tive to close-packed structures without 3-body forces [2]. In fact, the 3-body Keating model [3] which is fitted to small distortions of the diamond structure has had ...
173
Classical Two and Three-Body Interatomic Potentials for Silicon Simulations R. BISWAS AND D. R. HAMANN AT&T Bell Laboratories, 600 Mountain Avenue, Murray Hill, NJ 07974 ABSTRACT We develop two and three-body classical interatomic potentials that model structural energies for silicon. These potentials provide a global fit to a database of firstprinciples calculations of the energy for bulk and surface silicon structures which spans a wide range of atomic coordinations and bonding geometries. This is accomplished using a new "separable" form for the 3-body potential that reduces the 3-body energy to a~product of 2-body sums and leads to computations of the energy and atomic forces in n steps as opposed to n 3 for a general 3-body p6tential. Simulated annealing is performed to find globally minimum energy states of Si-atom clusters with these potentials using a Langevin molecular dynamics approach. The computer-based microscopic description of the structure and properties of materials, is an area of much current interest. First - principles quantum-mechanical calculations, for example, have recently enjoyed great success in predicting the properties of simple structures [I]. However, the computational complexity of quantummechanical energies and forces precludes their use for applications with more than 10-20 atoms per unit cell. Such applications include crystal growth and epitaxy, melting, laser annealing, defect motion and materials strength, amorphous structures, surface and interface reconstruction. These problems 2are 3of much current interest and generally require molecular dynamics studies of 10 -10 atoms evolving through 104-106 configurations. Simulated annealing techniques are very useful in determining low energy states of such complex systems. With a view towards studying some of the above dynamical problems for semiconductor materials, we explore in this paper the extent to which a classical potential model can succeed in providing an accurate global description of its structural energetics. Pair potentials, for example, have been extensively used in simulations of simple systems such as rare gas solids and liquids. However for covalent materials such as Si, pair forces alone are inadequate, since the equilibrium diamond lattice is unstable relative to close-packed structures without 3-body forces [2]. In fact, the 3-body Keating model [3] which is fitted to small distortions of the diamond structure has had much success in describing local distortions and phonons, but has also been extrapolated to compute energies of complex Si structures, sometimes far beyond the range of its validity. To investigate the feasibility of a global model of classical potentials, Si is the material of choice because of the huge computational effort that has produced accurate quantum-mechanical energies for simple Si structures spanning a wide range of atomic coordinations, bond lengths, and bond angles [1,4]. While only a few of these structures have low enough energies to be experimentally accessible in extended form, their features may occur as local distortions or dynamic intermediates associated with complex structures or processes. There are neither a priori arguments nor empirical tests to suggest that a classical model can accomplish such a global fit without arbitrarily many multi-atom potentials. The computational complexity of such a model would render it impractical, so we have confined our explorations to 2 and 3-body potentials. We find that it is possible to achieve excellent qualitative separation of structural groups, and quantitative accuracy for individual structures which should be acceptable for many purposes.
Mat. Res. Soc. SyMp. Proc. Vol. 63. C1985 Materials Research Society
174
A significant innovation in the present work is a new "separable" form for the 3-body potential which permits the energy to be calculated in n 2 computational steps instead of the n 3 steps generally required, where n is the number of interacting atoms. This reduction in the computational complexity of the 3-body model is a considerable aid for simulation purposes. This is particularly true for the simulated annealing of Siclusters, with the present 2 and 3-body potentials, discussed in this paper. Our 2 and 3-body potential model is defined by the following expression for the structural energy, E=- -L'V 2 (1,2) + I 'V3 0,2,3), 1,2
(1)
1,2,3
where primes indicate that all summation indices are distinct. There is no explicitly volume dependent term, since atomic volume is not a useful physical concept for inhomogenous structures. The assumption of neglecting 4-body and higher terms is used and tested in the present work. Any 3-body potential V3 (1,2,3) may be expressed as a function of 2 lengths r12,rI 3 and the included angle 01. This potential is symmetrized over the 3 particles in the sums in (1). Without losing generality we can expand the angular dependence of this potential in the complete set of Legendre polynomials. The coefficients in this expansion are functions Fe of bond lengths multiplied by linear coefficients Ce, V3 (r12,r13,01)
-
• CeFt(r 12,r13)Pe(cos0d).
(2)
Our key simplification is to assume that the functions F, are separable and symmetric products of functions (pi of each bond length. This leads to the symmetric separable form V3 (r1 2,r1 3,01d)
• Ce4e(r12)'0e(rI 3)Pe(cosOi)
(3)
.
Generally, separability is consistent with a local picture of the atomic bonding interactions. The addition theorem for spherical harmonics now reduces the 3-body energy to a rotationally invariant scalar product of vectors 4em that are simple 2-body sums i.e.: ZV3(rl2,rl3,01)
=
ZCe
2,3
U
4w 1 m. _
b
(4)
Mt
where •tm=-2e 2
(5)
(rj2)YVm(fj2) •
The 4e•m vectors are the moments of the structure around atom j, that describe its local environment. The calculation of the energy in (1) requires the sum in Eq. 4 to be corrected for the case when indices 2,3 are identical. This introduces a modification of the 2-body interaction, Z'V 3 (r1 2,r1 3 ,01 ) 2,3
where
V3 (r1 2,r1 3,01) - 2f 3 (r12 )
=2,3
2
,
(6)
175
f 3 (r)
-
(7)
2Cte2(r) •
We have examined a few short range, monotonic functional forms for the 3-body functions 0 along with similar functions for the 2-body potential. Overall, our best results were obtained with the family of simple exponentials, .1t -
e-at.
In addition,
we used generalized Morse 2-body potentials, V2 (r) - AIe-X'r + A2e-x2r .
(8)
Very general functional forms for 0, often led to unphysical solutions that provided good fits but performed poorly on test structures. The nonlinear parameters of our model are the decays of the radial functions, whereas the coefficients Ce,A 1,A2 are linear variables. The parameters in the potential were least-squares fitted to the database of accurate quantum-mechanical energies of diamond, wurtzite, the high pressure #-tin and simple hexagonal structures, and simple hypothetical Si structures [1,4,51 as shown in Fig. I(A), and a 4-layer slab (Fig. 2). The nonlinear parameters were varied with a simplex routine, and for each nonlinear parameter set the least-squares fit was performed to obtain the linear variables. Because of symmetry, only moments 4ým with certain values of the angular momentum e are allowed for each structure, e.g. the e - 0,3,4,6 • moments are allowed in diamond. To account for bond breaking energies, we have added to the database our own Linear Augmented Plane Wave calculations for the energy of a 4-layer Si(1 It) slab as a function of the positions of the outermost atomic layers (Fig. 2). The (111) slab is the only structure in the above database that permits an t-1 moment and is therefore essential to the fit. Fits to slabs have the important effect of requiring the cohesive energy to decrease monotonically from bulk structures to low-dimensional structures. For the fits, we chose a set of structures from Fig. 1 that, together with the (011) slab, contained all the moments up to t--6. The results of our fit compare very well with the quantum-mechanical energies for the Si(1 11) slab, as shown in Fig. 2. Our global fit to the crystal structures shown in Fig. I (B), agrees with the quantum-mechanical results of Fig. 1(A), to within an rms error of 0.05 eV and displays the correct structural trends over a large range of atomic volumes. The absolute energies are plotted in Figs. 1 and 2. The first highpressure phase is correctly predicted to be #3-tin. In Fig. 1(B), bcc and si-hexagonal were test structures that were not fitted. Si-hexagonal is very close to fl-tin as it should be [4]. Wurtzite is higher energy than diamond-Si. hcp is not as well fitted as other phases and s-cubic is somewhat lower in energy than the quantum-mechanical result. Our diamond Si-phase has an equilibrium bond length of 2.32A (experimentally 2.351A), but the nonlinear decay parameters could be uniformly scaled to produce the experimental bulk bond-length if desired. Table 1 lists the parameters for our fitted potential. The coefficients in the angular expansion (CI) decay uniformly indicating the convergence of the solution and a well-defined angular decomposition. The solution is stable, and remains in the same region of parameter space when the structures in the fitting database are altered or weighted differently, or when the number of 3-body nonlinear parameters is altered by 1. The range of the potentials is comparable to that of the atomic valence wavefunction overlaps, indicating a physical reasonableness in the overall fit. A real space cutoff of 10A only causes an error of order 0.01 eV in the total energies. Other classical models, beyond the restricted Keating model [31 include that of Stillinger and Weber (SW) [6], and Pearson, Takai, Halicioglu and Tiller (PTHT) [71, and Tersoff [8]. SW used 3-body potentials that have a Keating [31 angular form (IC=C I=C 2; C3,C4 "."'0), and are separable (although the separability was not utilized). The potentials were confined within a very small cutoff radius of 3.78 A.
176
WURT>
D~ENSITY--\
ILPRESENT
DIAMON4D
Sw -3.75
I. -4,5
WORK
B) PRESENT WORK FCC B
-4
PTH
HCP SIHEX BETA-TIN
E -4.
0
1
8CC
C sw FCC-HCR
3.5
direction.
BETA-TIN SIHEX
-375
4
5
IDEAL
S, (•111
FIGURE 2. Energy of a 4 layer Si(l 11) slab. Starting from the ideal (111) geometry the two outermost layers are symmetrically displaced in the normal
DIAMOND
,325
3 ATOMPOSITION (OU)
ON-
s-CUBIC
- 475,
-
-C
WURTZ WURI Z
-425 -.5
>
4
TZ
-4.5
S-CUBIC
The distance of either layer
from the slab center is plotted. At 2.22 au, the slab reduces to two graphitic atomic Splanes.
-4 -4.25 DI1AMOND
Table
HC FCC D) PTHT
-425 -.
8SCC\HE
--
ISIHEX'BETA-TIN
-45 -475
I
The values of the parameters for the present 2 and 3-body potentials.
S-CUBIC
-55
l
tmA i
a~i'%x(el)
Ai'Ct(eV)
-5.25
2-body
DIAMOND -5,5 0.6 06.
08 08...
1
1210 12
VOLUME (V/VO)
FIGURE 1. Energies for simple silicon structures as a function of atomic volume. SW potentials are from Ref. 6, and PTHT from Ref. 7. Wurtzite could not be distinuihe iaondin(C frm in (C) an tinguished from diamond and (). (D).
1
3.946668
0.2682936 x 10'
2
1.191187
-0.4259863 x 102
0 1 1 2 43
3-body 1.246156 1.901049 1.901049 1.786959 1.786959 1.786959
5 6
1.786959 1.786959
0.9139775 0.1644013 0.1644013 0.9580299 0.3987710 0.6663147 0.2046722 0.7018867
x x x x x x x x
102
10' 104 104 104 104 104
10'
177
Their 9 parameters were adjusted to fit the bond length, cohesive energy and melting temperature of bulk diamond Si, and satisfy other qualitative criteria [6]. The structural energies for the SW potential are plotted in Fig. 1(C) and Fig. 2. We were unable to obtain satisfactory fits with functions as short-ranged as the SW potential. Alternatively, PTHT [7] used the nonseparable Axilrod-Teller 3-body potential [9] which is based on the generalization of Van der Waals fluctuating-dipole forces for 3 particles. These potentials were long ranged with algebraic decays (V3 (r) -= r- 9 ). The 3-parameter PTHT potential was obtained through an average fit to the bond lengths and cohesive energies of bulk diamond-Si and molecular Si2 , and the structural energies for this potential are shown in Fig. 1(D) and Fig. 2. Clearly from Figs. 1 and 2, the present potentials compare with the quantummechanical results much better than previous work. The grouping of structures into three energy classes and the values of their equilibrium volumes are improved compared to SW, and the compressibilities and energy differences between the classes is improved compared to PTHT. Recently, Tersoff [81 has developed a 2-body potential for silicon in which the strength of the attractive part is a complex function of the local bonding geometry, i.e., of the bond lengths, bond angles etc. This ansatz effectively couples in 3-body as well as higher-body correlations. The model is confined to nearest neighbors and its 7 free parameters are fitted to diamond-Si properties and cohesive energies for Si2, s-cubic and fcc silicon. Energies of models for the reconstructed Si(1 11) surface have been computed with this potential. The present model has a weak 2-body potential that has a strength of -1.09 eV at the equilibrium position of 2.77A. Clearly this pair potential is inappropriate for the multiply-bonded Si 2 dimer. These properties of our 2-body potential are consistent with work of Carlsson et al. [10L Our attempts to constrain our 2-body potential to fit Si 2 did not yield satisfactory fits. We believe that a model which simultaneously fits small clusters and extended systems must include N-body potentials beyond N - 3. Alternatively, the 3-body potential, exhibits (for r 12 -r 13 - 2.35A) a very weak angular dependence for 90"
