The Environment-Dependent Interaction Potential (EDIP) is a transferable em- pirical potential for carbon which is well suited for studying disordered sys- tems [20] [21]. The functional form of EDIP consists of three components; a two-body pair energy, a three-body angular penalty and a generalised co- ordination. The two-body and three-body terms have environment depen- dence controlled by the atomic co-ordination Z. Within this framework, the total energy is written as a sum of on-site energies, equation 2.22.
Ui =
∑
j U2 rij,Zi +∑
j<k U3 rij,rik,θjik,Zi (2.22)WhereU2(rij,Zi)is an interaction between atomsiandj(iis the on-site species
and jandkare neighbours). U3(rij,rik,θjik,Zi) is the interaction between atoms
i, j, and k centred at atom i representing angular forces [66]. The two types of interactions depend on the local environment of atomi. This depends on its lo- cal environment and is defined by its effective co-ordination number, equation 2.23 [65].
Zi =
∑
m6=ifc (2.23)
where fc is the cut-off function defined further in equation 2.24 [66].
f(r) = 1, if r <c exp α 1−x−3 , if c <r< a 0, if r>a (2.24) wherex = (r−c)/(a−c).
The cut-offs are constrained to reproduce the co-ordinations of important crys- tal structures, e.g., Zi = 4 for the diamond lattice [66].
The co-ordination Zi, equation 2.22, has a fractional value and consists of two
components, a spherical contribution zi and an aspherical term which is vi-
tal for describing distorted configurations involving π-electrons. In graphite
figurations of the structures. The spherical term counts precisely the number of nearest neighbours and the co-ordination then assumes its intuitive integer value.
The pair potential,U2resembles the SW potential (equation 2.21). The pair po- tential is short ranged and decays to zero at a distance set by the denominator in the exponential term. Equation 2.25 gives the functional form for distances smaller than the cut-off [73].
U2(r,Z) = e " B r 4 −e−βZ2 # exp σ r−a−a0Z (2.25)
The bond-order is described by the exponential−βZ2term. This term moder-
ates the balance between attraction and repulsion. The parameter a0 controls the variable cut-off, which was absent from the earlier silicon model.
The three-body term in EDIP uses a SW-like radial and angular terms, these terms have environmental dependence. Equation 2.26 describes the three-body term for distancesrij and rik less than the cut-off.
U3 rij,rik,θ,Z
=λ(Z)g rij,Z
g(rik,Z)h(θ,Z) (2.26)
The three components ofU3 can be defined as equations 2.27 to 2.29.
λ(Z) =λ0exp h −λ0(Z−Z0)2 i (2.27) and g(r,Z) =exp γ r−a−a0Z (2.28) and h(θ,Z) =1−exp n −q[cosθ+τ(Z)]2 o (2.29) where τ(Z) ≡ −cos(θ0) describes the variation in ideal bonding angle which follows the SW philosophy of an ideal angle θ0 for which there is no angular penalty [73].
function λ(Z). The middle terms of equation 2.26 (g(rij,Z),g(rik,Z)) are the
radial functions (equation 2.28). These equations are generalised to include a variable range for the pair potential and the cut-off in the exponential denom- inator. The final term in equation 2.26 is the angular term, h(θ,Z), this term
captures the energy cost of distortions away from the ideal angleθ0.
The spherical co-ordination contribution zi is a function of distance and is
determined by the sumzi =∑f(rij), where f(r)is a three-parameter function
which is unity for r < flow, zero for r > fhigh and between these limits varies
continuously according to the expression in equation 2.30.
f(r) = exp α 1−x−3 (2.30) Wherex = (r− flow)/(fhigh− flow).
There is a second important factor in the co-ordination term of carbon EDIP, the aspherical term applying for sites withZ<4 which exhibitπ-like bonding
[73]. There are two mechanisms within the aspherical term which must be considered, the dihedral rotation term and the non-bonded π-repulsion term.
The dihedral rotation term promotes a locally planar structure and the non- bonded π-repulsion term fixes the equilibrium spacing of graphite and thus
prevents the collapse of planarsp2units. The two mechanisms are included in the generalised co-ordinationZi which can be seen in equation 2.31.
Zi =zi+π3(zi)Xdihi +π3(zi)X rep3
i +π2(zi)X rep2
i (2.31)
Whereπ2andπ3are switching functions identifying two and three-co-ordinated atoms, and the Xi described dihedral rotation, π-repulsion at a threefold site
and π-repulsion at a twofold site.
Vector products describe the co-ordination increment. These capture the ap- propriate symmetries via the functions in equations 2.32 to 2.34.
Xidih =Zdih
∑
π3(zj) Rˆjm·Rˆik×Rˆil 2 Cijklmdih (2.32) and Xrepi 3= Zrep∑
π(zj) Rˆij·Rˆik×Rˆil 2 Cijklrep3 (2.33)and
Xirep2 =Zrep
∑
π(zj)h
1− Rˆij·Rˆik2i
Crepijk2 (2.34) Where jkl are neighbours ofi, m is a neighbour of j and π(z) selects for two-
or threefold sites [73].
The distance-based cut-off function C can be seen in equations 2.35 to 2.37. These equations help avoid various unwanted first- and second-neighbour in- teractions. Cijklmdih = p Rij p(Ril)p Rjm (2.35) and Cijklrep3= Rij−c0 2 1−p Rij p(Rik)p(Ril) (2.36) and Cijkrep2= Rij−c0 2 1−p Rij p(Rik) (2.37)
With the function p(r) equivalent to f(r) except for different end-points plow
and phigh [73]. Both repulsive cut-offs are identically zero for Rij > c0, while the two switching functions π(z) are polynomials of the form (z−x)2−12
for|z−x| <1 and zero otherwise. The third switching functionπ(z) is equal
toπ3(z) forz >3 and unity elsewhere.
The six pair potential parameters are effectively fitted to scaled Hartree-Fock cluster calculations, using data extracted from the SW pair potentials for graphite and diamond. Table 2.1 gives the parameters of EDIP as fitted to ab initiodata [73].
Two-Body e = 20.09 eV B = 0.9538 β = 0.0490 σ= 1.257 α = 1.892 α’ = 0.170
Three-Body Z0= 3.615 λ0 = 19.86 eV λ’ = 0.30
γ = 1.354 q = 3.5
Co-ordination flow = 1.547 fhigh = 2.27 α = 1.544
plow = 1.481 phigh = 2.0
Zdih = 0.30 eV Zrep = 0.06 eV c0 = 3.2
Table 2.1: Parameters of EDIP as fitted to theab initiodata.
WhereBandβhelp to describe the bond order, λ,γandqdefine the cut-off for
the three-body term;Z is the atomic co-ordination (dihis the dihedral rotation and rep is the π repulsion); α is the spherical co-ordination; flow and fhigh are
the co-ordination functions and are equal to p(r) except at plow and phigh, c
determines the repulsive cut-off; e is the depth of the potential well and σ is
the finite distance at which the inter-particle potential is zero [73].
Table 2.2 compares the elastic constants of the EDIP potential along experimen- tal data for graphite and diamond. One can deduce from table 2.1 that carbon EDIP produces pleasing results when compared to experimental data. This is partially due to its resemblance to the SW potential which for theoretical reasons describes well the elastic properties of tetrahedral materials. It is for this reason that the carbon EDIP potential has been selected for undertaking the simulations in Chapters 4 and 5 [73].
Expt. EDIP Diamond B 442 418 c11 - c12 951 958 c44 576 472 Graphite c11 +c12 1232 1233 c11 - c12 880 849
Table 2.2: Elastic constants of diamond and a graphite sheet, in units of GPa. The computed graphite values presume a c-spacing of 3.35 Å.