Stillinger-Weber

The function/c is a smoothing function with the value 1 up to some distance Yy (typically chosen to include just the first neighbour shell) and then smoothly tapers to zero at the cutoff distance, by is the bond-order term, which incorporates an angular term dependent upon the bond angle 6yk- The Tersoff pofenfial is more broadly applicable than the Stillinger-Weber potential, but does contain more parameters. 

Thermodynamic properties. A molecular-dynamics simulation method (using a steepest decent method) with Stillinger-Weber potential is employed to optimize structures and to obtain the cohesive en-... 

Fig. 4. Optimized toroidal structures (a) torus and (b) torus C24o Pentagons and heptagons are shaded. The diameters of the tube of the stable torus determined by optimization using molecular dynamics with Stillinger-Weber poiemial[211, is 8.8 A. The diameter of the hole is 7.8 A, which is quite close to the diameter of fullerence Qy.

A full-scale treatment of crystal growth, however, requires methods adapted for larger scales on top of these quantum-mechanical methods, such as effective potential methods like the embedded atom method (EAM) [11] or Stillinger-Weber potentials [10] with three-body forces necessary. The potentials are obtained from quantum mechanical calculations and then used in Monte Carlo or molecular dynamics methods, to be discussed below. 

Different results were obtained in the case of the Stillinger-Weber (SW) potential for silicon (10). In this case, material deposited on the (ill) orientation was disordered, without distinct layers of atoms. Growth on the (100) face did produce about ten distinct layers at sufficiently high temperatures, although it is not clear whether thicker deposits would retain this order since some degeneration was observed as successive layers were deposited. 

In a similar study, Khor and Das Sarma studied the diffusion of Si, Si2 and Si3 on the (001), (011), and (111) surfaces of silicon. In their study, the forces on all of the atoms were determined by the Stillinger-Weber potential. For single atoms on all surfaces, they report an upper bound for diffusion of 4.8 X 10 cm /sec at 1600K. This value is significantly less than both the range determined by NoorBatcha of 2.031 x 10" to 15.8 x 10" cm /sec ", and the experimental estimate of 10 cm /sec. They also report that diatomic Si2 molecules diffuse more readily than single atoms on the Si(l 11) surface. 

For atoms with sp3 hybridization (e.g., Si in Si02), the parameter cosdg = —1/3 corresponds to the angle in the tetrahedron. The three-body term with an angular dependence is also used in the Stillinger-Weber potential [63] for systems with the diamond lattice... 

The characteristic time of this diffusion was estimated by carrying out the molecular dynamic relaxation of the film surface within the limits of the above model at 500°C. In MD calculations, the pair interaction energy between atoms is approximated by the Buckingham pair potential (Zr O, O-O) (see Table 9.4). To describe covalent bonds more correctly, a three-body O-Zr-O term in the Stillinger-Weber form was introduced in addition to the Coulomb term. 

As can be seen, there are eight parameters in the Stillinger-Weber potential A,B,p, q, /r, Tc, Z, and a. These parameters are fitted to experimental data, such as lattice constants and cohesive energy, for the diamond stmcture. The angle is the angle centered on atom i. If djik is 1010.47°, cos 6jii = the angular function has a minimum of zero, which... 

After ealeulating discrete values of frequency and relaxation time for the polarization mode and different wave veetors, a continuous function for the relaxation time and dispersion relations can be established for eaeh mode. The results shown in Fig. 3 are obtained with the Lennard-Jones potential for Argon. An approach similar to that described above can be applied with the Stillinger-Weber potential for silieon. 

Silicon thin film thermal conductivities are predicted using equilibrium molecular dynamics and the Grccn-Kubo relation. Periodic boundary conditions are applied in the direetions parallel to the thin film surfaees (Fig. 5). Atoms near the surfaces of the thin film are subjeeted to the above-described repulsive potential in addition to the Stillinger-Weber potential [75]. Simulations were also performed adding to each surface four layers of atoms kept frozen at their crystallographic positions, in order to eompare the dependence of the predieted thermal eonduetivities on the surface boundary eonditions. We found that the thermal eonduetivities obtained using frozen atoms or the repulsive potential are identical within the statistical deviations, exeept for the in-plane thermal eonduetivity of films with thickness less than 10 nm [79]. Therefore, in the present study, we present only the predietions obtained with the repulsive potential. 

Ion-enhanced etching mechanisms using MD were addressed by Barone and Graves (1995a,b, 1996) in several papers. In the first paper, the Stillinger-Weber potentials were used for Si-Si, Si-F, and F-F interactions. Si-Ar... 

The model surfaces chosen for this study are shown in Figs. 15 and 16. A bare silicon surface, a silicon surface with a single monolayer of Cl chemisorbed (Fig. 15), and a silicon surface with about 2.3 monolayers of Cl mixed into the top 20 A were chosen (Fig. 16). The potentials were Feil-Stillinger-Weber, and the simulations are described by Helmer and Graves (1998). For the results shown here, Ar+ and C1+ are used as the incident ions. 

The novel feature associated with the Stillinger-Weber potentials is this angular term. To be more specific, the three-body terms have an angular dependence of the form... 

The Stillinger-Weber scheme lacks the explicit environmental dependence that was argued to be of importance in our discussion of pair functionals. However, environmental dependence has been accounted for in covalent materials as well. Tersoff (1988) was one of the first to suggest such a scheme, and his total energy... 

Stillinger-Weber Energy for Si in Diamond Cubic Structure... 

Forces Associated with Stillinger-Weber Potentials Obtain an expression for the forces between atoms in a system characterized by the Stillinger-Weber potential. An explicit expression may be found in Wilson etal. (1990). 

Use the Stillinger-Weber potential to determine the energy vs volume curves for Si in the diamond cubic, simple cubic, fee, bee and yS-Sn structures. 

Stillinger-Weber Tersoff Tight-Binding LDA Expt. 

Use the Stillinger-Weber potential to compute the unrelaxed vacancy formation energy in Si. 

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