Tersoff potential

The form of interatomic potential suggested by Tersoff [25] is an example of the wider family of bond-order potentials [26]. The total energy is written as a sum of pair like terms. 

The resulting potential is, in fact, a many-body potential, as the bond-order terms depend on the local environment. Bond-order potentials can also be derived from a 

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The Tersoff potential [Tersoff 1988] is based on a model known as the empirical bond-order potential. This potential can be written in a form very similar to the Finnis-Sinclair potential ... 

The Tersoff potential was designed specifically for the group 14 elements and extends the basic empirical bond-order model by including an angular term. The interaction energy between two atoms i and j using this potential is ... 

The interaction between particle and surface and the interaction among atoms in the particle are modeled by the Leimard-Jones potential [26]. The parameters of the Leimard-Jones potential are set as follows pp = 0.86 eV, o-pp =2.27 A, eps = 0.43 eV, o-ps=3.0 A. The Tersoff potential [27], a classical model capable of describing a wide range of silicon structure, is employed for the interaction between silicon atoms of the surface. The particle prepared by annealing simulation from 5,000 K to 50 K, is composed of 864 atoms with cohesive energy of 5.77 eV/atom and diameter of 24 A. The silicon surface consists of 45,760 silicon atoms. The crystal orientations of [ 100], [010], [001 ] are set asx,y,z coordinate axes, respectively. So there are 40 atom layers in the z direction with a thickness of 54.3 A. Before collision, the whole system undergoes a relaxation of 5,000 fsat300 K. 

Dodson has used molecular dynamics to study atom-surface dynamics for silicon atom energies ranging from 10 to lOOeV incident on a silicon (111) surface . In this study a modified form of the Tersoff potential was... 

Recently, Ishimaru et al. [1, 17, 26] showed that the /-Si and a-Si networks generated using the Tersoff potential could reproduce the features of the structural properties reported by ab initio calculations. The procedure applied by Ishimaru et al. [17], therefore, is adopted to prepare a realistic amorphous structure of silicon. The main features are 1) arrangement of atoms in the diamond structure 2) equilibration of /-Si at 3500 K and 3) quenching and solidification of the /-Si to 500 K at a cooling rate of 10 K/s. They showed that the structural and dynamical properties of a-Si... 

Ohira, T., Inamura, T, and Adachi, T., Molecular dynamics simulation of hydrogenated amorphous silicon with Tersoff potential. Mater. Res. Soc. Symp. Proc. 336,177-182 (1994). 

Low-energy boron bombardment of silicon has been simulated at room temperature by MD. Tersoff potential T3 was used in the simulation and smoothly linked up with the universal potential. The boron-silicon interaction was simulated according to Tersoff potential for SiC but modified to account for the B-Si interaction. Silicon crystal (Si-c) in the (001) direction, with (2x1) surface reconstruction, was bombarded with boron at 200 and 500 eV. Reasonably good statistics are obtained with 1000 impact points uniformly distributed over a representative surface area. The simulation size was 16x 16x 14 unit cells. Periodic boundary conditions were applied laterally. The temperature was kept at 300 K with a thermal bath applied to the more external cells in the crystal except the top surface. In these conditions the crystal was relaxed during 19 ps. In order to avoid direct channeling, the incidence was inclined 7° out of the normal, as usual in experiments, with random azimuthal direction. 

While these potentials are notationally complex, they are conceptually simple in terms of the basic physics they incorporate. In particular, potentials of this type invoke both of the new physical ideas presented in this section environmental dependence and angular forces. In particular, the quantity denoted bij guarantees that the interaction between two given atoms will depend upon both the number and disposition of neighboring atoms. For our purposes, the key advantage of the Tersoff potential is its attempt to capture effects which are outside of the purview of more traditional potentials. 

Fig. 9.22. Comparison of the atomic-level displacements associated with the Si(001) (2 x 1) surface reconstruction as obtained using alternative total energy schemes (adapted from Balamane et al. (1992)). Arrows illustrate the atomic displacements in angstroms in comparison with the (1 x 1) reference state. The numbers in each group correspond to the displacements as obtained using density functional theory and Biswas-Hamaim, StUhnger-Weber and Tersoff potentials, respectively.

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

Classically-based simulations do not work for metals because of the very delocalized bonding characteristic of nearly-free electrons in the substrate. Pair potentials also fail to woik for semiconductors that have strongly directional covalent bonding. For example, a pair potential would predict a close-packed structure for Si instead of the experimentally-observed diamond structure. Only three-body or n-body interatomic PES s can yield the diamond lattice structure, although that structure may contain zero-energy lattice defects [88Terl], Since n-body interatomic potentials contain too many free parameters, in practice potential ejq)ansions usually incorporate terms with no more than three bodies. The two most popular three-body interatomic PES s are the Stilhnger-Weber (SW) and Tersoff potentials. 

To model disorder, a 56-atom unit cell as described in Fig. 1(b) was used [38] and heated to 10 000 K using a very approximate Tersoff potential [12]. We chose... 

Fig. 1.9 Molecular models using Tersoff potentials allow for study of a wide range of static, dynamic and thermodynamics properties of carbon forms here carbon nanotubules are shown in several phases of buckling [389]. Image courtesy Prof. Junichiro Shiomi and Dn Takuma Shiga, Department of Mechanical Engineering, University of Tokyo...

Fig. 6.5 Force errors compared to DFT forces for GAP and the Tersoff potential. The silicon potentials are shown in the left and the germanium potentials in the right...

We show in Fig. 6.5 the results for our potentials which were developed to model the two other group IV semiconductors, silicon and germanium, compared to the Tersoff potential. 

Fig. 6.9 Phonon dispersion of silicon calculated by GAP solid lines), the Tersoff potential dotted lines) and PBE-DFT open squares)...

However, the activation energy which was deduced by using the activation-relaxation technique, with a Metropolis accept-reject criterion and a fictitious temperature of 0.5eV, ranged from 0.22 to l.OeV. It also exhibited a steep increase at low temperature. The very large pre-exponential factor suggested that the interatomic forces which resulted from the Tersoff potential were very strong. These predictions were consistent, to some extent, with recent experimental results for liquid Si. 

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