Molecular Dynamics simulations

Lecture 08:

Covalent interaction models

Dr. Olli PakarinenUniversity of Helsinki

Fall 2012

Lecture notes based on notes by Dr. Jani Kotakoski, 2010

Introduction

I This lecture deals with interaction models developed formaterials with covalent bonding; typically semiconductors andinsulators

I The most often encountered semiconductor lattices are diamondand zincblende

I Although there are only three elements which have the diamondstructure (C, Si and Ge), being able to model this lattice iscrucially important for, e.g., semiconductor industry

I Also, the most common compound semiconductors (GaAs, AlAs,InAs, etc.) exhibit the zincblende structure (two-elementdiamond structure)

I In the diamond structure, each atomhas four neighbors attached togethervia covalent bonding

I Hence, it requires a group IVelement with four valence electrons

I The zincblende compoundstructures are formed from onegroup III element (B, Al, Ga, . . . ) andone group V element (N, P, As, . . . )

I Diamond structure can be formed byaligning two FCC lattices on top ofeach other, one displaced by(1/4,1/4,1/4)

I Each atom is in the middle of a regular tetrahedron

I This arrangement leads to regular bond angles ofcos−1(−1/3) = 109.47◦ (furthest away from each other)

I Chemically, this corresponds to sp3 hybridization of electrons

I The hexagonal counterpart of the zincblende structure iswurtzite

I Note the analogy with FCC↔ HCP

I Because the electrons are localized between the atoms, covalentbonds have a strong directional dependency

I The difference between ionic and covalent bonding – samestructure, different bonds

Hexagonal Boron Nitride Graphene (carbon sp2)

Bonding angles

I The localized electrons in covalent bonds lead to preferredbonding angles

I F.ex. in case of sp1, sp2 and sp3 bonds, the bonds align so thatthey are as far from each other as possible

I Therefore, the energy of a system is not simply dependent onthe distances between the atoms but also on their anglesi jk rijrik rjkϑjik

ϑkjiϑikj

I The simplistic view is to consider Coulombic repulsion betweenthe negatively charged bonds (localized electrons)

I Because of the bond angle preferences, also the amorphouscovalent solids tend to be loosely packed (i.e., open)

I Typically, the atoms have 2–4 neighbors, as compared to 12 inclose-packed structures

I Packing factor of diamond is 0.34 (FCC/HCP have 0.74)

I Since the optimal angles are predetermined, it is tempting toimplement interaction models which have a minimum exactly atthe right angle

I This has the obvious problem that, e.g., carbon bond angles indiamond (sp3, 109.47◦) differ from those in graphitic structures(sp2, 120◦) but energies are similar

Cluster Potentials

I Some attempts to create potentials for covalently bondedmaterials are presented in [Balamane et al., PRB 46, 2250 (1992)]

I Cluster potentials are interaction models of the form

U =∑

ijV2(ri, rj) +

∑ijk

V3(ri, rj , rk) (1)

I In cluster functionals, the three-body term is hidden in thetwo-body part without an explicit three-body part

I The cluster functionals have the form

U = Vrepulsive(rij) + bijVattractive(rij) (2)

similar to bond order potentials

POTENTIALS FOR SI AND GE

I Both Si and Ge have the diamond structure as crystalline solids

I Both show a density anomaly at molten phase (as water):

molten phase is denser by 11.4 % (Si) and 5.2 % (Ge) than solid

I Liquid Si and Ge are metals

I They show a coordination number of 6, compared to 4 in solid

I This structure and bonding change makes constructing goodpotentials for all condensed phases difficult

STILLINGER-WEBER–POTENTIAL (SW)

[PRB 31, 5262 (1985)]

I The SW potential was developed to correctly describe bothcrystalline and liquid silicon

I It was constructed to give the correct melting temperature

I The SW potential has been very popular because it turned out todescribe well, e.g., point defects energies and surface properties

I SW is an explicit angular potential, and it has the form

U =∑

ijV2(rij) +

∑ijk

V3(rijk) (3)

V2(rij) = εf2(rij)/σ) (4)

V3(rijk) = εf3(ri/σ, rj/σ, rk/σ) (5)

I Here, V2 is a pair potential, and V3 is the three-body partI The fi are

f2(r) ={

A(Br−p − 1) exp[(r − a)−1], r < a0, r > a (6)

f3(ri, rj , rk) = h(rij , rik, θjik) + h(rji, rjk, θijk) + h(rki, rkj , θikj)(7)

I Function h is

h(rij , rik, θjik) =

λ exp[γ(rij − a)−1 + γ(rik − a)−1](cos θijk +13 )

2],where rij < a and rik < a

0, else(8)

I The explicitness of the angle is in the +1/3 term

I As mentioned, the explicit angle becomes a problem whendifferent phases or disordered systems are studied

I Actually, it’s rather surprising that the SW potential works forliquid silicon because of this

I The constants A, B, p, a, λ and γ are all positive and fitted to getthe diamond structure as the most stable lattice, as well as to themelting point, cohesive energy and lattice parameter

I According to rumors, the authors also tested the potentialagainst elastic constants, although this is not stated in the article

I Melting point was fixed by adjusting the cohesive energy to avalue 7% off

I Despite the explicit angle, SW provides a better description ofthe Si surface reconstruction than f.ex. the Tersoff potential(discussed below)

I Reconstruction of the Si(001) surface as given by differentpotentials [Nurminen et al., PRB 67, 035405 (2003)]

EDIP POTENTIAL

I Environment dependent interatomic potential (EDIP) is a clusterpotential derived from a database of ab initio calculations for Siin diamond and graphite phases

I Main difference to SW is in the environment-dependent term,which is now not fixed to the diamond angle

I The potential is available athttp://www-math.mit.edu/∼bazant/EDIP/

I Both FORTRAN and C codes are available

I EDIP gives a good description ofthe diamond phase, includingelastic constants, point defectenergetics, stacking fault anddislocation properties

I It also describes amorphous Siwell, including the melting point[Nord et al., PRB 65, 165329 (2002)]

I Thermodynamical properties ofEDIP have been reviewed in[Keblinski et al., PRB 66, 064104 (2002)]

I Description of other phases isnot equally good

MEAM POTENTIAL

I There also exist so called MEAM (modified EAM) models for Si[M. I. Baskes, Phys. Rev. B 46, 2727 (1992)]

I This is basically EAM to which an angular term has been added:

Utot =∑

iFi[ρi] +

12∑

ijVij(rij) (9)

ρi =∑j 6=iρa(rij) +

∑k,j 6=i

ρa(rij)ρa(rik)g(cosθijk) (10)

I Baskes has developed some models, but is apparently not quitesatisfied with them.

I Applied (in addition to metals) to e.g. silicides (TaSi, MoSi;electronic components!)

LENOSKY POTENTIAL

I Another relatively new potential is that by Lenosky at al. [Modelling

and Simulation in Materials Science and Engineering 8, 825 (2000)]

I This model combines EAM and SW

I Obtaines an excellent fit to a large number of elastic constants,different structures and defect properties

I However, this model contains some alarming features, such asnegative electron densities for certain r

I Therefore, the transferability outside the parameter databasewhich it was fitted to is questionable

Bond Order

I The above potentials had no physical basis for the environmentalterms

I Bond order is the number of chemical bonds between a pair ofatoms

I In molecular orbital theory,

B.O. =nbondinge− − nantibondinge−

2 (11)

which typically gives the same result, but for example stablemolecules H+

2 and He+2 have B.O. = 0.5

I In most simple terms, it is a measure of the strength of the bond

I The magnitude of the bond order is associated with the bondlength as well as the bond angle

I Linus Pauling devised an experimental expression for the bondorder

sij = exp[

d1 − dijb

](12)

where d1 is the single bond length, dij is the experimentallymeasured bond length, and b is a constant

I Pauling suggested a value of b = 0.353 Å, for carbon 0.3 Å isoften used:

Bond Order Potentials (BOP)

I In the simplest form, a bond order potential is a potential which isbased on the idea that the environment affects the strength of abond

I In this sense, also the second-moment tight-binding potentials(Finnis-Sinclair, Rosato Group potentials) are bond orderpotentials

I However, here we define a bond order potential to be a potentialwhich has a starting point in the dimer properties as in Pauling’sbond order concept, and has the form

Uij = Vrepulsive(rij) + bijkVattractive(rij) (13)

I It is possible to show that bijkVattractive(rij) is equivalent to theelectron density term −D√ρi from the Finnis-Sinclair / EAM /Cleri-Rosato–potentials [Brenner, PRL 63, 1022 (1989)]

BOP, BASIC ASSUMPTIONS

I [Albe et al., PRB 65, 195124 (2002)]

I Assuming a single valence orbital per atomic site, the totalenergy of a system can be written as

U =12∑i 6=jφ(Rij) +

∑i

∫Ef

−∞(E − εi)Ni(E)dE︸ ︷︷ ︸V i

B

(14)

where the first term accounts for the repulsion between theatomic cores, and the second is the bond energy calculated asan integral over the local electronic density of states Ni(E)

I εi is the effective atomic energy level

I Most structural quantities are insensitive to the details of the Ni,and depend rather on its average value (first moment) and width(second moment, µ2

i )

I As mentioned during the previous lecture, for d transition metals(e.g. Pt), the cohesive energy is dominated by the d-bandcontribution

I Therefore, a rectangular Ni can be assumed with a width Wi sothat the Ni per atom for a full d-band becomes 10/W

I Then, the bond energy per site i can be written as

V iB ≈ −

120WiNd(10 − Nd), (15)

where Nd is the number of electrons in the d-band

I W is related to the second moment of the Ni via

µ2i =

∫∞−∞ E2Ni(E)dE =

1012W2

i (16)

I On the other hand, the second moment can be given directly asa sum the two-center hopping integrals hij , which depend on thenext neighbor distance rij :

µ2i = 10

∑j 6=i

h2(rij) (17)

I Combining these, we get

112W2

i =∑j 6=i

h2(rij) (18)

I Applying this to the total energy expression, we get

U =12∑

i

∑j 6=iφ(rij) − D

√√√√√√∑j 6=i

h2(rij)︸ ︷︷ ︸ρi

(19)

I Indentifying the sum of the hopping integrals as the resultingelectron density ρi leads to the Rosato group potentials

I Defining an embedding function F(ρ) = D√ρ, on the otherhand, leads to the Finnis-Sinclair implementation of EAMproposed by Daw and Baskes

I The equivalence of the bond-order ansatz and the EAM arisesfrom the fact that within the TB approach the chemical bondingof d transition metals and semiconductors can be explained inthe same terms

TERSOFF POTENTIAL

[Tersoff, PRB 39, 5566 (1989)], [Tersoff, PRL 61, 2879 (1988)], [Tersoff, PRB 38,

9902 (1988)], [Tersoff, PRB 37, 6991 (1988)] , [Tersoff, PRL 56, 632 (1986)], [Tersoff,

PRL 64, 1757 (1990)]

I Jerry Tersoff defined the bond order term as

bij = (1 + βnζnij)

−1/2n, (20)

ζij =∑k 6=i,j

fC(rik)g(θijk) exp[λ33(rij − rik)

3], (21)

g(θ) = 1 + c2/d2 − c2/[d2 + (h − cos θ)2] (22)

aij = (1 + αnνnij)

−1/2, (23)

νij =∑k 6=i,j

fC(rik) exp[λ33(rij − rik)

3] (24)

I Tersoff was unable to obtain parameters which would haveworked with both for the elastic and surface properties

I Therefore, he published two sets of parameters: Si B forsurfaces and Si C for elastic properties (Si A proved to beunstable with a negative vacancy formation energy)

I Tersoff Si B is also known as Tersoff 2, and Si C as Tersoff 3

I Parameter λ3 does not affect theequilibrium properties at all, butis important for some studiesfar-from-equilibrium

I Tersoff himself suggesteddisregarding this parameter(λ3 = 0)

I Also, he suggested α = 0, butincluded the equation forcompleteness

Coordination

COMPARISON OF SI POTENTIALS

I Balamane et al. [PRB 46, 2250 (1992)] carried out an extensivecomparison of Si potentials (EDIP appeared later)

I They looked at, e.g., bulk,surface, defect and smallmolecule properties

I Included potentials were SW,Tersoff 2 and Tersoff 3,Biswas-Hamann potential (BH)[PRL 55, 2001 (1985)], [PRB 34, 895

(1986)], Tersoff-like Dodsonpotential (DOD) [PRB 35, 2795

(1987)]. and Pearson potential(PTHT) [Cryst. Growth. 70, 33 (1984)]

Pair interaction

Lattice structures

I These properties are far from the complete list of comparisonsdone by the study authors

I Their conclusion was that no single potential is clearly superior

I Instead, different potentials are good for different applications

I SW, T3 and DOD are good for elastic properties

I T3, SW, DOD, T2 and BH give fairly good values for point defects,especially taking into account the experimental uncertainties

I The (100) surface is described best by BH, SW and T3, but nopotential describes correctly reconstructions of the (111) surface

I EDIP was not included in this comparison, but would havescored well at least regarding elestic and defect properties

POTENTIALS FOR GE

I Ge has two almost identical SW parametrizations, and a Tersoffparametrization:

I [Ding & Andersen, PRB 34, 6987 (1986)]I [Wang & Stroud, PRB 38, 1384 (1988)]I [Tersoff, PRB 39, 5566 (1989)]

I These potentials are reasonable for the crystalline phase, butthey severely overestimate the melting point (2500–3000 K whencompared to 1210 K)

I Nordlund et al. tried to fix this by adjusting the cohesive energydown by 18% [PRB 57, 7556 (1998)]

I They got a melting point of 1230±50 K with reasonabledisplacement threshold value and mixing coefficient, which areboth important for irradiation physics

I However, this approach is obviously questionable

I Finally, another SW parametrization for Ge represents allcondensed phases well:

I [Posselt & and Gabriel, PRB 80, 045202 (2009)]

I Gives a reasonable crystalline phase, melting point (1360 K vsexperimental 1210 K), and amorphous phase.

I Velocity of solid phase epitaxial recrystallization is clearlyoverestimated, however

HYDROCARBONS - BRENNER POTENTIAL

I Brenner refined the Tersoff potential for modeling hydrocarbons

I In his approach, the angular dependency is introduced via g(θ)which is incorporated in the inner sum of the bond-order function

I Now, bij becomes

bij = (1 + χij)−1/2,

χij =∑

k 6=i,j fik(rik)gik(θijk) exp[2µik(rij − rik)](25)

I The angular function g(θ) is

g(θ) = γ(

1 +c2

d2 −c2

d2 + (1 + cos θ)2

)(26)

I If c = 0, this becomes a constant g(θ) = γ, and the potentialresembles an EAM potential

I It’s worth noting that the angular term is not only important formodeling covalent systems, but also of metals

I For example, it has been shown that shear constants can bedescribed in a first-nearest-neighbor potential only if the bondorder is angular dependent

I For convenience, the repulsive and attractive parts of thepotential can be written as

VR(r) =D0

S − 1 exp[−β√

2S(r − r0)] (27)

VA(r) =SD0

S − 1 exp[−β√

2/S(r − r0)] (28)

where D0 is the dimer binding energy and r0 the equilibriumdistance

I A specific feature of the Brenner potential is the overbindingterm, which corrects the unphysical interpolation of single- anddouble-bonds for some atomic configurations

I The overbinding term is incorporated in the bond-order term,which now becomes

bij =bij+bji

2 + Fij(N ti , N t

j , Nconjij )

bij =[1 +∑

k 6=i,j g(θijk)fij(rijk) exp[. . .] + Hij(N(H)i , N(C)

i )]−σ(29)

I The quantities N(C)i and N(H)

i are the number of carbon andhydrogen atoms (Brenner potential is developed forhydrocarbonds) bonded to atom i

I Nconjij depends on whether a bond between carbon atoms i and j

is part of a conjugated system (takes into account changes incarbon bonding)

I The bonding connectivity is defined again via a continuousfunction fij(r)

I Values for N(H)i , N(C)

i and N(t)i for each carbon atom i is given

byN(H)

i =∑

j=hydrogen fij(rij)

N(C)i =

∑j=carbon fij(rij)

(30)

and N(t)i = N(H)

i + N(C)i

I Values for the neighbors of the two carbon atoms involved in abond are used to determine whether the bond is a part of aconjugated system

Nconjij = 1 +

∑carbonsk 6=i,j

fik(rik)F(xik) (31)

+∑

carbonsl 6=i,jfjl(rjl)F(xjl) (32)

I Above,

F(xik) =

1, xik 6 2{1 + cos[π(xik − 2)]}/2, 2 < xik < 30, xik > 3

(33)

and xik = N totk − fik(rik)

I If any neighbors are carbon atoms that have a coordination ofless than 4, the bond is defined as a part of a conjugated system

I The functions yield a continuous value of Nconj as bonds breakand form, and as second-neighbor coordinations change

I With the overbinding and conjugation corrections, the potentialgives a remarkable description of various hydrocarbons

I However, the additional sums over the neighbors of both of theatoms make the potential somewhat computationally expensive

I In fact, even with the corrections removed, the carbon potentialdescribes well graphene and carbon nanotubes, even though itwas not fitted to sp2 bonded all-carbon structures

I The most important feature not described well with the potentialis the inter-layer van der Waals bonding between graphite layers

I Also many other carbon potentials exist

I The other much used one was developed by Tersoff [PRL 61, 2879

(1988)]

I However, the van der Waals interaction can not be properlydescribed within the BOP formalism(nearest-neighbor–interactions)

I At least two ways to account for the van der Waals interactionhas been proposed

I The first one is a rather poorly motivated hack of the potentialwhich includes a weak interaction for structures very close to theideal graphite lattice [Nordlund et al., PRL 77, 699 (1996)]

I A more well-founded approach was proposed by Stuart et al. in[J. Chem. Phys. 112, 6472 (2000)]

I They included a Lennard-Jones–like term which works also forother carbon structures (f.ex. polymers)

POTS FOR SEMICONDUCTOR ALLOYS AND COMPOUND

SEMICONDUCTORS

I A straightforward way to construct a compound potential is totake geometric averages of the mixed parameters of theelemental potentials

I The major difference comes from the fact that the parametersmust be fitted separately for each element

I For example, in Tersoff’s compound potential [PRB 39, 5566 (1989)],the functional form allows for mixing parameters, but Tersoff setthem to 1

I Tersoff SiC potential describes reasonably many properties

I The most problematic values are the shear modulus c44 andvacancy formation energies

I This is surprising considering the fact that there’s only oneparameter actually fitted to the compound structure

I Many potentials exist for the mixed systems, including SiC,GaAs, GaN, Pt-C, InAs, AlAs etc.

I However, listing the proposed potentials serves no use here –the potentials have to be tested for the specific application in anycase

I However, the Albe-formalism of the Abell-Tersoff-Brenner–typeABOP potentials, as described in [Albe et al., PRB 65, 195124 (2002)],has turned out to be one of the best ways to implementpotentials for transition metals and covalent systems

I For example:I PtC [Albe et al., PRB 65, 195124 (2002)]I GaAs [Albe et al., PRB 66, 035205 (2002)]I GaN [Nord et al., J. Phys.: Cond. Matter 15, 5649 (2003)]I SiC [Erhart et al., PRB 71, 035211 (2005)]I W-C-H [Juslin et al., J. Appl. Phys. 98, 123520 (2005)]I ZnO2 [Erhart et al., J. Phys.: Cond. Matter 18, 6585 (2006)]I Fe (BCC/FCC) [Müller et al., J. Phys.: Condens. Matter 19, 326220

(2007)]

SUMMARY

I Covalent bonds result from shared electrons between the atoms

I This results in localized negatively charged areas (bonds) whichrepel each other due to electrostatic interaction

I For each covalently bonded material, certain well-defined bondangles yield the minimum energy

I This can be taken into account via bond order formalism, whichassumes weakening of the bond from dimer to structures withhigher coordination

I Bond order formalism can be used as a basis for interatomicpotentials, which can be shown to be equivalent to EAM and todescribe also transition metals

I The Abell-Tersoff-Brenner–type potentials have been developedfor many elemental semiconductors as well as their compoundmaterials