
The nanoscale structure of the surfaces is determined by the chemical bonding of
56 Chapter 4. Properties of Diamond
(a)
1db
3db
O[001]
[110] [111]
[211]
-
(b)
PC
SC {110}
{001} {111}
Figure 4.2: Geometry of diamond surfaces. The low-index crystallographic planes are
shown in relation to each other. (a) unreconstructed (as-cleaved) surfaces, with crystal-
lographic directions, and (b) reconstructed {111} − (2×1)and {100} − (2×1)surfaces
(schematic); the {110}faces do not reconstruct. The inset shows the crystallographic planes.
The stubs on each surface atom represent dangling bonds. The labels differentiate the one-
and three-dangling bond cutting planes of the {111}faces (“1db” and “3db”) and their
Pandey- and Seiwatz chain (“PC” and “SC”) reconstructions, respectively. The cube in-
scribed within the bulk section indicates a unit cell of the bulk lattice (cf. Fig. 4.1).
4.3. Diamond surface structure 57
surface atoms. Due to missing bond partners the outer atoms of a surface are
severely perturbed from their bulk configurations and will therefore find new equi-
librium positions. The energy of surface atoms is reduced on the order of 1–3 eV by
two related mechanisms. The first effect, called relaxation, is a strengthening of the
bonds between the outer monolayers and the bulk, leading to a slightly decreased
layer distance. The other effect is due to newly formed bonds between surface atoms
or surface atoms and adsorbates and is called reconstruction. The driving force to-
wards reconstruction is a saturation of dangling bonds. In general, reconstruction
lowers the symmetry of the surface net. Deeper layers near the surface often also re-
lax up to about 0.1 ˚
A. In diamond, the influence of the surface more or less subsides
beyond the fifth or sixth monolayer.
The surfaces are crystallographically characterised by a two-dimensional Bravais
lattice, called net, with the basis vectors determined by the ideal, unreconstructed
surface. Reconstructions are labelled in simplified form as (m×n)Rα, where m
and n(not necessarily integers) are the mesh sizes relative to the original mesh and
αis an angle specifying a possibly rotated mesh. The Rαnotation is omitted in
the absence of mesh rotation. A more elaborate nomenclature is required (but not
relevant here) when the reconstructed mesh is sheared, see, e.g. [118, chapter 19].
It is difficult to obtain unambiguous atomic-scale structural information about sur-
faces from experiments. Commonly used techniques are:
LEED (low-energy electron diffraction). LEED data are essentially two-dimensional
Laue diagrams of electron waves. In fact, the wave nature of the electron was
discovered accidentally by Davisson and Germer using this technique [119].
LEED yields the crystallographic surface structure, i.e., the mesh type and
width. Often, several domains are present on a microscopic scale and mix
in the diffraction pattern.
HREELS (high-resolution electron-energy-loss spectroscopy). This technique mea-
sures the energy loss of electrons due to excitation of surface vibrations, see
e.g. Ref. [120]. It provides indirect structural information by comparing the
spectra with those known from molecules. Both HREELS and LEED require
periodic surface structures and average over them.
STM (scanning-tunnelling microscopy). STM and related techniques like AFM
(atomic force microscopy) locally probe the surface with a possibly atomic-
scale tip and are thus able to resolve surface structures directly on atomic
scales. They do not require surface periodicity and can thus distinguish do-
mains and even identify irregular structures. Scanning techniques paved the
way for major advances of surface science in the 1990s. Binnig and Rohrer
invented the instrument a decade earlier [121] and were awarded part of the
1986 Nobel Prize in Physics [20].
Because atomic scale information on surfaces is difficult to obtain at any rate, theo-
retical studies have played an important role to complement and interpret experi-
mental results.
58 Chapter 4. Properties of Diamond
4.3.2 The individual surfaces
In the following, a brief overview of the three main surfaces of diamond and their
dominant reconstruction will be given. An extensive review on experimental and
theoretical evidence can be found in Ref. [122]. A more detailed analysis of the {111}
and {100}surfaces in conjunction with STM and HREELS results was presented
earlier in Refs. [49,50]. Growth processes on the {110}face are studied in Chapter 5.
Hydrogen is often present in the feed gasses and plays an essential role in stabil-
ising the surfaces in the conventional growth scheme. For this reason, hydrogen-
terminated surfaces will be discussed alongside clean, uncovered ones. In general,
hydrogen may be desorbed by heating surfaces above approx. 1000 ◦C. In order to
prevent re-adsorption, the surfaces are usually cleaned and subsequently investi-
gated in ultrahigh vacuum (UHV) systems.
The {111}surfaces
Figure 4.3: {111}−(1×1)
surface net (hexagonal)
The unreconstructed, {111} − (1×1)surface net is
hexagonal. Of the two cutting planes, the lower-energy
one (1db), shows one dangling bond normal to the sur-
face per surface atom. The clean surface is unstable
against graphitisation and delaminates at temperatures
above approx. 3000 ◦C [123,124,125,126]. Delamina-
tion occurs preferentially at twin boundaries and near
step edges [127]. In contrast, when hydrogenated, the
{111}−(1×1):H surface is the most stable among all
other hydrogenated reconstructions and thus routinely found in LEED patterns.
Figure 4.4: {111}−(2×1)
surface net
In the absence of hydrogen, the {111}1db face is most
stable in a (2×1)reconstruction as observed by LEED.
Numerous studies have now confirmed this reconstruc-
tion to be due to π-bonded chains as suggested by
Pandey [128], illustrated in Fig. 4.2(b), labelled “PC”.
A bond switch between the atoms of the first and sec-
ond monolayer leads to characteristic five- and seven-
fold rings underlying the chains. The chains themselves
are unbuckled, i.e., perfectly flat. Note that on the silicon
{111}surface these chains occur in an identical topology,
yet are buckled. In fact, Pandey arrived at his suggestion for diamond by extrapo-
lating the silicon results [129], taking into account the stronger ability of carbon to
form πbonds.
At an additional monolayer coverage the surface corresponds to the {111}3db cut-
ting plane and reconstructs in a (2×1)pattern into so-called Seiwatz chains [130],
see Fig. 4.2(b), labelled “SC”. Note that such chains occur as structural units within
the Pandey chains. Other reconstructions often occur during and after growth
as a result of chemisorption. For example, a (√3×√3)R30◦reconstruction was
deduced from STM data and explained theoretically as a quite strained trimer
(C3/C3H3) configuration [131], possibly stabilised by hydrogen. Of similar mesh
4.3. Diamond surface structure 59
size, and thus difficult to discern in STM by its absolute length scale alone is a
(2×2):CH3methyl superstructure. Based on energetic considerations and by re-
lating the domains to nearby (2×1)domains in STM images some of the earlier as-
signments could be revised to be more likely (2×2):CH3, as shown in Ref. [50,117].
The {110}surface
Figure 4.5: {110}−(1×1)
surface net (rectangular)
The {110}surface is the least studied one experimen-
tally. The clean surface as obtained from bulk cleav-
age has one dangling bond per atom pointing at an an-
gle of 35.3◦away from the surface normal (i.e., 90◦mi-
nus half the tetrahedral angle, arccos(−1/3) = 109.47◦).
The atoms of the top monolayer are arranged in zigzag
chains flat on the surface directed along the [¯
110]di-
rection, see Fig. 4.2(a), bottom face. These chains are
quite like those of the Seiwatz- and Pandey chains on the
{111}−(2×1)reconstructed faces. The [1¯
10]chain axis
and therefore, the zig-zag step width, which is aD/√2=
2.522 ˚
A, cf. Table 4.1, are the same in all three cases, as is obvious from Fig. 4.2(b).
However, the periodicity across the chain direction is given by the cubic lattice con-
stant aDon the {110}face, but is slightly larger, namely, aDp3/2≈1.22 aD, on the
{111}face. This makes the patterns distinguishable in experiments, e.g. using LEED
or STM.
The {100}surface
Figure 4.6: {100}−(1×1)
surface net (square)
The unreconstructed {100} − (1×1)face presents a
square net with two dangling bonds per surface atom,
each one pointing 54.7◦away from the surface nor-
mal (i.e., half the tetrahedral angle). The clean face is
rarely seen in this configuration because the dangling
bonds easily combine, leading to a (2×1)reconstruc-
tion (see below). Hydrogen saturation of both dangling
bonds (i.e., a two-monolayer coverage) has been sug-
gested [132,133] but this leads to a strong steric repul-
sion between the hydrogens. A herringbone structure of the dihydrogenated units
may relieve some of the stress [134]. Oxygen is variously able to singly bond to
surface atoms or bridge two dangling bonds maintaining the (1×1)pattern [135].
Figure 4.7: {100}−(2×1)
surface net
After annealing at moderate temperatures the {100}sur-
face reconstructs easily into (2×1)patterns by bonding
between neighbouring carbon atoms. The resulting car-
bon dimer and dimer rows are a fingerprint feature of
this surface. There is general agreement that on the clean
surface the dimer bond is a weak πdouble bond. How-
ever, upon hydrogenation, the dimer bond transforms
into a rather stretched σ-type single bond.
60 Chapter 4. Properties of Diamond
Table 4.2: Pseudo-atom parameters used for the subsequent SCC-DFTB calculations: r0and
r%
0– compression radii for wave function and density (see section 2.3.1 on page 21), Esand Ep
– onsite energies, Epsat – sum of occupied onsite energies, U– chemical hardness parameter.
The onsite energies have been calculated for free atoms with the PBE exchange-correlation
functional [139], and the Uparameter using eq. (2.28).
Type r0(aB)r%
0(aB)Es(H) Ep(H) Epsat (H) U(H/e2)
Ha2.4 3.3 −0.238600 −0.238600 0.406483
Cb2.7 7.0 −0.504892 −0.194355 −1.398494 0.364302
Nc2.2 11.0 −0.681969 −0.260728 −2.146122 0.43
Sid3.3 6.7 −0.395725 −0.150314 −1.092078 0.247609
aParameter file: hh sieck.spl (A. Sieck)
bParameter file: cc7.0 2.7.spl (M. Elstner/R. Gutierrez)
cParameter file: nn-epatch.spl (M. Elstner/M. Sternberg)
dParameter file: sisi6.7 3.30.spl (A. Sieck)
When the surface is cut at heights differing by one monolayer, the (2×1)recon-
struction occurs in two domains rotated by 90◦. As with the other surfaces, such
domains are difficult to distinguish by LEED, but relatively easy to resolve in STM,
see the review in Ref. [117] and more recent results in Ref. [136]. Between the do-
mains, characteristic step edges form, which are classified according to Chadi [137]
as SA,SB,DA, and DB(single- and double steps). The step edges are of utmost im-
portance during growth as adsorption sites and stopping point for diffusing species,
leading to layer expansion and thus promoting layer-by-layer growth [138].
Again, on the silicon {100}−(2×1)face the same topology occurs. As was the case
for the Pandey chains, the silicon dimer chains are buckled and the diamond ones
flat. The reason for the different morphology is the same, namely, carbon being able
to form πbonds.
4.4 Reference calculations on diamond bulk and surfaces
Numerous previous DFTB studies have been performed on diamond, but the SCC
variant of the method has not been extensively tested on diamond bulk and surfaces.
In this section, the DFTB results for various carbon bulk systems are presented, with
the aim of obtaining reference values for total and cohesive atomic energies, and to
verify the influence of the cell sizes for supercell calculations. The calculations also
serve as a more detailed overview of the main surface reconstructions.
4.4.1 Atomic and diatomic energies
First, Tables 4.2 and 4.3 gives the energies for the elements which enter the subse-
quent calculations. The atomic energies for H and N will be used as chemical poten-
tial in the calculation of formation energies for surfaces and substitutional defects.
The atomic energy for carbon is the reference point for cohesive energies.
4.4. Reference calculations on diamond bulk and surfaces 61
Table 4.3: SCC-DFTB reference calculations for diatomic molecules of the total energy Etot,
excluding zero-point vibrational corrections, and the equilibrium bond length d.
Type Etot (H) Etot (eV/atom) d(˚
A)
H2−0.71703 −9.7557 0.743
C2−3.16820 −43.1055 1.241
N2−4.89390 −66.5849 1.118
CNa−4.01033 −54.5634 1.166
aMulliken populations: qC=3.9188 e,qN=5.0812 e.
4.4.2 Bulk systems
Calculations of bulk diamond and graphite are summarised in Tables 4.4 and 4.5, re-
spectively. For diamond, supercells in various orientations with respect to the cubic
crystal directions were used, primarily in order to provide the necessary geome-
tries for subsequent surface calculations. As a side effect, the rundown allows to
judge the accuracy of the Brillouin-zone Γ-point sampling. The directions and side
lengths of the primitive cells and the number of repetitions making up the supercell
are given in the table in terms of the cubic lattice directions and its lattice constant
aD. The (111)-oriented supercell shows hexagonal symmetry along its zdirection.
As is customary, a conventional orthorhombic primitive cell containing two hexago-
nal cells was used in this case. The extent of the primitive cell along the (hexagonal)
[111]axis is given by three (111)double layers in ABC stacking. The (110)- and
(100)-oriented cells share the [001]axis, but differ by a 45◦rotation about this axis,
exposing either {110}or {100}faces on the other sides, respectively.
The equilibrium lattice constant for the used parameter files (i.e., the repulsive po-
tential) was obtained by sampling various scaled supercells of type (110)and (111)
containing 256 and 288 atoms, respectively. The result for the equilibrium diamond
lattice constant is aD=3.562 ˚
A. The same calculations allowed an estimate of the
bulk modulus via the definition:
B0=−V0∂2Ecoh
∂V2V0
. (4.1)
The result thus obtained by taking numerical derivatives is (541 ±10)GPa, which
is 22% above the experimental value of 442 GPa as determined by ultrasonic
waves [141,142]. A more involved fitting procedure of the total energy curve to
the Murnaghan equation of state [143,144] yielded the same value while a non-SCC
DFTB parameter set gives 487 GPa (10% above the experimental value) [145].
The total energy is converged for cells with Nat ≥216, corresponding to supercell
sizes above approx. 10 ˚
A in either extent. This supercell size is about twice the ef-
fective range of the Slater-Koster integrals for carbon. This is not only well-known
behaviour for group-IV semiconductors but also a required criterion for the nearest-
box approximation generally applied for molecular dynamics simulations here. The
cohesive (or binding) energy per atom is 9.278 eV. This is slightly higher than in
several other theoretical works [146], but lies well within the interval of variation of
about 1 eV by which even the more accurate methods differ.
For graphite, the same supercell type as for (111)diamond was used, but with
different stacking of the (now flat) graphene layers. Since the interaction between
62 Chapter 4. Properties of Diamond
Table 4.4: Reference energies of diamond bulk systems calculated with DFTB in the Γ-point
approximation at the equilibrium lattice constant aD=3.562 ˚
A. Atom parameters as in
Table 4.2. The cohesive energy Ecoh is defined as the total energy per atom minus the atomic
energy Epsat in Table 4.2; (spin polarisation, 1.125 eV/atom [46, sec. 4.3], is not included).
primitive cells
Cell type nxnynzNat Eat (H) Ecoh (eV)
[1¯
10] [11¯
2] [111]a
aD/√2aDp3/2aD/√3
3 2 3 72 −1.74173 −9.340
3 2 6 144 −1.74230 −9.355
(111) 4 3 6 288 −1.74304 −9.375
5 3 6 360 −1.74315 −9.378
[110] [1¯
10] [001]
aD/√2aD/√2aD
4 4 2 128 −1.74268 −9.366
(110) 4 4 4 256 −1.74292 −9.372
[100] [010] [001]
aDaDaD
2 2 2 64 −1.74178 −9.341
(100) 3 3 3 216 −1.74310 −9.377
a{111}double layer unit.
Table 4.5: Reference energies of graphite calculated with DFTB in the Γ-point approxi-
mation at the equilibrium hexagonal lattice constant of aG=2.473 ˚
A, i.e., at bond length
aG/√3=1.428 ˚
A. Atom parameters as in Table 4.2. The cohesive energy Ecoh is defined as
the total energy per atom minus the atomic energy Epsat in Table 4.2.
Primitive cells Layer sep.
Cell type nxnynzdequil. (˚
A) Nat Eat (H) Ecoh (eV)
[1¯
100] [11¯
20] [0001]a
aGaG√3d
single sheet 4 2 1 32 −1.73928 −9.273
single sheet 5 3 1 60 −1.74010 −9.295
AB stacking 5 3 4 ∞240 −1.74009 −9.295
AB+disp.b5 3 4 3.1 240 −1.74475 −9.422
AA+disp.b5 3 4 3.2 240 −1.74460 −9.418
asingle sheet units
bRef. [147].
graphene sheets is small at best, the calculation was started with single sheets. The
minimum reliable lateral size of a supercell for a single graphene sheet is again ap-
prox. 10 ˚
A, as for bulk diamond. The equilibrium lattice constant for graphite was
determined by sampling as aG=2.473 ˚
A.
A drawback of the DFTB method as used is that it does not correctly take into ac-
count the van der Waals interaction between individual graphene sheets. In DFTB,
the interaction between two such sheets is very weakly repulsive, as is evident
4.4. Reference calculations on diamond bulk and surfaces 63
from Table 4.5, column Eat, the atomic energy for a single sheet (60 atoms) is lower
than for hexagonal graphite (4 sheets, 240 atoms). Furthermore, the cohesive en-
ergy for carbon atoms in graphite is 9.295 eV, or about 80 meV above diamond.
Zero-point vibrational energies are not included; they would contribute (Ref. [146])
Evib(diamond) =0.1809 eV/atom and Evib(graphite) =0.1659 eV/atom. Experi-
mentally [3], graphite is more stable than diamond by approx. 30 meV. The recently
suggested inclusion of London-type dispersion interactions [147] appears to im-
prove this behaviour qualitatively, as the preliminary calculations in the last lines of
Table 4.5 indicate. The dispersion interaction lowers the cohesive energy per atom
by 120 meV, making graphite more stable than diamond by approx. 40 meV/atom.
The dispersion interaction correctly reproduces the hexagonal AB stacking to be
more stable than rhombohedral AA stacking, and roughly estimates the correct layer
distance (3.1 ˚
A, vs. 3.37 ˚
A from experiment). Note that the corrections have been in-
cluded only for these reference calculations, but not in other parts of this work.
4.4.3 Surfaces
With the bulk calculations secured, the subject of surfaces is finally at hand. The
surface models discussed here are generated from the bulk supercells presented in
the previous section by extending their periodic boundary in the zdirection to a
sufficiently large value, here, 80 ˚
A. This distance ensures that no direct interaction
takes place between adjacent surfaces thus exposed. As is customary in this type
of simulation, the dangling bonds on the new “bottom” faces were saturated with
hydrogen; the impact of this practice is minimal as we will see.
To furnish a comparison of the DFTB performance with ab initio methods, a con-
sistent set of first-principle calculations covering all the relevant bulk and surfaces
needed to be selected. The results published in Refs. [133,148,149,150] (using the
Vienna ab initio simulation package or VASP) provide such a set, and themselves
contain extensive comparisons with other ab initio methods and semi-empirical
ones, including previous non-SCC DFTB results.
In Table 4.6 total energy data are collected for the low- index surfaces and their main
reconstructions, both clean and hydrogenated. The details of this table are discussed
in the following sections.
DFTB results for the {111}surfaces
A detailed discussion of the geometries of the particularly rich set of reconstructions
on the {111}face is beyond the scope of the present work. Four surfaces were
chosen, since they represent typical cases: (a) the clean (1×1)face, resembling the
bulk cleavage situation, (b) the hydrogenated (1×1)face representing the most
stable surface among the hydrogenated ones, (c) the clean Pandey chain as most
stable clean surface, and finally, (d) the hydrogenated Pandey chain, for consistency.
The geometries of these surfaces are given in Table 4.7. For comparison with the
Pandey chains, Table 4.6 also contains the energetics of the Seiwatz chains, while a
discussion of their geometry has been omitted since it is rather similar to the Pandey
chain.
64 Chapter 4. Properties of Diamond
Table 4.6: Reference energies of relaxed diamond surfaces in two-dimensional slab geome-
try calculated with SCC-DFTB in the Γ-point approximation at the bulk equilibrium lattice
constant aD. Cell geometry as in Table 4.4.Esurf is the surface energy per site, defined as
Esurf = (Etot −NCEC−NHEH)/Nsite using the atomic energy for bulk diamond and the
hydrogen atomic energy in H2, Table 4.3.∆Erec is the reconstruction energy per site.
Surface Cells NCNHEtot Esurf ∆Erec Esurfa∆Ereca
x×y×z(H) (eV) (eV) (eV) (eV)
{111}surfaces, relaxed, saturated at bottom by hydrogen
unreconstructed, clean and hydrogenated
(1×1)5×3×6b360 30 −635.9754 2.098 2.151
(1×1):H 5 ×3×6 360 60 −649.0324 0.005 −2.826
Pandey chain, clean and hydrogenated
(2×1)−PC 5 ×3×6 360 30 −636.5737 1.555 −0.54 1.356 −0.80
(2×1)−PC:H 5 ×3×6 360 60 −648.1864 0.772 0.77 −2.132 0.69
Seiwatz chain, clean and hydrogenated
(2×1)−SC 3 ×2×6 132 12 −233.0172 2.873 0.77 2.689 0.54
(2×1)−SC:H 3 ×2×6 132 24 −238.3894 0.445 0.44 −2.403 0.42
{110}surfaces, relaxed, saturated at bottom by hydrogen
(1×1)5×3×3 180 30 −322.5128 1.743 1.66
(1×1):H 5 ×3×3 180 60 −335.1901 0.021 −2.68
{100}surfaces, relaxed, saturated at bottom by two pseudo-hydrogen per carbon
(1×1):2H04×4×4 256 64 −469.3359 −0.087c3.63
(2×1)4×4×4 256 32 −456.4161 2.288 2.12
(2×1):H 4 ×4×4 256 32 −463.2214 0.469 −2.42
aFrom Refs. [148,149,150]; Hydrogen accounted for at its atomic energy, EH=13.004 eV.
bThe numbering refers to the same cell size as the (2×1)reconstruction, i.e., a rectangular primitive
cell containing 2 hexagonal unit cells, cf. Table 4.4. The slab thickness along z={111}is given in units
of double layers.
cPseudo-dihydrogenated on both top and bottom surface.
First, one sees that the geometry of the hydrogen-saturated (1×1):H face is remark-
ably bulklike. Laterally, all atoms are on their hexagonal positions. The outer carbon
monolayer merely relaxes slightly inwards by 0.014 ˚
A. The relaxations of the other
ten monolayers in the 12-monolayer slabs are less than 0.002 ˚
A away from their ideal
separations. Equally remarkable is the surface energy for this model, given in Ta-
ble 4.6.
Subtracting from the total energy of the surface models the diamond bulk energy
(using the model of matching size in Table 4.4) as well as the appropriate multiple of
the energy of a hydrogen atom in a reservoir of H2molecules (Table 4.3), one obtains
a surface energy of just 5 meV per (111):H site. Both the absence of relaxation and
the near-vanishing surface energy clearly establish that hydrogen-passivation on
one side of a {111}slab of this type has virtually no influence on the geometry and
energetics for reconstructions on the other side.
Now broadening the view to the Pandey- and Seiwatz chain reconstructions one can
clearly deduce from their energetics (Table 4.6) that in the absence of hydrogen the
diamond (111)face is most stable in the (2×1)Pandey chain (PC) reconstruction
4.4. Reference calculations on diamond bulk and surfaces 65
DFTB Ref. [148] DFTB Ref. [148]
aD3.531 3.562
d01.529 1.542
(1×1) (1×1):H
dH1 1.12 1.12
d12 1.49 1.46b1.53 1.52
d34 1.54 1.52b1.54 1.53
∆z12 0.34 0.26 0.50
∆z23 1.64 1.67 1.54 1.54
∆z34 0.49 0.48 0.51
∆z45 1.55 1.55 1.54 1.53
(2×1)PC (2×1)PC:H
dHa 1.12 1.11
daba1.44 1.43 1.56 1.55
dac 1.54 1.53 1.58 1.57
dbd 1.54 1.53 1.58 1.57
dcd 1.55 1.54 1.59 1.58
ddf 1.67 1.63 1.59 1.59
dce 1.62 1.60 1.56 1.55
dfh 1.53 1.52 1.53 1.53
deh 1.52 1.51 1.53 1.52
deg 1.54 1.53 1.54 1.53
dfg 1.57 1.57 1.57 1.56
dgi 1.60 1.60 1.60 1.59
dhj 1.50 1.49 1.50 1.48
ϑ1121.7 107.7
ϑ2108.3 104.2
∆z10.01 0.01 0.01 0.01
∆z20.00 0.01 0.01 0.01
∆z30.04 0.03 0.04 0.04
∆z40.17 0.17 0.18 0.18
∆z50.07 0.06 0.07 0.07
∆z60.02 0.02 0.02 0.02
aundimerised, 5 ×3×6 primitive cells
bcalculated from available data
Table 4.7: Calculated geometry of
diamond {111}surfaces. d0is the
equilibrium bulk bond length, dab
etc., are the bond lengths between
atoms labelled as in the inset fig-
ure; ∆zij are the interlayer separa-
tions, and ∆ziis the intralayer buck-
ling for monolayer i.ϑiis the intra-
chain bond angle within layer i. The
units are ˚
A and degrees.
{111}−(1×1):H
H
1
2
3
4
5
6
7
-
[110]
-
[112]
[111]
{111}−(2×1)PC:H
H H
a b
c c d
e e f
g h
i j
k l
H
1
2
3
4
5
6
7
-
[110]
-
[112]
[111]
as predicted by Pandey. The surface energy is 1.6 eV per site. The surface energies
of the unreconstructed surface and the Seiwatz chain are decidedly higher, at 2.1 eV
and 2.9 eV, respectively.
The situation is very different for the hydrogen-covered surfaces. In this case, the
unreconstructed {111}−(1×1):H surface is the most stable one, with an extremely
low surface energy of 5 meV per site, as noted above, followed by the hydrogenated
SC and PC chains with 0.4 eV and 0.8 eV per site, respectively. This means that
hydrogenation will induce de-reconstruction of the Pandey chains as is indeed ob-
served [122].
The comparison of the surface energies with the ab initio results, given in the last
columns of Table 4.6, is highly satisfactory. The energies of the respective recon-
structions agree with those obtained from DFTB to within about 0.2 eV per site. Un-
66 Chapter 4. Properties of Diamond
fortunately, in the cited works, the energy of isolated hydrogen atoms rather than
bonded ones was used to renormalise energies of systems with unequal numbers of
atoms. For this reason, the energies for hydrogenated and unhydrogenated surfaces
are not directly comparable; nonetheless, those for the same numbers of atoms but
different reconstructions are.
The four typical surfaces discussed above were chosen for a statistical comparison of
geometries. Because of disparate equilibrium bond lengths in DFTB and VASP, the
geometry parameters of VASP, i.e., bond lengths and monolayer separations, must
be scaled proportionately to the same bulk bond length, to reach a common footing.
This is equivalent to expressing all reconstructed geometry parameters in terms of
the equilibrium bond length of the respective method.
With the exception of the clean surface, the root-mean-square (rms) deviation of the
bond lengths and layer separations for the uppermost layers, as given in Table 4.7, is
below 0.01 ˚
A, after scaling to the DFTB bondlength. The largest discrepancy, 0.08 ˚
A
or +30 %, occurs in the buckling of the clean {111}−(1×1)face, see ∆z12 in Ta-
ble 4.7. Given that this face is unstable at high temperatures, the equilibrium geom-
etry at zero temperature is bound to be a sensitive indicator of accuracy. At any rate,
the DFTB geometries are in general astoundingly accurate.
Various methods disagree on whether the zigzag chains of the clean Pandey chain
reconstruction are dimerised, i.e., show a Peierls distortion with alternate bond
lengths. Ref. [148] reports a tendency for dimerisation depending on k-point sam-
pling. The same tendency occurs in DFTB under the Γ-point approximation. At 4
primitive cells along the [1¯
10](chain) direction, the bond lengths d11 along the chain
are 1.39 ˚
A and 1.50 ˚
A, or ±3.8 % in terms of their average. At 5 primitive cells along
the chain direction, no dimerisation is found, as given in the table. Unequivocal
experimental evidence has yet to be brought forward [151], while theoretically, a
reliable prediction of this phenomenon requires many-particle theories.
DFTB results for the {110}surfaces
Table 4.8 compares the geometries of the clean and hydrogenated {110}surfaces,
calculated ab initio [150] and with DFTB.
The hydrogenated surface shows hardly any relaxation and is therefore equally suit-
able for hydrogen passivation as the {111}face. This is supported by the low surface
energy of just 21 meV per site, see Table 4.6. Both the {111}and {110}surface en-
ergies are of course close to the uncertainty of the DFTB method, and are subject to
the somewhat debatable normalisation for hydrogen as the total energy per atom in
H2. With the normalisation being the same, however, the meaning of the values is
that (a) the surface energy per site on the hydrogenated {111}and {110}diamond
surfaces are essentially equal and (b) hydrogen saturation is possible on either face
without significant relaxation effects in energy and geometry.
On the clean surface the relaxations are more pronounced than on the hydrogenated
one, but still rather small. The upper carbon monolayer moves inward by ∆z=
−0.15 ˚
A, and the next layer outward by ∆z=0.02 ˚
A. The chains in the top layer
are slightly straightened by a relaxation of 0.09 ˚
A towards the chain axis, resulting
in a bond angle of 121.7◦and a bond length of 1.44 ˚
A. For the set of bondlengths
4.4. Reference calculations on diamond bulk and surfaces 67
DFTB Ref. [150] DFTB Ref. [150]
aD3.566 3.531
d01.544 1.529
(1×1) (1×1):H
dH1 1.117 1.106
d11 1.444 1.419 1.526 1.508
d12 1.491 1.467 1.534 1.520
d22 1.505 1.490 1.542 1.526
d23 1.586 1.576 1.545 1.533
d33 1.541 1.526 1.545 1.530
ϑ1121.7 123.3 111.4 111.8
ϑ2113.7 113.8 109.7 109.8
ϑH1z 33.6 33.5
∆zbulk 1.26 1.25 1.26 1.25
∆z12 1.09 1.11 1.24 1.23
∆z23 1.29 1.28 1.26 1.25
∆y1±0.09 ±0.10 ±0.02 ±0.02
∆y2±0.03 ±0.03 ±0.00 ±0.00
Table 4.8: Calculated geometry of
diamond {110}surfaces. Nomencla-
ture as in the previous Table 4.7. In
addition, ∆yiis the chain straighten-
ing relaxation along y= [001], i.e.,
transversal to chains. The units are
˚
A and degrees.
{110}−(1×1):H
H
1
2
3
4
5
-
[110]
[110]
[001]
labelled dij, the rms deviations after matching the lattice constant are 0.008 ˚
A and
0.002 ˚
A, for the clean and hydrogenated surface, respectively, again an astoundingly
satisfactory agreement.
While there is general consensus on the zigzag chains of the {110}surface be-
ing slightly straightened, there has been some controversy in the literature about
a possible dimerisation or buckling [132,152]. Both of these modifications main-
tain a (1×1)surface net, which is identical to the unrelaxed surface. Maier et
al. [153] were able to conclude from LEED the prevalence of symmetric chains on
both the clean and hydrogenated surface by considering extinction rules, i.e., the
two-dimensional structure factor. UHV STM experiments [154] as well as ab initio
studies[150] also indicate a symmetric, undimerised, flat (1×1)surface, which is
just the most stable morphology in DFTB.
DFTB results for the {100}surfaces
The {100}surface presents a simple and typical (2×1)reconstruction. The DFTB
results on the geometry and a comparison to ab initio values are given in Table 4.9.
The most critical measure is the length of the characteristic surface dimer and, again,
whether it buckles or not. The DFTB result are well in line with the recent ab initio
data. On the clean surface, the dimer is rather short and similar to a double bond
in C2H4(1.33 ˚
A), while on the hydrogenated face it is elongated beyond the single-
bond analogue in C2H6(1.50 ˚
A). The dimers are symmetric and unbuckled, as in
most other theoretical and experimental studies of this surface [117,133,136].
The surface dimer causes a pinchlike strain near the surface which leads to a rather
strong buckling of the third and even fourth monolayer. The subdimer atoms along
the [110]chain direction of these layers are lower than the sub-gap atoms in the par-
allel row by 0.25 ˚
A and 0.15 ˚
A, respectively. There is hardly any buckling on the next
monolayers, but their intralayer second-neighbour distances (measured parallel to
68 Chapter 4. Properties of Diamond
DFTB Ref. [133] DFTB Ref. [133]
aD3.562 3.531
d01.544 1.529
(2×1) (2×1):H
dHa 1.114 1.10
daa 1.398 1.37 1.604 1.61
dab 1.517 1.50 1.544 1.53
dbc 1.571 1.57 1.533 1.52
dbd 1.569 1.55 1.574 1.56
dce 1.514 1.50 1.520 1.50
ddf 1.574 1.56 1.565 1.55
deg 1.523 1.527
dfg 1.570 1.562
dgh 1.542 1.542
∆y5±0.05 ±0.03
∆y6±0.03 ±0.03
∆zbulk 0.891 0.88 0.891 0.88
∆z12 0.704 0.67 0.814 0.80
∆z30.249 0.26 0.185 0.19
∆z40.145 0.16 0.108 0.11
(1×1):H0
dH0a1.086
dab 1.519
dbc 1.547
dcd 1.541
∆yi±0.00
∆z12 0.849
∆z23 0.899
Table 4.9: Calculated geometry
of diamond {100}surfaces. The
nomenclature is the same as in the
previous tables, 4.7 and 4.8.∆yiis
the lateral displacement along y=
[¯
110]. The length unit is ˚
A.
{100}−(2×1):H
a a
b b
d d c
f f e
g g
h h
H
1
2
3
4
5
6
7
-
[001]
[110]
[110]
{100}−(1×1):H0
a
b
c
d
e
f
H
1
2
3
4
5
6
7
-
[001]
[110]
[110]
the surface dimer) are still somewhat affected by the strain field of the surface.
The as-cleaved {100}face is difficult to passivate by hydrogen in the same spirit as
the {111}and {110}faces. High hydrogen coverage of the {100}face has been the
subject of several studies [133,134], but the goal here is the saturation of both dan-
gling bonds per surface atom with minimal strain. A simple placement of H atoms
at a C–H distance of 1.03 ˚
A would result in an H–H separation of 0.83 ˚
A, and thus
cause significant repulsion [Fig. 4.2 on page 56]. Fortunately, in DFTB it is possi-
ble to circumvent the problem. One usually defines a pseudohydrogen species with
its homoatomic interactions switched off while leaving all heteroatomic interactions
the same as for regular hydrogen. The resulting geometry, given in Table 4.9, is
essentially bulklike from the second monolayer inwards, with very small vertical
relaxations. This is clear indication on the viability of this saturation concept.
4.4.4 Summary
This chapter collected various data on the atomic structure of diamond surfaces. By
a systematic survey of bulk and surfaces using SCC-DFTB the method under the
given parameter set was confirmed to be quite accurate (to within about 0.2 eV/site)
and thus sufficient to describe the formation energies of the various reconstructions
4.4. Reference calculations on diamond bulk and surfaces 69
Table 4.10: Summary of stable reconstructions on diamond surfaces as calculated using
SCC-DFTB; cf. Tables 4.1 on page 55 and 4.6 on page 64.
Face clean surface hydrogenated surface
structure Esurf (eV) Esurf (%) structure Esurf (eV)
{111}1db (2×1)−PC 1.555 100 (1×1):H 0.005
{111}3db (2×1)−SC 2.873 185 (2×1)−SC:H 0.445
{110}(1×1)1.743 112 (1×1):H 0.021
{100}(2×1)2.288 147 (2×1):H 0.469
as compared to ab initio data [133,148,149,150]. Furthermore, the geometry de-
scription is, with rms deviations in the 0.02 ˚
A range, fabulously accurate. Some
weaknesses exist in the description of metallic surfaces such as the zigzag chains on
the {111}−(2×1)and {110}−(1×1)surfaces, which is partly due to problems in
the one-electron description and partly due to numerical problems with scc conver-
gence. Furthermore, the parameter set used has the slight blemish of overestimating
the bulk modulus by about 20%.
To close the circle to the bond scission model Table 4.10 summarises the most sta-
ble diamond surface reconstructions and their formation energies. Even though the
bond scission model does not account for reconstructions it does predict the correct
energetic ordering. However, since reconstructions in general reduce the number of
dangling bonds, especially on the {111}3db surface where only one bond remains
after reconstruction, the high-energy values estimated by the simpler model are sig-
nificantly off the mark.
The clean surfaces show considerable geometrical reconstructions with appreciable
reconstruction energies around 2 eV. In contrast, the hydrogenated surfaces show
smaller reconstructions and have reconstruction energies below 1 eV. Therefore,
hydrogen saturation is suitable for surface passivation, especially for the {111}
and {110}surfaces. Hydrogen saturation also induces a de-reconstruction on the
{111}1db surface.
Chapter 5
Growth of (110) Diamond Using
Pure Dicarbon
5.1 Introduction
The growth of ultrananocrystalline diamond (UNCD) films from C2precursors pro-
duced by C60 fragmentation in hydrogen-poor plasmas [4,155,156,157] has recently
attracted attention because of the high growth rates and resulting good mechanical
and electronic properties of the films. The physical properties of these films, which
are mainly determined by their microstructure, must be tuned for each of these
applications. Conventional diamond films as grown from H/CH3mixtures are
characterised by crystallites in the micrometer size range. In several recent experi-
ments [158,159,160,161] it was confirmed that the addition of argon to the plasma
allows continuous control over the crystallite size. Most importantly, at argon con-
centrations in a narrow window around 95%, the resulting films are ultrananocrys-
talline with a typical crystallite size of just 3–10 nm. Fig. 5.1 illustrates the dramatic
change in morphology as a function of the argon percentage in the feed gasses. The
transition is accompanied by a marked increase in the concentration of C2near the
surface during growth. The presence of C2and its likely contribution to the growth
process have been confirmed experimentally through characteristic intense Swan-
band radiation in both microwave plasma chemical vapour deposition [158,160]
(MPCVD) and hot-filament chemical vapour deposition [161] (HFCVD). The result-
ing UNCD films exhibit a smooth surface and a uniform morphology throughout
the film, which has a thicknesses of at least 20µm. The films are found to be hard,
have low friction, and are surprisingly wear resistant [155,162,163]. The small size
of the crystallites implies that a relatively high percentage (up to 10%) of all atoms
are located at grain boundaries, where they are πbonded [164,165] and contribute
to electrical conductivity. The UNCD films thus exhibit a combination of useful
properties typical for diamond films supplemented by electrical conductivity and
electron emissivity [166], which makes them very attractive for device applications.
There is evidence that diamond growth proceeds mainly on the (110) face [167].
Previous studies [168,169] explored initial growth stages with dicarbon on the
hydrogen-terminated (110) face without hydrogen abstraction by way of insertion
71
72 Chapter 5. Growth of (110) Diamond Using Pure Dicarbon
(%) Ar H2CH4
(a) 2 97 1
(b) 80 19 1
(c) 90 9 1
(d) 97 2 1
Figure 5.1: Cross-section SEM images of as-grown UNCD films prepared from microwave
plasmas with varying volume percentages of feed gasses, as given in the adjacent table
(reprinted from Ref.[158] with permission from Am. Inst. Phys.).
of C2into C–H bonds on the surface. In the present work, we consider deposition
steps onto the clean diamond (110) surface without hydrogen participation. We ne-
glect hydrogen because firstly, its concentration in the feed gasses is much lower
than in the conventional process and secondly, growth generally proceeds on sur-
face sites which have become reactive only after hydrogen abstraction.
Starting out from a clean surface we investigate the local atomic configuration aris-
ing from the adsorption of a C2molecule. Subsequently, more C2molecules are
deposited in the vicinity of a previous adsorbate cluster. By comparing the total en-
ergy of these structures, we identify preferred growth stages as those arising from
C2nclusters in the form of zigzag chains running along the [¯
110]direction parallel
to the surface.
We also perform simulations of the diffusion of C2molecules on the surface in
the vicinity of existing adsorbate clusters using a constrained minimisation tech-
nique [170,171,172]. The barrier heights and pathways indicate that the growth
from gaseous dicarbon proceeds either by direct adsorption onto clean sites or after
migration above existing C2nchains.
The simulation approach is briefly outlined in Sec. 5.2. The initial adsorption steps
for diamond growth are studied in Sec. 5.3, followed by an evaluation of diffusion
5.2. Simulation setup 73
barriers for C2on diamond (110) in Sec. 5.4. We then analyse some molecular dy-
namics trajectories in Sec. 5.5.
5.2 Simulation setup
The surfaces are simulated using a two-dimensional slab geometry with varying
thickness. The bottom layer is saturated by a fixed monolayer of pseudo hydrogen
atoms which do not mutually interact. The lateral extent of the cell varies from 3 ×3
unit cells up to 8 ×3 unit cells, where the bigger extent is along the chain direction.
This direction is assigned as the xaxis. For all atomic structure calculations, we used
the Γ-point approximation to sample the Brillouin zone, which amounts to a k-point
sampling with as many points as real-space unit cells. We let the atoms relax in a
conjugate gradient scheme.
The diffusion and adsorption runs are done using a constrained conjugate gradient
method. The forces during the conjugate gradient minimisation are modified by the
method described by Ciccotti et al. and by Ryckaert [170,171,172]. The centre of
mass of the C2molecule is moved in a given direction with steps of 0.1 ˚
A. Because
the constraint is to the centre of mass movement, the C2may rotate or dissociate
freely.
For initial studies we used six bulk monolayers of carbon. We find that only neg-
ligible relaxations take place in the lower two layers and therefore we use only
four carbon monolayers for the remaining calculations. The applicability of this
size of supercell was tested by comparing the minimum energy geometries of a C4
adsorbate on the (110) surface with supercells consisting of four and six monolay-
ers, respectively. The comparison between fully relaxed structures calculated with
both numbers of monolayers yields very small differences in the atomic positions of
about 0.015 ˚
A in both the lateral and vertical positions of all corresponding atoms,
including those of the adsorbate.
5.3 Adsorption and energetics of small carbon clusters on
(110) diamond
The energetically most favourable cluster configurations after repeated C2additions
are summarised in Fig. 5.2 up to C8on (110). In the following, a detailed description
of the individual deposition steps and the resulting surface cluster geometries will
be given.
5.3.1 Initial C2deposition
In order to sample the energy landscape above the clean (110) diamond surface for
C2adsorption we place a C2molecule in a vertical orientation near the surface on
a hexagonlike set of points above the atoms and bond centres of the two topmost
monolayers. By symmetry, only seven positions remain unique. The lower atom
74 Chapter 5. Growth of (110) Diamond Using Pure Dicarbon
[001]y
[110]z
[110]z
[¯
110]x
1.45
1.91
1.55
1.45
1.51
1.50
2.42
1.60
1.96
2.34
1.67
1.62
1.43
1.47
1.93
1.41
1.91
1.51
1.62
0.05
1.51
1.0
2.42
1.3
0.3
1.40 1.47
1.38
1.52
1.42 1.46 1.44 1.45
1.54 1.48
1.48
1.54
1.52
2.30
1.54 1.49 2.46
1.47
1.49 (b) C
(e) C(d) C
(c) C
(a) C2
6
88
4
Figure 5.2: Overview of relaxed structures from subsequent depositions of C2molecules
onto a clean diamond (110) surface. (a) C2, (b) C4, (c) C6, (d) C8, all along the y= [001]
direction, and (e) C8, along the x= [¯
110]direction. The numbers given are distances in
˚
A. Atoms of the adsorbate and the surface layer are shown larger. Red (dark) atoms in-
dicate the adsorbate cluster, and magenta (medium grey) atoms its first and ring-forming
second neighbours within the surface layer. Small spheres at the bottom indicate hydrogen
saturation. For (c)–(e), only partial models are shown.
−5.0 eV
−6.1 eV
−8.1 eV
(0.1 eV)
( < 0.1 eV)
Figure 5.3: Initial steps for de-
position of a C2molecule onto a
clean diamond (110) surface. The
energies are given relative to a
clean surface and a distant C2.
Energies in parentheses indicate
barriers. Atom designation is the
same as in Fig. 5.2.
5.3. Adsorption and energetics of small carbon clusters 75
Table 5.1: Energy barriers (Ebarr) and adsorption energies (Eads) of C2(and C) at the initial and final stages of growth, for varying target sites. The
“top” position means that the initial position of the C2is above a C2ncluster; “end” refers to an initial C2position above the edge along the [¯
110]
direction of the C2ncluster; “same” and “other” express whether the added C2is above the same or the adjacent (110) trough as the existing C2n
cluster on the surface. Negative indices indicate missing C atoms in an otherwise continuous chain or on the (110) surface.
Row Initial configuration Final configuration Figure Ebarr (eV) Eads (eV) Eads (eV), (Ref. [169])
1. C2+ (110) (110):C2Figs. 5.2(a), 5.3 0.1 −8.1a−7.8
2. C2+ (110):C2top (110):C4Figs. 5.2(b), 5.5 0.1 −10.3a−8.8
3. C2+ (110):C2same (110):C2,C2same Fig. 5.4(a) 0.2 −7.2
4. C2+ (110):C2other (110):C2,C2other Fig. 5.4(b) 0.0 −8.3 −7.8
5. C2+ (110):C4end (110):C6Fig. 5.2(c) 0.1 −9.6a
6. C2+ (110):C4top (110):C4,C2– 0.7 −6.5
7. C2+ (110):C6end (110):C8Fig. 5.2(d) 0.5 −8.8a
8. C2+ (110):C6top (110):C6,C2Fig. 5.6(b) 0.0 −2.0
9. C2+ (110):C6top (110):C8defect Fig. 5.6(c) 1.8 −4.9
10. C2+ (110):C6other (110):C6,C2other – 0.0 −7.8
11. C2+ (110):(2×1):C−3(110):(2×1):C−1Fig. 5.7(b) 0.6 −6.3
12. C2+ (110):(2×1):C−2(110):(2×1)Fig. 5.7(d) 0.3 −10.2a
13. C2+ (110):(2×1)(110):(2×1):C2Fig. 5.8 0.0 −7.2
14. C2+ (110):C−3(110):C−1– 0.5 −6.9
15. C2+ (110):C−2(110) – 0.4 −8.1
16. C2+ (110):C−1(110):C – 0.0b/2.6 −6.8b/−8.3
17. C + (110):C−1(110) – 0.0 −10.4
aC2nchain growth and coalescence processes
bMetastable state
76 Chapter 5. Growth of (110) Diamond Using Pure Dicarbon
of the molecule is placed about 2 ˚
A away from the nearest surface atom. A conju-
gate gradient relaxation from each of the lateral starting positions shows that the
molecule is either reflected from or adsorbed onto the surface. The reflections oc-
cur for starting positions directly above the atoms and bonds of the top monolayer.
From all of the starting positions not directly above a top-layer chain the C2molecule
is bonded and forms a bridge above the “trough” between two adjacent top-layer
chains. The deposition proceeds in two stages (see Fig. 5.3).
Initially, the C2sticks with one end to the nearest surface atom with an inclination
of about 45◦to the surface normal. There is a very low energy barrier of order 0.1 eV
towards the final adsorption stage in which the molecule bonds symmetrically in an
orientation corresponding to the diamond lattice. At this final stage, both adsorbate
atoms are 1.0 ˚
A above the top monolayer and are threefold coordinated with a bond
length of 1.38 ˚
A between them. Each adsorbate atom forms two bonds towards the
surface, one bond corresponding to the diamond lattice with a length of 1.49 ˚
A, the
other being a stretched bond of 1.91 ˚
A towards the adjacent atom in the same sur-
face chain [see Fig. 5.2(a)]. The result is the formation of two fivefold rings with a
common bond formed by the adsorbed molecule. There is a slight lateral pinch con-
traction of the surface monolayer. All its atoms remain bonded to the subsurface,
which is indented below the adsorbate by about 0.05 ˚
A.
The energetics of all depositions discussed in this chapter are summarised in Ta-
ble 5.1. For the first deposition, the binding energies of the C2molecule to the clean
(110) surface are fairly high, as can be seen in the fifth column of Table 5.1. The
energy gain is mainly the result of forming four bonds to the surface, which yields
about 2 eV for each bond; this is plausible considering its similarity to the atomic
binding energy of 2.3 eV/bond as obtained by the DFTB method for bulk diamond.
Breaking the initial triple bond within the C2molecule and the stretch of surface
bonds near the adsorbate offsets the result to give the adsorption energy of 8.1 eV,
listed in the table.
5.3.2 Addition of C2
As a next step, we studied the effect of adding another C2molecule near an existing
C2adsorbate. Obviously, the deposition onto a site more than one surface lattice
spacing away from the initial adsorbate results in two isolated clusters of similar
configuration as discussed above, unless a topological mismatch prevents the com-
pletion of the ring formation for a nearby site, as shown in Fig. 5.4(a). In this case,
the strain field introduced by the second adsorbate results in a bond switch for one
of the backbonds, with an accordingly high adsorption energy of -7.2 eV. For deposi-
tion onto a site of the neighbouring trough, the geometry and energy are essentially
the same as for the isolated case [Fig. 5.4(b)]. We note that in this case the low bar-
riers found in the initial adsorption process are no longer present, probably due to
the small local strain field induced. This effect supports the idea of a rapid spread
of such C2adsorption sites across the surface.
5.3. Adsorption and energetics of small carbon clusters 77
−8.3 eV
(0.0 eV)(0.2 eV)
−7.2 eV
(a) (b)
Figure 5.4: Continued deposition of a C2molecule onto a diamond (110) surface on sites
next to an existing C2adsorbate. The targeted neighbouring site is (a) along the x= [¯
110]
direction, and (b) along the y= [001]direction. The total energies are given relative to
initially separated components. Energies in brackets indicate barriers. Atom designation is
the same as in Fig. 5.2. Additional small markers indicate the target location.
5.3.3 C4clusters
The highest gain in energy for the second C2deposition is obtained at a site directly
above the first C2molecule [see Figs. 5.2(b) and 5.5]. The second molecule bonds
at the neighbouring diamond lattice site along the [¯
110]valley, in a diamondlike
configuration next to the first one, and forms a four-atom-long zigzag chain which
amounts to a seed for the next monolayer. At the ends of the new chain, fivefold
rings are formed similar to the ones in the C2case. The ridge of the adsorbate is
a z-shaped symmetric chain of three bonds with lengths of 1.40 ˚
A at the ends and
1.47 ˚
A in the centre. The central two atoms are raised 1.3 ˚
A above the top mono-
layer, which corresponds to a slightly outward relaxation with respect to the ideal
lattice sites. The end atoms are 0.3 ˚
A closer to the surface. The centre bond is the
common side of two adjacent sixfold rings connecting the new chain to the surface
monolayer. Topologically, the rings supporting the C4adsorbate above the surface
form a pyracylene structure, which is the basic structural element of a C60 fullerene.
A local pinch contraction of the surface monolayer occurs as in the previous case.
However, the outer bonds of the adsorbate are here, at 1.6 ˚
A, closer in length to ac-
tual bonds, at the expense of the transition to the uncovered parts of the surface, for
which the bonds are now stretched to a distance of 1.9–2.0 ˚
A. Furthermore, we ob-
serve the breaking of backbonds in the middle of the aggregate, resulting in a cluster
of sp2-like coordinated atoms arranged in a domelike configuration. The similarity
78 Chapter 5. Growth of (110) Diamond Using Pure Dicarbon
(b)
(0.1 eV)
−10.3 eV
(a)
(c)
Figure 5.5: Continued deposition of a C2molecule onto a diamond (110) surface on top of an
existing C2adsorbate, with the transition state shown from the top and side. The resulting
adsorbate cluster is topologically similar to a C60 fragment. The panels show (a) the initial,
(b) the transition, and (c) the final state. The total energies are given relative to initially
separated components. Atom designation is the same as in Fig. 5.2.
to such a highly stable configuration as a fullerene explains the fact that this C4clus-
ter represents the highest energy gain for an approaching C2molecule among the
structures considered in Table 5.1.
5.3.4 C6and C8clusters
The next C2adsorption to the C4pyracylenelike adsorbate results in a C6adsorbate,
shown in Fig. 5.2(c), with an adsorption energy of 9.6 eV. Further C2adsorption
yields an adsorption energy of 8.8 eV and a surface C8cluster, shown in Fig. 5.2(d).
Thus, the energy gain is at least 8 eV for the repeated C2surface chain addition as
shown up to C8on the surface. We expect the adsorption gain to level off at about
8.5 eV for longer chains.
The C6adsorbate has four sixfold rings in the middle and a fivefold ring at each end.
The C8adsorbate is essentially identical to it, but has of course two more sixfold
rings along its length. As in the case of the C4adsorbate, the underlying substrate
atoms are raised above the surface and flattened to sp2-like coordination together
with the adsorbed atoms.
5.3. Adsorption and energetics of small carbon clusters 79
−4.9 eV
−2.0 eV
(1.8 eV)
(c)
(b)
(a)
Figure 5.6: Continued deposition of a C2molecule onto a diamond (110) surface over a C6
adsorbate with high insertion energy. The panels show (a) the initial, (b) the transition, and
(c) the final state. The total energies are given relative to initially separated components.
Atom designation is the same as in Fig. 5.2, with the added C2molecule shown in white.
5.3.5 Adsorption barriers
The energy barriers for C2adsorption (Table 5.1, fourth column) are either zero or
very low. The highest energy barrier in the case of C2addition at the end of C8is
probably a finite size effect.
The initial C2addition at a diamond site in a “valley” on the surface makes the bonds
shorter in the neighbouring “valley.” This makes the adsorption of an additional C2
easier at the neighbouring “valley” at a diamond site, as can be seen by comparing
the energies in the third and fourth rows in Table 5.1.
5.3.6 Surface defect formation
There is a metastable energy minimum when adsorbing a C2molecule on the top of
C4or C6adsorbates (rows 6 and 8 in Table 5.1). In the metastable state (C2+ C2n)
one of the C2atoms is singly bonded to the surface C2ncomplex [see Fig. 5.6(b)]. The
bonded C2is only 2.0 eV lower in energy than free C2. We believe this metastable
minimum configuration plays a key role in the growth of (110) diamond. It enables
the diffusion of the C2molecule to the end of an existing growing C2ncomplex in
the diamond configuration. If the C2molecule is forced deeper onto a C6, there is
another metastable minimum energy structure consisting of one seven-membered
ring, one six-membered ring and six five-membered rings (row 9 in Table 5.1). The
80 Chapter 5. Growth of (110) Diamond Using Pure Dicarbon
energy gain from the gaseous C2to this non-diamond-growth favouring configura-
tion is 4.9 eV and the energy barrier towards the final metastable minimum is 1.8 eV.
We believe that similar metastable defect structures form when a C2adsorbs on C2n
with too high kinetic energy, approximately EK>3–5 eV, taking into account kinetic
contributions to the energy barriers.
5.3.7 Surface vacancy filling
When the growth proceeds further, different C2nclusters along the same (110) sur-
face trough will eventually meet and coalesce. Given that growth proceeds by C2
addition, the critical stage is reached just before coalescence, when there will be a
gap between two cluster ends corresponding to either three or two missing atoms.
Assuming the clusters are seeded at random sites, both cases have equal probability
but quite different energetics for subsequent C2additions.
In either case we see the approaching C2first in a metastable bridging configuration
from a chain end to a bare surface site, and directly between the two chain ends,
for the three- and two-site-wide gap, respectively [see Figs. 5.7(a) and 5.7(c)]. Both
added atoms remain just twofold coordinated. After overcoming barriers of 0.6 eV
and 0.3 eV, respectively, the adsorbate extends the existing C2nchain by another
atom pair [see Figs. 5.7(b) and 5.7(d)].
For the original three-atom gap, a single-atom vacancy remains next to a still just
twofold coordinated atom. Accordingly, the gain in adsorption energy is rather low
at 6.3 eV. However, the remaining single-atom gap remains reactive, and may be
filled at a later stage.
The two-atom gap yields a much higher adsorption energy of 10.2 eV, comparable to
the high gains found in the initial adsorption stages. The final stable configuration is
a continuous chain with broken backbonds for atoms on either side of the top ridge.
This structure is a bent graphene sheet with a bending radius of about 3 ˚
A. Before
discussing its properties in the next section, we briefly sketch the other variants for
surface vacancy filling.
For the final stage of surface coalescence, we considered a nearly complete surface
monolayer, with up to three consecutive atoms along the [¯
110]direction removed.
The filling of these surface vacancies results in energy gains between 6.9 and 10.4 eV,
as listed in the last rows of Table 5.1. The filling of the last single-atom vacancy can
take place either by a single C atom adsorption at the vacancy, with an energy gain
of 10.4 eV without a barrier, or by a C2addition process, which has a rather high
barrier from a metastable minimum at a gain of 6.8 eV to its completion at 8.4 eV.
Furthermore, it leaves a singly bonded C atom on an otherwise perfect (110) surface.
The energy required to desorb the extra C atom is of order 8 eV.
5.3.8 Graphitisation and rebonding
As shown in the preceding sections, the growth by the C2chain addition mech-
anism will eventually lead to coalescing chains, with broken backbonds on either
side. In order to investigate the consequences of the broken backbonds for the sur-
5.3. Adsorption and energetics of small carbon clusters 81
[001]y
[110]z
[110]z
[¯
110]x
(0.3 eV)(0.6 eV)
−6.2 eV−6.0 eV
−6.3 eV −10.2 eV
3.0 Å
(d)
(a) C (c) C−2
(b)
−3
Figure 5.7: Final stages of [¯
110]chain growth and coalescence on a diamond (110) surface:
(a) and (b) for a three-site vacancy, (c) and (d) for a two-site vacancy. Panels (a) and (c) show
the metastable adsorption phases, and panels (b) and (d) the relaxed minimum configura-
tion. Atom designation is the same as in Fig. 5.2, with the added C2molecule shown in
white. The total energies are given relative to initially separated substrate and added C2.
Energies in parentheses indicate barriers. The dashed line in (b) indicates a cut used for the
alternative view along [¯
110]in this panel. The arrow in (d) indicates an empirical bending
radius for the graphene sheet.
face stability during growth we have generated and relaxed a model in which ev-
ery other trough along [¯
110]was covered with a continuous chain, resulting in a
C(110):(2×1)reconstruction, shown in Fig. 5.8(a). The relaxed structure shows
multiple bent graphene sheets along [¯
110]. The bending orientation is that of a car-
bon nanotube of the (n,n)type, known as the armchair tube [173,174]. The bend-
ing radius of the sheets is about 3 ˚
A, which corresponds to a (4,4) tube, illustrated
in Fig. 5.8(c). Nanotubes as small as this are energetically in competition with flat
graphene sheets. However, tubes with diameters as small as 5 ˚
A have been observed
recently [175].
The structural similarity of the bent graphene sheet to a single-wall carbon nanotube
allows us to deduce the electronic structure of the bent sheet. The common feature
between the bent sheets and the armchair tubes is an atomic zigzag chain of carbon
atoms running parallel to the bending axis. Atoms contributing to these chains are
sp2+xhybridised, with x=0 for flat graphene and 0 <x1 for nanotubes. Along
either chain, the overlap of carbon porbitals normal to the sheet and tube wall, re-
spectively, results in an extended π-bonded system which forms a one-dimensional
band. This band has a negligible gap because we find here that Peierls distortion
82 Chapter 5. Growth of (110) Diamond Using Pure Dicarbon
(a)
(b)
(c)
Figure 5.8: (a) Graphitisation on a 50% covered diamond (110) surface in (2×1)reconstruc-
tion, and (b) induced rebonding after deposition of C2. Dark, grey and white atoms indicate
atoms in the top three monolayers, which are also shown larger than the remaining atoms.
The bottom monolayer is the hydrogen termination. (c) For comparison, a (4,4) single-wall
carbon nanotube with similar structure and atom designation as (a).
does not occur, quite similar to the situation in armchair nanotubes [174]. Therefore,
the atomic and electronic structure will be susceptible to distortions which break the
symmetry.
The question of stability of the bent sheets and therefore the diamond surface it-
self during growth naturally arises. It is known that graphitisation on clean dia-
mond surfaces, namely, on (111) and near (111) twin boundaries, leads to delam-
ination [123,125,176]. We find for the present configuration that it is stable and
does not debond. Furthermore, continued adsorption of C2in the valley between
two arches is possible without a barrier and, more importantly, it causes the sp2-
like atoms near the adsorbate to return to an sp3configuration and rebond in the
diamond structure, as is illustrated in Fig. 5.8(b).
Considering the electronic structure of the sheet, we can deduce the reason for the re-
bonding. Clearly, an approaching dimer will lead to a disturbance of the π-electron
system near the graphene sheet. Furthermore, at the terminus of the graphene
sheets near the surface, the π-electron system is imperfect to begin with because
sp2-hybridised atoms of the sheets are bonded to sp3-hybridised, fourfold coordi-
nated atoms at the diamond surface. Since ideal sp2- and sp3-hybridised carbon
atoms are energetically close, as the cohesive energies of graphite and diamond in-
dicate, the rebonded sp3-hybridised configuration will be lower in energy than the
disturbed sp2system near the C2molecule.
We thus reach the important conclusion that, in the C2growth regime established
here, intermediate graphitisation may occur, but the diamond growth process is sta-
bilised against extended graphitisation and delamination.
5.4. Surface diffusion of C283
Table 5.2: The diffusion barriers Ebarr and the change in total energy along the diffusion
path. The low energy barriers (rows 4 and 5) are associated with the diffusion of a vertically
aligned C2on top of a C2ncluster.
Path Ebarr (eV) (Efinal −Einit)(eV)
C2along valley 3.8 0.0
C2to other valley 3.3 0.0
C2to C4along →C63.7 −1.8
C2to end of C6→C81.0a/0.6b−3.7
C2to side of C6→C81.0a/0.6c−3.1
aNear centre of C6.
bNear end of C2n.
cFrom metastable state to final energy minimum.
5.4 Surface diffusion of C2
In order to estimate the influence of surface diffusion of C2on the growth mecha-
nism, we investigated some diffusion paths, as summarised in Table 5.2. The asso-
ciated diffusion barriers along the various C2-related diffusion paths are shown in
the second column of this table, and the gain in energy in the last column.
Generally, on the clean diamond (110) surface, the diffusion barriers are rather high,
and exceed 3 eV (Table 5.2, rows 1–3). This is easily understood from the strong
covalent bond that is formed between a C2adsorbed species and the surface once the
molecule reaches the surface. The only exceptions to such high barriers are for sites
above existing adsorbates, where the binding energy for further adsorbed species
is low to begin with. C2has the lowest diffusion barriers when it starts diffusion
on top of an existing C2ncomplex. In this case, the barrier for diffusion along the
adsorbate ridge is of the order of 1 eV. The C2remains nearly vertically oriented,
with one of its atoms bonded to one or two surface atoms throughout the diffusion
path. The energy barriers are decreasing when the chain end is approached. The
last energy barrier towards completing a chain addition step is only 0.6 eV.
We note that the energy gain attainable for a C2molecule by diffusion to the end of
an existing C2nadsorbate is considerable [see Table 5.2, last column]. These gains,
when added to the adsorption energies found for the “top” deposition sites, as listed
in Table 5.1, naturally result in the same total adsorption energies as those for the
“end” sites. We have thus found two different growth channels, converging to the
same growth mechanism. One is adsorption-dominated growth on nearly clean
surfaces with deposition directly into diamond lattice sites, and the other is diffusion
driven on surfaces densely covered with adsorption clusters.
In the diffusion studies, we found that when the C2is directed in either major dif-
fusion direction, along the chain direction of C2nor perpendicular to it, it passes
metastable minima before reaching the lowest energy configurations at the chain
ends. Along the parallel diffusion path, the last metastable minimum consists of
a C2fragment singly bonded to the surface, similar to the configuration shown
in Fig. 5.5(b). For the perpendicular direction away from the chain axis towards
84 Chapter 5. Growth of (110) Diamond Using Pure Dicarbon
a neighbouring surface valley, the metastable minimum is such that the C2atoms
complete a fivefold ring, with each one being singly bonded to the surface.
While both intermediate and final diffusion barriers are incidentally the same for
diffusion parallel and orthogonal to the C2nchain direction, the ultimate energy gain
is higher by 0.6 eV for the growth-favouring parallel diffusion; this indicates high
adsorption rates in either case, with a preference towards crystalline growth. How-
ever, the introduction of defects is quite easily possible under this regime, which
helps to explain the rather small grain size found in the final material.
5.5 Molecular dynamics depositions
Inspired by the adsorption and diffusion results discussed above we simulated the
deposition of C2on top of a C6complex on the surface directly using molecular dy-
namics (MD), though within a rather short time span of just 0.1–0.5 ps. The time
step in the MD simulations was 10 a.u., i.e., 0.24 fs, and the surface atoms (but not
the deposited C2) were coupled to a heat bath at 1000 K by scaling their velocities
with a probability of 0.1 per time step. All the atoms except the terminating hydro-
gen atoms on the bottom of the surface slab were allowed to follow the Newtonian
equations of motion. The goal of these runs was to investigate a diamond growth
reaction as follows:
Cgas
2+Csurf
2n→Csurf
2n+2. (5.1)
In a first set of experiments, each molecule was initially aligned orthogonal to the
surface and shot with a kinetic energy of 2–9 eV along the direction of the [¯
110]sur-
face chains at an angle of 80◦to the surface normal vector. The motivation of this
particular choice was that the molecule may get adsorbed at the lowest energy posi-
tion at the end of a C2ncluster, as identified in the diffusion studies. The molecules
with up to 7 eV kinetic energy resulted in a C2fragment on top of a C2ncluster, sim-
ilar to the structure in Fig. 5.5(b). At 9 eV kinetic energy the molecule was deflected
from the C2ncluster but was subsequently adsorbed as a lone C2on a neighbouring
clean surface site. While the deflection at higher impact energy seems counterin-
tuitive at first sight, it must be recalled that the incident angle is high, so that the
nature of the process is rather one of a steady dissipation of kinetic energy from the
approaching C2into the substrate until the molecule is slowed down enough to be
deposited.
In a second set of runs, a C2was aligned parallel to the surface with an initial kinetic
energy of 0.1–2 eV and a starting position on top of a C6cluster. This leads again
to the C2fragment being singly bonded to the C6surface cluster, a configuration
from which diffusion to either end is possible, as established before. There is a small
region above the edge of a C2n(n=2, 3)cluster, from where a C2, if given an initial
velocity towards the surface, can bond to the metastable minimum which precedes
the diamond position, shown in Fig. 5.6. However, in the molecular dynamics simu-
lations the barrier towards growth completion is too high on our timescale, and we
obtained solely the metastable minimum.
Chapter 6
Structure and Impurities in
Ultrananocrystalline Diamond
Grain Boundaries
6.1 Introduction
The preceding chapter presented a model for the growth of diamond surfaces by di-
carbon in the absence of hydrogen. Now, the question of the nature of grain bound-
aries in the interior of UNCD films comes into focus.
Ultrananocrystalline diamond (UNCD) films are characterised by crystallites with
an average size of just 3–10 nm, c.f. [159]. While submicron grain sizes have been
observed in conventional diamond films1the average grain size is an order of mag-
nitude above the one for UNCD films. This leads to qualitative differences and
justifies their separate classification. Most importantly, the small grain size invari-
ably leads to a significantly higher fraction of atoms at or near grain boundaries. In
UNCD films, this fraction is estimated to be as high as 10% [159], derived from sim-
ple grain surface to volume ratio considerations. Many of the physical properties of
the films show a strong dependence on this ratio, on the concentration of impurities
and on the crystallite size in general, all of which can be influenced during growth
in a controlled fashion. This opens an opportunity to design materials for specific
applications such as tribology and electronic devices.
There have been a number of studies for microcrystalline diamond grain bound-
aries [177,178,179], that have reported on the structures of twins, stacking faults
and tilt boundaries. Most of these defects have relatively low formation energies
and largely maintain tetrahedral bonding within the interface layer. They are also
quite ordered, evidenced by relatively small unit cells or the repetition of closely
repeated structural units [178,179]. In contrast, crystallites in UNCD films are ran-
domly oriented because of a much higher nucleation rate precluding the growth of
larger grains.2
1So-called nanodiamonds are also produced for abrasives applications by high-pressure high-
temperature synthesis. These are not relevant here.
2Nucleation rate, growth rate, and crystallite size are connected by the following phenomenologi-
85
86 Chapter 6. Structure and Impurities in UNCD Grain Boundaries
The focus in this study will be on high-energy high-angle (100) twist grain bound-
aries. Such twist grain boundaries are of interest for two reasons: (a) they involve
one of the growth termination faces and are thus likely to occur, and (b) they are rep-
resentative of generic grain boundaries in diamond, since the interface plane results
in two broken bonds per atom, as would be the case for most randomly oriented
planes in the diamond crystal (see Fig. 4.2 on page 56), save for vicinal planes.
The structure of high-energy high-angle (100) twist grain boundaries in diamond
and silicon was previously addressed by Keblinski et al. [164] using Tersoff inter-
atomic potentials in extensive Monte-Carlo simulations. It was found that (100)
grain boundaries are more stable against decohesion than (110)and (111)grain
boundaries. A particularly interesting observation was that up to 80% of the Σ29
grain boundary carbon atoms were threefold coordinated. The (100)Σ29 twist grain
boundary of diamond was also studied by Cleri et al. [181] using molecular dynam-
ics within a tight-binding model, allowing to discuss the electronic structure of the
grain boundaries. This study indicated that only about 40% of the grain boundary
atoms are threefold coordinated and that the electronic band structure has a broad
spectrum of gap states due to dangling bonds and double bonds. It was shown that
the states are localised, but can participate in hopping conduction. Finally, a calcula-
tion on the density-functional level was performed by Zapol et al. [182] on a stacking
fault which models a diamond grain boundary with pure sp2bonding across the in-
terface. The electronic structure of this model is characterised by the presence of
π-states in the forbidden gap. Those states are localised on the interfacial atoms.
All these theoretical studies, while disagreeing on some aspects of the coordination
statistics, unanimously agree on the fact that the width of the grain boundaries in
UNCD is extremely small, close to 0.2 nm, as is indeed confirmed experimentally by
transmission electron microscopy data [159].
In comparing the diamond results with the analogous grain boundaries for silicon,
the latter were found to be slightly thicker (relative to the lattice constant) and show
more extensive interfacial disorder. The difference is attributed to silicon being re-
stricted to sp3-type local bonding, whereas carbon is flexible between sp2and sp3-
type bonding, resulting in a higher coordination disorder at lower structural disor-
der [164,181].
The effect of impurities such as nitrogen, silicon, and hydrogen on the properties of
UNCD is of great interest. Impurities can be accidentally or intentionally introduced
into diamond during the growth process. On the one hand, small amounts of hydro-
gen and nitrogen are unavoidably present in plasmas. On the other hand, substrate
materials such as silicon contribute impurities by diffusion processes. Sometimes,
nitrogen is added to control electrical properties. These impurities have already
been well studied in bulk CVD and natural diamonds, as reviewed by Mainwood
[112]. As compared to impurities in bulk diamond, impurities in the disordered
grain boundaries can have a different concentration, geometry, and electronic struc-
cal equation (cf. [159]):
Nucleation rate [s−1cm−2]=Growth rate [cm s−1]
Crystallite volume [cm3].
Both UNCD and conventional growth rates are of the order of 1µm/h. UNCD crystallite sizes are
around 10nm, and nucleation rates are of the order of 1010 cm−2s−1.
6.1. Introduction 87
ture and may drastically change the mechanical and electrical properties of UNCD.
The distribution of impurities between grains and grain boundaries within UNCD
has not yet been determined unequivocally in experiments. The main goal of the
present study is to provide information on the geometry and electronic structure
of impurities in high-energy disordered diamond grain boundaries using the DFTB
method.
Before proceeding any further, however, a brief introduction to the relevant grain
boundary nomenclature is in order. Beginning with section 6.2, the results on chem-
ical bonding, structure, and electronic density of states of three selected twist (100)
grain boundaries are presented and discussed, followed by results on bonding, sub-
stitution energies, geometries, and energy levels of N, Si, and H impurities.
6.1.1 Grain boundary primer
Classification
Grain boundaries are a class of materials interfaces defined as the region showing
appreciable structural perturbation between otherwise more or less perfect crystal-
lites of the same material. Grain boundaries can thus be considered extended planar
defects. The crystallography of grain boundaries is derived from the considerably
more complicated case of general crystalline interfaces, which was comprehensively
described by Wolf [183].
In general, there are eight degrees of freedom (DOF) for an interface formed between
two semi-infinite crystals. Five of them are classified as macroscopic or rotational
DOF describing the direction of the interfacial plane in each semicrystal and a mu-
tual twist angle. The other three DOF are microscopic or translational ones which
describe residual atomic-scale translations between the crystal lattices. Of these, an
interface-normal translation is particularly important since it accounts for a thermo-
dynamically relevant volume change of the interface region due to stress relaxation.
All degrees of freedom, except for that of the volume change are subject to symme-
try operations of the crystals. Ordinarily, this leads to a relatively small irreducible
phase space for these DOF. If the directions of the interface planes are identical, the
interface is termed symmetrical and the number of macroscopic DOF reduces from
five to three.
Several subtly overlapping nomenclatures are in use for the description of crys-
talline interfaces, most of them concentrating on the macroscopic DOF by describ-
ing the interface in terms of twist and tilt angles and the Miller indices of the in-
terface planes. Grain boundaries in the narrow sense of homophase interfaces are
traditionally described in terms of a coincident site lattice (CSL), however. The CSL
scheme assumes a superlattice common to the two semicrystals, prior to a possible
translation. For this reason, the scheme is restricted to commensurate interfaces, i.e.,
interfaces possessing a planar unit cell. The CSL notation for a grain boundary con-
tains the Miller indices of the two interface planes and the twist angle ϑ. A twist
angle of 180◦leads to a pure tilt between the semicrystals, except for special cases
where this operation is a symmetry element of the semicrystal. A suitably adapted
nomenclature is used in the pure tilt situation, naming just tilt axis and angle. The
88 Chapter 6. Structure and Impurities in UNCD Grain Boundaries
(a)
PSfrag replacements
Σ2Σ5Σ13
Σ29
(a)
(b)
(c)
(d)
a
b
ϑ
/2
(b)
Figure 6.1: Crystallographic relation for twist grain boundaries on a square Bravais lattice
(see text). (a) Various supercells before rotation. (b) Σ13 cells after conjugate rotation and
overlay. The added thick short lines indicate the direction of bonds on a diamond (100)
interface plane.
twist or tilt angle is commonly augmented or even replaced by the inverse volume
density of CSL sites, denoted Σ.
Since most calculations performed on interfaces apply periodic boundary condi-
tions in at least the lateral directions, they require commensurability to generate a
simulation cell and therefore the CSL concept is a convenient way to select and de-
scribe supercells. Limitations in computing power restrict the selection of tractable
supercells to those with relatively small Σ.
Twist (100) grain boundaries
The grain boundaries of interest here are of the (100) twist type. The diamond (100)
face exhibits a square grid of atomic sites. Fig. 6.1 illustrates the CSL formation for
this case. Any pair of lattice vectors g=b±aleads to conjugate square superlattices
with a unit cell area Σ=a2+b2and a mutual twist angle ϑ. For uniqueness it shall
be required that aand bare incommensurate. The twist angle is trivially obtained
6.1. Introduction 89
repeat unit
PSfrag replacements
Σ2
Σ5
Σ13
Σ29
(a)
(b)
(c)
(d)
c
c/2
c/2∆c
∆c
Figure 6.2: Setup of a grain bound-
ary simulation cell with the middle
layer twisted and additional interfa-
cial separation ∆cinserted. The pe-
riodic unit in the vertical direction is
z=2(c+∆c).
from
tan ϑ
2=a
b, (6.1)
and the supercell size in terms of the diamond lattice constant is (cf. Table 4.1 on
page 55):
d=aDqΣ/2 . (6.2)
The CSL lattice for the twist grain boundary is formed by matching the two conju-
gate superlattices on top of each other, as shown in Fig. 6.1(b). Note that the broken
bonds from the diamond lattice introduce non-equivalent directions on the original
lattice and consequently reduce the symmetry of the CSL. For the same reason, the
case ϑ=90◦(Σ2) is more properly described as a stacking fault. This case has been
studied in detail by Zapol et al. [182].
The CSL scheme leads to a certain lateral periodic cell for a given surface. To achieve
periodicity in the third dimension, the twist operation must be restricted to a fi-
nite slab rather than extending through the two semicrystals. This generates two
related grain boundaries per periodic cell as illustrated in Fig. 6.2. Effectively, a
three-dimensional periodic model of the crystal with planar repeating grain bound-
aries is constructed. An alternative setup would be to use hydrogen passivation as
discussed in Chapter 4. The disadvantage of such a setup is that it would add a
separate species and introduce alien electronic states.
It should be stressed at this point that due to relatively large unit cells for the grain
boundaries studied here, the actual coincidence of atomic sites is rather unimpor-
tant, as are minute in-plane translations [164]. The reason for using the CSL de-
scription is that only certain twist angles yield two-dimensional periodic boundary
conditions with sufficiently small unit cells, both of which are prerequisite to super-
cell electronic structure calculations.
90 Chapter 6. Structure and Impurities in UNCD Grain Boundaries
Table 6.1: Structure and energetics of twist grain boundary supercells from density func-
tional tight binding calculations. aand bare the components of the generating surface cell
vector [see Fig. 6.1(a)], ϑis the twist angle, dthe lateral size of the supercell, NCis the num-
ber of carbon atoms in the periodic cell used for the calculations, ∆V=∆c/aDis the increase
in volume of the grain boundary layer per unit area as a fraction of the lattice constant, and
EGB is the grain boundary energy discussed on page 96.
Grain boundary Geometry parameters EGB
a b ϑ(deg.) d(˚
A) NC∆V(%) (J/m2) (eV/atom)
Σ5(2×2)2 4 53.1 11.28 320 14 7.80 1.55
Σ13 2 3 67.4 9.09 208 14 7.91 1.57
Σ29 2 5 43.6 13.58 464 10 7.96 1.58
6.1.2 Grain boundary supercell setup
We have selected three grain boundaries for our inquiries: Σ5, Σ13, and Σ29. Their
geometry parameters are listed in Table 6.1. The Σ5 unit cell had to be doubled in
both lateral directions to make the cell viable for minimum-image conventions and
to allow more freedom for bond reorientations. As is evident from the table and
Fig. 6.1(a), the three selected cells are fairly representative for general high-angle
twist grain boundaries.
For all models, we have chosen 16 monolayers per supercell, resulting in a separa-
tion of about 7.5 ˚
A between the two grain boundaries in the cell. The interface struc-
tures of selected models constructed with 24 monolayers essentially agreed with
those from the models with 16 monolayers, which were therefore used for all subse-
quent calculations. We recall that the O(N3)scaling of the standard DFTB formalism
would have incurred a more than threefold increase in calculation time.
An additional separation ∆cin the interface-normal direction was applied in order
to account for a volume increase in the grain boundary region. The value was op-
timised by discrete sampling with a step size for ∆cof 2% of the lattice constant.
For this setup procedure and all analyses which follow, a thermal equilibration of
the initial structures was performed at 1500 K for 0.5 ps. Subsequently, simulated
annealing was performed at gradually lower temperatures and the final structure
was optimised by a conjugate gradient method. It was verified that an increase in
equilibration temperature up to 5000 K does not qualitatively change the results (see
Table 6.2 on page 93). In particular, the GB cohesion energy remains practically the
same and statistical distributions of structural and bonding parameters are similar.
The same procedure was used in the simulations of the Σ13 twist (100) grain bound-
aries with impurities. Substitutional Si or N atoms or interstitial H atoms were
placed in the GB followed by equilibration, annealing and relaxation of the resulting
structure.
6.2. Twist (100) grain boundaries without impurities 91
Figure 6.3: Side view of the
periodic cell for a relaxed dia-
mond Σ13 grain boundary. Two
grain boundaries are shown,
oriented as in Fig. 6.2. Dark
shading indicates threefold
coordinated atoms and light
shading fourfold coordination.
Atoms in the first monolayers
of the interfaces are shown as
larger spheres. Bonds extend-
ing across the cell boundary are
shown as half bonds (interface
layer only).
6.2 Twist (100) grain boundaries without impurities
6.2.1 Structure and bonding
As detailed in the preceding section, DFTB molecular dynamics calculations and
simulated annealing were done on Σ5, Σ13, Σ29 twist (100) grain boundaries in
diamond. The final minimum-energy structure for Σ13 is shown in Fig. 6.3. The
next figure, Fig. 6.4 on the next page, gives an overview of all three selected grain
boundary structures.
First, we notice that the structural disorder in all of the grain boundaries consid-
ered is confined to two atomic monolayers constituting the interface while the rest
of the diamond crystal remains ordered. Since carbon can form both single and dou-
ble bonds, energy minimisation is achieved by rehybridisation rather than signifi-
cant atomic displacements. Therefore, a major component of disorder in the grain
boundary is manifested by the different bond-order of the carbon atoms. Our results
are consistent with TEM studies which give a grain boundary width of the order of
0.3 nm in the UNCD without impurities [158].
Calculated coordination and bond length distributions are given in Table 6.2.
Threefold-coordinated atoms (denoted by C3p and C3np in Table 6.2;p= planar
and np = nonplanar) constitute from 23% to 35% of atoms in the two grain bound-
ary planes. About one half, denoted C3p, of the three-coordinated atoms have bond
angles close to 120◦with their neighbours, forming π-bonds, and the other half,
92 Chapter 6. Structure and Impurities in UNCD Grain Boundaries
(d)(c)
(f)(e)
(b)(a)
Σ13
ϑ = 67.4°
ϑ = 53.1°
Σ5 (2×2)
ϑ = 43.6°
Σ29
Figure 6.4: Comparison of relaxed periodic cells of grain boundaries: (a,b) Σ13, (c,d) Σ5(2×
2), and (e,f) Σ29. The left panels show a side view containing two grain boundaries, and the
right panels a top-view of the four interface layers for the upper grain boundary in each
periodic cell. Atom appearance is the same as in Fig. 6.3.
6.2. Twist (100) grain boundaries without impurities 93
Table 6.2: Coordination distribution and average bondlengths for carbon atoms in two
grain boundary planes. C3p: threefold planar coordinated carbon, C3np: threefold nonpla-
nar coordinated carbon, C4: fourfold coordinated carbon, and C3/4: fourfold coordinated
carbon with the fourth bond longer than 1.84 ˚
A (but less than 1.96 ˚
A). Tis the annealing
temperature.
Grain T(K) Coordination (%) Average bond distance ( ˚
A)
boundary C3p C3np C3/4 C4 C3p-C3p C3p-C4 C4-C3np C4-C4
Σ5(2×2)1500 11.3 23.8 16.2 48.8 1.40 1.53 1.54 1.56
Σ29 1500 19.8 9.5 22.4 31.0 1.38 1.51 1.52 1.56
Σ13 1500 23.1 7.7 15.4 53.8 1.43 1.51 1.51 1.57
Σ13 3000 19.2 7.4 19.2 53.8 1.42 1.50 1.53 1.57
Σ13 5000 11.5 11.5 28.8 48.1 1.41 1.51 1.58 1.57
denoted C3np, have a tetrahedral arrangement of their three bonds, leading to a ge-
ometry configuration which is typical for a carbon atom with a dangling bond. The
results in Table 6.2 indicate that the fraction of threefold coordinated carbons does
not change drastically for the three twist angles studied, although the ratio of C3p to
C3np changes. Fourfold-coordinated carbons, C4, constitute about 50% of atoms in
the grain boundaries. This is in good agreement with 40% of threefold-coordinated
carbons in the Σ29 grain boundaries reported in previous tight-binding [181] stud-
ies. A larger fraction of 80% reported in an earlier atomistic [164] study is likely
to originate from the known tendency of the Tersoff potential to overbind radi-
cals, and thus stabilise threefold coordinated carbons. The remaining atoms in our
case, denoted C3/4, have a tetrahedral arrangement of bonds, but the fourth bond
is stretched beyond 1.84 ˚
A. The radial distribution function J(r)shown in Fig. 6.5
features these bonds next to the first maximum. Spin-unrestricted B3LYP density
functional calculations on a C2H6molecule with the C– C bond stretched up to 2.1 ˚
A
indicate that the electrons prefer energetically to remain paired and, therefore, there
will be no dangling bonds formed at these interatomic distances. The presence of
such bonds was not reported in previous calculations of carbon grain boundaries,
but similar weak carbon bonds were reported before in rapidly quenched amor-
phous tetrahedral carbon [184].
Estimates from UV Raman spectra indicate that the concentration of sp2carbon in
the UNCD films is about 5%. If one assumes a model where all of the sp2carbon
from the Raman estimates is located in the grain boundaries and none in the grains,
the resulting sp2percentage in the grain boundaries is consistent with the one found
here in the simulations.
Although UNCD and tetrahedral amorphous carbons have predominantly tetrahe-
dral bonding, there are two important differences between these materials. First,
the ordered diamond structure is still the dominant phase in UNCD (more than
90%) and the fractions reported above are related to disordered grain boundaries
only, i.e., 30% of threefold-coordinated carbons in the grain boundary translates into
less than 3% in the material. Conversely, structures reported for amorphous carbon
give an overall fraction of threefold-coordinated atoms (often summarily called sp2)
typically around 15 – 30%. Second, force-unbiased structural relaxation in the grain
boundaries of UNCD is more difficult than in amorphous carbon because the former
94 Chapter 6. Structure and Impurities in UNCD Grain Boundaries
1.4 1.6 1.8 2.0 2.2 2.4
J(r) (arb. units)
r (Å)
D1
G1G2
all atoms
GB atoms
Figure 6.5: Radial distribution function J(r)for a Σ13 grain boundary and a partial contri-
bution to J(r)of the first interface layers. For reference, nearest and next nearest neighbour
distances of diamond and graphite are indicated by arrows labelled D and G, respectively.
has constraints on the geometry imposed by the presence of extremely rigid grains
having diamond structure. About 20% of the atoms in the grain boundaries have
one of their bonds weakened and stretched, which might be the consequence of the
geometry constraints.
Across the interface, most of the bonds are single bonds between fourfold coordi-
nated carbons and bonds involving a strongly deformed four-coordinated carbon
and a three-coordinated carbon. A small number of double bonds are formed be-
tween sp2-like carbons (C3p). The structures of the interfacial planes for the Σ13
model are shown in Fig. 6.6. It is evident that in addition to bonds formed across
the interface, there are some bonds between interface atoms belonging to the same
(100) plane. These bonds between pairs of four (three)-coordinated carbons are sim-
ilar to surface dimers formed on a monohydrided (free) diamond (100) surface (see
Chapter 4). The rest of the in-plane bonds are between three-coordinated and four-
coordinated carbons, the latter having a bond across the interface. The number of
in-plane dimers appears to be somewhat higher in our studies, especially for the
Σ5 model [Fig. 6.4(d)], than found in the previous atomistic and tight-binding sim-
ulations for general high-angle grain boundaries; only at lower angles (≈10◦) do
in-plane dimers occur on the (100)faces in these simulations and are then more
properly described as dislocations [164,181]. The fact that we used a twofold en-
larged Σ5(2×2)supercell may explain the disparity. Obviously, the formation of
surface dimers within a single Σ5 mesh is incommensurate with the odd number of
atoms. While the same assertion holds for the other grain boundaries we simulated,
it has lesser impact there because of the larger meshes, see Fig. 6.1 on page 88.
An analysis of bond length distributions in the Σ13 grain boundary (cf. Table 6.2)
shows that the average bond length between two sp2atoms (C3p) is about 1.40 ˚
A
6.2. Twist (100) grain boundaries without impurities 95
(c)
(a) (b)
(d)
Figure 6.6: Overview of the individual interface planes for both grain boundaries in the
diamond Σ13 grain boundary cell shown in Fig. 6.3. (a)-(d) are the top and bottom faces of
the first and the top and bottom faces of the second grain boundary interface, respectively.
Atom appearance is the same as in Fig. 6.3.
(double bond) and the average bond length between sp2and sp3(C4) atoms is
1.51 ˚
A. The latter bonds are slightly shorter than single bonds, which on average
have a length of 1.54 ˚
A. It was found previously by Cleri et al. [181] for the case of
the Σ29 twist grain boundary that π-bonds across the grain boundary are distorted
by a dihedral torsion around the interfacial C– C bond arising from the twist angle of
the grain boundary. This distortion is stronger in the case of Σ13, which has a twist
angle of 67.4◦, vs. 43.6◦for Σ29. Larger twist angles weaken the π-bonds across the
interface and lead to a larger number of in-plane dimers, i.e., bonds between carbon
atoms belonging to the same interface plane. In the study of the π-bonded stacking
fault [182], where all C– C interfacial bonds are double bonds, a strong repulsion
between carbon atoms forming these bonds was identified to be due to the geome-
try restrictions imposed by the diamond lattice spacings. Because of this repulsion,
the formation of some in-plane dimers will lead to an energetically more favourable
geometry.
96 Chapter 6. Structure and Impurities in UNCD Grain Boundaries
6.2.2 Energetics
The grain boundary energy per atom is defined as
EGB = (Ecell −EdiamNcell)/NGB (6.3)
Here, Ecell is the total energy for the entire grain boundary periodic cell, Ediam the
diamond bulk cohesive energy per atom [Tab. 4.4 on page 62], Ncell is the number of
atoms in the grain boundary periodic cell and NGB the number of grain boundary
atoms involved (in both grain boundary layers). To convert to the more general
quantity of energy per area, denoted EGB, the value per atom must be taken twice,
apart from the normalisation for the area of a (100)−(1×1)mesh.3
The grain boundary energies thus calculated were already included in Table 6.1 on
page 90. The grain boundary energies are very similar for the three grain bound-
aries considered here. This indicates that the misorientation angle for high angle
grain boundaries is not crucial in determining its energy, as is well known from
other high-angle grain boundaries, both twist (e.g., Wang et al. [185]) and tilt (Shen-
derova et at. [178,179]). Therefore, any of the angles can be taken as a representa-
tive for a disordered grain boundary. Our grain boundary energy of about 7.9 J/m2
can be compared to about 6 J/m2obtained by Wang et al. [185] for a Σ5 diamond
grain boundary using energy minimisation and to 6.15 J/m2obtained by Keblinski
et al. [164] for a high-temperature relaxed Σ29 grain boundary. Both studies used
Tersoff potentials. Energy values obtained with Tersoff potentials are known to be
somewhat lower than tight binding results [185]. At any rate, the formation energies
for diamond are much higher than those for silicon on twist [164] and tilt [178,179]
GB.
The calculated energy of about 1.6 eV/atom is higher than typical energies of low-
angle and special grain boundaries; nevertheless it is much lower than the sur-
face energy of 2.3 eV/atom of a (100)−(2×1)reconstructed surface [Table 4.6 on
page 64]. Therefore, the high angle grain boundaries are relatively stable.
The grain boundary energy is the result of energy competition among different lo-
cal configurations in the interface; the major competing configurations are two dis-
torted single bonds and one double bond. If we imagine that any pair of atoms
across the interface is connected by a double bond instead of two single bonds in
the bulk, the resulting loss in energy per atom will be twice the energy of a sin-
gle bond (Ecoh/4=2.34 eV in the present calculations, Tab. 4.4) less the energy of
the double bond which is 4/3 times stronger. The result is Ecoh/6=1.56 eV which
matches quite well the calculated grain boundary energies. Because of the large mis-
orientation between the two planes in the interface, two single bonds will come out
strongly distorted or, if the distortion is too strong, they will compete with a more
stable and flexible double bond configuration. Some of the distorted tetrahedral ar-
rangements that we have commonly seen in the resulting structures are similar to
surface carbons on the monohydrided (100) surface, i.e., one of the initially broken
bonds belongs to a dimer formed in the interface plane whereas the other one is
directed across the interface.
3The energy conversion for the diamond (100)face is: 1 eV/atom = 2 ×2.518 J/m2, cf. Table 4.1
and Appendix A.
6.2. Twist (100) grain boundaries without impurities 97
−8 −6 −4 −2 0 2 4 6 8 10
DOS (arb. units)
E−EF (eV)
Σ13
Σ5 (2×2)
Σ29
Diamond
Figure 6.7: Electronic density of states for diamond, Σ13, Σ5(2×2)and Σ29 grain bound-
aries. Each plot is normalised by the number of atoms in the periodic cell. Energies are given
relative to the Fermi energy. Note that the fraction of grain boundary atoms is higher in the
calculation compared to experimental structures giving density of states peaks associated
with the grain boundary atoms higher than expected weight.
6.2.3 Energy levels
It is well known that one-electron Kohn-Sham energies within LDA produce forbid-
den gaps which are too narrow (known as overbinding). On the other hand, use
of our DFTB approach has the opposite effect mainly due to the use of a minimal
basis set. Thus, the LDA underestimation of the gap width is partially compensated
in our calculations. Therefore, the results reported here should be considered semi-
quantitative. For example, the gap in diamond calculated using our DFTB method
is 6.4 eV compared to the experimental value of 5.45eV.
When topological disorder is introduced and is mostly confined to the two interface
planes in the grain boundary region, the electronic structure changes profoundly. A
number of electronic states appear in the band gap, which are πand π∗states on
sp2carbon atoms as well as σ∗states associated with dangling bonds and distorted
tetrahedral arrangement of four-coordinated carbons. The identification of states by
their hybridisation (sp2,sp2+xand sp3) is based here on the arrangement of neigh-
bours relative to the carbon atom rather than a rigorous analysis of the correspond-
ing orbitals. For instance, if an atom has four neighbours, the orbitals of this atom
are assumed to be sp3hybridised. In certain situations, the hybridisation might not
be clearly sp2or sp3because of distortions of bonds and angles induced by geome-
try restrictions. Nevertheless, we retain the common hybridisation terminology for
the sake of discussion.
The density of states plots for Σ5(2×2),Σ13, Σ29 and diamond are given in Fig. 6.7.
98 Chapter 6. Structure and Impurities in UNCD Grain Boundaries
−8 −6 −4 −2 0 2 4 6 8 10
DOS (arb. units)
E−EF (eV)
sp2
sp2+x
sp3
Figure 6.8: Local density of states for a diamond Σ13 twist grain boundary. Contributions
from planar (sp2) and nonplanar (sp2+x) three-coordinated atoms and four-coordinated
atoms (sp3) in the grain boundary region are shown. Contributions from bulk atoms are
omitted.
N ,
bNc
Na
EF
EF
EF
Valence bandValence bandValence band
Dangling bonds Dangling bonds
π∗, some N, d.b.
πC , N lone pair
π∗
c
N ,
c
N ,
d
N ,
ππ
Conduction bandConduction band
π
Conduction band
π∗
C
π∗
, some Nσ∗
σ∗
σ∗
σ∗
σ∗
σ∗
(eV)
3
1
F
E−E
5
4
2
0
−1
−2
−3
(c)(a) (b)
Figure 6.9: Schematic band structure for a diamond Σ13 grain boundary: (a) without ni-
trogen impurities, (b) with one nitrogen per cell, and (c) with sixteen nitrogens per cell.
Subscripts on nitrogen levels correspond to different substitution sites (see text).
6.3. Nitrogen substitutional impurities 99
Several features in the band gap can be distinguished: (a) tails near the valence and
conduction band edges, (b) a peak at the Fermi level (EF) and (c) broad features
from 0.7 eV to 2.5 eV and from 3 eV to 4.5 eV. To distinguish contributions of orbitals
on sp2(C3p), sp2+x(C3np and C3/4) and sp3(C4) carbons, we have calculated
the local density of states for each of these types of atoms. The local densities of
states for these different contributions are shown in Fig. 6.8 for Σ13. A scheme of
the different energy levels is shown in Fig. 6.9. The states below EF−1.4 eV are de-
localised and are in the diamond valence band. Threefold coordinated atoms with
planar bond configurations (sp2,C3p) produce πstates from 1.2 eV to 0.8 eV below
the Fermi level and π∗states from 1.2 eV to 1.5 eV above the Fermi level. These π
and π∗levels are illustrated in Fig. 6.9(a). For comparison, electronic structure cal-
culations of a diamond (100) surface, which agree with experimental photoemission
data, produce πand π∗surface states in the ranges −1 to 0 eV and 2 to 4 eV, re-
spectively [133]. The states near the Fermi level (−0.5 eV to 0.6 eV) are identified
predominantly with threefold coordinated atoms in the nonplanar configuration
(C3np,sp2+x). These atoms have only fourfold-coordinated neighbours and thus
are associated with dangling bonds. The orbitals of the C3/4 atoms give rise to the
σ∗peak from 2.65 eV to 3.1 eV above the Fermi level in Fig. 6.8. The four-coordinated
atoms in the grain boundary layers as well as in the adjacent layers contribute to the
σ∗feature at 3.5 eV to 4 eV. The states above 5.3 eV are delocalised at the bulk-like
atoms and are in the diamond conduction band. As a result, the valence band and
conduction band mobility edges are 6.7 eV apart, somewhat similar to the 7.9 eV re-
ported in a previous tight-binding study [181]. The separation between the Fermi
level and the bulk-like conduction band is 5.3 eV, which is 1.1 eV lower than the cal-
culated diamond band gap. However, some of the states in the gap might be quite
delocalised and could form a new band similar to an impurity band with its own
mobility edges. In a future publication we will present a more detailed analysis of
the amount of delocalisation of the wave function in the grain boundaries.
6.3 Nitrogen substitutional impurities
6.3.1 Structure and bonding
For the study of nitrogen impurities, carbon atoms at various sites in the interface
layer of the annealed grain boundary were substituted by nitrogen and the anneal-
ing procedure was fully repeated to find the optimised position of the nitrogen atom.
The choice of the substitutional site in the disordered structure is not unique be-
cause local disorder creates a number of chemically diverse carbon atom arrange-
ments. The sites themselves are no longer strictly determined by the translational
symmetry of the system, since carbon atoms in the interface layers are significantly
displaced from their initial positions in the coincident site lattice geometry. The po-
sitions of the grain boundary carbon atoms are determined by energy minimisation
of the structure and any of the positions can be considered for impurity substitution
in the simulations.
Four different initial positions for a single nitrogen atom substitution in the grain
boundary were considered: C3np,C3p,C4 in the grain boundary layer and C4 in
100 Chapter 6. Structure and Impurities in UNCD Grain Boundaries
21
3
4
Figure 6.10: Substitution sites
selected in a Σ13 grain bound-
ary. The sites are: 1. C3np
[threefold non-planar], 2. C3p
[threefold planar], 3. C4 [four-
fold] within the grain boundary,
and 4. C4 [fourfold] in the sec-
ond layer.
Table 6.3: Coordination distribution for nitrogen and carbon atoms in two grain boundary
planes of Σ13. nNis the number of nitrogen atoms per cell, Tthe initial annealing temper-
ature, and Eis the nitrogen substitution energy (see text). The notation for carbon atoms is
the same as in Table 6.2.C2 is twofold coordinated carbon, and C3/4 is included in C3np.
nNT(K) Initial N C-coordination (%) N-coordination (%) E(eV)
position C2 C3p C3np C4 N2 N3p N3np
1 1500 C3np 0 23.1 21.2 53.8 0 0 1.9 −0.6
1 1500 C3p 0 19.2 28.8 50.0 0 1.9 0 0.7
1 1500 C4 0 23.1 25.0 50.0 1.9 0 0 1.7
1 1500 2nd layer 0 25.0 23.1 50.0 0 0 1.9 2.6
16 1500 Random 3.8 9.6 19.2 32.7 13.5 9.6 7.7
16 5000 Random 5.8 11.5 36.5 15.4 13.5 9.6 7.7
the second layer, as shown in Figure 6.10.
A summary of the nitrogen and carbon coordinations is given in Table 6.3. The
most stable nitrogen position is a three-coordinated site with nonplanar arrange-
ment of the bonds identified originally as a carbon with a dangling bond. This site
changes its geometry only slightly after the substitution. The local structure of the
optimised nitrogen geometry is shown in Fig. 6.11(a). The nitrogen atom is located
0.57 ˚
A out of the plane of its three neighbours. The small changes in the geometry
upon relaxation are an expected result considering that the nonplanar threefold co-
ordinated nitrogen configuration is favoured for substitutional nitrogen in diamond
bulk [186,187,188]. It will be shown later that the nitrogen saturates the original
dangling bond and forms a lone pair instead. As opposed to the bulk case, no bond
breaking is required. In the bulk, substitutional nitrogen moves from a Tdposition
along the [111]direction away from a nearest neighbour to a threefold coordinated
position of C3vsymmetry.
Annealed and relaxed local structures for the other three positions are shown in
6.3. Nitrogen substitutional impurities 101
(c) (d)
(b)(a)
1.50
1.58
1.48 1.54
1.56
1.49
1.96
1.45
1.46 1.90
1.47
Figure 6.11: Relaxed local structure around a substitutional nitrogen atom at various sites
in a Σ13 grain boundary. The panels correspond to the substitution sites shown in Fig. 6.10,
i.e. (a) C3np, (b) C3p, (c) C4 in grain boundary, and (d) C4 in second layer. The molecular
analogies of nitrogen bonding are shown schematically in the insets. Bondlengths in ˚
A.
Fig. 6.11(b-d). Substitution in the planar threefold-coordinated site (C3p,sp2car-
bon), shown in Fig. 6.11(b), results in a slightly puckered nitrogen configuration
with the nitrogen atom located 0.27 ˚
A out of the plane. Thus, it is similar to the
resulting geometry of substitution into a C3np site discussed above. However, its
carbon bond partner across the interface moves from a previously planar configu-
ration into a considerably more pyramidal one. In the final structure, this carbon
atom has moved to 0.42 ˚
A out of plane, compared to 0.12 ˚
A originally. The bond to
the nitrogen lengthens from 1.42 ˚
A to 1.56 ˚
A. Both processes indicate a change of the
bonding character from a double bond to a single bond and a new dangling bond
on the carbon atom. The next substitution site for nitrogen substitution, a fourfold-
coordinated site (C4) in the grain boundary layer becomes twofold coordinated
upon relaxation and the third nearest neighbour is 1.96 ˚
A away. Finally, the con-
figuration of nitrogen upon substitution in the second layer, shown in Fig. 6.11(d),
is nonplanar threefold coordinated and the fourth neighbour is 1.90 ˚
A away, quite
similar to that in the bulk diamond. Generally, the carbon-nitrogen bond lengths are
in the range 1.45 ˚
A–1.56 ˚
A. The distortions of bonds are caused by the geometrical
constraints due to the surrounding carbon lattice. These restrictions are weaker for
nitrogen in the grain boundary compared to the diamond bulk because of the local
disorder. Therefore, it is easier to accommodate nitrogen in the grain boundary.
102 Chapter 6. Structure and Impurities in UNCD Grain Boundaries
6.3.2 Energetics
The nitrogen defect formation energies were calculated as the work to bring a nitro-
gen atom from an N2reservoir to the system minus the work to remove a carbon
atom from the system to a diamond bulk reservoir:
E=EN
cell −Ecell −Ediam +1
2EN2, (6.4)
where EN
cell (Ecell) is a total energy of the grain boundary periodic cell with (without)
substituted nitrogen, Ediam the diamond total energy per atom, and EN2the total
energy of the nitrogen dimer.
The calculated formation energy of a substitutional defect in the grain boundary
for the most stable configuration is −0.64 eV compared to 4.9 eV in the diamond
single crystal (see Table 6.3). The formation energies for other nitrogen substi-
tutions are slightly higher: the planar three-coordinated site energy is 0.7 eV, the
four-coordinated carbon site energy is 1.7 eV and the energy of the site in the grain
boundary second layer is 2.6 eV. All of the grain boundary sites are preferred over
the diamond bulk site for nitrogen substitution. Since the substitution energy in the
bulk of the grain should be close to that in a diamond crystal, under equilibrium
conditions the nitrogen concentration in the grain boundary will be many orders of
magnitude higher than in the grains. Typical growth conditions are usually far from
equilibrium; however, in view of the large energy difference it is likely that nitrogen
will be concentrated in the grain boundaries.
It was found experimentally that the nitrogen concentration saturates in the UNCD
at 0.2% with increasing nitrogen content in the plasma [189]. This concentration is
much higher than in microcrystalline diamond and lower than the nitrogen satura-
tion limit in amorphous carbon. The saturation of the UNCD nitrogen concentration
is consistent with the calculated lower formation energy for nitrogen defects in the
grain boundary compared to the bulk because if nitrogen were incorporated into the
grains during the growth, the saturation limit would be much higher. Furthermore,
if some sites in the grain boundaries are more favourable for nitrogen incorporation,
their limited number would further limit nitrogen concentration in the UNCD.
6.3.3 Energy levels
Nitrogen in bulk diamond
The electronic structure of nitrogen defects in diamond has been the subject of many
theoretical and experimental studies. Single substitutional nitrogen in diamond (the
P1 centre) gives rise to a well-known carbon dangling bond state in the diamond
band gap and a nitrogen state below the valence band maximum (VBM). Our DFTB
calculations on diamond find the dangling bond state at 3.0 eV above the valence
band top compared to 3.8 eV from experiment [114] (1.7 eV below the diamond con-
duction band minimum). This state is localised mostly on the threefold-coordinated
carbon atom which is 2.04 ˚
A away from the nitrogen, which also has about 25% par-
ticipation. The atomic charge distribution of this state is illustrated in Fig. 6.12(a). A
6.3. Nitrogen substitutional impurities 103
385
C
N
384
(a) EVBM +2.96 eV =EF(b) EVBM +0.11 eV
383
382
(c) EVBM (degenerate) (d) EVBM (degenerate)
381
377
(e) EVBM −0.62 eV (f) EVBM −1.36 eV
Figure 6.12: Localisation of electronic states around a substitutional nitrogen atom in dia-
mond (P1 centre), as calculated in SCC-DFTB and discussed in the text. Mulliken population
per state, qi
afrom eq. (2.35), are visualised as sphere volume. All states are normalised. A
(110) type supercell with 192 atoms was used. The panels show the charge distribution for
states near the valence band maximum which have a significant N participation. The inset
labels denote state numbers.
104 Chapter 6. Structure and Impurities in UNCD Grain Boundaries
similar antibonding combination of the carbon dangling bond and nitrogen orbitals
was found at 1.9 eV below the conduction band by Kajihara et al. [190] in their Car-
Parinello study. The nitrogen state which Kajihara et al. reported to be at 0.15 eV
above the valence band top was ascribed to the nitrogen lone pair with an admix-
ture of valence band states. In our study, we find a similar state near the top of the
valence band [Fig. 6.12(b)]. However, a closer look reveals that there is a bonding
combination of the lone pair with the orbitals of the unique carbon neighbour (as
verified by a molecular orbital analysis of a cluster model) which is strongly hy-
bridised with other delocalised carbon states near the VBM, as can be seen clearly in
the figure from the nearly equal charges associated with all atoms. As a result, we
observed a number of states [Fig. 6.12(b-f)] with appreciable nitrogen participation
near the top of the valence band. A resonance which has the largest nitrogen con-
tribution was found at 1.4 eV below the VBM and this state will be referred to as a
lone pair.
Nitrogen in diamond grain boundaries
The DFTB calculations of this study indicate that some nitrogen substitutions in
the Σ13 grain boundary give rise to electronic states similar to those for nitrogen
defects in diamond. However, the positions and occupancies of the levels in the
case of GB substitutions are more diverse and in some cases have no analogy with
the bulk substitution. The states for different nitrogen sites are shown schematically
in Fig. 6.9(b). Substitution into the three-coordinated sites in the grain boundary
(sites aand b) gives a nitrogen lone pair state about 1.5 to 2 eV below the Fermi
level. If nitrogen is substituted into a fourfold coordinated site (site c), the nitrogen
orbitals give contributions to the carbon πstates at −1 eV and to the π∗states at
0.5 eV relative to the Fermi level, in addition to the nitrogen lone pair. Finally, the
substitution site in the second layer (site d) gives rise to an unoccupied level at about
0.8 eV above the Fermi level, which has strong carbon participation. In this case, the
Mulliken charge on the nitrogen is reduced by about 0.4 e. This charge is transferred
to the state at the Fermi level associated with a carbon dangling bond, forming a
carbon lone pair. Whereas there is no evidence of shallow donor levels near the σ∗
conduction band, the σ∗carbon dangling bond levels which are above the Fermi
level will donate electrons to the states at the Fermi level. This will lead to the
increase of the electron density at the Fermi level. The Fermi level will shift towards
the conduction band and this could result in the increase in n-type conductivity in
ultrananocrystalline diamond.
To investigate the possible shift of the Fermi level due to nitrogen substitution as
well as the interaction of nitrogen impurities, we performed a calculation with a
higher concentration of nitrogen in the grain boundaries. Thirty percent of the grain
boundary carbons were substituted by nitrogen resulting in an overall nitrogen con-
centration of about eight atomic percent. After annealing and relaxation, the nitro-
gen distribution in the grain boundaries with respect to configuration is about 50%
sp1, i.e., nitrogen with two bonds to carbons (N2), and the rest nearly equally di-
vided between sp2(N3p) and sp2+x(N3np) geometries. The sp1configuration ap-
pears only at the increased nitrogen concentration and it is similar in bonding to the
nitrogen in a pyridine molecule. However, no aromatic rings or pyridine rings were
6.3. Nitrogen substitutional impurities 105
−8 −6 −4 −2 0 2 4 6 8 10
DOS (arb. units)
E−EF (eV)
Nitrogen partial
Σ13 with Nitrogen
Σ13
∆EF
Figure 6.13: Densities of states for a diamond Σ13 grain boundary without and with 30%
nitrogen impurities. The local density of states for nitrogen atoms is given in lower panel.
found in the resulting structure. Nitrogen atoms are found to avoid each other in
the grain boundaries, which was also found in amorphous carbon nitride studies
using non-SCC-DFTB [191,192].
The electronic states associated with nitrogen are most prominent just above the top
of the valence band, near the Fermi level and in the σ∗carbon dangling bond region
around 3 eV above the Fermi level. This is illustrated schematically in Fig. 6.9(c),
as well as in the density of states plot in Fig. 6.13. The Fermi energy is shifted to-
wards the conduction band by about 0.4 eV. The states near the top of the valence
band originate from nitrogen lone pair electrons and they are significantly mixed
with carbon πstates. Due to the finite size of the periodic cell in our model, it is
difficult to determine the mobility edge since these states are not strongly localised.
There is some admixture of nitrogen states in the carbon dangling bond states at
the Fermi level and at 3 eV above. The latter states are quite similar to the states
which appear due to the interaction between a carbon dangling bond and nitrogen
in diamond bulk, where the electron is localised on the carbon with some nitrogen
participation. Since the orbitals at 3 eV are no longer the highest occupied ones as
in the case of nitrogen defects in bulk diamond, the electrons are transferred to the
lowest unoccupied states right above the Fermi level, which explains its shift rela-
tive to the nitrogen-free case. The highest occupied state was found to be associated
with a threefold coordinated carbon atom which has a Mulliken charge of −0.6 e,
i.e., a carbon lone pair. The tail of the conduction band has been significantly ex-
tended, primarily due to distortions of tetrahedral bond geometries that in this case
involve not only the immediate grain boundary atomic layer but the second atomic
layer as well. The states in the band gap are visibly more delocalised in the presence
of impurities in the grain boundaries.
106 Chapter 6. Structure and Impurities in UNCD Grain Boundaries
We propose that the conduction in UNCD occurs via grain boundaries based on the
results of our calculations. The conclusion that nitrogen is predominantly in the
grain boundaries makes it highly unlikely that the UNCD conductivity is due to ni-
trogen doping in the grains. The DFTB calculations of nitrogen substitution show
that new electronic states associated with carbon and nitrogen in the grain bound-
aries are introduced into the diamond fundamental gap. The carbon dangling bond
states hybridised with nitrogen lone pairs are above the Fermi level and donate
electrons to the carbon defect states near the Fermi level causing it to shift upward,
towards the delocalised π∗carbon band. Thus, it is reasonable to imagine that vari-
able range hopping or other thermally activated conduction mechanisms can occur
in the grain boundaries and result in enhanced electron transport. Note that this
mechanism does not require a true doping nitrogen state (fourfold-coordinated ni-
trogen or threefold-coordinated nitrogen with a double bond). Furthermore, an in-
crease in nitrogen concentration could lead to a semimetallic behaviour because of
the increase in the connectivity of sp2bonded carbon, higher delocalisation of the
π∗electronic states and broadening of the π∗band. A similar conduction mecha-
nism was discussed by Veerasamy et al. [193] based on their experimental results on
tetrahedral amorphous carbon.
6.4 Silicon substitutional impurities
Silicon impurities were generated using the same substitution procedure as for ni-
trogen. The optimised structure of a grain boundary with a substituted silicon atom
is shown in Fig. 6.14. Silicon substitution into a grain boundary site always results
in the Si atom being four-coordinated independent of the initial configuration. The
carbon – silicon bond lengths are typically 1.78 ˚
A to 2.01 ˚
A. The bond angles are dis-
torted from tetrahedral angles, with values between 84◦and 110◦. The formation en-
ergies of the substitutional Si are −0.85 eV to 1.45 eV compared to bulk reservoirs as
a reference. The energy required to insert Si into the grain boundary is much lower
than the 4.45 eV required to insert it into the diamond crystal. A grain boundary has
a much lower local density than the ordered diamond structure and, therefore, it is
easier to accommodate a larger silicon atom. The number of four-coordinated car-
bon atoms in the vicinity of Si increases. In the timescale of a simulation (about 1 ps)
neither silicon nor nitrogen atoms move between different sites. In our calculations,
we have not found electronic levels due to silicon in the diamond band gap. There-
fore, we believe that Si will not influence the electrical and electronic properties of
UNCD.
6.5 Hydrogen addition
Hydrogen incorporation into UNCD was simulated by adding four hydrogen atoms
per grain boundary unit cell at random positions in the interface of the annealed
structure. As in the previous simulations, the annealing procedure was repeated.
In the final structure, all hydrogens are bonded to carbon atoms, which, as a re-
sult, become four-coordinated. The average carbon hydrogen bond length is about
6.5. Hydrogen addition 107
Si
Si
(b)(a)
Figure 6.14: Side view (a) and top view (b) of the four interface layers of the periodic cell
for an optimised diamond Σ13 grain boundary with a silicon impurity.
1.83 1.78
1.88
1.78
1.87 1.88
1.97
1.81
1.80 1.84
1.86 1.78
2.01
1.88 1.93
1.86
(c) (d)
(b)(a)
Figure 6.15: Relaxed local structure around a substitutional silicon atom at various sites in
aΣ13 grain boundary, analogous to Fig. 6.11.
108 Chapter 6. Structure and Impurities in UNCD Grain Boundaries
4
8
2
3
4
1
2
31
576
(b)(a)
Figure 6.16: Relaxed structure of a Σ13 grain boundary with added hydrogen. (a) side view
of the periodic cell showing four added hydrogens in each of the two grain boundaries, and
(b) top view of one grain boundary. The numbers enumerate the added hydrogen atoms.
1.10 ˚
A. The average hydrogen binding energy is 3.5 eV. The number of carbon atoms
with a dangling bond (C3np) decreased from 21% to 4%. Thus, hydrogens saturate
dangling bonds of three-coordinated carbons, as expected. The hydrogen concen-
tration in UNCD films is smaller than the concentration of carbon atoms in the grain
boundaries because the films are grown under hydrogen-poor conditions and also
the overall fraction of atoms in the grain boundaries is high. We have not found any
EPR active hydrogen-related defects since the hydrogen concentration in our calcu-
lations is lower than the density of the dangling bonds. Such defects, labelled H1
and H2 centres, are known in microcrystalline CVD diamond [194,195]. Since the
total amount of hydrogen used in the calculation was lower than the number of dan-
gling bonds in the initial structure, some dangling bonds are still present. The main
change in the electronic structure is the reduction of the density of states near the
Fermi level. This finding is consistent with the fact that hydrogen is known to pas-
sivate surfaces, grain boundaries and other lattice imperfections in CVD diamond.
Small shifts of other carbon levels and a broadening of the peak at about 4 eV above
the Fermi level were found.
A particular observation from the molecular dynamics on the mobility of hydrogen
within the grain boundaries supports the passivation conclusion. At the beginning
of the simulation (at about 1200 K) the hydrogens were found to move among dif-
ferent carbon sites within the grain boundary plane, whereas at lower temperatures
(about 1000 K) their (thermal) motion is restricted to the vicinity of their carbon bond
partner. A similar migration pattern was observed in a tight-binding study of hy-
drogen in diamond [196] where real hydrogen diffusion was observed above 1700 K
and jumps between equivalent sites around the same C– C bond were observed at
1200 K. Our observation of a lower temperature of the diffusion onset might indi-
cate that grain boundaries have lower barriers for diffusion compared to the bulk
diamond. It was found experimentally that hydrogen plasma treatment suppresses
the electrochemical activity of the UNCD films [197]. This is consistent with the
6.6. Summary 109
4
0.4ps
0.1ps8
(d)(c)
(b)(a)
Figure 6.17: Molecular dynamics path of two selected hydrogen atoms during the initial
annealing phase (0.5 ps at 1200 K, and 0.4 ps at 1000 K). The MD time step is 1 fs, but one
path segment covers three steps. Characteristic transition times for site hops are indicated.
The colour of the path indicates elapsed time, progressing from red to blue over the course
of 0.9 ps. For clarity, the customary atom colouring has been subdued. (a) and (b) show
atom 4 of Fig. 6.16, and (c) and (d) atom 8. The selected atoms belong to different interfaces.
calculated saturation of the dangling bonds by hydrogen and the corresponding de-
crease of the sp2/sp3ratio of the grain boundaries.
6.6 Summary
This concludes the study of high-energy twist-angle (100)grain boundaries in di-
amond. In summary, the grain boundaries were confirmed to be atomically nar-
row, spanning only the two immediate interface monolayers. About 40%–50% of
all atoms in the interface at the grain boundaries are threefold coordinated. The
amount of three-coordinated carbons is very similar for all of the twist angles stud-
ied. The formation energies for the Σ5, Σ13 and Σ29 (100)twist grain boundaries
are also about the same, namely, 1.6 eV per interface atom.
110 Chapter 6. Structure and Impurities in UNCD Grain Boundaries
The electronic structures of clean GB’s are quite similar, being characterised by a
smaller band gap than in bulk diamond and the presence of electronic levels at and
above the Fermi level. Those gap states are localised on grain boundary atoms.
The incorporation of nitrogen impurities into the grain boundary is easier than into
bulk diamond. The nitrogen substitution energy for the GB is lower than for bulk
diamond by 2.6 eV to 5.6 eV. Nitrogen increases the amount of three-coordinated
carbon atoms in the grain boundary. A shift in the Fermi energy toward the con-
duction band of about 0.4 eV at larger nitrogen concentrations was observed. We
propose that GB conduction involving carbon π-states in the GB is responsible for
the high electrical conductivities in these films.
Silicon substitution into the grain boundary is more favourable by 3.0 eV to 5.3 eV
than into the bulk and always results in a four-coordinated Si atom in the grain
boundary. Finally, hydrogen saturates dangling bonds of carbon atoms in the grain
boundary and an average hydrogen binding energy of 3.5 eV. Similar to clean sur-
faces, hydrogen removes electronic states associated with dangling bonds from the
band gap and so lowers the film conductivity.
Chapter 7
Summary and Conclusions
The main contributions of this work are a methodological development of the
density-functional based tight-binding method and the application of the standard
SCC DFTB method to calculate the growth and structure of ultrananocrystalline di-
amond films.
The theoretical background on density functional theory was reviewed in Chap-
ter 1. This was followed by a closer look on the DFTB method in Chapter 2, where
the variational background was reviewed. The developments over the years of the
DFTB implementations up to the SCC formulation were also reviewed and juxta-
posed.
In Chapter 3, a linear-scaling implementation of the DFTB total energy calculation
including charge self-consistency was presented. Specific problems solved in com-
parison with existing schemes were of quantitative and qualitative nature. Firstly,
the interaction ranges between support functions in O(N)-DFTB are much big-
ger than in usual tight binding O(N)schemes. In the present implementation,
the longer ranging TB interaction pushes the breakeven point with diagonalisation
rather far out. Stability problems were addressed by the adaptation of existing root
finders to a nonlinear optimisation problem. Secondly, the present method handles
heteroatomic systems with a species dependent number of LCAO basis functions.
Finally, bond charges were introduced due to the overlapping basis functions. The
bond charges and, trivially, onsite charges enter the calculation of atomic charges as
well as energy and gradient corrections due to charge self-consistency contributions,
approximately doubling the time for a wave function minimisation. The implemen-
tation was verified with several numerical examples to satisfactorily reproduce en-
ergies and forces and to scale linearly for the major energy and force contributions,
apart from the quantum Coulomb problem, which plays a subordinate role. The
scheme presented is potentially suitable for both condensed matter and molecular
systems containing multiple species and naturally couples to molecular dynamics
simulations. Nonetheless, practical applications will require significantly extended
effort.
Chapter 4offered a review on more general aspects on diamond, followed by ex-
perimental data for bulk and surfaces, and closes with DFTB calculations on these
systems. The calculations serve as benchmarks corroborating the reliability of the
111
112 Chapter 7. Summary and Conclusions
Γ-point- and nearest-box approximations in DFTB. Given their fundamental impor-
tance these issues have of course been studied before, but a systematic review has
not been performed within SCC DFTB. The primary goal of the chapter was to estab-
lish reference data for bulk and surface energies. A comparison to relevant ab initio
results is quite satisfactory, yielding deviations of about ±0.2 eV/site for energies
and 0.02 ˚
A rms deviations for the geometry, relative to the respective equilibrium
lattice constants. However, the bulk modulus is overestimated by about 20%.
In Chapter 5, growth steps on a (110) diamond surface were modelled by simulating
successive depositions of C2molecules onto the surface. The initial C2adsorption
onto a clean (110) diamond surface proceeds with small barriers (0.1–0.2 eV) into the
diamond lattice site. The growth mechanism and energetics of this insertion are sim-
ilar to those on hydrogenated surfaces, as suggested previously [169]. Subsequent
C2additions on and around the initial adsorbate preferably lead to C2nchains form-
ing along the [¯
110]direction on the surface. The adsorption energies, as listed in
Table 5.1, are in the range of 7–10 eV per C2molecule at adsorption sites which lead
to chain growth, and slightly smaller, 5–7 eV, for sites leading to defected growth.
Some backbonds at the C2nchains are broken, leading to a graphenelike morphol-
ogy, if 50% coverage is reached for the added monolayer. However, the surface
remains stable at this point. If the C2deposition continues, it induces healing of the
broken backbonds due to re-formation of sp3bonds at the terminus of the graphene
sheets.
Some metastable C2defects were also found to occur during growth, which may
be responsible for starting new nucleation sites, a tendency that would explain the
rather small grain size in the experimental studies which motivated this work. Low
energy growth may be possible, if the approaching molecule has a kinetic energy
within a window of 3–5 eV in order to overcome barriers and avoid defect formation.
Direct adsorption into a diamond lattice position is possible only at the end of a C2n
cluster or onto a clean site of the surface. Upon coalescence of different C2nchains,
the remaining vacancies can be filled by the same growth species, although with
slightly higher barriers (0.3–0.6 eV) than in the initial stages.
Finally, surface diffusion studies were carried out. Because the C2adsorptions
normally result in tightly bonded adsorbate structures, interisland diffusion of C2
molecules is rather unlikely. However, an intraisland diffusion path exists, where
an added C2molecule diffuses on top of a C2nchain until it reaches its end and is
incorporated there. This diffusion behaviour supports the C2addition model that
was evident from the deposition energetics. An implication of this fact is that an
enhancement of surface diffusion rates would result in an increase in growth rate,
by virtue of diffusion of migrating C2species to and eventual incorporation into
growth sites.
Chapter 6reported on density-functional based tight-binding molecular dynamics
calculations of high-energy twist-angle (100)grain boundaries in diamond. Grain
boundaries with and without impurities were investigated. In agreement with pre-
vious atomistic and tight-binding simulations, all grain boundaries were confirmed
to be atomically narrow, spanning only the two immediate interface monolayers.
Within the grain boundaries about 40%–50% of all atoms in the interface are three-
fold coordinated. The fraction of three-coordinated carbons does not depend on the
A look ahead 113
twist angles. Formation energies of Σ5, Σ13 and Σ29 (100)twist grain boundaries
are also about the same, namely, 1.6 eV per interface atom, or 7.9 J/m2. The elec-
tronic structures are quite similar. It is characterised by a smaller band gap than in
bulk diamond and the presence of electronic levels at and above the Fermi level that
are localised on the grain boundary atoms.
Further, the incorporation of nitrogen impurities into the grain boundary is eas-
ier than into bulk diamond. The nitrogen substitution energy for the GB is lower
than for bulk diamond by 2.6 eV to 5.6 eV. Nitrogen increases the amount of three-
coordinated carbon atoms in the grain boundary. A shift in the Fermi energy toward
the conduction band of about 0.4 eV at larger nitrogen concentrations was observed.
A grain boundary conduction mechanism involving carbon π-states in the GB is
suggested to be responsible for the high electrical conductivities in these films.
Silicon substitution into the grain boundary is more favourable by 3.0 eV to 5.3 eV
than into the bulk and always results in a four-coordinated Si atom in the grain
boundary. Hydrogen saturates dangling bonds of carbon atoms in the grain bound-
ary and the average hydrogen binding energy is 3.5 eV. Hydrogen removes elec-
tronic states associated with dangling bonds from the band gap and so lowers the
film conductivity.
A look ahead
The issue of conductivity in the UNCD material is one of the major forces which
drives materials research in this area. Recent evidence underpins the pivotal role
of nitrogen during growth and for the conductivity mechanism. These issues could
only be scratched here. Work is under way towards a closer look of the role of
nitrogen during growth [198].
The actual conduction mechanisms could only be addressed indirectly here. A fur-
ther development will attempt to investigate them employing tools from the field of
disordered electronic systems [199,200].
Appendix A
Atomic Units Reference
In molecular dynamics simulations one deals with physical quantities on atomic
scales. Under such circumstances, the use of units from the Syst`eme International
d’Unit´es (SI), which are defined in terms of macroscopic quantities, would lead
to unwieldy numerical values.1Appropriate decimal fractions of SI units, e.g.,
the nanometre, or non-SI units like the electron volt (which is officially accepted),
or the angstrom (which is merely tolerated), serve much better to present results
in digestible numbers. However, these units do not fulfil the practical demand
that numerical calculations be decoupled from specific values of fundamental con-
stants [12]. In order to achieve this, one divides the governing equations by reference
quantities formed from fundamental constants. In non-relativistic quantum theory,
these reference quantities form the body of atomic units2(a.u.). Bohr’s model of the
hydrogen atom provides the atomic units relevant for dynamical simulations:
Length: aB(also a0), the radius of the first Bohr orbit,
Time: τB, the inverse of the (circular) frequency of this orbit,
Mass: me, the mass of an electron,
Energy: Eh, the Coulomb energy of two elementary charges at distance aB.
The last quantity is called Hartree energy. Historically, the value of the ionisation
energy for the hydrogen ground state has been named a Rydberg, and differs from
the Coulomb energy in Bohr’s model by the kinetic energy of revolution, which is
half of the former’s magnitude, such that 1 Ry = 0.5 H. The abbreviation ‘a.u.’ is
used as common designation for either one of these units. To avoid confusion, the
unit Ry should be discouraged.
The tables in this section give the recommended values of constants (Tab. A.1), de-
rived atomic units (Tab. A.2), and energy conversion factors (Tabs. A.3 and A.4). A
compact version of these tables is also available on the web [201].
1The trend in national standards laboratories of actually realising SI units by quantum objects does
not remedy this incongruity in the least.
2Basic quantities like length, time and energy may be formed from a set of fundamental constants
either including or excluding the speed of light. Those variants in which the speed of light does enter
are called natural units. They play a role mainly in cosmology.
115
116 Appendix A. Atomic Units Reference
Table A.1: List of selected fundamental constants of physics and chemistry based on the
CODATA recommended values of the 1998 adjustment [202]. For brevity, the accuracy has
been reduced in general to 7 significant digits and uncertainty data has been omitted. The
values as shown are unaffected by uncertainty, except for k(±2 in the last digit). Full data,
including uncertainties, are to be found in Ref. [202] and on the web [203].
Quantity Symbol Numerical value Unit Note
speed of light in vacuum c299 792 458 m/s (per definition)
magnetic constant µ04π×10−7V s/A m (per definition)
electric constant, 1/µ0c2ε08.854 188×10−12 A s/V m
Planck constant h6.626 069×10−34 J s 4.135 667×10−15 eV s
h/2π¯h1.054 572×10−34 J s 6.582 119×10−16 eV s
elementary charge e1.602 176×10−19 A s
Boltzmann constant k1.380 650×10−23 J/K 8.617 342×10−5eV/K
Avogadro constant NA6.022 142×1023 mol−1
atomic mass unit u 1.660 539×10−27 kg 1 822.888 me
electron mass me9.109 382×10−31 kg 510.999 keV/c2
proton mass mp1.672 622×10−27 kg 938.272 MeV/c2
neutron mass mn1.674 927×10−27 kg 939.565 MeV/c2
fine structure constant α7.297 353×10−31/137.036 00
Rydberg constant R∞10 973 731.6 m−1(13.6057 eV)/hc
Table A.2: Values of some atomic units in SI and non-SI units. Source, accuracy and uncer-
tainty as in the previous table.
Quantity Symbol Numerical value Unit Note
Bohr radius (bohr) aB0.529 177×10−10 m 0.529 177 ˚
A
Hartree energy (hartree, H) Eh4.359 744×10−18 J 27.2114 eV
a.u. of time, τB¯h/Eh2.418 884×10−17 s 0.024 189 fs
In Bohr’s model, the radial force Frad =merω2for an electron circling around a
proton arises from their Coulomb attraction Fel =e2/4πε0r2. Further, the angular
momentum l=mer2ω=n¯his quantised. From these postulates, one easily obtains
the radius an, the angular frequency ωnand the total (Coulomb plus kinetic) energy
Enof the nth Bohr orbit as:
an=4πε0n2¯h2
mee2=n2α
4πR∞
;aB=a1
ωn=n¯h
mea2
n
=4πR∞c
n3;τB=1
ω1
En=1
2
e2
4πε0an
=R∞hc
n2;Eh=2E1
(A.1)
Historically, these units have been derived spectroscopically from the Rydberg con-
stant R∞=mee4/8h3ε2
0c=α2mec/2hand the Sommerfeld fine structure constant
α=e2/4πε0¯hc, as shown.
117
Table A.3: Conversion between units of energy and values of energy equivalents derived
from the relations E=hc/λ=hν=kT. Source, accuracy and uncertainty as in Tab. A.1. ‘H’
is the Hartree unit. The non-SI unit ‘kcal/mol’ is obsolete but included here for reference.
Frequently consulted values are highlighted in bold.
J eV H kcal/mol∗
1 J 1 6.241 51×1018 2.293 71×1017 1.438 36×1020
1 eV 1.602 18×10−19 1 3.674 93×10−223.0451
1 H 4.359 75×10−18 27.2114 1 627.090
1 kcal/mol∗6.952 34×10−21 4.339 31×10−21.594 67×10−31
(1 cm−1)hc 1.986 45×10−23 1.239 84×10−44.556 34×10−62.857 23×10−3
(1 Hz)h6.626 07×10−34 4.135 67×10−15 1.519 83×10−16 9.530 70×10−14
(1 K)k1.380 65×10−23 8.617 34×10−53.166 82×10−61.985 88×10−3
∗1 calIT = 4.1868 J (International Table calorie, exact) continued .. .
(continued)
cm−1Hz K
1 J 5.034 12×1022 1.509 190×1033 7.242 96×1022
1 eV 8 065.54 2.417 989×1014 1.160 45×104
1 H 2.194 75×1056.579 684×1015 3.157 75×105
1 kcal/mol1349.989 1.049 241×1013 503.556
(1 cm−1)hc 1 2.997 925×1010 1.438 78
(1 Hz)h3.335 64×10−11 1 4.799 24×10−11
(1 K)k6.950 36×10−12.083 664×1010 1
Table A.4: Energy equivalents for electromagnetic radiation of selected wave lengths for
visible light, and selected energies for infrared (IR) and ultraviolet (UV) [1, “Colour”].
Energy Wave number Wave length Colour
E(eV) λ−1(cm−1)λ(nm) (typical)
1.00 8 066 1240 IR
1.77 14 286 700 Red (limit)
1.91 15 385 650 Red
2.07 16 667 600 Orange
2.25 18 182 550 Green
2.48 20 000 500 Cyan
2.76 22 222 450 Blue
3.10 25 000 400 Violet (limit)
5.00 40 328 248 UV
Appendix B
Approximations for Exchange and
Correlation Energies
Practical applications of the DFT apparatus hinge on suitable approximations of the
exchange-correlation functional Exc[n(r)]. It should be recognised that the contribu-
tion of Exc to the total energy (1.11) is by design a rather small, albeit significant one.
Without it, chemical bonds would be much weaker [204].
To study the electron interaction effects, Harris and Jones suggested in 1974 [205]
a fictitious many-particle Hamiltonian b
Hλas interpolation between non-interacting
and fully interacting electrons. The Coulomb-interaction is scaled by a coupling
parameter 0 ≤λ≤1 as λ/|ri−rj|, with λ=0 representing the non-interacting
system and λ=1 the physical system. An additional potential Vλ(r)is introduced
such that the corresponding density nλ(r)always equals the physical density, n(r):
nλ(r)≡nλ=1(r) = n(r). (B.1)
The exchange-correlation energy is the result of the interaction of an electron with
its xc hole around it. The xc hole describes the reduction of the average density n(r0)
due to the presence of an electron at r:
nxc(r,r0;λ) = g(r,r0;λ)−n(r0), (B.2)
where g(r,r0;λ)is the pair correlation function for the system with interaction pa-
rameter λ, i.e., the conditional density at r0given that one electron is at r. Conse-
quently, the xc hole is normalised:
Znxc(r,r0;λ)dr=−1 . (B.3)
Several authors [205,206,207] have found an exact description for Exc in the form of
the Coulomb interaction:
Exc[n(r)] = 1
2ZZdrdr0n(r)¯
nxc(r,r0)
|r−r0|, (B.4)
118
119
where ¯
nxc is a λ-averaged xc hole,
¯
nxc(r,r0) = Z1
0
dλnxc(r,r0;λ). (B.5)
Kohn pointed out in 1996 [208] that a many-electron system is “near-sighted”, i.e.,
correlation effects have a microscopically limited range, typically of the order of the
Fermi wavelength λF= [3π2n(r)]−1/3. This fact a posteriori justifies many attempts
at series expansions and parametrisations for eq. (B.4), e.g. [25,27,207,209,210].
The simplest and unexpectedly accurate expansion is the Local Density Approximation
(LDA):
ELDA
xc [n(r)] = Zεxc(n(r))n(r)dr. (B.6)
where εxc(n)is the exchange-correlation energy per electron in a uniform electron
gas of density n(r) = const. The energy εxc(n)is a function of only the local density
value, and no longer a functional of the global density distribution. εxc(n)may be
split into an exchange and a correlation contribution:
εxc(n(r)) = εx(n(r)) +εc(n(r)) . (B.7)
The exchange part follows directly from Hartree-Fock theory for the homogeneous
electron gas as follows (in atomic units):
εx(n(r)) = −3
43
πn(r)1/3
(B.8)
For the effective potential (1.20) we have, by eq. (1.22) from (B.6):
Vx(r) = εx(n(r)) + n(r)dεx(n(r))
dn(r)r
=−3
πn(r)1/3
. (B.9)
The correlation part is much more involved and is only available numerically from
many-body calculations of the homogeneous electron gas as energy residue after all
known contributions have been subtracted. Very accurate interpolation expressions
(up to 0.1%) have been given early by Hedin and Lundqvist [211] and later, interpo-
lating quantum Monte-Carlo simulations of Ceperley and Alder [212], by Perdew
and Zunger [209]. These expressions are conveniently summarised in a recent in-
troductory review [34]. To illustrate a simple case, the Hedin-Lundqvist expression
reads for metallic densities [41]:
VHL
xc (r) = (1+0.0368 3
4πn(r)1/3
ln "1+21 4π
3n(r)1/3#)Vx(r). (B.10)
LDA is only the first step in the expansion of Exc. This is followed by parametrisa-
tions which retain the spin degrees of freedom as independent variables throughout
the theory. This Local Spin Density Approximation (LSDA1) reads:
ELSDA
xc [n↑(r),n↓(r)] = Zεxc(n↑(r),n↓(r))n(r)dr. (B.11)
1also abbreviated as LSD
120 Appendix B. Approximations for Exchange and Correlation Energies
LSDA is only slightly more complicated than LDA and shows marked improve-
ments or is even essential when the systems considered contain unpaired spins [34].
Despite its gross simplification, L(S)DA works remarkably well. While ionisation
energies of atoms, dissociation energies for molecules and cohesive energies for
solids are reproduced with errors in the 1 eV range, geometry parameters like bond
lengths are described particularly well. This is even more surprising given the fact
that the electron density in those systems does not at all represent a slowly varying
function, as would be deemed necessary. For a long time, there had been no strin-
gent explanation for the success of this approach, which was the reason for its re-
luctant appreciation by the quantum chemistry community. Only recently, Burke et
al. [213,214] pointed out that LSDA fulfils several conditions for the xc hole, among
them the sum rule (B.3) and certain gradient conditions. Furthermore, as is evident
from eq. (B.4), Exc only depends on the spherical average of the xc hole ¯
nxc, which is
well-approximated in LSDA.
Taking into account gradient information, and thus leaving the domain of strictly
local functions, leads to Generalised Gradient Approximations (GGA). Commonly ac-
cepted GGA’s are spin-dependent and employ a function fof the spin densities
coupled with the magnitude of their gradients,
EGGA
xc =Zfn↑(r),n↓(r),|∇n↑(r)|,|∇n↓(r)|dr. (B.12)
Early parametrisations of GGA’s have been rather disappointing. Recent improve-
ments in their functional form, up to essentially parameter-free expressions by
Perdew, Burke and Ernzerhof [139] (known as PBE, or “GGA made simple”), have
led to considerable improvements in the description of total energies. The remain-
ing errors are about twice as high as those obtained from the best wave function
methods. The relative simplicity of GGA’s have made them an accepted compan-
ion to wave function methods in quantum chemistry for systems in which their size
prohibits accurate calculations of the Hartree-Fock type.
It must be noted that neither LSDA nor GGA is applicable in systems with sepa-
rated components which typically interact via van der Waals forces. For instance,
the seemingly simple problem of the inter-layer interaction of graphite is not well
described by either of these methods. Likewise, some details of the adsorption of
aromatic molecules on graphite are difficult to model satisfactorily. In most practical
cases, however, minor empirical or first-principle [58] corrections suffice to remedy
such shortcomings.
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Acknowledgements
It’s is not, it isn’t ain’t, and it’s it’s, not its, if you mean it is. If you don’t,
it’s its. Then too, it’s hers. It isn’t her’s. It isn’t our’s either. It’s ours, and
likewise yours and theirs.
– Oxford University Press, Edpress News
This place is customarily reserved for notes of a more private nature. They are es-
sential and will duly follow below. First, however, professional acknowledgements
are in order. I would like to thank all persons involved in the preparation of this
manuscript.
The material presented in Chapter 3has been published as Ref. [54]. I am most
grateful to Giulia Galli for her interest, welcoming support, and guidance in the
preparation of this material. I thank the Institut Romand de Recherche Num´
erique
en Physique des Mat´
eriaux (IRRMA) for enabling several visits to Lausanne, not in
the least allowing me to pick up a bit of French from the locals.
The work presented in Chapter 5has been published in Ref. [215] and has been
carried out with the support of Helsinki University of Technology and The Center
for Scientific Computing in Finland. I thank my colleague Markus Kaukonen from
the HUT Laboratory of Physics for the chance to bring in my experience on diamond
surfaces and apply DFTB to the problem of growth and his significant contributions
to the barrier calculations for the adsorption and diffusion studies. I also thank
Prof. Risto Nieminen and Markus for welcoming and supporting me in Helsinki and
the discussions we had there. While in Helsinki, among quite a few more things on
Finish culture, I learned to appreciate modern art in the then newly opened Kiasma.
I am also grateful to Dieter Gruen and Gotthard Seifert for many fruitful discussions
concerning the transient role of the graphene sheets. Gotthard always answered my
theory questions thoroughly and with patience.
Last, but not least, Chapter 6is the result of a very stimulating and ongoing collab-
oration with my colleague Peter Zapol and the groups of Larry Curtiss and Dieter
Gruen at the Chemistry Division of Argonne National Laboratory. I am indebted
to these distinguished individuals for calling me aboard their undertaking in the
exciting field of ultrananocrystalline diamond. Material in this chapter has been
presented in various meetings [198,216,217] and will presently appear as publica-
tion [218].
Most of the activity over the years would have been impossible without the financial
135
support from the Deutsche Forschungsgemeinschaft. Specifically, the roots of my
interest in diamond go back to being part of the tri-national D-A-CH collaboration
1991–2000 on Superhard Materials established by the DFG and its sister organisa-
tions in Austria (A) and Switzerland (CH).
Finally, these collaborations would not have come into being without the instiga-
tion and support by my thesis advisor Prof. Thomas Frauenheim. He helped push
forward the use of the methodology into the areas here presented and provided the
prerequisite machinery and contacts. I am grateful to Thomas for setting me on this
track and allowing me to pursue own interests in the field of computing, even if
they consumed considerable resources occasionally.
I would like to express gratitude to Walter Lambrecht and John Angus of Case West-
ern Reserve University who allowed me to share in their theoretical and practical
experience in the very early stages of this undertaking. It is by the suggestion of
John that Fig. 4.1 on page 53 came into being some years ago (quite a few, actually!)
I also wish to thank Walter and his family for their warm welcome and re-kindling
in me an appreciation for the silver screen.
Much of the methodological work was performed while the group was located in
Chemnitz. I thank Prof. Schreiber and all my former colleagues there for providing
a good working atmosphere during that time and beyond.
The final stages of this work were performed in Paderborn. I thank my office neigh-
bour Uwe Gerstmann not only for the admirable tangram he performed with our fur-
niture and so organising the space but also for enduring my presence over the years
and being a fountain of insight extending far beyond the field of defects in semi-
conductors. Prof. Overhof was always responsive when I approached him about
physical and meta-physical issues, for which I express my gratitude. H.-J. Wagner
left a lasting impression on me by tactfully pointing out quite a number of literature
resources which helped to refine not only the bibliography in this work but also my
personal library and interests about the far distant past. Our secretary, Astrid Can-
isius quickly and efficiently took the reigns of office into her hands once she joined
us. I highly appreciate her help in expertly relieving the pain in organising official
matters. I also express my gratitude to Christof and Thomas K¨
ohler for their crit-
ical reading of the manuscript, helping to dissolve some obscure verbal constructs
of mine. I would also like to thank Zolt´
an Hajnal specifically for relieving me of
some of the responsibilities in maintaining our machine pool and being vigilant of
its running.
My thanks to all those whom I forgot to mention but influenced the work nonethe-
less by being around and enlightening the atmosphere.
Ein besonders herzlicher Dank gilt meinen Eltern, die trotz der Entfernung immer
f¨
ur mich da waren und mir Verst¨
andnis und Unterst¨
utzung entgegenbrachten.
136
Colophon
T
EX is potentially the most significant invention in typesetting in this
century. It introduces a standard language for computer typography, and
in terms of importance could rank near the introduction of the Gutenberg
press.
– Gordon Bell, in [219]
This work was prepared using the free L
A
T
EXand pdfL
A
T
EXtypesetting system on a
machine considered modern at the time of writing—a laptop with a Pentium III pro-
cessor running the Linux operating system. The main text is typeset in the Palatino
font family accompanied by matching mathematical fonts provided with the math-
pple package. Extensive cross-referencing was facilitated by the package hyperref.
The bibliography was prepared using BIBT
EXwith the REVT
EX4 citation style pro-
vided by the American Physical Society and modified here to show article titles and
include hyperlinks to local and external resources.
Data plots were prepared with a recent version of Gnuplot (v3.7.1). Atomic structure
images were produced using RasMol (v2.6.4) by Roger Sayle, which I have extended
to provide perspective views. Auxiliary images were generated using the vector
graphics program XFig. The scripting facilities offered by all these programs helped
towards maintaining a consistent style.
The text has been edited with the vim text editor and has been version-controlled
using CVS.
Because of my past experience and a fondness of mine for that particular accent
British spelling is used throughout. It is hoped that this gets little in the way of
readers who are accustomed otherwise.
The quote in the introduction is due to Charles Babbage, known chiefly as pioneer
of information technology in association with Lady Ada Lovelace. His rather witty
autobiography [220] attests that he was well versed in other fields as well, among
them economics and street musicians.
137