Progress in Materials Science 52 (2007) 196–229
www.elsevier.com/locate/pmatsci
Analytic bond-order potentials for modelling
the growth of semiconductor thin films
R. Drautz
b
a,*
, X.W. Zhou b, D.A. Murdick b, B. Gillespie b,
H.N.G. Wadley b, D.G. Pettifor a
a
Department of Materials, University of Oxford, Oxford OX1 3PH, UK
Department of Materials Science and Engineering, School of Engineering and Applied Science,
University of Virginia, Charlottesville, VA 22904-4745, USA
Abstract
Interatomic potentials for modelling the vapour phase growth of semiconductor thin films must
be able to describe the breaking and making of covalent bonds in an efficient format so that molecular dynamics simulations of thousands or millions of atoms may be performed. We review the derivation of such potentials, focusing upon the emerging role of the bond-based analytic bond-order
potential (BOP). The BOP is derived through systematic coarse graining from the electronic to
the atomistic modelling hierarchies. In a first step, the density functional theory (DFT) electronic
structure is simplified by introducing the tight-binding (TB) bond model whose parameters are determined directly from DFT results. In a second step, the electronic structure of the TB model is coarse
grained through atom-centered moments and bond-centered interference paths, thereby deriving the
analytic form of the interatomic BOP. The resultant r and p bond orders quantify the concept of
single, double, triple and conjugate bonds in hydrocarbon systems and lead to a good treatment
of radical formation. We show that the analytic BOP is able to predict accurately structural energy
differences in quantitative agreement with TB calculations. The current development of these potentials for simulating the growth of Si and GaAs thin films is discussed.
2006 Elsevier Ltd. All rights reserved.
*
Corresponding author. Tel.: +44 1865 273700; fax: +44 1865 273789.
E-mail address: [email protected] (R. Drautz).
0079-6425/$ - see front matter 2006 Elsevier Ltd. All rights reserved.
doi:10.1016/j.pmatsci.2006.10.013
R. Drautz et al. / Progress in Materials Science 52 (2007) 196–229
197
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Coarse graining I: from DFT to TB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1. The reduced TB model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1. Repulsive energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2. Promotion energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.3. Covalent energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.4. Ionic energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.5. Electron counting rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.6. Magnetic energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.7. Van der Waals energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2. Parameterization of the reduced TB model from first principles . . . . . . . . . . . . .
2.2.1. The homovalent dimer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2. Screening the dimer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Coarse graining II: from TB to BOP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1. Exact many-atom expansion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2. Analytic BOP for covalent bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1. Simplified expression for the r bond order with half-full valence . . . . . .
3.2.2. The p bond order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.3. Analytic expression for the promotion energy. . . . . . . . . . . . . . . . . . . . .
3.2.4. Generalization of the analytic bond order to non-half-full valence . . . . .
3.2.5. Structural prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Analytic bond-order potentials for Si and GaAs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1. Si potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2. GaAs potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Atomic assembly of Si film growth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1. Properties of Si bulk structures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2. Properties of Si surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3. Si growth simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Atomic assembly of GaAs film growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1. Properties of the GaAs bulk structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2. Properties of GaAs surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3. GaAs growth simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix A. Expressions for hopping paths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Contents
1.
2.
3.
4.
5.
6.
7.
1. Introduction
Molecular dynamics simulations provide a powerful method for exploring the mechanisms of atomic assembly of thin films during vapour phase growth. Recently embedded
atom method potentials [1] have been successfully applied to model the growth of metal
multilayers [2–4], and by including charge transfer [5,6], extended to the simulation of
the growth of metal oxide multilayers [7]. However, these embedded atom method potentials are unable to model the growth of semiconductor thin films since the directional character of the covalent bonds is not taken into account.
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Robust simulations of the vapour phase growth of covalently bonded semiconductor
materials would be of significant technological importance, for example, in developing
processing routes for synthesizing spintronic materials [8] or in the design of novel hard
coatings [9]. Molecular dynamics studies of the mechanisms of atomic assembly of covalently bonded thin films require an interatomic potential that is able to describe the gas
phase interactions as well as the interatomic forces in the solid state, and most importantly, the transitions of the atoms between the gas and the solid phase, including the formation and breaking of bonds in small clusters, at surfaces and in the bulk. En passant, the
potentials must capture the intuitive concepts used in chemistry and material science to
understand and to explain the complex processes in simple terms, such as bond formation
and breaking, saturated and unsaturated bonds, dangling or radical bonds, and single,
double or triple bonds.
The most widely used class of interatomic potentials for simulating covalent materials
are the reactive empirical bond-order (REBO) potentials of Tersoff [10] and Brenner [11].
Within this class the energy is approximated as the sum of a repulsive pair potential and an
attractive pairwise contribution that depends on the bond order, which measures the difference in the number of electrons associated with the bonding and anti-bonding states.
The bond order within the REBO potentials is calculated as an empirical function that
depends on the number and types of atoms surrounding a given bond. Despite numerous
successful applications of Tersoff–Brenner potentials, the ad hoc expressions for the bond
order have been found to suffer from serious shortcomings when used to model the growth
of thin films. For example, Albe et al. [12] found that the description of the As-rich surfaces in GaAs was unphysical, being unable to modify their fitting parameters to agree
with experiments or ab initio predictions. We have shown elsewhere that the available
parameterizations of the Tersoff–Brenner type potentials for GaAs predict either unrealistic forces between the As–As dimer bond, or underestimate the sticking probability of the
As2 molecule upon impact on the surface during vapour deposition [13,14]. A detailed discussion of Tersoff–Brenner potentials for carbon and hydrocarbons is given by Mrovec
et al. [9] in this issue.
An alternative approach to developing robust interatomic potentials for covalently
bonded materials has been to extend classical valence force fields to handle bond breaking
and remaking explicitly. These so-called reactive force fields (ReaxFFs) were initially
developed for the hydrocarbons [15,16] but have been extended to cover a wide range
of sp-valent elements and some transition metals [17]. The analytic form of the ReaxFFs
is essentially empirical requiring nearly fifty fitting parameters for each element. The many
parameters are required in the ReaxFF framework in order that the empirical bond-order
function is able to describe the complex chemistry of covalent bond formation. The accuracy of ReaxFFs for modelling surface reconstructions and the growth of compound semiconductor thin films such as GaAs, has yet to be demonstrated.
In this review we show how interatomic bond-order potentials (BOPs) can be derived
from quantum mechanics by systematically coarse graining the electronic structure at
two levels of approximation as illustrated in Fig. 1.
1. In the first step, the expression for the binding energy of a material within the effective
one-electron density functional theory (DFT) formalism [18] is re-written in terms of
physically and chemically intuitive contributions within the tight-binding (TB) bond
model [19]. This TB approximation is sufficiently accurate to predict structural trends
R. Drautz et al. / Progress in Materials Science 52 (2007) 196–229
199
Fig. 1. Illustration of electronic and atomistic modelling hierarchies and the derivation of analytic interatomic
bond-order potentials through two steps of coarse graining, firstly from DFT to TB and secondly from TB to
BOP.
across the sp-valent elements, as well as sufficiently simple to allow a physically meaningful interpretation of the bonding in terms of r and p contributions. We demonstrate
how the unknown TB parameters can be obtained from ab initio calculations in a systematic way.
2. In the second step, the TB electronic structure is coarse grained through atom-centered
moments and bond-centered interference paths as discussed in the first paper of this
issue [20]. This allows the bond order to be related to the local topology and coordination of the material. In this way the functional form of the bond order is derived as a
function of positions and types of atoms that surround a given bond. We argue that
these analytic bond-order potentials [21,22] should overcome many of the shortcomings
of empirical bond-order parameterizations.
The outline of this review is as follows. We begin in Section 2 by discussing the coarse
graining of the DFT electronic structure and the parameterization of the TB bond integrals and repulsive contributions. In Section 3 we discuss the coarse graining of the TB
electronic structure in order to derive analytic expressions for the TB r and p bond orders
in terms of the local environment. We illustrate the accuracy of the derived BOP by predicting the known structural trends across the sp-valent elements (Section 3.2.4). In Section 4 we fit BOPs for Si and GaAs. In Sections 5 and 6 these potentials are applied to
the simulation of the growth of Si and GaAs thin films. In Section 7 we conclude.
2. Coarse graining I: from DFT to TB
2.1. The reduced TB model
The tight-binding bond model [19] justifies the functional form of the TB approximation by deriving it from density functional theory as discussed in detail in the first paper
of this issue [20]. It follows from Eq. (69) of [20] that the total binding energy EB of a multicomponent sp-valent system within the orthogonal TB bond model may be written as a
sum of several physically based contributions,
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EB ¼ Erep þ Eprom þ Econv þ Eion þ Emag þ EvdW :
ð1Þ
We discuss each of the terms in the following subsections.
2.1.1. Repulsive energy
The first term Erep contains the overlap and electrostatic repulsion and is often approximated by a simple pair potential U,
1 X
Erep ¼
UðRij Þ;
ð2Þ
2 i;j;i6¼j
where Rij is the distance between atoms. Recent, more accurate TB schemes express Erep in
the form of a many-body potential by taking into account the screening of both the overlap [23] and electrostatic [24] repulsive contributions in the local atomic environment
about a bond.
2.1.2. Promotion energy
The second term Eprom is the energy of promotion that arises from the change in the
hybridization state when sp-valent atoms are brought together from infinity,
X prom X
Eprom ¼
Ei
¼
ðlp ls ÞDN ls :
ð3Þ
i
i
As shown in Section 2.2.1 the level splitting dl ¼ ðlp ls Þ of the atomic s and p states is
approximately constant and independent of bond length for a given atomic species l, so
that it may be assumed to take its free atom value. The difference in the number of electrons in the s(p) orbital with respect to the occupancy of the s(p) state of the free atom is
expressed as DN lsðpÞ ¼ N lsðpÞ N l;0
sðpÞ . For covalently bonded materials the charge transfer
between atoms is often small. Thus, if we assume local charge neutrality (LCN), the
change in the number of s and p electrons will satisfy,
DN ls ¼ DN lp :
ð4Þ
2.1.3. Covalent energy
The third term Ecov is the attractive covalent bond energy. It can be written in the form
X Z F
ð a Þna ðÞd;
ð5Þ
Econv ¼
a¼s;p
where F is the Fermi energy and ns(p) is the local s(p) electronic density of states. The
covalent bond energy may be decomposed into contributions from individual bonds
1 X conv lm
ðE Þij
ð6Þ
Econv ¼
2 i;j;i6¼j
with
lm
ðEconv Þij ¼ 2
X
lm
H lm
im;jm0 Hjm0 ;im :
ð7Þ
m;m0
The matrix elements of the Hamiltonian H and the bond-order H are evaluated within the
Slater–Koster two center approximation [25]. If the z-axis of the local coordinate system is
R. Drautz et al. / Progress in Materials Science 52 (2007) 196–229
201
chosen to point in the direction of the bond ij, then the matrix elements of H are written as
lm
ssrlm
ij , sprij , etc.
We simplify the expression for the bond energy by making the reduced TB approximation [26,27]
lm
lm
lm
sprlm
ij psrij ¼ ssrij pprij :
ð8Þ
The reduced TB approximation imposes the physically intuitive picture of a single r bond
order, i.e., it ensures the simple rule of chemistry that a sp-valent material may form only
one fully saturated r bond but two saturated p bonds is obeyed. The reduced TB approximation is valid to within 16% for Harrison’s canonical TB parameterization of sp-valent
elements [28] and to within 12% for Xu et al.’s parameterization of carbon [29].
A chemically intuitive expression for the covalent bond energy may then be derived by
making a basis transformation from atomic orbitals to bonding hybrids that point into the
bond
pffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffi
jilri ¼ 1 plr jilsi þ plr jilzi;
ð9Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffi
m
m
ð10Þ
jjmri ¼ 1 pr jjmsi pr jjmzi;
and non-bonding hybrids that point away from the bond,
pffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffi
jilr i ¼ plr jilsi 1 plr jilzi;
pffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffi
jjmr i ¼ pmr jjmsi þ 1 pmr jjmzi:
plr
ð11Þ
ð12Þ
pmr
and
give the relative admixture of p character in the bonding hybrid
The prefactors
and determine the directional character of the bond. Within the reduced TB approximation they take the values
2
2
2
ð13Þ
2
2
2
ð14Þ
lm
lm
plr ¼ ðpprlm
ij Þ =½ðsprij Þ þ ðpprij Þ ;
and
lm
lm
pmr ¼ ðpprlm
ij Þ =½ðpsrij Þ þ ðpprij Þ :
With respect to the new basis, the 2 · 2 r-block in the Hamiltonian matrix entering Eq. (6)
takes the diagonal form
lm
br;ij 0
lm
H r;ij ¼
;
ð15Þ
0
0
where the bond integral between the bonding and the hybrids is given by
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
lm
ð1 plr Þð1 pmr Þ:
¼
ssr
blm
ij
r;ij
ð16Þ
The zero eigenvalue in Eq. (15) reflects the absence of bonding between the non-bonding
hybrids, as expected. Substituting Eq. (15) into Eq. (6) the covalent bond energy can be
written explicitly as
lm
lm
ml
ml
ml
ðEconv Þij ¼ 2blm
r;ij Hr;ij þ 2bp;ij ðHpþ ;ij þ Hp ;ij Þ;
ð17Þ
where
lm
blm
p;ij ¼ pppij :
ð18Þ
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R. Drautz et al. / Progress in Materials Science 52 (2007) 196–229
We see, therefore, that the reduced TB approximation allows us to write the bond energy
in terms of contributions from a single r bond order and two p bond orders, as taught in
standard chemistry textbooks.
2.1.4. Ionic energy
The fourth term Eion represents the ionic energy. For Si we can assume that there is no
net charge transfer between the atoms whose on-site atomic energy levels adjust to maintain local charge neutrality. This neglect of the ionic contribution in Eq. (1) is a good
approximation for homovalent semiconductors, but begins to break down for III–V heterovalent compounds such as GaAs. In the molecular dynamics simulations presented in
this paper we implicitly model the charge transfer between the Ga and As dangling bonds
at the GaAs surface by using the electron counting rule which we now discuss.
2.1.5. Electron counting rule
Nearly a dozen surface reconstructions have been observed experimentally on the (0 0 1)
GaAs surface [30,31]. These surfaces often have special surface stoichiometry. A typical
example is the As-terminated b(2 · 4) surface reconstruction, which requires that one
As dimer is missing for every four As dimers in the [1 1 0] dimer row direction such that
the number of the surface As dimers MAs and surface Ga dimers MGa satisfies
MAs = 0.75MGa. Density functional theory calculations show that this surface reconstruction has a low surface free energy over a wide range of atmospheric conditions when compared with many competing surface reconstructions [32–34]. The low energy of the b(2 · 4)
surface reconstruction may be explained by counting the number of dangling Ga and As
bonds. The ratio of Ga to As dangling bonds is such that all the electrons from high energy
Ga surface dangling bonds can be redistributed into low energy As dangling bonds. Interatomic potentials that do not explicitly treat this redistribution of electrons from Ga to As
dangling bonds have been found to predict incorrectly the relative energies of the surface
structures [13,35].
Pashley [36] successfully explained the surface reconstructions by using the electron
counting rule (ECR). The ECR assumes that low energy reconstructions are obtained
when the low energy Ga–As, Ga–Ga, As–As and the As dangling bonds are fully occupied
by two electrons while the high energy Ga dangling bonds are left empty. This rule is consistent with most of the known surface reconstructions in GaAs.
The condition of local charge neutrality that we assume in the derivation of the BOPs
does not allow the environment-dependent occupation of dangling bonds. We therefore
have developed a separate model to incorporate ECR into molecular dynamics simulations [13]. The basic concept is explained as follows. Suppose that each atom, i, has a
valence Ni (Ni = 5 for an As atom and Ni = 3 for a Ga atom), then the total number of
electrons Ntot in a system with n atoms is given by
N tot ¼
n
X
N i:
ð19Þ
i¼1
Assume further that the bond between atoms i and j is occupied by Nij electrons, and atom
i contains ai electrons per dangling bond. The total number of electrons in all interatomic
and dangling bonds can be written as
R. Drautz et al. / Progress in Materials Science 52 (2007) 196–229
N ECR ¼
X
1 X
N ij þ
ai d i ;
2 i;j;j6¼i
i
203
ð20Þ
where di is the total number of dangling bonds on atom i. The ECR criterion that the As
dangling bonds as well as all other nearest-neighbour bonds are occupied and that the Ga
dangling bonds are empty, means that ai = 2 for As atoms, ai = 0 for Ga atoms, Nij = 2
for nearest-neighbour bonds and Nij = 0 for bonds beyond the nearest-neighbour distance.
According to the ECR the low energy reconstructions are such that the electrons may be
distributed into low energy dangling bonds, hence Ntot = NECR. When the number of electrons Ntot does not match the number of available low energy electron states NECR, high
energy Ga dangling bonds are occupied with electrons, thereby increasing the surface energy. By expanding the surface energy in the vicinity of Ntot NECR to second order, the
energy increase for violating the ECR by occupying high energy Ga dangling bonds may
be written
DEECR ¼ wðN tot N ECR Þ2 ;
ð21Þ
where w is a parameter that is defined by the energy required to occupy Ga dangling
bonds. Eq. (21) can be added to Eq. (1) to define the total potential energy of the system.
It essentially introduces an electronic degree of freedom into the interatomic potential. To
retain the fidelity of the BOP for modelling GaAs, the term DEECR must drop to zero within a bulk crystal. It only becomes positive at a surface that violates the ECR. Two examples are used to illustrate this.
Consider a zinc-blende GaAs crystal that does not have dangling bonds,
hence di = 0. From Eqs. (19,20) we see that the number of electrons Ntot equals
the number of low energy electron states NECR, therefore DEECR = 0. The addition
of the ECR modification does not affect the BOP potential for GaAs bulk
lattices.
As a second example, consider an As-terminated b(2 · 4) surface. We consider only
the top As and its bonds with the adjacent underlying Ga atomic planes. Assume that
the top As plane contains MAs As dimers, and the Ga plane contains MGa Ga dimers.
Because one As surface dimer is missing for every four dimers in the b(2 · 4) surface
structure, we have MAs = 0.75MGa. From Eq. (19), the total number of electrons is
Ntot = 2NAsMAs + NGaMGa=10.5MGa, where we took into account that half of the Ga
electrons occupy bonds that are formed with layers below the first two surfaces layers.
The b(2 · 4) structure can be created by adding As dimers to a Ga-terminated (0 0 1) surface. The addition of each As dimer converts four Ga dangling bonds into four Ga–As
bonds, creates two As dangling bonds (one per As adatom) and one As–As dimer bond.
According to Eq. (20), we obtain NECR = 7 · 2 · MAs = 10.5MGa, which means that
DEECR = 0.
Eqs. (19) and (20) do not constitute an interatomic potential because they are not continuous functions of atomic positions. The ECR modification has recently been cast into
the form of an interatomic potential [37]. This potential successfully predicts many of the
(0 0 1) GaAs surface reconstructions including the a(2 · 4), a(4 · 2), a2(2 · 4), a2(4 · 2),
b(2 · 4), b(4 · 2), b2(2 · 4), b2(4 · 2), c(4 · 4) 75%, c(2 · 4), and f(4 · 2) surfaces
[37,38].
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R. Drautz et al. / Progress in Materials Science 52 (2007) 196–229
2.1.6. Magnetic energy
We do not include the magnetic contribution to the binding energy. For the non-magnetic semiconductor bulk and surfaces, this is a very good approximation. However, it
should be noted that magnetism contributes significantly to the energy of sp-valent materials in situations where the atoms are pulled apart towards their non-singlet free atom
states. For example, the magnetic energy contributions to C and Si free atoms are between
1 and 2 eV per atom.
2.1.7. Van der Waals energy
The weak, long-ranged van der Waals energy EvdW due to fluctuation-induced dipole
moments is of significance only when the atoms are too far apart to form a covalent bond.
These dipole fluctuations are not included within the LDA and GGA approximations to
DFT. However, it is straightforward to model the van der Waals contribution with a longranged effective pair interaction, should this be required [11].
2.2. Parameterization of the reduced TB model from first principles
The free parameters of the reduced TB model described above can be obtained from
first-principles DFT calculations in a step-by-step procedure. We start from the simplest
possible molecule, the homovalent dimer, and then take into account the screening of
the TB matrix elements by other atoms surrounding the bond.
2.2.1. The homovalent dimer
The TB eigenspectrum, which comprises four non-degenerate r and two degenerate p
levels, can be expressed analytically in terms of the six reduced TB parameters br, pr, bp,
and s, pz and px;y . We have included the effect of the crystal field which splits the on-site
degeneracy of the atomic p levels pz and px;y , corresponding to the pz and (px, py) orbitals
respectively. The six reduced TB parameters for the first to fourth row sp-valent dimers are
obtained from their non-spinpolarized DFT eigenspectra. As an example, the solid curves
in the left hand panel of Fig. 2 show the four r and the two p eigenvalues for Si2 as a function of bond length. The resultant values of the reduced TB parameters, from the inversion
of this spectrum, are shown in the right hand panel of Fig. 2. The degeneracy of the pz and
px,y atomic p levels is split by the crystal field and non-orthogonality overlap contributions. Nevertheless, this splitting remains small compared to the total s-p splitting. Since
the overlap repulsion affects the upwards shift of both the s and p atomic energy levels
in a similar way, the relative energy p s is found to change only slowly as the atoms
are brought together from infinity. Thus we approximate pz ¼ px;y and d = p s = d0,
see Fig. 2. We see that an interpolation of the bond integrals with a simple exponential
function bexp
rðpÞ is able to capture the behaviour of the TB parameters for distances larger
than the equilibrium bond length.
The repulsive energy of a homovalent dimer may now be obtained by calculating analytically the covalent bond and promotion energies of the sp-valent dimer using the
reduced TB bond parameters obtained from the eigenspectrum and subtracting them from
the DFT binding energy in Eq. (1), namely
Erep ¼ EB Econv Eprom :
ð22Þ
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205
10.0
8.0
0.0
6.0
-2.0
4.0
-4.0
2.0
E [eV]
E [eV]
2.0
-6.0
-8.0
δ
0.0
-2.0
-4.0
-10.0
-6.0
-12.0
-8.0
-14.0
-10.0
3
4
5
6
7
8
9
10
3
4
5
6
R [a.u.]
7
8
9
10
R [a.u.]
Fig. 2. (a) Eigenspectrum of Si2 as calculated within the local density approximation to DFT using DMol3 [39].
(b) Reduced TB parameters from the DFT eigenspectrum: diamond symbols correspond to data points from the
eigenspectrum, solid curves correspond to interpolation of the data with a simple analytic function. The
exp
exponential tail of the bond integral interpolation is indicated by bexp
p and br . The dotted vertical line indicates
the equilibrium bond length.
Fig. 3 shows the behaviour of EB, Erep, Eprom and Ecov as the Si atoms are brought
together to form the dimer. The fact that Eq. (22) defines a strictly positive repulsive
energy shows that the reduced TB model is indeed a physically sound model of the
DFT electronic structure. As can be seen from Fig. 3, the repulsive energy decays faster
than the bond energy as a function of distance.
The total repulsive energy within DFT contains contributions of different origins [40]
that we separate into two parts
Erep ¼ Eover þ Ecore ;
ð23Þ
where the first term describes the repulsion due to the overlap of the non-orthogonal atomic orbitals and the second term Ecore contains electrostatic interaction between the atoms,
including their ion-core repulsion (see Fig. 3 of [40]).
20.0
15.0
rep
[eV]
10.0
E [eV]
E
10.0
5.0
5.0
0.0
3
4
5
6
7
8
9
R [a.u.]
0.0
-5.0
-10.0
-15.0
3
4
5
6
7
8
9
10
R [a.u.]
Fig. 3. The repulsive, covalent and promotion energy contributions to the binding energy EB for the Si2 dimer.
The inset shows the interpolation of the repulsive energy as the sum of an overlap repulsion and a short-ranged
hard-core potential (dashed curves). The exponent k of the overlap repulsion for Si is found to be k = 1.9. The
hard-core potential becomes important only at distances smaller than the dimer bond length.
206
R. Drautz et al. / Progress in Materials Science 52 (2007) 196–229
If one assumes a decay of the overlap matrix elements Olm
ij proportional to the Hamillm
tonian matrix elements, Olm
/
H
,
then
to
first
order
the
overlap
repulsion scales as the
ij
ij
square of the bond integrals [23]. The functional form of the overlap repulsion
lm k
ðEov Þlm
ij ¼ aðbr;ij Þ
with k 2
ð24Þ
is able to describe the decay of the repulsion for distances larger than the equilibrium
dimer bond lengths. The short-ranged core repulsion Ecore is interpolated with a generalized Yukawa-type potential, see Fig. 3.
2.2.2. Screening the dimer
Interatomic potentials for modelling the growth of semiconductors must be able to
describe the gas phase, the surface and the bulk with the same set of parameters. In order
that the orthogonal reduced TB model becomes transferable to different surroundings, the
environmental dependence of the bond integrals on the surrounding atoms must be taken
into account. Starting from a non-orthogonal TB representation, effective orthogonal TB
Hamiltonian matrix elements can be derived [23] in the form
ð0Þ;lm
H lm
ij ¼ H ij
ð1 S lm
ij Þ;
ð25Þ
ð0Þ;lm
refers to the unscreened Hamiltonian matrix element and S lm
where H ij
ij represents the
screening matrix element. If the z-axis of the local coordinate system is pointing along the
axis of the bond, the unscreened Hamilton matrix element is given by the appropriate
dimer bond integral. Keeping terms to second order in Eq. (11) of [23], the screening function S lm
ij may be written in terms of the unscreened Hamiltonian matrix elements and the
overlap matrix elements Olm
ij ¼ hiljjmi,
X
1 1
ð0Þlj jm
ð0Þjm
S lm
ðH
Okj þ Olj
Þ:
ð26Þ
ik H kj
ij ¼
2 H ijð0Þlm k6¼i;j ik
Fig. 4 shows the first nearest-neighbour r bond integrals for Si in different structures
which are obtained by self-consistently solving the TB-LMTO equations [41]. As expected
0.0
-1.0
βσ [eV]
-2.0
-3.0
dimer
diamond
sc
bcc
fcc
-4.0
-5.0
4.0
4.5
5.0
5.5
6.0
6.5
R [a.u.]
Fig. 4. Screened nearest-neighbour r bond integral for different structures of Si. The bond integrals represented
by symbols for diamond, sc, bcc and fcc structures were calculated from screened TB-LMTO [41]. Solid curves
are fitted using the screening expression Eq. (26). The dashed curve corresponds to the predicted second nearestneighbour bond in bcc.
R. Drautz et al. / Progress in Materials Science 52 (2007) 196–229
207
the bond integrals are considerably weakened in close-packed environments when more
atoms are surrounding the bond. This screening results in more distant neighbour interactions being very small, thereby explaining why Tersoff–Brenner-type potentials have been
so successful in modelling many bulk properties with short-ranged interactions. Clearly,
however, the interatomic potentials will be much longer ranged at the more open surfaces
or in the gas phase, which is reflected in Fig. 4 by the much weaker screening. The curves
in Fig. 4 were obtained by evaluating the screening function Eq. (26) with a parameterization of the overlap integrals that has the same functional form and similar decay lengths
as the unscreened matrix elements.
The repulsive pairwise contributions for the dimer Eq. (23) should also be screened. The
overlap repulsion may be screened within the formalism of Ref. [23], whereas the core
repulsion may be screened using an environment-dependent Yukawa potential [42,43].
We are currently investigating these screening functions for sp-valent elements.
3. Coarse graining II: from TB to BOP
3.1. Exact many-atom expansion
The term ‘‘bond order’’ was introduced by the chemists [44] as one half the difference
between the numbers of electrons in the bonding and anti-bonding states (see Ref. [20] in
this issue for a detailed discussion of the history of the term bond order). That is
1
Hlm
ij ¼ ðN þ N Þ;
2
ð27Þ
where N+() gives the number of electrons in the bonding (anti-bonding) state. Thus, for
example, the hydrogen dimer with two electrons in the bonding state but none in the antibonding state forms a saturated bond with the bond order H = 1.
The starting point for the expansion of the bond order is its relation to the intersite
Green’s function through Eq. (36) of [20] (we omit superscripts lm in the remainder of this
section)
Z F
2
Hij ¼ Im
Gij ðÞd;
ð28Þ
p
where F is the Fermi energy. Within bond-based BOP theory the off-diagonal matrix elements of the Green’s function are calculated from diagonal elements Gk00 ,
b 1 juk i
Gk00 ¼ huk0 j½ H
0
ð29Þ
1
juk0 i ¼ pffiffiffi ½jii þ expði cos1 ðkÞÞjji:
2
ð30Þ
with
By taking the derivative of Gk00 with respect to k the intersite Green’s function Gij may be
calculated
Gij ¼
o k
G :
ok 00
ð31Þ
208
R. Drautz et al. / Progress in Materials Science 52 (2007) 196–229
The above equation becomes the starting point for the derivation of an interatomic analytic BOP by using the Lanczos recursion algorithm and writing Gk00 in the form of a continued fraction [45,46],
Gk00 ¼
1
ak0 ;
ðbk1 Þ2
ð32Þ
ðbk Þ2
2
ak1 ðbk Þ2
3
ak 2 ak 3
where the recursion coefficients {an, bn} are the matrix elements of the semi-infinite onedimensional Lanczos chain (see Ref. [20]). The relation of the matrix elements to the moments of the density of states can be worked out by evaluating the moments along the
Lanczos chain. We give the relation for the first four moments explicitly
l1 ¼ a 0 ;
l2 ¼ a20 þ b21 ;
l3 ¼ a30 þ 2a0 b21 þ a1 b21 ;
l4 ¼ a40 þ 3a20 b21 þ 2a0 a1 b21 þ a21 b21 þ b41 þ b21 b22 :
ð33Þ
By substituting the continued fraction representation of Gk00 into Eq. (31), Gij becomes a
function of the recursion coefficients
Gij ¼
1
1
X
oGk00 oakn X
oGk00 obkn
þ
:
k
oakn ok
ok
n¼0
n¼1 obn
ð34Þ
The derivatives of the Green’s function with respect to the recursion coefficients may be
b 1 juk i [47],
replaced by products of Green’s function matrix elements Gknm ¼ hukn j½ H
m
oGk00
¼ Gk0n Gkn0 ;
oan
oGk00
¼ Gk0n Gkn1;0 þ Gk0;n1 Gkn0 :
obn
ð35Þ
By evaluating this equation for k = 0 an exact expansion for the intersite Green’s function
is found,
!
1
1
X
X
Gij ¼ 2
G0n Gn0 dan þ 2
G0n1 Gn0 dbn
ð36Þ
n¼0
n¼1
with
2X
nþ1
oakn oakn ¼
dan ¼
ðnlþ1 Þ;
ok k¼0
olkl l¼1
k¼0
2n
X
obkn obkn dbn ¼
¼
ðnlþ1 Þ;
ok k¼0
olkl l¼1
ð37Þ
ð38Þ
k¼0
1
b l jii þ hjj H
b l jji þ knlþ1 ;
lkl ¼ ½hij H
2
b l jji:
nlþ1 ¼ hij H
ð39Þ
ð40Þ
R. Drautz et al. / Progress in Materials Science 52 (2007) 196–229
209
By inserting Eq. (36) into Eq. (28) the bond order can be written as a non-linear function of
the moments ll and a linear function of the interference paths nl.
3.2. Analytic BOP for covalent bonds
The starting point for the derivation of an analytic BOP for covalent materials is the
observation that saturated covalent bonds only exist in structures where each atom has relatively few neighbours, i.e., in open structures, whereas in close-packed structures the
atoms are unable to form saturated bonds with all their neighbours. In most open structures there are no three-membered self-returning hopping paths, which means that the
third moment is zero if we assume that d = p s = 0. For the derivation of the analytic
BOP we will assume that all the odd moments l2l+1 vanish, which implies that the density
of states is symmetric
nðÞ ¼ nðÞ:
ð41Þ
b , shows that
Inspection of the poles of Eq. (32), which correspond to the eigenvalues of H
the eigenvalues form a symmetric spectrum only if all the terms an vanish identically,
an ¼ 0:
ð42Þ
For the derivation of an analytic BOP for the r bond, the sum in Eq. (36) is limited to the
first four sites along the Lanczos chain (b4 = 0) [21]. The Green’s function, Eq. (32) with
k = 0, then takes the form
G00 ¼
P 00 ðÞ
3 þ A00 2 þ B00 þ C 00
¼
;
D00 ðÞ ð 1 Þð 2 Þð 3 Þð 4 Þ
ð43Þ
with coefficients
A00 ¼ 0;
B00 ¼ ðb22 þ b23 Þ;
C 00 ¼ 0:
ð44Þ
(The more general case for an 5 0 is given in Refs. [48,49].) The four poles are located at
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
¼ j j ¼ ð ðb1 þ b3 Þ2 þ b22 ðb1 b3 Þ2 þ b22 Þ:
ð45Þ
2
For the calculation of Gij from Eq. (34), a pairwise multiplication of Green’s function elements according to Eq. (35) has to be carried out. This means that Gij will appear as a
ð2n2Þ
ðn1Þ
function of polynomials Gij ðÞ ¼ ppð2nÞ ðÞðÞ when G00 ¼ ppðnÞ ðÞðÞ for n recursion levels, where
p(m) indicates a polynomial of leading order m. For the exact eigenspectrum, corresponding to n ! 1 recursion levels, the poles of G00 and Gij are of course identical. By constraining the poles of the intersite Green’s function Gij = Pij()/Dij() to be the same as
those of the average on-site Green’s function G00, i.e.,
Dij ðÞ ¼ D00 ðÞfij ðÞ;
P ij ðÞ ¼ Qij ðÞfij ðÞ;
ð46Þ
ð47Þ
where fij is a polynomial of order n and Qij is a polynomial of order n 2, the convergence
of the BOP expansion for a finite number of recursion levels is greatly improved. The new,
constrained form of Gij is written as
210
R. Drautz et al. / Progress in Materials Science 52 (2007) 196–229
Gij ¼
Qij
Aij 2 þ Bij þ C ij
¼
D00 ð 1 Þð 2 Þð 3 Þð 4 Þ
ð48Þ
with
Aij ¼ ðn2 Þij ;
C ij ¼ ðn4 Þij ½b21 þ b22 þ b23 ðn2 Þij :
Bij ¼ ðn3 Þij ;
ð49Þ
The bond order is now calculated from the residues of Gij below the Fermi energy,
Hij ¼ 2
4
X
HðnÞ H ðF n Þ;
ð50Þ
n¼1
where H is the Heaviside step function that equals 0 for F < n (i.e., the eigenstate n is not
occupied), but equals 1 for F > n (a fully occupied eigenstate). The weights H(n) are a
measure of how much an electron in eigenstate n contributes to the bond order. Due to
the symmetry of the eigenspectrum, the values for H(n) are asymmetric, H(1) = H(4) =
H(+), H(2) = H(3) = H() and are given by
2
3
b
b
C ij
C ij
1 61 þ ^þ^ 1 ^þ^ 7
ð51Þ
HðÞ ¼ 4
5:
4 ^þ þ ^ ^þ ^
b ij ¼ C ij =jðbr Þ j3 and ^ ¼ =jðbr Þ j.
with C
ij
ij
3.2.1. Simplified expression for the r bond order with half-full valence
The r bond order of Eqs. (50,51) simplifies for the case of a half-full sp-valent shell to
ð12Þ
b ij =^
Hij;r
¼ ð1 þ C
b1 ^
b3 Þ=ð^þ þ ^ Þ;
ð52Þ
b ij and the
bn =jðbr Þij j. It follows from Eqs. (33) and (49) that the coefficients C
where ^
bn ¼ ^
recursion coefficients are given by
^2 ^
b ij ¼ 1 ðb
C
b22 ^
b23 Þ Rij4r ;
1
^2 ¼ 1 þ U2r ;
b
ð53Þ
ð54Þ
1
^b2 ¼ ðU2r U2 þ U4r þ
2
2r
Rij4r Þ=ð1
þ U2r Þ;
ð55Þ
Rij4r
to a 4-atom ring-type interference
where U2r corresponds to a 2-hop contribution,
path linking atoms i and j, and U4r refers to self-returning hopping paths of length four
as illustrated in Fig. 5. By neglecting the ring contributions to ^b3 it is found that ^b3 can
be approximated by [22,49]
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
^
b1 ^
ð56Þ
b3 ¼ DU4r þ Ui2r Uj2r :
The definitions of Ui2r , U2r, U4r, and DU4r used in the equations above are given in
Appendix A. Substituting Eqs. (53)–(56) into Eq. (52) and after some algebra, one obtains
the explicit expression for the r bond order for half-full valence,
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
,v
u
u
g
^2
1
e ðjÞ
e ðiÞ U
2U2r þ Rij4r þ U
ð 2Þ
2r 2r ð2 þ DU 4r Þ þ d
Hij;r
;
ð57Þ
¼ 1 t1 þ c r
2
ð1 þ g
DU 4r Þ
R. Drautz et al. / Progress in Materials Science 52 (2007) 196–229
i
i
j
j
i
i
j
211
j
i
j
Fig. 5. Illustration of the hopping paths that are taken into account for the evaluation of the simplified analytic
BOP. The top row illustrates the contributions Uj2r and Rji4r , the bottom row illustrates the different contributions
to Uj4r .
where
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
j
i ej
i
e
U 2r U 2r ¼ U2r U2r
DU4r þ Ui2r Uj2r ;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
g
DU4r þ Ui2r Uj2r :
DU 4r ¼ DU4r
ð58Þ
ð59Þ
The term ^
d in Eq. (57) is given by ^
d2 ¼ 12 ½d2i þ d2j 4pr ð1 pr Þ=b2ij;r (cf. Ref. [9] in this issue
2
^
where a modified d is introduced and discussed). It accounts for the loss of covalent bonding due to the s and p atomic energy level mismatch. The constant cr 1 is a fitting
parameter that can be used to improve the comparison between BOP and TB r bond
orders for a given parameterization of the TB model. We note that unlike Eqs. (43) and
(48), Eq. (57) explicitly ensures that the bond order is correctly bounded by Hr 6 1.
3.2.2. The p bond order
The p bond order must be invariant to the particular choice of the x and y coordinate
axes normal to the z-direction along the bond. This invariance of the expression for the p
bond order at any level of approximation is guaranteed by using the matrix Lanczos recursion algorithm [21,50,51] where the recursion coefficients an and bn become 2 · 2 matrices
instead of scalar variables. Carrying out the matrix Lanczos iteration to two levels, assuming a symmetric spectrum, and constraining the poles of Gij to be the same as the poles of
G00, one obtains analytic expressions for the p+ and p bond orders [21,48],
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffi
ð 1Þ
Hij2 ; p ¼ 1
1 þ cp ðU2p U4p Þ:
ð60Þ
Explicit formula for the two- and four-hop contributions U2p and U4p are given in Appendix A. The fitting parameter cp is introduced to improve the comparison between BOP and
TB p bond orders and is expected to take a value close to 1. Table 1 shows that the analytic
expressions for the p bond order (with cp = 1) capture the chemistry of bond formation in
the hydrocarbons [52].
3.2.3. Analytic expression for the promotion energy
The s and p atomic level separation in carbon and silicon is approximately 7 eV.
This means that there is an energy penalty of about 7 eV to promote the free atom
212
R. Drautz et al. / Progress in Materials Science 52 (2007) 196–229
Table 1
Bond orders of C-C bonds in hydrocarbon molecules as predicted by analytic BOPs [52]
C2
C2H2
C2H4
C6H6
Graphite
C2H5
C2H6
Diamond
Hr
Hp+
Hp
Htot
Bond character
0.936
0.974
0.955
0.953
0.951
0.929
0.917
0.915
1.000
1.000
1.000
0.577
0.477
0.214
0.149
0.126
1.000
1.000
0.194
0.141
0.121
0.145
0.149
0.126
2.936
2.974
2.149
1.671
1.549
1.288
1.215
1.167
Triple
Triple
Double
Conjugate
Conjugate
Radical
Single
Single
The analytic BOPs correctly capture the chemistry of bond formation in the hydrocarbon systems, including the
formation of single, double, triple, conjugate, and radical bonds.
configuration s2p2 to the sp3 hybrid configuration of the diamond structure. Hence, the
promotion energy cannot be neglected for the calculation of realistic binding energies. Following Refs. [21,27], the promotion energy can be approximated by
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
0
,v
u
X br;ij 2
u
prom
t
@
A;
ðE
Þi ¼ d 1 1
1þ
A
ð61Þ
d
j;j6¼i
where d is the s–p atomic energy level separation and A is a fitting parameter.
3.2.4. Generalization of the analytic bond order to non-half-full valence
For the derivation of the analytic expressions for the bond order in the previous sections, the integration of the Green’s function was carried out for a half-full eigenspectrum
corresponding to a fractional bond occupancy of f = 1/2. In this section we extend the
bond-order expressions to a general fractional bond occupancy 0 6 f 6 1. This enables
the simulation of III–V semiconductor systems such as GaAs, where Ga has 3 valence electrons (f = 3/8), As has five valence electrons (f = 5/8), and GaAs has an average of four
valence electrons per atom (f = 1/2).
Following Ref. [53] we assume that the bond order of a symmetric eigenspectrum can be
approximated by a third-order polynomial in the symmetric function f(1 f), whereby
Hs ðf Þ ¼ as f ð1 f Þf1 bs f ð1 f Þ½1 cs f ð1 f Þg:
ð62Þ
Fitting the unknown coefficients to the fact that the bond order is bounded by the envelope function
2f
for 0 6 f < 1=2;
jHj 6
ð63Þ
2ð1 f Þ for 1=2 6 f 6 1;
we find that Eq. (62) can be written in the form
8
2f
>
<
Hs ðf Þ ¼ 2f 0 þ as Fð1 2f 0 Þ½1 bs Fð1 cs FÞ
>
:
2ð1 f Þ
for 0 6 f < f0 ;
for f 0 6 f < 1 f0 ;
for 1 f0 6 f 6 1;
ð64Þ
R. Drautz et al. / Progress in Materials Science 52 (2007) 196–229
213
where
4
1
5
H ð 2Þ ;
3
8
8
<0
cs ¼
: 32 5 Hð12Þ
8
f0 ¼
F¼
for Hð2Þ > 58 ;
1
for Hð2Þ 6 58 ;
1
ð65Þ
f ð1 f Þ f0 ð1 f0 Þ
2
ð1 2f 0 Þ
with as = 2, bs = 1.
We then assume that the bond order of an asymmetric eigenspectrum is obtained by
skewing the symmetric bond order about f = 1/2 by writing
(
"
3
5 # )
1
1
1
^ 3 Hs ðf Þ;
R
f þ k3
f þ k5
f
Hðf Þ ¼ 1 k 1
ð66Þ
2
2
2
^ 3 is proportional to the 3-member
where k1, k3 and k5 are fitting parameters and where R
^ 3 is given
ring contribution that links the two ends of the bond. An explicit expression for R
in the Appendix (cf. Eq. (A.10)). It is demonstrated in Ref. [53] that this polynomial
approximation, which contains no arbitrary parameters for the symmetric case, reproduces very well the r and p TB bond orders as a function of fractional bond occupancy
or valence.
3.2.5. Structural prediction
This BOP is the only interatomic potential that includes the valence dependence of the
bond order explicitly within its framework. This gives the BOP the ability to predict the
known structural trend across the sp-valent elements as the group number changes from
1 to 7 [53]. This is illustrated by Fig. 6. The left hand panel shows the TB structural energy
curves as a function of valence for structures with local coordinations ranging from z = 1
(dimer) to z = 12 (close-packed). We see that the correct structural trend from closepacked to more open structure types is found. Moreover, since the BOP has been derived
by coarse graining the TB electronic structure, the right hand panel of Fig. 6 shows that it
too predicts correctly these changes in structure as the number of valence electrons is changed from 1 to 7.
These trends are driven by the third and fourth moments of the density of states [53].
The third moment drives the skewing of the eigenspectrum and accounts for the observed
switch from close-packed to open structures as the fractional bond occupancy moves
through f = 1/2 (cf. Eq. (66)). The fourth moment determines the bimodality of the eigenspectrum which accounts for diamond with the lowest fourth moment being most stable
for f = 1/2. Second moment potentials such as the Tersoff potentials (or the 2-level BOP
expression, BOP2) cannot predict these structural trends (although, of course, the ground
state structure of any particular element can be fitted by suitably adjusting the parameters
within the given model).
This can be demonstrated by considering the binding energy per atom within the second
moment approximation, which from Eq. (1) can be written
214
R. Drautz et al. / Progress in Materials Science 52 (2007) 196–229
Tight-Binding
Analytic BOP
40
20
ΔE [%]
,
2
,
2
3
6
0
, 3
-20
0
1
2
3
2
3
8
8
1
4
12
4
5
N
6
7
4
12
80
1
2
3
4
6 ,
3
5
2
6
1
7
8
N
Fig. 6. Comparison of the structural energies of reduced TB (in the left column) and analytic BOP (in the right
column) for pr = 2/3 and bp/br = 1/6. The energies have been normalized with respect to the energy of a half-full
rectangular band with identical second moment. Shown are the dimer (1), the linear chain (2 0 ), the helical chain
(2, dashed line), the graphene sheet (3 0 ), the puckered graphene sheet (3, dashed line), cubic diamond (4), simple
cubic (6), simple hexagonal (8) and face-centered cubic (12). The fitting parameters take values cr = 1.27 and
cp = 1.
1
EzB ðRÞ ¼ z½UðRÞ 2Hð2Þ
ð67Þ
r bðRÞ;
2
if the promotion energy and p bond contributions are neglected. The first contribution
represents the repulsion between the z first nearest-neighbours, the second contribution
gives the attractive covalent bond energy within the second moment approximation to
the bond order, namely,
.pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Hrð2Þ ¼ 1
1 þ U2r :
ð68Þ
The former contribution will, therefore, vary as z, the latter contribution for the threedimensional diamond, simple cubic and face-centered cubic lattices, as z1/2 [54]. Assuming that the overlap repulsion U(R) / [br(R)]2 (cf. Eq. (24)), Eq. (67) becomes
pffiffi
2
EzB ðRÞ ¼ zA½br ðRÞ zBbr ðRÞ;
ð69Þ
where A and B are constants for a particular element. It is trivial to show that at equilibrium the binding energy is given by
EzB ðReq Þ ¼
1 B2 1 B2
1 B2
¼
:
4 A 2 A
4 A
ð70Þ
This is independent of coordination and hence the second moment approximation cannot predict relative structural stability of the diamond, simple cubic and fcc lattices. This
is in contrast to either TB or the 4-level approximation where the correct structural trend
is found in Fig. 6 (for the case of U(R) / [br(R)]2). Although second-moment-type Tersoff potentials can be fitted to reproduce the energy differences between different structures, they cannot explain in a systematic way the origin of structural stability for a
given element nor the structural trends across the periodic table as the group number
changes.
R. Drautz et al. / Progress in Materials Science 52 (2007) 196–229
215
4. Analytic bond-order potentials for Si and GaAs
The growth of Si and GaAs films have been simulated using our current parameterizations of the analytic BOP, apart from screening the bond integrals. Instead of introducing
an explicit screening function as in Eq. (25), they have been screened implicitly through the
use of pairwise Goodwin–Skinner–Pettifor (GSP) functions [55].
4.1. Si potential
The analytic Si BOP presented here ignores the electrostatic, magnetic, and van der
Waals energies. Therefore it follows from Eqs. ((1)–(3), (6), (17)) that
X prom
1 X
1 X
ESi
UðRij Þ ½2br ðRij ÞHr;ij þ 2bp ðRij ÞHp;ij þ
Ei ;
ð71Þ
B ¼
2 i;j;i6¼j
2 i;j;i6¼j
i
where U(Rij), br(Rij), and bp(Rij) are assumed to be pairwise functions approximated by
the GSP equations [55]
UðRÞ ¼ U0 ½hU ðRÞnU ;
nr
br ðRÞ ¼ br;0 ½hr ðRÞ ;
bp ðRÞ ¼ bp;0 ½hp ðRÞ
np
with hX (X = U, r, p) expressed as
n
nc;X R0
R0 c;X
R
exp
:
hX ðRÞ ¼
Rc;X
R
Rc;X
ð72Þ
ð73Þ
ð74Þ
ð75Þ
Here U0, br,0, bp,0, R0, nX, nc,X, and Rc,X (X = U, r, p) are parameters. In order to cutoff
the potential smoothly at a chosen cutoff distance Rcut, Eqs. (72)–(74) are used only when
the atomic separation distance R is within a chosen value R1. When R falls in the range of
R1 6 R 6 Rcut, a polynomial spline function SX(R) = aX + bXR + cXR2 + dXR3 (X = U,
br, and bp) is used. The parameters aX, bX, cX, and dX for functions U(R), br(R), and
bp(R) are determined in such a way that the value and the first derivative of the spline
function SX(R) match those of the corresponding [U(R), br(R), or bp(R)] function at
R = R1, and the value and the first derivative of the spline function SX(R) drops to zero
at R = Rcut. R1 and Rcut are two parameters for the cutoff. The r and p bond orders
are directly calculated from Eqs. (57) and (60).
A complete Si bond-order potential requires 13 parameters in the pair functions, two
parameters for the cutoff function, two parameters in the angular function, two parameters for the r bond order, one parameter in the p bond order, and two parameters in the
promotion energy. These 22 parameters were determined by fitting the predicted cohesive
energy, atomic volume, and bulk modulus to those obtained from either experiments or
ab initio calculations for a variety of structures including dimer, diamond cubic (dc),
face-centered-cubic (fcc), body-centered-cubic (bcc), and simple-cubic (sc) structures.
Extended test simulations of high temperature annealing and vapour deposition were also
carried out to ensure that no other phases had lower energy than the dc structure. The
(2 · 1) reconstruction on the (0 0 1) surface and its surface energy were also used to
aid the selection of the parameters. A complete set of parameters can be found in
Ref. [56].
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R. Drautz et al. / Progress in Materials Science 52 (2007) 196–229
4.2. GaAs potential
The parameterization of BOPs for a heterovalent system like GaAs requires the fitting
of the elemental subsystems as well as the description of the compounds. This is a much
more complex and tedious exercise than the parameterization of BOP for Si. As a first
step, we have simplified the analytic BOP for GaAs by neglecting both the promotion
energy contribution and the four-hop and ring contributions to the r bond order. Without
the promotion energy, the total energy is written as
1 X
1 X
EGaAs
¼
UðRij Þ ½2b ðRij ÞHr;ij þ 2bp ðRij ÞHp;ij :
ð76Þ
B
2 i;j;i6¼j
2 i;j;i6¼j r
The pair functions U(R), br(R), and bp(R) are approximated by the GSP function Eqs.
(72)–(74). The functions and their parameters depend on the species lm of the pair of
atoms ij [27]. The same cutoff function that was used for the Si pair functions is used to
smoothly cutoff the above pair functions for the Ga–As system.
Without the four-hopping and ring terms, the r bond order for half-full valence is
expressed by
ð12Þ
1
Hr;ij
¼ ð1 þ 2cr U2r Þ 2 :
ð77Þ
Here the parameter cr as well as the angular function gr Eq. (A.7) is species dependent. If
the species of atoms i, j, and k forming the bond angle hjik are l, c, and m, respectively, the
two parameters pr and br used in Eq. (A.7) are assumed to depend on the six bond
angle types clm (clm = GaGaGa, AsAsAs, AsGaAs, GaAsGa, GaGaAs/AsGaGa, and
AsAsGa/GaAsAs, where the bond angle types clm and mlc are equivalent). Note that
pr is used here instead of pr in Eq. (A.7) because the three-body dependent pr is no longer
equivalent to the pr used in the p bond hopping paths, Eqs. (A.13)–(A.15). The introduction of the species dependence increases the flexibility of the angular function.
The dependence of the r bond order on the fractional bond occupancy was calculated
using Eq. (66). In Eq. (66), the bond filling parameter f and the asymmetric skewing
parameter k1, k3, and k5, are assumed to depend on the bond type lc between atoms i
and j. The effect of bond filling on the p bond interaction has been neglected in this work.
This is justified by noting that the p bond contribution in bulk GaAs is small. The p bond
order, therefore, is calculated using Eq. (60) with the parameter cp being dependent on the
species of the pair i and j.
In summary, the complete GaAs bond-order potential presented here requires three sets
(lc = GaGa, AsAs, GaAs/AsGa) of 13 pairwise parameters, three sets of two pairwise cutoff parameters, six sets (clm = GaGaGa, AsAsAs, AsGaAs, GaAsGa, GaGaAs/AsGaGa,
and AsAsGa/GaAsAs) of two angular parameters, three sets of one pairwise parameter in
the half-full valence r bond order, three sets of two pairwise parameters for general r
bond order, two sets (l = Ga, As) of the species dependent parameter pr and three sets
of the pairwise parameter cp for the p bond order. These 71 parameters were determined
by fitting the predicted properties to those obtained from either experiments or DFT calculations for a wide range of structures.
It can be proven that the BOP pair functions alone completely define the relationship
between the equilibrium bond energy, bulk modulus and bond length for simple crystal
structures [38]. This allows us to determine completely all the pair functions by fitting
R. Drautz et al. / Progress in Materials Science 52 (2007) 196–229
217
the bond energy/bulk modulus as a function of bond length trend defined by the target
values of cohesive energy, bulk modulus, and lattice constant for a variety of selected simple phases spanning a wide range of local environments (chemistry, coordination, and
bond angles). The knowledge of the bond energy/bulk modulus versus equilibrium bond
length also facilitates the selection of specific target data sets from a large collection of
experimental and DFT data. Additional constraints were imposed during parameterization of the pair functions to ensure that they were smoothly cutoff. Once the pair functions
were determined, the angular function parameters were optimized in a second step to best
match the properties of various Ga, As, and GaAs phases. This approach was found to
significantly improve the transferability of the potential compared with other parameterization methods. The parameterization was published in Ref. [38].
5. Atomic assembly of Si film growth
5.1. Properties of Si bulk structures
During vapour deposition, a variety of surface configurations nucleate due to the adatom condensation at random locations. These configurations are often associated with
high energies, mismatch stresses and defects. They therefore will evolve towards lower
energy, reconstructed crystalline surfaces if permitted by the growth kinetics. Thus, the
accurate description of the cohesive energies, the atomic volumes, the elastic constants,
and the defect energies by the interatomic potential for a variety of configurations is essential for robust molecular dynamics simulations of growth.
Fig. 7 compares bulk property predictions of the analytic BOP with those of a Stillinger–Weber (SW) potential [57], two parameterizations of the Tersoff potential T2 [58] and
T3 [59], using our and published DFT data [60,61] and experimental measurements [62] as
the reference. In Fig. 7 the structures are arranged along the horizontal axis in decreasing
order of the cohesive energy from the DFT calculations as shown by the black monotonic
DFT curve in Fig. 7b. The BOP parameterization fits the relative stability of the different
phases well, apart from the structure st12. In general, the trend of phase stability predicted
by the other three potentials is less satisfactory. We see in Fig. 7a that the equilibrium
atomic volumes predicted by the BOP and the Tersoff potentials reproduce the DFT data
rather well, whereas the SW potential deviates significantly for close-packed systems.
Fig. 7c shows that the bulk moduli predicted by the BOP for different phases are significantly improved over those calculated by the other three potentials. In addition, the shear
elastic constants of the ground state dc structure are well reproduced by BOP. In Fig. 7d
we show the energies of four types of defects: vacancy, tetrahedral interstitial, hexagonal
interstitial, and x-split interstitial [63–67]. It indicates that the predictions of the overall
energies across different defects by the BOP are much closer to those of the DFT calculations [63–66] than those using the SW, T2, and T3 potentials [67].
5.2. Properties of Si surfaces
The Si surfaces commonly used for growth are the (1 0 0) and (1 1 1) surfaces. These surfaces undergo surface reconstructions. The surface reconstruction energies for the widely
observed (0 0 1) (2 · 1), (1 1 1) (7 · 7) [68], (110) (2 · 1)-adatom [69] and the (113) (3 · 2)
[70–72] surfaces predicted by the BOP are compared with data obtained from ab initio
218
R. Drautz et al. / Progress in Materials Science 52 (2007) 196–229
Fig. 7. (a) Atomic volume, (b) cohesive energy, and (c) elastic constants for a variety of Si phases. (d) Defect
energy of the (dc) Si structure.
Table 2
Surface reconstruction energies for four Si surface reconstructions (eV/Å2)
Surface
(1 0 0)
(1 1 1)
(1 1 0)
(1 1 3)
(2 · 1)
(7 · 7)
(2 · 1)-adatom
(3 · 2)
Ab initio
TB
0.054 [74]
0.403 [73]
0.190 [69]
0.036 [70–72]
BOP
SW
T2
T3
0.046
0.379
0.131
0.139
0.061
0.028
0.085
0.009
0.051
0.033
[69–74], SW, T2, and T3 calculations in Table 2. It can be seen that the reconstruction
energies predicted by the BOP are much closer to the ab initio data than those predicted
by the SW, T2 and T3 potentials. For the (7 · 7) surface, the SW and T3 potentials even
predict positive reconstruction energies. The BOP predictions of the surface properties are
hence superior to the other interatomic potentials [75].
5.3. Si growth simulation
The growth of (dc) Si films in the (0 0 1) direction has been simulated using molecular
dynamics [2–4] with analytic forces obtained from BOP. The initial substrate had the
predicted bulk equilibrium lattice constant. It was oriented in the ð1 1 0Þ x-direction,
(0 0 1) y-direction, and ð
1
1 0Þ z-direction. Periodic boundary conditions were used in both
R. Drautz et al. / Progress in Materials Science 52 (2007) 196–229
219
normal incidence
E = 0.17 eV
R = 0.4 nm/ns
y [001]
z [110]
x [110]
initial substrate
Fig. 8. Simulated atomic structure of the Si films. (a) T = 600 K, (b) T = 800 K, and (c) T = 900 K.
x- and z-directions while free boundary condition were used for the y-(growth) direction.
Growth was simulated by continuously injecting Si adatoms to the surface at random locations. The adatoms had a remote incident kinetic energy of 0.17 eV and their incident
direction was normal to the surface plane. To prevent the crystal from shifting during adatom impacts, atoms in the two lowest y-planes were fixed during simulations. Heating of
the film due to dissipation of adatom kinetic energy and the latent heat release during adatom condensation was prevented by maintaining a sub-surface region above the fixed
atoms at the desired growth temperature using the Nose–Hoover thermostat algorithm
[76]. Newton’s equation of motion was then used to evolve the positions of the atoms
in the system. An accelerated growth rate of 0.4 nm/ns was used to grow films that were
sufficiently thick for a further analysis of the film structures.
As an example the atomic structures of the films simulated at three different substrate
temperatures are shown in Fig. 8. It can be seen that increasing the temperature results in
an improvement of the film crystallinity, in good agreement with experiments [77–79].
BOP-based molecular dynamics simulations therefore provide a valuable tool to explore
the detailed atomic assembly mechanisms during the growth of Si films.
6. Atomic assembly of GaAs film growth
6.1. Properties of the GaAs bulk structures
Zinc-blende (zb) GaAs films were grown using molecular As2 and atomic Ga vapour
sources. The growth mechanisms for the binary GaAs film are more complex than for
Si, mainly due to two reasons. First, the two elements that condense at random positions
of the film surface must find their way in the correct sublattice of the zb crystal. This
requires an accurate description of the energies of point defects. Second, experiments show
that crystalline (zb) GaAs films usually grow when the As:Ga flux ratio is significantly
higher than unity [34,80–82]. This is due to the fact that the binding energy between
As2 molecules is much weaker than that within an As2 molecule. As a result, As2 molecules
readily evaporate in experiments. To capture these effects, the potential must be able to
model the transition between As2 molecules and As atoms on the surface, including the
relatively strong binding energy of As2 molecules and the relatively low binding energies
of larger As clusters.
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R. Drautz et al. / Progress in Materials Science 52 (2007) 196–229
Fig. 9 compares bulk property predictions of the BOP with those of a Stillinger–Weber
(SW) potential [83,84] and a Tersoff potential [12] using the reference data compiled from
our and published DFT calculations [12,85–92] as well as experiments [80,93–98]. Fig. 9a
and b show that the BOP and the Tersoff potential predict well the atomic volumes and
cohesive energies. They capture correctly the relatively large binding energy of the As2
dimer, and therefore are more likely to predict the evaporation of As2 molecules during
As deposition. In sharp contrast, both the cohesive energies and the atomic volumes of
different phases predicted by the SW potential [83,84] are significantly different from those
of the reference data. More importantly, the SW potential significantly underestimates the
cohesive energy of the As2 dimer. It therefore prohibits the evaporation of the As2 molecules during deposition and cannot be used to study the effects of As:Ga vapour flux ratio
on the structure of the GaAs films. Fig. 9c indicates that the overall bulk modulus for different phases predicted by the BOP are significantly improved compared with the Tersoff
and the SW potentials.
The characteristic neutral defect formation energies [99,100] in the zinc-blende GaAs
lattice are compared in Fig. 9d. From all possible point defects, defects that are important
for characterizing the potential were selected. These include Ga and As vacancies (VGa and
VAs), Ga and As antisites (GaAs and AsGa), Ga and As interstitials at the tetrahedral site
(Gai,tet and Asi,tet) and the <1 1 0> dumbbell site [92,101] (Gai,<110> and Asi,<110>). Fig. 9d
Fig. 9. (a) Atomic volume, (b) cohesive energy, and (c) elastic constants for a variety of Ga, As, and GaAs
phases. (d) Defect energy of the (zb) GaAs structure.
R. Drautz et al. / Progress in Materials Science 52 (2007) 196–229
221
shows that defect formation energy calculated by the BOP match very well with the reference data for most defects except the Astet where the predicted value is lower than the reference value. In sharp contrast, the defect energies predicted by both the Tersoff and the
SW potentials deviate significantly from the reference data.
6.2. Properties of GaAs surfaces
The GaAs (0 0 1) surface exhibits many reconstructed structures [33,34,102,103]. These
include the experimentally validated As-terminated b2(2 · 4) [104–106], As-terminated
a2(2 · 4) [107], As-rich c(4 · 4) [108,109], and Ga-rich f(4 · 2) [32] surface reconstructions.
The surface reconstructions are affected by temperature, vapour composition and deposition rate [31].
The occupancy of dangling bonds is not treated explicitly in the BOP or other empirical
potentials. However, the electron redistribution in dangling bonds has been shown to play
a significant role in stabilizing GaAs (0 0 1) surface reconstructions [36,110,111]. To
address this, the electron counting approach described in Section 2.1.5 was superimposed
upon the various potentials to calculate the surface free energy c [13,112]. The results for
the relative energy (with respect to the ca2(2·4) surface) of the minimum energy surfaces is
plotted in Fig. 10 as a function of the relative As chemical potential normalized with
respect to the heat of formation of zb–GaAs. DFT data [32,113] is included in the figure
for comparison.
The charge build-up effects that destabilize the b(2 · 4) surface reconstruction with
respect to the b2(2 · 4) reconstruction were not included in the model due to the absence
of Coulombic electrostatic interactions [106]. Therefore, the free energy predictions by the
Fig. 10. Relative surface energy of the lowest energy (0 0 1) GaAs surfaces as a function of the relative,
normalized As chemical potential.
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R. Drautz et al. / Progress in Materials Science 52 (2007) 196–229
potentials cannot differentiate between the b(2 · 4) and b2(2 · 4) surface reconstructions.
As can be seen in Fig. 10, the surface phase diagram predicted by BOP is closer to the DFT
results than the surface phase diagram obtained from SW or Tersoff potentials. At Ga-rich
conditions (low As chemical potential), all three potentials predict that the b(4 · 2) and the
b2(4 · 2) surface reconstructions are most stable, in contradiction to the f(4 · 2) reconstruction shown in the DFT calculations. When the As chemical potential is increased,
both the BOP and the Tersoff potential predict that the a(2 · 4) and a2(2 · 4) surfaces
are most stable, in agreement with the DFT data. In contrast, the SW potential predicts
the a(4 · 2) and a2(4 · 2) to be the most stable surfaces. When the As chemical potential
is further increased, all three potentials show that the b(2 · 4) and the b2(2 · 4) surfaces
are most stable, in agreement with the DFT results. Finally, under As-rich condition (high
As chemical potential), the BOP predicts that the c(4 · 4) reconstruction is most stable, as
it is also found from the DFT calculations. No stable or metastable c(4 · 4) surface was
predicted by either the Tersoff or the SW potential when coupled with the electron counting rule as this surface reconstruction dissociated during energy minimization.
6.3. GaAs growth simulation
Following the molecular dynamics approach described for Si above, the BOP was used
to simulate the growth of GaAs films from As2 and Ga vapour fluxes using a wide range of
deposition conditions that cover substrate temperatures T between 500 K and 1500 K and
As:Ga flux ratios R between 0.9 and 3.4. Examples of the atomic structures grown at various substrate temperatures and flux ratios are shown in Fig. 11. We assumed that the
vapour particle incident direction is normal to the growth surface, the deposition rate
was 0.125 nm/ns and the adatoms have a thermalized kinetic energy. Fig. 11a and b indicate that at a near constant flux ratio of R = 1.1 1.2, the film crystallinity improves as
substrate temperature is increased from 500 K to 800 K. Further increase in temperature
resulted in a further improvement of the film crystallinity, Fig. 11c and d. It is interesting
that the best quality film shown in Fig. 11 was obtained at a flux ratio of 3.14, which is
significantly higher than the unity of the stoichiometric film composition. The observed
effects of substrate temperature and vapour flux ratio on the crystallinity of the films
are in good agreement with the experiments [34,80–82]. The excess As:Ga ratio is due
to the experimentally observed evaporation of As2 molecules from the As-rich surface.
Clearly, the effect was correctly captured during simulations because the relative low
energy of the As2 molecules with respect to the isolated and condensed As atoms was well
predicted, Fig. 9. It should be pointed out that to our knowledge, this is the first time that
both the substrate temperature and flux ratio effects on the atomic structure of the GaAs
film has been demonstrated.
Analysis of extensive BOP molecular dynamics simulations revealed a clear relationship
between GaAs film structure and deposition conditions. As2 evaporation increased as the
growth temperature was increased. As a result, Ga-rich surfaces were observed during simulations at high temperatures and near unity As:Ga flux ratios. When the flux ratio was
increased at high growth temperatures, excessive As atoms that initially condensed on
the surface were found to later desorb, resulting in stoichiometric films. Excessive As
was incorporated into the film only at a low growth temperature and high As:Ga flux
ratio, until a temperature-dependent solubility limit was reached. All these observations
are in good agreement with experiments [34,114–118].
R. Drautz et al. / Progress in Materials Science 52 (2007) 196–229
223
Fig. 11. Simulated atomic structures of the GaAs films after 10 ns of deposition at different substrate
temperatures T and As:Ga flux ratios R. (a) T = 500 K, R = 1.14; (b) T = 800 K, R = 1.19; (c) T = 1100 K,
R = 1.67, and (d) T = 1500 K, R = 3.14.
7. Conclusions
The simulation of the growth of thin semiconductor films provides a stringent testbed
for interatomic potentials. The BOP framework achieves the chemical flexibility that is
required to describe bonding from the dimer through to the bulk by systematically coarse
graining the electronic structure, thereby deriving the format of the potential. The derived
format enables a straightforward interpretation of the physical meaning of the parameters
and subsequently a direct calculation of the numerical values of the parameters from first
principles. This approach has led to preliminary interatomic potentials for Si and GaAs
that are suitable for molecular dynamics simulation of thin film growth.
Our simulations for Si and GaAs show that the BOPs are able to model the growth of
thin semiconductor films in a more realistic description than other potentials. Particularly,
the effects of growth temperature on the crystallinity of Si films, and the effects of both
growth temperature and As:Ga vapour flux ratio on the crystallinity and defect population
of GaAs films are correctly described.
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R. Drautz et al. / Progress in Materials Science 52 (2007) 196–229
Acknowledgements
We are grateful to the Defense Advanced Research Projects Agency and Office of Naval
Research (C. Schwartz and J. Christodoulou, program managers) for support of this work
through grant N00014-03-C-0288. We also thank S.A. Wolf for numerous helpful
discussions.
Appendix A. Expressions for hopping paths
In Section 3.2.1 the following shorthand notation has been used for the r bond:
and
1
U2r ¼ ðUi2r þ Uj2r Þ;
2
1
2
2
U22r ¼ ½ðUi2r Þ þ ðUj2r Þ ;
2
1
U4r ¼ ðUi4r þ Uj4r Þ;
2
ðA:2Þ
DU4r ¼ U4r U22r =U2r :
ðA:4Þ
ðA:1Þ
ðA:3Þ
The expressions for the 2- and 4-hop self-returning hopping paths are given by
2
X
b ðRik Þ
Ui2r ¼
g2r ðhjik Þ r
;
br ðRij Þ
k6¼i;j
ðA:5Þ
and
Ui4r
¼
X
g2r ðhjik Þ
k6¼i;j
br ðRik Þ
br ðRij Þ
4
2 2
br ðRik Þ br ðRik0 Þ
br ðRij Þ
br ðRij Þ
k;k 0 6¼i;j
2 2
X
b ðRik Þ br ðRkk0 Þ
þ
g2r ðhjik Þg2r ðRikk0 Þ r
:
br ðRij Þ
br ðRij Þ
k;k 0 6¼i;j
þ
X
gr ðhjik Þgr ðhkik0 Þgr ðhk0 ij Þ
ðA:6Þ
The angular function gr(hjik) is expressed as
gr ðhjik Þ ¼
1 pr þ pr cos hjik þ br cos 2hjik
;
1 þ br
ðA:7Þ
where hjik is the bond angle between atoms j and k centered on atom i. br is a fitting
parameter that has been introduced to give additional flexibility to the curvature of the
angular function, since this controls the bond-bending force constant. For br = 0, the
angular function is determined solely by pr, which we have seen from Eqs. (9) and (10)
determines the amount of p character in the bonding hybrid. As expected, for only s orbitals with pr = 0, the angular function takes the constant value of unity, whereas for only p
orbitals with pr = 1, the angular function varies as cos h.
R. Drautz et al. / Progress in Materials Science 52 (2007) 196–229
225
The corresponding 3-atom and 4-atom ring-type interference contributions (cf. Fig. 5)
are given by
X
b ðRik Þ br ðRjk Þ
Rij3r ¼
;
ðA:8Þ
gr ðhjik Þgr ðhkji Þgr ðhikj Þ r
br ðRij Þ br ðRij Þ
k;k6¼i;j
and
Rij4r ¼
X
gr ðhjik Þgr ðhikk0 Þgr ðhkk0 j Þgr ðhijk0 Þ k;k 0 6¼i;j;k6¼k 0
br ðRik Þbr ðRkk0 Þbr ðRjk0 Þ
:
br ðRij Þbr ðRij Þbr ðRij Þ
ðA:9Þ
^ 3 between atoms i and j that enters Eq. (66) is defined
The normalized ring contribution R
by
b 3 ¼ Rij3r =ð1 þ U2r Þ
R
ðA:10Þ
for r bonds. Similarly, for the p bond we have used the notation
1
U2p ¼ ðUi2p þ Uj2p Þ;
2
1
U4p ¼ ðUi4p þ Uj4p Þ:
2
ðA:11Þ
ðA:12Þ
The self-returning 2-hop and 4-hop contributions are given by
"
#
2
2
X
b
ðR
Þ
b
ðR
Þ
ik
ik
þ ð1 þ cos2 hjik Þ p2
pr sin2 hjik r2
Ui2p ¼
;
bp ðRij Þ
bp ðRij Þ
k;k6¼i;j
1 X
2
2
2 2
^2 b
^2
0^ ^
½sin2 hjik sin2 hjik0 b
Ui4p ¼
ik ik 0 þ sin hjik sin hijk bik bjk 0 cos 2ð/k /k 0 Þ;
2 k;k0 6¼i;j
ðA:13Þ
ðA:14Þ
^2 is defined by
b
2
2
^2 ¼ p br ðRik Þ bp ðRik Þ :
b
r
ik
b2p ðRij Þ b2p ðRij Þ
ðA:15Þ
The dihedral angle contribution may be expressed as
cos 2ð/k /k0 Þ ¼
2ðcos hkik0 cos hjik0 cos hjik Þ2
sin2 hjik sin2 hjik0
1:
ðA:16Þ
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