analytic bond-order potentials for modelling the …...the atomistic modelling hierarchies. in a...

Progress in Materials Science 52 (2007) 196–229

www.elsevier.com/locate/pmatsci

Analytic bond-order potentials for modellingthe growth of semiconductor thin films

R. Drautz 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, UKb 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 mustbe able to describe the breaking and making of covalent bonds in an efficient format so that molec-ular dynamics simulations of thousands or millions of atoms may be performed. We review the der-ivation of such potentials, focusing upon the emerging role of the bond-based analytic bond-orderpotential (BOP). The BOP is derived through systematic coarse graining from the electronic tothe atomistic modelling hierarchies. In a first step, the density functional theory (DFT) electronicstructure is simplified by introducing the tight-binding (TB) bond model whose parameters are deter-mined directly from DFT results. In a second step, the electronic structure of the TB model is coarsegrained through atom-centered moments and bond-centered interference paths, thereby deriving theanalytic form of the interatomic BOP. The resultant r and p bond orders quantify the concept ofsingle, double, triple and conjugate bonds in hydrocarbon systems and lead to a good treatmentof radical formation. We show that the analytic BOP is able to predict accurately structural energydifferences in quantitative agreement with TB calculations. The current development of these poten-tials for simulating the growth of Si and GaAs thin films is discussed.� 2006 Elsevier Ltd. All rights reserved.

0079-6425/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.pmatsci.2006.10.013

* Corresponding author. Tel.: +44 1865 273700; fax: +44 1865 273789.E-mail address: [email protected] (R. Drautz).

mailto:[email protected]

R. Drautz et al. / Progress in Materials Science 52 (2007) 196–229 197

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1972. Coarse graining I: from DFT to TB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

2.1. The reduced TB model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
2.1.1. Repulsive energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2002.1.2. Promotion energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2002.1.3. Covalent energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2002.1.4. Ionic energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2022.1.5. Electron counting rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2022.1.6. Magnetic energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2042.1.7. Van der Waals energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
2.2. Parameterization of the reduced TB model from first principles . . . . . . . . . . . . . 204
2.2.1. The homovalent dimer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2042.2.2. Screening the dimer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
3. Coarse graining II: from TB to BOP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
3.1. Exact many-atom expansion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2073.2. Analytic BOP for covalent bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
3.2.1. Simplified expression for the r bond order with half-full valence . . . . . . 2103.2.2. The p bond order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2113.2.3. Analytic expression for the promotion energy. . . . . . . . . . . . . . . . . . . . . 2113.2.4. Generalization of the analytic bond order to non-half-full valence . . . . . 2123.2.5. Structural prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

4. Analytic bond-order potentials for Si and GaAs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
4.1. Si potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2154.2. GaAs potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
5. Atomic assembly of Si film growth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
5.1. Properties of Si bulk structures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2175.2. Properties of Si surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2175.3. Si growth simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
6. Atomic assembly of GaAs film growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
6.1. Properties of the GaAs bulk structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2196.2. Properties of GaAs surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2216.3. GaAs growth simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
7. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224Appendix A. Expressions for hopping paths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

1. Introduction

Molecular dynamics simulations provide a powerful method for exploring the mecha-nisms of atomic assembly of thin films during vapour phase growth. Recently embeddedatom method potentials [1] have been successfully applied to model the growth of metalmultilayers [2–4], and by including charge transfer [5,6], extended to the simulation ofthe growth of metal oxide multilayers [7]. However, these embedded atom method poten-tials are unable to model the growth of semiconductor thin films since the directional char-acter of the covalent bonds is not taken into account.

198 R. Drautz et al. / Progress in Materials Science 52 (2007) 196–229

Robust simulations of the vapour phase growth of covalently bonded semiconductormaterials would be of significant technological importance, for example, in developingprocessing routes for synthesizing spintronic materials [8] or in the design of novel hardcoatings [9]. Molecular dynamics studies of the mechanisms of atomic assembly of cova-lently bonded thin films require an interatomic potential that is able to describe the gasphase interactions as well as the interatomic forces in the solid state, and most impor-tantly, the transitions of the atoms between the gas and the solid phase, including the for-mation and breaking of bonds in small clusters, at surfaces and in the bulk. En passant, thepotentials must capture the intuitive concepts used in chemistry and material science tounderstand and to explain the complex processes in simple terms, such as bond formationand 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 materialsare 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 anattractive pairwise contribution that depends on the bond order, which measures the dif-ference 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 thatdepends on the number and types of atoms surrounding a given bond. Despite numeroussuccessful applications of Tersoff–Brenner potentials, the ad hoc expressions for the bondorder have been found to suffer from serious shortcomings when used to model the growthof thin films. For example, Albe et al. [12] found that the description of the As-rich sur-faces in GaAs was unphysical, being unable to modify their fitting parameters to agreewith experiments or ab initio predictions. We have shown elsewhere that the availableparameterizations of the Tersoff–Brenner type potentials for GaAs predict either unrealis-tic forces between the As–As dimer bond, or underestimate the sticking probability of theAs2 molecule upon impact on the surface during vapour deposition [13,14]. A detailed dis-cussion of Tersoff–Brenner potentials for carbon and hydrocarbons is given by Mrovecet al. [9] in this issue.

An alternative approach to developing robust interatomic potentials for covalentlybonded materials has been to extend classical valence force fields to handle bond breakingand remaking explicitly. These so-called reactive force fields (ReaxFFs) were initiallydeveloped for the hydrocarbons [15,16] but have been extended to cover a wide rangeof sp-valent elements and some transition metals [17]. The analytic form of the ReaxFFsis essentially empirical requiring nearly fifty fitting parameters for each element. The manyparameters are required in the ReaxFF framework in order that the empirical bond-orderfunction is able to describe the complex chemistry of covalent bond formation. The accu-racy of ReaxFFs for modelling surface reconstructions and the growth of compound semi-conductor thin films such as GaAs, has yet to be demonstrated.

In this review we show how interatomic bond-order potentials (BOPs) can be derivedfrom quantum mechanics by systematically coarse graining the electronic structure attwo levels of approximation as illustrated in Fig. 1.

1. In the first step, the expression for the binding energy of a material within the effectiveone-electron density functional theory (DFT) formalism [18] is re-written in terms ofphysically and chemically intuitive contributions within the tight-binding (TB) bondmodel [19]. This TB approximation is sufficiently accurate to predict structural trends

Fig. 1. Illustration of electronic and atomistic modelling hierarchies and the derivation of analytic interatomicbond-order potentials through two steps of coarse graining, firstly from DFT to TB and secondly from TB toBOP.


across the sp-valent elements, as well as sufficiently simple to allow a physically mean-ingful interpretation of the bonding in terms of r and p contributions. We demonstratehow the unknown TB parameters can be obtained from ab initio calculations in a sys-tematic way.

2. In the second step, the TB electronic structure is coarse grained through atom-centeredmoments and bond-centered interference paths as discussed in the first paper of thisissue [20]. This allows the bond order to be related to the local topology and coordina-tion of the material. In this way the functional form of the bond order is derived as afunction of positions and types of atoms that surround a given bond. We argue thatthese analytic bond-order potentials [21,22] should overcome many of the shortcomingsof empirical bond-order parameterizations.

The outline of this review is as follows. We begin in Section 2 by discussing the coarsegraining of the DFT electronic structure and the parameterization of the TB bond inte-grals and repulsive contributions. In Section 3 we discuss the coarse graining of the TBelectronic structure in order to derive analytic expressions for the TB r and p bond ordersin terms of the local environment. We illustrate the accuracy of the derived BOP by pre-dicting the known structural trends across the sp-valent elements (Section 3.2.4). In Sec-tion 4 we fit BOPs for Si and GaAs. In Sections 5 and 6 these potentials are applied tothe 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 approxima-tion by deriving it from density functional theory as discussed in detail in the first paperof this issue [20]. It follows from Eq. (69) of [20] that the total binding energy EB of a mul-ticomponent sp-valent system within the orthogonal TB bond model may be written as asum of several physically based contributions,


EB ¼ Erep þ Eprom þ Econv þ Eion þ Emag þ EvdW: ð1ÞWe discuss each of the terms in the following subsections.

2.1.1. Repulsive energyThe first term Erep contains the overlap and electrostatic repulsion and is often approx-

imated by a simple pair potential U,

Erep ¼ 1

2

Xi;j;i6¼j

UðRijÞ; ð2Þ

where Rij is the distance between atoms. Recent, more accurate TB schemes express Erep inthe form of a many-body potential by taking into account the screening of both the over-lap [23] and electrostatic [24] repulsive contributions in the local atomic environmentabout a bond.

2.1.2. Promotion energy

The second term Eprom is the energy of promotion that arises from the change in thehybridization state when sp-valent atoms are brought together from infinity,

Eprom ¼X

i

Epromi ¼

Xi

ð�lp � �ls ÞDNls : ð3Þ

As shown in Section 2.2.1 the level splitting dl ¼ ð�lp � �ls Þ of the atomic s and p states isapproximately constant and independent of bond length for a given atomic species l, sothat it may be assumed to take its free atom value. The difference in the number of elec-trons in the s(p) orbital with respect to the occupancy of the s(p) state of the free atom isexpressed as DN l

sðpÞ ¼ N lsðpÞ � Nl;0

sðpÞ. For covalently bonded materials the charge transfer

between atoms is often small. Thus, if we assume local charge neutrality (LCN), thechange in the number of s and p electrons will satisfy,

DNls ¼ �DN l

p: ð4Þ

2.1.3. Covalent energy

The third term Ecov is the attractive covalent bond energy. It can be written in the form

Econv ¼Xa¼s;p

Z �F

ð�� aÞnað�Þd�; ð5Þ

where �F is the Fermi energy and ns(p) is the local s(p) electronic density of states. Thecovalent bond energy may be decomposed into contributions from individual bonds

Econv ¼ 1

2

Xi;j;i 6¼j

ðEconvÞlmij ð6Þ

with

ðEconvÞlmij ¼ 2

Xm;m0

Hlmim;jm0H

lmjm0 ;im: ð7Þ

The matrix elements of the Hamiltonian H and the bond-order H are evaluated within theSlater–Koster two center approximation [25]. If the z-axis of the local coordinate system is


chosen to point in the direction of the bond ij, then the matrix elements of H are written asssrlm

ij , sprlmij , etc.

We simplify the expression for the bond energy by making the reduced TB approxima-tion [26,27]

sprlmij psrlm

ij ¼ ssrlmij pprlm

ij : ð8ÞThe reduced TB approximation imposes the physically intuitive picture of a single r bondorder, i.e., it ensures the simple rule of chemistry that a sp-valent material may form onlyone fully saturated r bond but two saturated p bonds is obeyed. The reduced TB approx-imation is valid to within 16% for Harrison’s canonical TB parameterization of sp-valentelements [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 bymaking a basis transformation from atomic orbitals to bonding hybrids that point into thebond

jilri ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi1� pl

r

pjilsi þ

ffiffiffiffiffipl

r

pjilzi; ð9Þ

jjmri ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi1� pm

r

pjjmsi �

ffiffiffiffiffipm

r

pjjmzi; ð10Þ

and non-bonding hybrids that point away from the bond,

jilr�i ¼ffiffiffiffiffipl

r

pjilsi �

ffiffiffiffiffiffiffiffiffiffiffiffiffi1� pl

r

pjilzi; ð11Þ

jjmr�i ¼ffiffiffiffiffipm

r

pjjmsi þ

ffiffiffiffiffiffiffiffiffiffiffiffiffi1� pm

r

pjjmzi: ð12Þ

The prefactors plr and pm

r give the relative admixture of p character in the bonding hybridand determine the directional character of the bond. Within the reduced TB approxima-tion they take the values

plr ¼ ðpprlm

ij Þ2=½ðsprlm

ij Þ2 þ ðpprlm

ij Þ2�; ð13Þ

and

pmr ¼ ðpprlm

ij Þ2=½ðpsrlm

ij Þ2 þ ðpprlm

ij Þ2�: ð14Þ

With respect to the new basis, the 2 · 2 r-block in the Hamiltonian matrix entering Eq. (6)takes the diagonal form

Hlmr;ij ¼

blmr;ij 0

0 0

� �; ð15Þ

where the bond integral between the bonding and the hybrids is given by

blmr;ij ¼ ssrlm

ij

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1� pl

rÞð1� pmrÞ

q�: ð16Þ

The zero eigenvalue in Eq. (15) reflects the absence of bonding between the non-bondinghybrids, as expected. Substituting Eq. (15) into Eq. (6) the covalent bond energy can bewritten explicitly as

ðEconvÞlmij ¼ 2blm

r;ijHmlr;ij þ 2blm

p;ijðHmlpþ ;ijþHml

p�;ijÞ; ð17Þ

where

blmp;ij ¼ ppplm

ij : ð18Þ


We see, therefore, that the reduced TB approximation allows us to write the bond energyin terms of contributions from a single r bond order and two p bond orders, as taught instandard chemistry textbooks.

2.1.4. Ionic energyThe 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 main-tain local charge neutrality. This neglect of the ionic contribution in Eq. (1) is a goodapproximation for homovalent semiconductors, but begins to break down for III–V het-erovalent compounds such as GaAs. In the molecular dynamics simulations presented inthis paper we implicitly model the charge transfer between the Ga and As dangling bondsat 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 (001)GaAs surface [30,31]. These surfaces often have special surface stoichiometry. A typicalexample is the As-terminated b(2 · 4) surface reconstruction, which requires that oneAs dimer is missing for every four As dimers in the [110] dimer row direction such thatthe number of the surface As dimers MAs and surface Ga dimers MGa satisfiesMAs = 0.75MGa. Density functional theory calculations show that this surface reconstruc-tion has a low surface free energy over a wide range of atmospheric conditions when com-pared 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 Asbonds. The ratio of Ga to As dangling bonds is such that all the electrons from high energyGa surface dangling bonds can be redistributed into low energy As dangling bonds. Inter-atomic potentials that do not explicitly treat this redistribution of electrons from Ga to Asdangling bonds have been found to predict incorrectly the relative energies of the surfacestructures [13,35].

Pashley [36] successfully explained the surface reconstructions by using the electroncounting rule (ECR). The ECR assumes that low energy reconstructions are obtainedwhen the low energy Ga–As, Ga–Ga, As–As and the As dangling bonds are fully occupiedby two electrons while the high energy Ga dangling bonds are left empty. This rule is con-sistent with most of the known surface reconstructions in GaAs.

The condition of local charge neutrality that we assume in the derivation of the BOPsdoes not allow the environment-dependent occupation of dangling bonds. We thereforehave developed a separate model to incorporate ECR into molecular dynamics simula-tions [13]. The basic concept is explained as follows. Suppose that each atom, i, has avalence Ni (Ni = 5 for an As atom and Ni = 3 for a Ga atom), then the total number ofelectrons Ntot in a system with n atoms is given by

N tot ¼Xn

i¼1

N i: ð19Þ

Assume further that the bond between atoms i and j is occupied by Nij electrons, and atomi contains ai electrons per dangling bond. The total number of electrons in all interatomicand dangling bonds can be written as


NECR ¼1

2

Xi;j;j 6¼i

N ij þX

i

aidi; ð20Þ

where di is the total number of dangling bonds on atom i. The ECR criterion that the Asdangling bonds as well as all other nearest-neighbour bonds are occupied and that the Gadangling bonds are empty, means that ai = 2 for As atoms, ai = 0 for Ga atoms, Nij = 2for 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 bedistributed into low energy dangling bonds, hence Ntot = NECR. When the number of elec-trons Ntot does not match the number of available low energy electron states NECR, highenergy Ga dangling bonds are occupied with electrons, thereby increasing the surface en-ergy. By expanding the surface energy in the vicinity of Ntot � NECR to second order, theenergy increase for violating the ECR by occupying high energy Ga dangling bonds maybe written

DEECR ¼ wðN tot � N ECRÞ2; ð21Þ

where w is a parameter that is defined by the energy required to occupy Ga danglingbonds. 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. Toretain the fidelity of the BOP for modelling GaAs, the term DEECR must drop to zero with-in a bulk crystal. It only becomes positive at a surface that violates the ECR. Two exam-ples 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 equalsthe number of low energy electron states NECR, therefore DEECR = 0. The additionof the ECR modification does not affect the BOP potential for GaAs bulklattices.

As a second example, consider an As-terminated b(2 · 4) surface. We consider onlythe top As and its bonds with the adjacent underlying Ga atomic planes. Assume thatthe 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) surfacestructure, we have MAs = 0.75MGa. From Eq. (19), the total number of electrons isNtot = 2NAsMAs + NGaMGa=10.5MGa, where we took into account that half of the Gaelectrons 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 (001) sur-face. The addition of each As dimer converts four Ga dangling bonds into four Ga–Asbonds, 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 thatDEECR = 0.

Eqs. (19) and (20) do not constitute an interatomic potential because they are not con-tinuous functions of atomic positions. The ECR modification has recently been cast intothe form of an interatomic potential [37]. This potential successfully predicts many of the(001) 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].


2.1.6. Magnetic energy

We do not include the magnetic contribution to the binding energy. For the non-mag-netic semiconductor bulk and surfaces, this is a very good approximation. However, itshould be noted that magnetism contributes significantly to the energy of sp-valent mate-rials in situations where the atoms are pulled apart towards their non-singlet free atomstates. For example, the magnetic energy contributions to C and Si free atoms are between1 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 dipolemoments 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 toDFT. However, it is straightforward to model the van der Waals contribution with a long-ranged 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 fromfirst-principles DFT calculations in a step-by-step procedure. We start from the simplestpossible molecule, the homovalent dimer, and then take into account the screening ofthe 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 plevels, 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 �pzand �px;y

, corresponding to the pz and (px,py) orbitalsrespectively. The six reduced TB parameters for the first to fourth row sp-valent dimers areobtained from their non-spinpolarized DFT eigenspectra. As an example, the solid curvesin the left hand panel of Fig. 2 show the four r and the two p eigenvalues for Si2 as a func-tion of bond length. The resultant values of the reduced TB parameters, from the inversionof this spectrum, are shown in the right hand panel of Fig. 2. The degeneracy of the pz andpx,y atomic p levels is split by the crystal field and non-orthogonality overlap contribu-tions. Nevertheless, this splitting remains small compared to the total s-p splitting. Sincethe overlap repulsion affects the upwards shift of both the s and p atomic energy levelsin a similar way, the relative energy �p � �s is found to change only slowly as the atomsare brought together from infinity. Thus we approximate �pz

¼ �px;yand d = �p � �s = d0,

see Fig. 2. We see that an interpolation of the bond integrals with a simple exponentialfunction bexp

rðpÞ is able to capture the behaviour of the TB parameters for distances largerthan the equilibrium bond length.

The repulsive energy of a homovalent dimer may now be obtained by calculating ana-lytically the covalent bond and promotion energies of the sp-valent dimer using thereduced TB bond parameters obtained from the eigenspectrum and subtracting them fromthe DFT binding energy in Eq. (1), namely

Erep ¼ EB � Econv � Eprom: ð22Þ

3 4 5 6 7 8 9 10R [a.u.]

-14.0

-12.0

-10.0

-8.0

-6.0

-4.0

-2.0

0.0

2.0

E [

eV]

3 4 5 6 7 8 9 10R [a.u.]

-10.0

-8.0

-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0

8.0

10.0

E [

eV]

δ

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 theeigenspectrum, solid curves correspond to interpolation of the data with a simple analytic function. Theexponential tail of the bond integral interpolation is indicated by bexp

p and bexpr . 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 broughttogether to form the dimer. The fact that Eq. (22) defines a strictly positive repulsiveenergy shows that the reduced TB model is indeed a physically sound model of theDFT electronic structure. As can be seen from Fig. 3, the repulsive energy decays fasterthan 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; ð23Þwhere the first term describes the repulsion due to the overlap of the non-orthogonal atom-ic orbitals and the second term Ecore contains electrostatic interaction between the atoms,including their ion-core repulsion (see Fig. 3 of [40]).

3 4 5 6 7 8 9 10R [a.u.]

-15.0

-10.0

-5.0

0.0

5.0

10.0

15.0

20.0

E [

eV]

3 4 5 6 7 8 9R [a.u.]

0.0

5.0

10.0

Ere

p [eV

]

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-rangedhard-core potential (dashed curves). The exponent k of the overlap repulsion for Si is found to be k = 1.9. Thehard-core potential becomes important only at distances smaller than the dimer bond length.


If one assumes a decay of the overlap matrix elements Olmij proportional to the Hamil-

tonian matrix elements, Olmij / Hlm

ij , then to first order the overlap repulsion scales as thesquare of the bond integrals [23]. The functional form of the overlap repulsion

ðEovÞlmij ¼ aðblm

r;ijÞk with k � 2 ð24Þ

is able to describe the decay of the repulsion for distances larger than the equilibriumdimer bond lengths. The short-ranged core repulsion Ecore is interpolated with a general-ized Yukawa-type potential, see Fig. 3.

2.2.2. Screening the dimer

Interatomic potentials for modelling the growth of semiconductors must be able todescribe the gas phase, the surface and the bulk with the same set of parameters. In orderthat the orthogonal reduced TB model becomes transferable to different surroundings, theenvironmental dependence of the bond integrals on the surrounding atoms must be takeninto account. Starting from a non-orthogonal TB representation, effective orthogonal TBHamiltonian matrix elements can be derived [23] in the form

Hlmij ¼ H ð0Þ;lm

ij ð1� Slmij Þ; ð25Þ

where H ð0Þ;lmij refers to the unscreened Hamiltonian matrix element and Slm

ij represents thescreening matrix element. If the z-axis of the local coordinate system is pointing along theaxis of the bond, the unscreened Hamilton matrix element is given by the appropriatedimer bond integral. Keeping terms to second order in Eq. (11) of [23], the screening func-tion Slm

ij may be written in terms of the unscreened Hamiltonian matrix elements and theoverlap matrix elements Olm

ij ¼ hiljjmi,

Slmij ¼

1

2

1

H ð0Þlmij

Xk 6¼i;j

ðH ð0Þljik Ojm

kj þ Oljik H ð0Þjm

kj Þ: ð26Þ

Fig. 4 shows the first nearest-neighbour r bond integrals for Si in different structureswhich are obtained by self-consistently solving the TB-LMTO equations [41]. As expected

4.0 4.5 5.0 5.5 6.0 6.5

R [a.u.]

-5.0

-4.0

-3.0

-2.0

-1.0

0.0

β σ [eV

]

dimerdiamondscbccfcc

Fig. 4. Screened nearest-neighbour r bond integral for different structures of Si. The bond integrals representedby symbols for diamond, sc, bcc and fcc structures were calculated from screened TB-LMTO [41]. Solid curvesare fitted using the screening expression Eq. (26). The dashed curve corresponds to the predicted second nearest-neighbour bond in bcc.


the bond integrals are considerably weakened in close-packed environments when moreatoms are surrounding the bond. This screening results in more distant neighbour interac-tions being very small, thereby explaining why Tersoff–Brenner-type potentials have beenso successful in modelling many bulk properties with short-ranged interactions. Clearly,however, the interatomic potentials will be much longer ranged at the more open surfacesor in the gas phase, which is reflected in Fig. 4 by the much weaker screening. The curvesin Fig. 4 were obtained by evaluating the screening function Eq. (26) with a parameteri-zation of the overlap integrals that has the same functional form and similar decay lengthsas the unscreened matrix elements.

The repulsive pairwise contributions for the dimer Eq. (23) should also be screened. Theoverlap repulsion may be screened within the formalism of Ref. [23], whereas the corerepulsion 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 differencebetween the numbers of electrons in the bonding and anti-bonding states (see Ref. [20] inthis issue for a detailed discussion of the history of the term bond order). That is

Hlmij ¼

1

2ðNþ � N�Þ; ð27Þ

where N+(�) gives the number of electrons in the bonding (anti-bonding) state. Thus, forexample, the hydrogen dimer with two electrons in the bonding state but none in the anti-bonding 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 intersiteGreen’s function through Eq. (36) of [20] (we omit superscripts lm in the remainder of thissection)

Hij ¼ �2

pIm

Z �F

Gijð�Þd�; ð28Þ

where �F is the Fermi energy. Within bond-based BOP theory the off-diagonal matrix ele-ments of the Green’s function are calculated from diagonal elements Gk

00,

Gk00 ¼ huk

0j½�� bH ��1juk0i ð29Þ

with

juk0i ¼

1ffiffiffi2p ½jii þ expði cos�1ðkÞÞjji�: ð30Þ

By taking the derivative of Gk00 with respect to k the intersite Green’s function Gij may be

calculated

Gij ¼o

okGk

00: ð31Þ


The above equation becomes the starting point for the derivation of an interatomic ana-lytic BOP by using the Lanczos recursion algorithm and writing Gk

00 in the form of a con-tinued fraction [45,46],

Gk00 ¼

1

�� ak0 �

ðbk1Þ2

��ak1�

ðbk2Þ2

��ak2�ðbk

3Þ2

��ak3��

; ð32Þ

where the recursion coefficients {an,bn} are the matrix elements of the semi-infinite one-dimensional Lanczos chain (see Ref. [20]). The relation of the matrix elements to the mo-ments of the density of states can be worked out by evaluating the moments along theLanczos chain. We give the relation for the first four moments explicitly

l1 ¼ a0;

l2 ¼ a20 þ b2

1;

l3 ¼ a30 þ 2a0b2

1 þ a1b21;

l4 ¼ a40 þ 3a2

0b21 þ 2a0a1b2

1 þ a21b2

1 þ b41 þ b2

1b22: ð33Þ

By substituting the continued fraction representation of Gk00 into Eq. (31), Gij becomes a

function of the recursion coefficients

Gij ¼X1n¼0

oGk00

oakn

oakn

okþX1n¼1

oGk00

obkn

obkn

ok: ð34Þ

The derivatives of the Green’s function with respect to the recursion coefficients may bereplaced by products of Green’s function matrix elements Gk

nm ¼ huknj½�� bH ��1juk

mi [47],

oGk00

oan¼ Gk

0nGkn0;

oGk00

obn¼ Gk

0nGkn�1;0 þ Gk

0;n�1Gkn0: ð35Þ

By evaluating this equation for k = 0 an exact expansion for the intersite Green’s functionis found,

Gij ¼ 2X1n¼0

G0nGn0dan þ 2X1n¼1

G0n�1Gn0dbn

!ð36Þ

with

dan ¼oak

n

ok

��k¼0

¼X2nþ1

l¼1

oakn

olkl

��k¼0

ðnlþ1Þ; ð37Þ

dbn ¼obk

n

ok

��k¼0

¼X2n

l¼1

obkn

olkl

��k¼0

ðnlþ1Þ; ð38Þ

lkl ¼

1

2½hij bH ljii þ hjj bH ljji� þ knlþ1; ð39Þ

nlþ1 ¼ hij bH ljji: ð40Þ


By inserting Eq. (36) into Eq. (28) the bond order can be written as a non-linear function ofthe 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 theobservation that saturated covalent bonds only exist in structures where each atom has rel-atively few neighbours, i.e., in open structures, whereas in close-packed structures theatoms are unable to form saturated bonds with all their neighbours. In most open struc-tures there are no three-membered self-returning hopping paths, which means that thethird moment is zero if we assume that d = �p � �s = 0. For the derivation of the analyticBOP we will assume that all the odd moments l2l+1 vanish, which implies that the densityof states is symmetric

nð�Þ ¼ nð��Þ: ð41Þ

Inspection of the poles of Eq. (32), which correspond to the eigenvalues of bH , shows thatthe 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 thefirst four sites along the Lanczos chain (b4 = 0) [21]. The Green’s function, Eq. (32) withk = 0, then takes the form

G00 ¼P 00ð�ÞD00ð�Þ

¼ �3 þ A00�2 þ B00�þ C00

ð�� 1Þð�� 2Þð�� 3Þð�� 4Þ; ð43Þ

with coefficients

A00 ¼ 0; B00 ¼ �ðb22 þ b2

3Þ; C00 ¼ 0: ð44Þ

(The more general case for an 5 0 is given in Refs. [48,49].) The four poles are located at

� ¼ �j��j ¼ �1

2ðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðb1 þ b3Þ2 þ b2

2

q�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðb1 � b3Þ2 þ b2

2

qÞ: ð45Þ

For the calculation of Gij from Eq. (34), a pairwise multiplication of Green’s function ele-ments according to Eq. (35) has to be carried out. This means that Gij will appear as a

function of polynomials Gijð�Þ ¼ pð2n�2Þð�Þpð2nÞð�Þ when G00 ¼ pðn�1Þð�Þ

pðnÞð�Þ for n recursion levels, where

p(m) indicates a polynomial of leading order m. For the exact eigenspectrum, correspond-ing to n!1 recursion levels, the poles of G00 and Gij are of course identical. By con-straining the poles of the intersite Green’s function Gij = Pij(�)/Dij(�) to be the same asthose of the average on-site Green’s function G00, i.e.,

Dijð�Þ ¼ D00ð�Þfijð�Þ; ð46ÞP ijð�Þ ¼ Qijð�Þfijð�Þ; ð47Þ

where fij is a polynomial of order n and Qij is a polynomial of order n � 2, the convergenceof the BOP expansion for a finite number of recursion levels is greatly improved. The new,constrained form of Gij is written as


Gij ¼Qij

D00

¼ Aij�2 þ Bij�þ Cij

ð�� 1Þð�� 2Þð�� 3Þð�� 4Þð48Þ

with

Aij ¼ ðn2Þij; Bij ¼ ðn3Þij; Cij ¼ ðn4Þij � ½b21 þ b2

2 þ b23�ðn2Þij: ð49Þ

The bond order is now calculated from the residues of Gij below the Fermi energy,

Hij ¼ 2X4

n¼1

HðnÞHð�F � �nÞ; ð50Þ

where H is the Heaviside step function that equals 0 for �F < �n (i.e., the eigenstate n is notoccupied), but equals 1 for �F > �n (a fully occupied eigenstate). The weights H(n) are ameasure of how much an electron in eigenstate n contributes to the bond order. Due tothe 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

Hð�Þ ¼ 1

4

1þ bC ij

�þ ��

�þ þ ��

1� bC ij

�þ ��

�þ � ��

264375: ð51Þ

with bCij ¼ Cij=jðbrÞijj3 and �� ¼ ��=jðbrÞijj.

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

H12ð Þ

ij;r ¼ ð1þ bCij=b1b3Þ=ð�þ þ ��Þ; ð52Þ

where bn ¼ bn=jðbrÞijj. It follows from Eqs. (33) and (49) that the coefficients bCij and therecursion coefficients are given bybCij ¼ 1� ðb2

1 � b22 � b2

3Þ � Rij4r; ð53Þ

b21 ¼ 1þ U2r; ð54Þ

b22 ¼ ðU2r � U2

2r þ U4r þ Rij4rÞ=ð1þ U2rÞ; ð55Þ

where U2r corresponds to a 2-hop contribution, Rij4r to a 4-atom ring-type interference

path linking atoms i and j, and U4r refers to self-returning hopping paths of length fouras illustrated in Fig. 5. By neglecting the ring contributions to b3 it is found that b3 canbe approximated by [22,49]

b1b3 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDU4r þ Ui

2rUj2r

q: ð56Þ

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 obtainsthe explicit expression for the r bond order for half-full valence,

H12ð Þ

ij;r ¼ 1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ cr

2U2r þ Rij4r þ eUðiÞ2r

eUðjÞ2rð2þgDU4rÞ þ d2

ð1þgDU4rÞ2

vuut,; ð57Þ

ji ji

ji ji ji

Fig. 5. Illustration of the hopping paths that are taken into account for the evaluation of the simplified analyticBOP. The top row illustrates the contributions Uj

2r and Rji4r, the bottom row illustrates the different contributions

to Uj4r.


whereeUi2reUj

2r ¼ Ui2rUj

2r

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDU4r þ Ui

2rUj2r

q�; ð58Þ

gDU4r ¼ DU4r

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDU4r þ Ui

2rUj2r

q�: ð59Þ

The term d in Eq. (57) is given by d2 ¼ 12½d2

i þ d2j �4prð1� prÞ=b2

ij;r (cf. Ref. [9] in this issuewhere a modified d2 is introduced and discussed). It accounts for the loss of covalent bond-ing due to the s and p atomic energy level mismatch. The constant cr � 1 is a fittingparameter that can be used to improve the comparison between BOP and TB r bondorders 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 coordinateaxes normal to the z-direction along the bond. This invariance of the expression for the pbond order at any level of approximation is guaranteed by using the matrix Lanczos recur-sion algorithm [21,50,51] where the recursion coefficients an and bn become 2 · 2 matricesinstead of scalar variables. Carrying out the matrix Lanczos iteration to two levels, assum-ing a symmetric spectrum, and constraining the poles of Gij to be the same as the poles ofG00, one obtains analytic expressions for the p+ and p� bond orders [21,48],

H12ð Þ

ij ; p� ¼ 1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ cpðU2p �

ffiffiffiffiffiffiffiU4p

pÞ

q�: ð60Þ

Explicit formula for the two- and four-hop contributions U2p and U4p are given in Appen-dix A. The fitting parameter cp is introduced to improve the comparison between BOP andTB p bond orders and is expected to take a value close to 1. Table 1 shows that the analyticexpressions for the p bond order (with cp = 1) capture the chemistry of bond formation inthe 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

Table 1Bond orders of C-C bonds in hydrocarbon molecules as predicted by analytic BOPs [52]

Hr Hp+ Hp� Htot Bond character

C2 0.936 1.000 1.000 2.936 TripleC2H2 0.974 1.000 1.000 2.974 TripleC2H4 0.955 1.000 0.194 2.149 DoubleC6H6 0.953 0.577 0.141 1.671 ConjugateGraphite 0.951 0.477 0.121 1.549 ConjugateC2H5 0.929 0.214 0.145 1.288 RadicalC2H6 0.917 0.149 0.149 1.215 SingleDiamond 0.915 0.126 0.126 1.167 Single

The analytic BOPs correctly capture the chemistry of bond formation in the hydrocarbon systems, including theformation of single, double, triple, conjugate, and radical bonds.


configuration s2p2 to the sp3 hybrid configuration of the diamond structure. Hence, thepromotion energy cannot be neglected for the calculation of realistic binding energies. Fol-lowing Refs. [21,27], the promotion energy can be approximated by

ðEpromÞi ¼ d 1� 1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ

Xj;j 6¼i

Abr;ij

d

� �2vuut,0@ 1A; ð61Þ

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 sec-tions, the integration of the Green’s function was carried out for a half-full eigenspectrumcorresponding to a fractional bond occupancy of f = 1/2. In this section we extend thebond-order expressions to a general fractional bond occupancy 0 6 f 6 1. This enablesthe simulation of III–V semiconductor systems such as GaAs, where Ga has 3 valence elec-trons (f = 3/8), As has five valence electrons (f = 5/8), and GaAs has an average of fourvalence electrons per atom (f = 1/2).

Following Ref. [53] we assume that the bond order of a symmetric eigenspectrum can beapproximated by a third-order polynomial in the symmetric function f(1 � f), whereby

Hsðf Þ ¼ asf ð1� f Þf1� bsf ð1� f Þ½1� csf ð1� f Þ�g: ð62Þ

Fitting the unknown coefficients to the fact that the bond order is bounded by the enve-lope function

jHj 62f for 0 6 f < 1=2;

2ð1� f Þ for 1=2 6 f 6 1;

ð63Þ

we find that Eq. (62) can be written in the form

Hsðf Þ ¼2f for 0 6 f < f0;

2f 0 þ asFð1� 2f 0Þ½1� bsFð1� csFÞ� for f 0 6 f < 1� f0;

2ð1� f Þ for 1� f0 6 f 6 1;

8><>: ð64Þ


where

f0 ¼4

3H

12ð Þ � 5

8

� �;

cs ¼0 for H

12ð Þ > 5

8;

32 58�H

12ð Þ

�for H

12ð Þ 6 5

8;

8<: ð65Þ

F ¼ f ð1� f Þ � f0ð1� f0Þð1� 2f 0Þ

2

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

Hðf Þ ¼ 1� k1

1

2� f

� �þ k3

1

2� f

� �3

þ k5

1

2� f

� �5" #

R3

( )Hsðf Þ; ð66Þ

where k1, k3 and k5 are fitting parameters and where R3 is proportional to the 3-memberring contribution that links the two ends of the bond. An explicit expression for R3 is givenin the Appendix (cf. Eq. (A.10)). It is demonstrated in Ref. [53] that this polynomialapproximation, which contains no arbitrary parameters for the symmetric case, repro-duces very well the r and p TB bond orders as a function of fractional bond occupancyor valence.

3.2.5. Structural predictionThis 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 theknown structural trend across the sp-valent elements as the group number changes from1 to 7 [53]. This is illustrated by Fig. 6. The left hand panel shows the TB structural energycurves 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 close-packed to more open structure types is found. Moreover, since the BOP has been derivedby coarse graining the TB electronic structure, the right hand panel of Fig. 6 shows that ittoo predicts correctly these changes in structure as the number of valence electrons is chan-ged 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 observedswitch from close-packed to open structures as the fractional bond occupancy movesthrough f = 1/2 (cf. Eq. (66)). The fourth moment determines the bimodality of the eigen-spectrum which accounts for diamond with the lowest fourth moment being most stablefor f = 1/2. Second moment potentials such as the Tersoff potentials (or the 2-level BOPexpression, BOP2) cannot predict these structural trends (although, of course, the groundstate structure of any particular element can be fitted by suitably adjusting the parameterswithin the given model).

This can be demonstrated by considering the binding energy per atom within the secondmoment approximation, which from Eq. (1) can be written

0 1 2 3 4 5 6 7 8

N

-20

0

20

40

Tight-Binding

0 1 2 3 4 5 6 7 8

N

Analytic BOP

2,

6

8 3, 3

12 4 12

1

3

48

2,

12 23,6

ΔE [

%]

Fig. 6. Comparison of the structural energies of reduced TB (in the left column) and analytic BOP (in the rightcolumn) for pr = 2/3 and bp/br = 1/6. The energies have been normalized with respect to the energy of a half-fullrectangular 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), simplecubic (6), simple hexagonal (8) and face-centered cubic (12). The fitting parameters take values cr = 1.27 andcp = 1.


EzBðRÞ ¼

1

2z½UðRÞ � 2Hð2Þr bðRÞ�; ð67Þ

if the promotion energy and p bond contributions are neglected. The first contributionrepresents the repulsion between the z first nearest-neighbours, the second contributiongives the attractive covalent bond energy within the second moment approximation tothe bond order, namely,

Hð2Þr ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ U2r

p.: ð68Þ

The former contribution will, therefore, vary as z, the latter contribution for the three-dimensional diamond, simple cubic and face-centered cubic lattices, as z�1/2 [54]. Assum-ing that the overlap repulsion U(R) / [br(R)]2 (cf. Eq. (24)), Eq. (67) becomes

EzBðRÞ ¼ zA½brðRÞ�

2 �ffiffizp

BbrðRÞ; ð69Þ

where A and B are constants for a particular element. It is trivial to show that at equilib-rium the binding energy is given by

EzBðReqÞ ¼

1

4

B2

A� 1

2

B2

A¼ � 1

4

B2

A: ð70Þ

This is independent of coordination and hence the second moment approximation can-not predict relative structural stability of the diamond, simple cubic and fcc lattices. Thisis in contrast to either TB or the 4-level approximation where the correct structural trendis found in Fig. 6 (for the case of U(R) / [br(R)]2). Although second-moment-type Ters-off potentials can be fitted to reproduce the energy differences between different struc-tures, they cannot explain in a systematic way the origin of structural stability for agiven element nor the structural trends across the periodic table as the group numberchanges.


4. Analytic bond-order potentials for Si and GaAs

The growth of Si and GaAs films have been simulated using our current parameteriza-tions of the analytic BOP, apart from screening the bond integrals. Instead of introducingan explicit screening function as in Eq. (25), they have been screened implicitly through theuse of pairwise Goodwin–Skinner–Pettifor (GSP) functions [55].

4.1. Si potential

The analytic Si BOP presented here ignores the electrostatic, magnetic, and van derWaals energies. Therefore it follows from Eqs. ((1)–(3), (6), (17)) that

ESiB ¼

1

2

Xi;j;i 6¼j

UðRijÞ �1

2

Xi;j;i6¼j

½2brðRijÞHr;ij þ 2bpðRijÞHp;ij� þX

i

Epromi ; ð71Þ

where U(Rij), br(Rij), and bp(Rij) are assumed to be pairwise functions approximated bythe GSP equations [55]

UðRÞ ¼ U0½hUðRÞ�nU ; ð72ÞbrðRÞ ¼ br;0½hrðRÞ�nr ; ð73ÞbpðRÞ ¼ bp;0½hpðRÞ�np ð74Þ

with hX (X = U,r,p) expressed as

hX ðRÞ ¼R0

Rexp

R0

Rc;X

� �nc;X

� RRc;X

� �nc;X� �

: ð75Þ

Here U0, br,0, bp,0, R0, nX, nc,X, and Rc,X (X = U,r,p) are parameters. In order to cutoffthe potential smoothly at a chosen cutoff distance Rcut, Eqs. (72)–(74) are used only whenthe atomic separation distance R is within a chosen value R1. When R falls in the range ofR1 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), andbp(R) are determined in such a way that the value and the first derivative of the splinefunction SX(R) match those of the corresponding [U(R), br(R), or bp(R)] function atR = R1, and the value and the first derivative of the spline function SX(R) drops to zeroat R = Rcut. R1 and Rcut are two parameters for the cutoff. The r and p bond ordersare directly calculated from Eqs. (57) and (60).

A complete Si bond-order potential requires 13 parameters in the pair functions, twoparameters for the cutoff function, two parameters in the angular function, two parame-ters for the r bond order, one parameter in the p bond order, and two parameters in thepromotion energy. These 22 parameters were determined by fitting the predicted cohesiveenergy, atomic volume, and bulk modulus to those obtained from either experiments orab 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 alsocarried out to ensure that no other phases had lower energy than the dc structure. The(2 · 1) reconstruction on the (001) surface and its surface energy were also used toaid the selection of the parameters. A complete set of parameters can be found inRef. [56].


4.2. GaAs potential

The parameterization of BOPs for a heterovalent system like GaAs requires the fittingof the elemental subsystems as well as the description of the compounds. This is a muchmore complex and tedious exercise than the parameterization of BOP for Si. As a firststep, we have simplified the analytic BOP for GaAs by neglecting both the promotionenergy contribution and the four-hop and ring contributions to the r bond order. Withoutthe promotion energy, the total energy is written as

EGaAsB ¼ 1

2

Xi;j;i 6¼j

UðRijÞ �1

2

Xi;j;i6¼j

½2brðRijÞHr;ij þ 2bpðRijÞHp;ij�: ð76Þ

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 ofatoms ij [27]. The same cutoff function that was used for the Si pair functions is used tosmoothly 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 isexpressed by

H12ð Þ

r;ij ¼ ð1þ 2crU2rÞ�12: ð77Þ

Here the parameter cr as well as the angular function gr Eq. (A.7) is species dependent. Ifthe species of atoms i, j, and k forming the bond angle hjik are l, c, and m, respectively, thetwo parameters p�r and br used in Eq. (A.7) are assumed to depend on the six bondangle types clm (clm = GaGaGa, AsAsAs, AsGaAs, GaAsGa, GaGaAs/AsGaGa, andAsAsGa/GaAsAs, where the bond angle types clm and mlc are equivalent). Note thatp�r is used here instead of pr in Eq. (A.7) because the three-body dependent p�r is no longerequivalent to the pr used in the p bond hopping paths, Eqs. (A.13)–(A.15). The introduc-tion of the species dependence increases the flexibility of the angular function.

The dependence of the r bond order on the fractional bond occupancy was calculatedusing Eq. (66). In Eq. (66), the bond filling parameter f and the asymmetric skewingparameter 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 bondorder, therefore, is calculated using Eq. (60) with the parameter cp being dependent on thespecies 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 cut-off parameters, six sets (clm = GaGaGa, AsAsAs, AsGaAs, GaAsGa, GaGaAs/AsGaGa,and AsAsGa/GaAsAs) of two angular parameters, three sets of one pairwise parameter inthe half-full valence r bond order, three sets of two pairwise parameters for general rbond order, two sets (l = Ga, As) of the species dependent parameter pr and three setsof the pairwise parameter cp for the p bond order. These 71 parameters were determinedby fitting the predicted properties to those obtained from either experiments or DFT cal-culations for a wide range of structures.

It can be proven that the BOP pair functions alone completely define the relationshipbetween the equilibrium bond energy, bulk modulus and bond length for simple crystalstructures [38]. This allows us to determine completely all the pair functions by fitting


the bond energy/bulk modulus as a function of bond length trend defined by the targetvalues of cohesive energy, bulk modulus, and lattice constant for a variety of selected sim-ple phases spanning a wide range of local environments (chemistry, coordination, andbond angles). The knowledge of the bond energy/bulk modulus versus equilibrium bondlength also facilitates the selection of specific target data sets from a large collection ofexperimental and DFT data. Additional constraints were imposed during parameteriza-tion of the pair functions to ensure that they were smoothly cutoff. Once the pair functionswere determined, the angular function parameters were optimized in a second step to bestmatch the properties of various Ga, As, and GaAs phases. This approach was found tosignificantly improve the transferability of the potential compared with other parameter-ization 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 ada-tom condensation at random locations. These configurations are often associated withhigh energies, mismatch stresses and defects. They therefore will evolve towards lowerenergy, reconstructed crystalline surfaces if permitted by the growth kinetics. Thus, theaccurate 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 essen-tial for robust molecular dynamics simulations of growth.

Fig. 7 compares bulk property predictions of the analytic BOP with those of a Stillin-ger–Weber (SW) potential [57], two parameterizations of the Tersoff potential T2 [58] andT3 [59], using our and published DFT data [60,61] and experimental measurements [62] asthe reference. In Fig. 7 the structures are arranged along the horizontal axis in decreasingorder of the cohesive energy from the DFT calculations as shown by the black monotonicDFT curve in Fig. 7b. The BOP parameterization fits the relative stability of the differentphases well, apart from the structure st12. In general, the trend of phase stability predictedby the other three potentials is less satisfactory. We see in Fig. 7a that the equilibriumatomic volumes predicted by the BOP and the Tersoff potentials reproduce the DFT datarather 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 signifi-cantly improved over those calculated by the other three potentials. In addition, the shearelastic constants of the ground state dc structure are well reproduced by BOP. In Fig. 7dwe show the energies of four types of defects: vacancy, tetrahedral interstitial, hexagonalinterstitial, and x-split interstitial [63–67]. It indicates that the predictions of the overallenergies across different defects by the BOP are much closer to those of the DFT calcula-tions [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 (100) and (111) surfaces. These sur-faces undergo surface reconstructions. The surface reconstruction energies for the widelyobserved (001) (2 · 1), (111) (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

Table 2Surface reconstruction energies for four Si surface reconstructions (eV/A2)

Surface Ab initio TB BOP SW T2 T3

(100) (2 · 1) �0.054 [74] �0.046 �0.061 �0.085 �0.051(111) (7 · 7) �0.403 [73] �0.379 0.028 �0.009 0.033(110) (2 · 1)-adatom �0.190 [69] �0.131(113) (3 · 2) �0.036 [70–72] �0.139

Fig. 7. (a) Atomic volume, (b) cohesive energy, and (c) elastic constants for a variety of Si phases. (d) Defectenergy of the (dc) Si structure.


[69–74], SW, T2, and T3 calculations in Table 2. It can be seen that the reconstructionenergies predicted by the BOP are much closer to the ab initio data than those predictedby the SW, T2 and T3 potentials. For the (7 · 7) surface, the SW and T3 potentials evenpredict positive reconstruction energies. The BOP predictions of the surface properties arehence superior to the other interatomic potentials [75].

5.3. Si growth simulation

The growth of (dc) Si films in the (001) direction has been simulated using moleculardynamics [2–4] with analytic forces obtained from BOP. The initial substrate had thepredicted bulk equilibrium lattice constant. It was oriented in the ð1�10Þ x-direction,(001) y-direction, and ð�1�10Þ z-direction. Periodic boundary conditions were used in both

y [001]

x [110]z [110]

E = 0.17 eVR = 0.4 nm/ns

normal incidence

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 loca-tions. The adatoms had a remote incident kinetic energy of 0.17 eV and their incidentdirection was normal to the surface plane. To prevent the crystal from shifting during ada-tom impacts, atoms in the two lowest y-planes were fixed during simulations. Heating ofthe film due to dissipation of adatom kinetic energy and the latent heat release during ada-tom condensation was prevented by maintaining a sub-surface region above the fixedatoms 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 atomsin the system. An accelerated growth rate of 0.4 nm/ns was used to grow films that weresufficiently thick for a further analysis of the film structures.

As an example the atomic structures of the films simulated at three different substratetemperatures are shown in Fig. 8. It can be seen that increasing the temperature results inan improvement of the film crystallinity, in good agreement with experiments [77–79].BOP-based molecular dynamics simulations therefore provide a valuable tool to explorethe 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 vapoursources. The growth mechanisms for the binary GaAs film are more complex than forSi, mainly due to two reasons. First, the two elements that condense at random positionsof the film surface must find their way in the correct sublattice of the zb crystal. Thisrequires an accurate description of the energies of point defects. Second, experiments showthat crystalline (zb) GaAs films usually grow when the As:Ga flux ratio is significantlyhigher than unity [34,80–82]. This is due to the fact that the binding energy betweenAs2 molecules is much weaker than that within an As2 molecule. As a result, As2 moleculesreadily evaporate in experiments. To capture these effects, the potential must be able tomodel the transition between As2 molecules and As atoms on the surface, including therelatively strong binding energy of As2 molecules and the relatively low binding energiesof larger As clusters.


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 fromour and published DFT calculations [12,85–92] as well as experiments [80,93–98]. Fig. 9aand b show that the BOP and the Tersoff potential predict well the atomic volumes andcohesive 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 duringAs deposition. In sharp contrast, both the cohesive energies and the atomic volumes ofdifferent phases predicted by the SW potential [83,84] are significantly different from thoseof the reference data. More importantly, the SW potential significantly underestimates thecohesive energy of the As2 dimer. It therefore prohibits the evaporation of the As2 mole-cules during deposition and cannot be used to study the effects of As:Ga vapour flux ratioon the structure of the GaAs films. Fig. 9c indicates that the overall bulk modulus for dif-ferent phases predicted by the BOP are significantly improved compared with the Tersoffand the SW potentials.

The characteristic neutral defect formation energies [99,100] in the zinc-blende GaAslattice are compared in Fig. 9d. From all possible point defects, defects that are importantfor characterizing the potential were selected. These include Ga and As vacancies (VGa andVAs), Ga and As antisites (GaAs and AsGa), Ga and As interstitials at the tetrahedral site(Gai,tet and Asi,tet) and the <11 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 GaAsphases. (d) Defect energy of the (zb) GaAs structure.


shows that defect formation energy calculated by the BOP match very well with the refer-ence data for most defects except the Astet where the predicted value is lower than the ref-erence value. In sharp contrast, the defect energies predicted by both the Tersoff and theSW potentials deviate significantly from the reference data.

6.2. Properties of GaAs surfaces

The GaAs (001) surface exhibits many reconstructed structures [33,34,102,103]. Theseinclude the experimentally validated As-terminated b2(2 · 4) [104–106], As-terminateda2(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 deposi-tion rate [31].

The occupancy of dangling bonds is not treated explicitly in the BOP or other empiricalpotentials. However, the electron redistribution in dangling bonds has been shown to playa significant role in stabilizing GaAs (001) surface reconstructions [36,110,111]. Toaddress this, the electron counting approach described in Section 2.1.5 was superimposedupon the various potentials to calculate the surface free energy c [13,112]. The results forthe relative energy (with respect to the ca2(2·4) surface) of the minimum energy surfaces isplotted in Fig. 10 as a function of the relative As chemical potential normalized withrespect to the heat of formation of zb–GaAs. DFT data [32,113] is included in the figurefor comparison.

The charge build-up effects that destabilize the b(2 · 4) surface reconstruction withrespect to the b2(2 · 4) reconstruction were not included in the model due to the absenceof Coulombic electrostatic interactions [106]. Therefore, the free energy predictions by the

Fig. 10. Relative surface energy of the lowest energy (001) GaAs surfaces as a function of the relative,normalized As chemical potential.


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 DFTresults than the surface phase diagram obtained from SW or Tersoff potentials. At Ga-richconditions (low As chemical potential), all three potentials predict that the b(4 · 2) and theb2(4 · 2) surface reconstructions are most stable, in contradiction to the f(4 · 2) recon-struction 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) surfacesare most stable, in agreement with the DFT data. In contrast, the SW potential predictsthe a(4 · 2) and a2(4 · 2) to be the most stable surfaces. When the As chemical potentialis further increased, all three potentials show that the b(2 · 4) and the b2(2 · 4) surfacesare most stable, in agreement with the DFT results. Finally, under As-rich condition (highAs chemical potential), the BOP predicts that the c(4 · 4) reconstruction is most stable, asit is also found from the DFT calculations. No stable or metastable c(4 · 4) surface waspredicted by either the Tersoff or the SW potential when coupled with the electron count-ing 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 usedto simulate the growth of GaAs films from As2 and Ga vapour fluxes using a wide range ofdeposition conditions that cover substrate temperatures T between 500 K and 1500 K andAs:Ga flux ratios R between 0.9 and 3.4. Examples of the atomic structures grown at var-ious substrate temperatures and flux ratios are shown in Fig. 11. We assumed that thevapour particle incident direction is normal to the growth surface, the deposition ratewas 0.125 nm/ns and the adatoms have a thermalized kinetic energy. Fig. 11a and b indi-cate that at a near constant flux ratio of R = 1.1 � 1.2, the film crystallinity improves assubstrate temperature is increased from 500 K to 800 K. Further increase in temperatureresulted in a further improvement of the film crystallinity, Fig. 11c and d. It is interestingthat the best quality film shown in Fig. 11 was obtained at a flux ratio of 3.14, which issignificantly higher than the unity of the stoichiometric film composition. The observedeffects of substrate temperature and vapour flux ratio on the crystallinity of the filmsare in good agreement with the experiments [34,80–82]. The excess As:Ga ratio is dueto the experimentally observed evaporation of As2 molecules from the As-rich surface.Clearly, the effect was correctly captured during simulations because the relative lowenergy of the As2 molecules with respect to the isolated and condensed As atoms was wellpredicted, Fig. 9. It should be pointed out that to our knowledge, this is the first time thatboth the substrate temperature and flux ratio effects on the atomic structure of the GaAsfilm has been demonstrated.

Analysis of extensive BOP molecular dynamics simulations revealed a clear relationshipbetween GaAs film structure and deposition conditions. As2 evaporation increased as thegrowth temperature was increased. As a result, Ga-rich surfaces were observed during sim-ulations at high temperatures and near unity As:Ga flux ratios. When the flux ratio wasincreased at high growth temperatures, excessive As atoms that initially condensed onthe surface were found to later desorb, resulting in stoichiometric films. Excessive Aswas incorporated into the film only at a low growth temperature and high As:Ga fluxratio, until a temperature-dependent solubility limit was reached. All these observationsare in good agreement with experiments [34,114–118].

Fig. 11. Simulated atomic structures of the GaAs films after 10 ns of deposition at different substratetemperatures 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 testbedfor interatomic potentials. The BOP framework achieves the chemical flexibility that isrequired to describe bonding from the dimer through to the bulk by systematically coarsegraining the electronic structure, thereby deriving the format of the potential. The derivedformat enables a straightforward interpretation of the physical meaning of the parametersand subsequently a direct calculation of the numerical values of the parameters from firstprinciples. This approach has led to preliminary interatomic potentials for Si and GaAsthat 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 ofthin 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 bothgrowth temperature and As:Ga vapour flux ratio on the crystallinity and defect populationof GaAs films are correctly described.


Acknowledgements

We are grateful to the Defense Advanced Research Projects Agency and Office of NavalResearch (C. Schwartz and J. Christodoulou, program managers) for support of this workthrough grant N00014-03-C-0288. We also thank S.A. Wolf for numerous helpfuldiscussions.

Appendix A. Expressions for hopping paths

In Section 3.2.1 the following shorthand notation has been used for the r bond:

U2r ¼1

2ðUi

2r þ Uj2rÞ; ðA:1Þ

U22r ¼

1

2½ðUi

2rÞ2 þ ðUj

2rÞ2�; ðA:2Þ

U4r ¼1

2ðUi

4r þ Uj4rÞ; ðA:3Þ

and

DU4r ¼ U4r � U22r

�=U2r: ðA:4Þ

The expressions for the 2- and 4-hop self-returning hopping paths are given by

Ui2r ¼

Xk 6¼i;j

g2rðhjikÞ

brðRikÞbrðRijÞ

� �2

; ðA:5Þ

and

Ui4r ¼

Xk 6¼i;j

g2rðhjikÞ

brðRikÞbrðRijÞ

� �4

þX

k;k0 6¼i;j

grðhjikÞgrðhkik0 Þgrðhk0ijÞbrðRikÞbrðRijÞ

� �2 brðRik0 ÞbrðRijÞ

� �2

þX

k;k0 6¼i;j

g2rðhjikÞg2

rðRikk0 ÞbrðRikÞbrðRijÞ

� �2 brðRkk0 ÞbrðRijÞ

� �2

: ðA:6Þ

The angular function gr(hjik) is expressed as

grðhjikÞ ¼1� pr þ pr cos hjik þ br cos 2hjik

1þ br; ðA:7Þ

where hjik is the bond angle between atoms j and k centered on atom i. br is a fittingparameter that has been introduced to give additional flexibility to the curvature of theangular function, since this controls the bond-bending force constant. For br = 0, theangular 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 orbi-tals with pr = 0, the angular function takes the constant value of unity, whereas for only porbitals with pr = 1, the angular function varies as cosh.


The corresponding 3-atom and 4-atom ring-type interference contributions (cf. Fig. 5)are given by

Rij3r ¼

Xk;k 6¼i;j

grðhjikÞgrðhkjiÞgrðhikjÞbrðRikÞbrðRijÞ

brðRjkÞbrðRijÞ

; ðA:8Þ

and

Rij4r ¼

Xk;k0 6¼i;j;k 6¼k0

grðhjikÞgrðhikk0 Þgrðhkk0jÞgrðhijk0 Þ brðRikÞbrðRkk0 ÞbrðRjk0 ÞbrðRijÞbrðRijÞbrðRijÞ

: ðA:9Þ

The normalized ring contribution R3 between atoms i and j that enters Eq. (66) is definedby bR3 ¼ Rij

3r=ð1þ U2rÞ ðA:10Þfor r bonds. Similarly, for the p bond we have used the notation

U2p ¼1

2ðUi

2p þ Uj2pÞ; ðA:11Þ

U4p ¼1

2ðUi

4p þ Uj4pÞ: ðA:12Þ

The self-returning 2-hop and 4-hop contributions are given by

Ui2p ¼

Xk;k 6¼i;j

pr sin2 hjikb2

rðRikÞb2

pðRijÞþ ð1þ cos2 hjikÞ

b2pðRikÞ

b2pðRijÞ

" #; ðA:13Þ

Ui4p ¼

1

2

Xk;k0 6¼i;j

½sin2 hjik sin2 hjik0 b2ikb

2ik0 þ sin2 hjik sin2 hijk0 b

2ikb

2jk0 � cos 2ð/k � /k0 Þ; ðA:14Þ

b2 is defined by

b2ik ¼ pr

b2rðRikÞ

b2pðRijÞ

� b2pðRikÞ

b2pðRijÞ

: ðA:15Þ

The dihedral angle contribution may be expressed as

cos 2ð/k � /k0 Þ ¼2ðcos hkik0 � cos hjik0 cos hjikÞ2

sin2 hjik sin2 hjik0� 1: ðA:16Þ

References

[1] Daw MS, Baskes MI. Embedded-atom method: derivation and application to impurities, surfaces, andother defects in metals. Phys Rev B 1984;29(12):6443–53.

[2] Zhou XW, Wadley HNG, Johnson RA, Larson DJ, Tabat N, Cerezo A, et al. Atomic scale structure ofsputtered metal multilayers. Acta Mater 2001;49(19):4005–15.

[3] Zou W, Wadley HNG, Zhou XW, Johnson RA, Brownell D. Surfactant-mediated growth of giantmagnetoresistance multilayers. Phys Rev B 2001;64(17):174418.

[4] Zhou XW, Wadley HNG. Atomistic simulations of the vapor deposition of Ni/Cu/Ni multilayers: theeffects of adatom incident energy. J Appl Phys 1998;84(4):2301–15.

[5] Zhou XW, Wadley HNG, Filhol J-S, Neurock MN. Modified charge transfer-embedded atom methodpotential for metal/metal oxide system. Phys Rev B 2004;69(3):035402.


[6] Zhou XW, Wadley HNG. A charge transfer ionic-embedded atom method potential for the O–Al–Ni–Co–Fe system. J Phys Condens Matter 2005;17:054418.

[7] Zhou XW, Wadley HNG. Atomistic simulation of AlOx magnetic tunnel junction growth. Phys Rev B2005;71(5):3619–35.

[8] Wolf SA, Awschalom DD, Buhrman RA. Spintronics: a spin-based electronics vision for the future. Science2001;294(5546):1488–95.

[9] Mrovec M, Moseler M, Elsasser C, Gumbsch P. Atomistic modeling of hydrocarbon systems using analyticbond-order potentials, Prog Mater Sci, this issue, doi:10.1016/j.pmatsci.2006.10.012.

[10] Tersoff J. New empirical model for the structural properties of silicon. Phys Rev Lett 1986;56(6):632–5.[11] Brenner DW, Shenderova OA, Harrison JA, Stuart SJ, Ni B, Sinnott SB. A second-generation reactive

empirical bond-order (REBO) potential energy expression for hydrocarbons. J Phys Condens Matter2002;14(4):783–802.

[12] Albe K, Nordlund K, Nord J, Kuronen A. Modeling of compound semiconductors: analytical bond-orderpotential for Ga, As, and GaAs. Phys Rev B 2002;66(3):035205.

[13] Murdick DA, Zhou XW, Wadley HNG, Nguyen-Manh D. Modeling GaAs surfaces with interatomicpotentials and electron counting. J Phys Condens Matter 2005;17:6123–37.

[14] Murdick DA, Zhou XW, Wadley HNG. Low temperature atomic assembly of stoichiometric galliumarsenide from equiatomic vapor. J Cryst Growth 2006;286:197–204.

[15] van Duin A, Dasgupta S, Lorant F, Goddard III W. ReaxFF: a reactive force field for hydrocarbons. JPhys Chem A 2001;105:9396.

[16] Chen N, Lusk M, van Duin A, Goddard III W. Mechanical properties of connected carbon nanorings viamolecular dynamics simulation. Phys Rev B 2005;72:085416.

[17] van Duin A, Strachan A, Stewman S, Zhang Q, Xu X, Goddard III W. ReaxFF(SiO) reactive force field forsilicon and silicon oxide systems. J Phys Chem A 2003;107:3803.

[18] Kohn W, Sham LJ. Self-consistent equations including exchange and correlation effects. Phys Rev1965;140(4A):A1133–8.

[19] Sutton AP, Finnis MW, Pettifor DG, Ohta Y. The tight-binding bond model. J Phys C 1988;21:35.[20] M.W. Finnis, Bond-order potentials through the ages, Prog Mater Sci, this issue, doi:10.1016/j.pmatsci.

2006.10.003.[21] Pettifor DG, Oleinik II. Analytic bond-order potentials beyond Tersoff–Brenner: I. Theory. Phys Rev B

1999;59(13):8487–99.[22] Pettifor DG, Oleinik II. Analytic bond-order potential for open and close-packed phases. Phys Rev B

2002;65(17):172103.[23] Nguyen-Manh D, Pettifor DG, Vitek V. Analytic environment-dependent tight-binding bond integrals:

application to MoSi2. Phys Rev Lett 2000;85(19):4136–9.[24] Znam S, Nguyen-Manh D, Pettifor DG, Vitek V. Atomistic modelling of TiAl I. Bond-order potentials

with environmental dependence. Philos Mag A 2003;83(4):415–38.[25] Slater J, Koster G. Simplified lcao method for the periodic potential problem. Phys Rev 1954;94:1468.[26] Pettifor DG. Springer Proc Phys 1990;48:64.[27] Pettifor DG, Finnis MW, Nguyen-Manh D, Murdick DA, Zhou XW, Wadley HNG. Analytic bond-order

potentials for multi-component systems. Mater Sci Eng A – Struct 2004;365(1–2):2–13.[28] Harrison WA. Electronic structure and the properties of solids: the physics of the chemical bond. San

Francisco: Freeman; 1980.[29] Xu CH, Wang CZ, Chan CT, Ho KM. A transferable tight-binding potential for carbon. J Phys Condens

Matter 1992;4(28):6047–54.[30] Daweritz L, Hey R. Reconstruction and defect structure of vicinal GaAs(001) and AlxGa1�xAs(001)

surfaces during MBE growth. Surf Sci 1990;236(1–2):15–22.[31] Xue Q-K, Hashizume T, Sakurai T. Scanning tunneling microscopy of III–V compound semiconductor

(001) surfaces. Prog Surf Sci 1997;56(1–2):1–131.[32] Lee S-H, Moritz W, Scheffler M. GaAs(001) surface under conditions of low As pressure: evidence for a

novel surface geometry. Phys Rev Lett 2000;85(18):3890–3.[33] Paget D, Pulci O, Sauvage M, Garreau Y, Reining L, Chiaradia P, et al. Do we understand the structure

of the gallium-rich surface of GaAs(001)? Experimental and theoretical approaches. Surf Rev Lett2002;9(3–4):1497–510.

[34] Pristovsek M, Tsukamoto S, Ohtake A, Koguchi N, Orr BG, Schmidt WG, et al. Gallium-richreconstructions on GaAs(001). Phys Status Solidi B 2003;240(1):91–8.


[35] Murdick DA, Zhou XW, Wadley HNG. An assessment of interatomic potentials for molecular dynamicssimulations of GaAs deposition. Phys Rev B 2005;72:205340.

[36] Pashley MD. Electron counting model and its applications to island structures on molecular-beam epitaxygrown GaAs(001) and ZeSe(001). Phys Rev B 1989;40(15):10481–7.

[37] Zhou XW, Murdick DA, Wadley HNG. An electron counting modification to potentials for covalentlybonded surfaces. J Appl Phys 2006;99:064908.

[38] Murdick DA, Zhou XW, Wadley HNG, Nguyen-Manh D, Drautz R, Pettifor DG. An analytic bond-orderpotential for gallium arsenide. Phys Rev B 2006;73:045206.

[39] DMol3 is available from Accelrys, Inc. <http://www.accelrys.com>.[40] Skinner AJ, Pettifor DG. Transferability and the pair potential within the tight-binding bond model – an

analytic study for hydrogen. J Phys Condens Matter 1991;3:2029.[41] Tank RW, Arcangeli C, Dasgupta T. An introduction to the third-generation LMTO method. Phys Status

Solidi B 2000;89:217.[42] Nguyen-Manh D, Pettifor DG, Znam S, Vitek V. Negative Cauchy pressure within the tight-binding

approximation. In: Turchi PEA, Gonis A, Colombo L., editors. Tight-binding approach to computationalmaterials science, vol. 491, MRS symposium proceedings, 1998. p. 353–8.

[43] Mrovec M, Nguyen-Manh D, Pettifor DG, Vitek V. Bond-order potential for molybdenum: application todislocation behavior. Phys Rev B 2004;69:094115.

[44] Coulson C. The electronic structure of some polyenes and aromatic molecules vii. bonds of fractional orderby the molecular orbital method. Proc Roy Soc London A 1939;169:413.

[45] Lanczos C. An iteration method for the solution of the eigenvalue problem of linear differential and integraloperators. J Res Natl Bur Stand 1950;45:225.

[46] Haydock R. The recursive solution of the Schrodinger equation. Solid State Phys 1980;35:216.[47] Turchi P, Ducastelle F. The recursion method and its applications. Berlin: Springer Verlag; 1985. p. 104.[48] Pettifor DG, Oleinik II. Bounded analytic bond-order potentials for r and p bonds. Phys Rev Lett

2000;84(18):4124–7.[49] Pettifor DG, Oleynik II. Interatomic bond-order potentials and structural prediction. Prog Mater Sci

2004;49(3–4):285–312.[50] Jones R, Lewis MW. Philos Mag B 1984;49:95.[51] Inoue J, Ohta Y. J Phys C 1987;20:1947.[52] Oleinik II, Pettifor DG. Analytic bond-order potentials beyond Tersoff–Brenner: II. Application to the

hydrocarbons. Phys Rev B 1999;59(13):8500–7.[53] Drautz R, Murdick DA, Nguyen-Manh D, Zhou X, Wadley HNG, Pettifor DG. An analytic bond-order

potential for predicting structural trends across the sp-valent elements. Phys Rev B 2005;72:144105.[54] Nishitani SR, Alinaghian P, Hausleitner C, Pettifor DG. Angularly dependent embedding potentials and

structural prediction. Philos Mag Lett 1994;69(4):177–84.[55] Goodwin L, Skinner AJ, Pettifor DG. Generating transferable tight-binding parameters – application to

silicon. Europhys Lett 1989;9(7):701–6.[56] Gillespie BA, Zhou XW, Murdick DA, Drautz R, Pettifor DG, Wadley HNG. A bond-order potential for

silicon, unpublished.[57] Stillinger FH, Weber TA. Computer simulation of local order in condensed phases of silicon. Phys Rev B

1985;31(8):5262–71.[58] Tersoff J. New empirical approach for the structure and energy of covalent systems. Phys Rev B

1988;37(12):6991–7000.[59] Tersoff J. Modeling solid-state chemistry: interatomic potentials for multicomponent systems. Phys Rev B

1989;39(8):R5566–8.[60] Goedecker S, Deutsch T, Billard L. A fourfold coordinated point defect in silicon. Phys Rev Lett

2002;88(23):235501.[61] Needs RJ. First-principles calculations of self-interstitial defect structures and diffusion paths in silicon. J

Phys Condens Matter 1999;11:10437–50.[62] Simmons G, Wang H. Single crystal elastic constants and calculated aggregate properties: a handbook. 2nd

ed. Cambridge, MA: MIT Press; 1971.[63] Pandey KC. Diffusion without vacancies or interstitials: a new concerted exchange mechanism. Phys Rev

Lett 1986;57(18):2287–90.[64] Bar-Yam Y, Joannopoulos JD. Barrier to migration of the silicon self-interstitial. Phys Rev Lett

1984;52(13):1129–32.

http://www.accelrys.com


[65] Kelly PJ, Car R. Green’s-matrix calculation of total energies of point defects in silicon. Phys Rev B1992;45(12):6543–63.

[66] Seong H, Lewis LJ. First-principles study of the structure and energetics of neutral divacancies in silicon.Phys Rev B 1996;53(15):9791–6.

[67] Justo JF, Bazant MZ, Kaxiras E, Bulatov VV, Yip S. Interatomic potential for silicon defects anddisordered phases. Phys Rev B 1998;58(5):2539–50.

[68] Takayanagi K, Tanishiro Y, Takahashi M, Takahashi S. Structural analysis of Si(111)-7 · 7 by UHV-transmission electron diffraction and microscopy. J Vac Sci Technol A 1985;3(3):1502–6.

[69] Stekolnikov AA, Furthmuller J, Bechstedt F. Structural elements on reconstructed Si and Ge(110) surfaces.Phys Rev B 2004;70:045305.

[70] Ranke W. Atomic structure of Si and Ge surfaces: models for (113), (115), and stepped (001) vicinalsurfaces. Phys Rev B 1990;41(8):5243–50.

[71] Dabrowski J, Mussig HJ, Wolff G. Atomic structure of clean Si(113) surfaces: theory and experiment. PhysRev Lett 1994;73(12):1660–3.

[72] Wang J, Horsfield AP, Pettifor DG, Payne MC. Puckering models for the Si(113) surface reconstruction.Phys Rev B 1996;54(19):13744–7.

[73] Qian GX, Chadi DJ. Si(111)-7 · 7 surface: energy-minimization calculation for the dimer-adatom-stacking-fault model. Phys Rev B 1987;35(3):1288–93.

[74] Howe JM. Interfaces in materials. New York: Wiley-Interscience; 1997.[75] Balamane H, Halicioglu T, Tiller WA. Comparative study of silicon empirical interatomic potentials. Phys

Rev B 1992;46(4):2250–79.[76] Hoover WG. Canonical dynamics: equilibrium phase-space distributions. Phys Rev A 1985;31(3):1695–7.[77] Zalm PC, van de Walle GFA, Gravesteijn DJ, van Gorkum AA. Ge segregation at SiSi1�xGex interfaces

grown by molecular beam epitaxy. Appl Phys Lett 1989;55(24):2520/2522.[78] Bevk J, Mannaerts JP, Feldman LC, Davidson BA, Ourmazd A. Ge–Si layered structures: artificial crystals

and complex cell ordered superlattices. Appl Phys Lett 1986;49(5):286/288.[79] Pearsall TP, Bevk J, Feldman LC, Bonar JM, Mannaerts JP, Ourmazd A. Structurally induced optical

transitions in Ge–Si superlattices. Phys Rev Lett 1987;58(7):729/732.[80] Adachi S. Specific heat and Debye temperature of GaAs. In: Brozel MR, Stillman GE., editors. Properties

of gallium arsenide, 3rd ed., vol. 16, INSPEC, London, 1996. p. 27–31.[81] Arthur JR. Interaction of Ga and As2 molecular beams with GaAs surfaces. J Appl Phys 1968;39(8):4032–4.[82] Arthur JR. Surface stoichiometry and structure of GaAs. Surf Sci 1974;43(2):449–61.[83] Angelo JE, Mills MJ. Investigations of the misfit dislocation structure at a CdTe(001)/GaAs(001) interface

using Stillinger–Weber potentials and high-resolution transmission electron microscopy. Philos Mag A1995;72(3):635–49.

[84] Grein CH, Faurie JP, Bousquet V, Tournie E, Benedek R, de la Rubia T. Simulations of ZnSe/GaAsheteroepitaxial growth. J Cryst Growth 1997;178(3):258–67.

[85] Mujica A, Needs RJ, Munoz A. First-principles pseudopotential study of the phase stability of the III–Vsemiconductors GaAs and AlAs. Phys Rev B 1995;52(12):8881–92.

[86] Kresse G, Hafner J. Norm-conserving and ultrasoft pseudopotentials for first-row and transition elements.J Phys Condens Matter 1994;6:8245–57.

[87] Kresse G, Hafner J. Ab initio molecular dynamics for liquid metals. Phys Rev B 1993;47:R558–61.[88] Kresse G, Hafner J. Ab initio molecular-dynamics simulation of the liquid-metaluamorphous-semicon-

ductor transition in germanium. Phys Rev B 1994;49:14251–69.[89] Kresse G, Furthmuller J. Efficiency of ab-initio total energy calculations for metals and semiconductors

using a plane-wave basis set. Comput Mater Sci 1996;6:15–50.[90] Kresse G, Furthmuller J. Efficient iterative schemes for ab initio total-energy calculations using a plane-

wave basis set. Phys Rev B 1996;54:11169–86.[91] Zollo G, Tarus J, Nieminen RM. Reliability of analytical potentials for point-defect simulation in GaAs. J

Phys Condes Matter 2004;16(23):3923–32.[92] Zollo G, Nieminen RM. Small self-interstitial clusters in GaAs. J Phys Condes Matter 2003;15(6):843–53.[93] Lide DR, editor. CRC handbook of chemistry and physics. Boca Raton, FL: CRC Press; 2003.[94] Villars P, Calvert LD, editors. Pearson’s handbook of crystallographic data for intermetallic phases, 2nd

ed., vol. 1. Materials Park, OH: ASM International; 1991. p. 1134–5.[95] McSkimin HJ, Jayaraman A, Andreatch Jr P. Elastic moduli of GaAs at moderate pressures and the

evaluation of compression to 250 kbar. J Appl Phys 1967;38(5):2362–4.


[96] Donohue J. The structures of the elements. New York: Wiley; 1974.[97] Lyall KR, Cochran JF. Velocity of sound and acoustic attenuation in pure gallium single crystals. Can J

Phys 1971;49(9):1074.[98] Beister HJ, Strossner K, Syassen K. Rhombohedral to simple-cubic phase transition in arsenic under

pressure. Phys Rev B 1990;41(9):5535–43.[99] Zhang S, Northrup J. Chemical potential dependence of defect formation energies in GaAs: application to

Ga self-diffusion. Phys Rev Lett 1991;67(17):2339–42.[100] Northrup J, Zhang S. Dopant and defect energetics: Si in GaAs. Phys Rev B 1993;47(11):R6791–4.[101] Chadi DJ. Self-interstitial bonding configurations in GaAs and Si. Phys Rev B 1992;46(15):9400–7.[102] Xue Q-K, Hashizume T, Sakurai T. Scanning tunneling microscopy study of GaAs(001) surfaces. Appl

Surf Sci 1999;141(3–4):244–63.[103] Tsukamoto S, Pristovsek M, Ohtake A, Orr BG, Bell GR, Ohno T, et al. Ga-rich GaAs(001) surfaces

observed by STM during high-temperature annealing in MBE. J Cryst Growth 2003;251(1–4):46–50.[104] Chadi DJ. Atomic structure of GaAs(100)-(2 · 1) and (2 · 4) reconstructed surfaces. J Vac Sci Technol A

1987;5(4):834–7.[105] Northrup JE, Froyen S. Energetics of GaAs(100)-(2 · 4) and -(4 · 2) reconstructions. Phys Rev Lett

1993;71(14):2276–9.[106] Northrup JE, Froyen S. Structure of GaAs(001) surfaces: the role of electrostatic interactions. Phys Rev B

1994;50(3):R2015–8.[107] Laukkanen P, Kuzmin M, Perala RE, Ahola M, Mattila S, Vayrynen IJ, et al. Electronic and structural

properties of GaAs(100)(2 · 4) and InAs(100)(2 · 4) surfaces studied by core-level photoemission andscanning tunneling microscopy. Phys Rev B 2005;72:045321.

[108] Biegelsen DK, Bringans RD, Northrup JE, Swartz LE. Surface reconstructions of GaAs(100) observed byscanning tunneling microscopy. Phys Rev B 1990;41(9):5701–6.

[109] Ohtake A, Koguchi N. Two types of structures for the GaAs(001)-c(4 · 4) surface. Appl Phys Lett2003;83(25):5193–5.

[110] Harrison WA. Theory of polar semiconductor surfaces. J Vac Sci Technol 1979;16(5):1492–6.[111] Harbison JP, Farrell HH. Molecular-beam epitaxial growth mechanisms on the GaAs(001) surface. J Vac

Sci Technol B 1988;6(2):733–5.[112] Qian G-X, Martin RM, Chadi DJ. First-principles study of the atomic reconstructions and energies of Ga-

and As-stabilized GaAs(100) surfaces. Phys Rev B 1988;38(11):7649–63.[113] Ohno T. Theoretical study of atomic structures and dynamics on semiconductor surfaces. Thin Solid Films

1996;272(2):331–44.[114] Tok ES, Neave JH, Zhang J, Joyce BA, Jones TS. Arsenic incorporation kinetics in GaAs(001)

homoepitaxy revisited. Surf Sci 1997;374(1–3):397–405.[115] Evans R. High temperature MBE growth of GaAs. In: Brozel, MR, Stillman, GE., editors. Properties of

gallium arsenide, 3rd ed., vol. 16, INSPEC, London, 1996. p. 655–61.[116] Kaminska M, Weber ER, Liliental-Weber Z, Leon R, Rek ZU. Stoichiometry-related defects in GaAs

grown by molecular-beam epitaxy at low temperatures. J Vac Sci Technol B 1989;7(4):710–3.[117] Missous M. Low temperature GaAs: growth dynamics and effects on As:Ga flux ratio. In: Brozel MR,

Stillman GE, editors. Properties of gallium arsenide, 3rd ed., vol. 16, INSPEC, London, 1996. p. 679–83.[118] Suda A, Otsuka N. Surface atomic process of incorporation of excess arsenic in molecular-beam epitaxy of

GaAs. Surf Sci 2000;458:162–72.

analytic bond-order potentials for modelling the …...the atomistic modelling hierarchies. in a...

Documents