Tersoff model

The MD calculations using the Tersoff model are carried out under constant-volimie and -temperature conditions. The melting temperatures predicted by tiie Tersoff model are much hi er tiim tiie real phase-change temperatures. The model is, however, widely applied in tiie recent simulations, because it can reproduce well the structural and dynamical properties of elemental semiconductors such as silicon md cM bon ranging from crystalline to amorphous structures In tiie Tersoff model, tiie potential energy between two neighboring atoms i and j can be expressed as... 

The parameters in the Tersoff model have been calculated for the compound semiconductor alloys by fitting to a database of experimental and theoretical values available for cohesive energies, lattice parameters, bond lengths, elastic constants, etc. (Ashu et al. 1995 Powell et al. 2007). Then, MD simulations have been used to predict properties including cohesive energies, elastic constants. 

Adhikari and Kumar (2007) have validated the use of the Monte Carlo simulations with the Tersoff model for III-V binary alloys by comparing the simulation predicted values for lattice constants, thermal expansion coefficients and bond lengths with experimental data available for the GaAs binary alloy. It was found that good agreement exists between the experimental data and simulation results as seen in Table 12.6. 

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. 

Tersoff, J., Modeling Solid-State Chemistry Interatomic Potentials for Multicomponent Systems," Phys. Rev. B, Vol. 39, No. 8,1989, pp. 5566-5568. 

In the near future, the expansion of the covalent-bonding formalisms developed to model silicon to other systems appears promising. Very recently the extension of the Abell-Tersoff covalent-bonding formalism to few-body reactive systems has been demonstrated by the development of an accurate potential energy expression for In the determination of an analytic... 

In the s-wave-tip model (Tersoff and Hamann, 1983, 1985), the tip was also modeled as a protruded piece of Sommerfeld metal, with a radius of curvature R, see Fig. 1.25. The solutions of the Schrodinger equation for a spherical potential well of radius R were taken as tip wavefunctions. Among the numerous solutions of this macroscopic quantum-mechanical problem, Tersoff and Hamann assumed that only the s-wave solution was important. Under such assumptions, the tunneling current has an extremely simple form. At low bias, the tunneling current is proportional to the Fermi-level LDOS at the center of curvature of the tip Pq. 

Fig. 1.25. The s-wave-tip model. The tip was modeled as a spherical potential well of radius R. The distance of nearest approach is d. The center of curvature of tip is To, at a distance (R + d) from the sample surface. Only the 5-wave solution of the spherical-potential-well problem is taken as the tip wavefunction. In the interpretation of the images of the reconstructions on Au(llO), the parameters used are R = 9 A, d = 6 A. The center of curvature of the tip is 15 A from the Au surface. (After Tersoff and Hamann, 1983.)...

Similar to the. y-wave model, the Na-atom-tip model predicts a poor resolution. The agreement of the Na-atom-tip model with the y-wave-tip model does not mean that the s-wave-tip model describes the actual experimental condition in STM. According to the analysis of Tersoff and Lang (1990), real tips are neither Na or Ca, but rather transition metals, probably contaminated with atoms from the surface (for example. Si and C are common sample materials). For a Si-atom tip, the p state dominates the Fermi-level LDOS of the tip. For a Mo-atom tip, while the p contribution is reduced, this is more than compensated by the large contribution from states of d like symmetry. The STM images from a Si, C, or Mo tip, as predicted by Tersoff and... 

See Surface states Tersoff-Hamann approximation See s-wave-tip model Tetragonal symmetry 128 Tip annealing 286, 288 Tip preparation 281—285 cutting 282... 

An expression of the type (7.101), which gives the bond order explicitly in terms of the positions of the neighbouring atoms, is called a bond order potential (BOP). Angularly dependent bond order potentials were first derived heuristically for the elemental semiconductors by TersofF (1988). We will see in the next chapter that a many-body expansion for the bond order may be derived exactly within the model. 

The purpose of doing STM is to learn about surface structures, and the tip as such is regarded as an uninteresting probe. In this sense, it is a problem that the electronic structure of the tip is contained in the formula for the tunnel current in the original work by Bardeen 58). Tersoff and Hamann 59,60), however, extended Bardeen s formalism and showed by simple, yet relevant approximations that the impact of the unwanted electronic structure of the tip is in many cases less pronounced for typical tunneling parameters. Fortunately, the Tersoff-Hamann model provides a simple conceptual framework for interpreting STM images, and therefore it is still the most widely used model. 

The difficulty of evaluating the effect on the tunneling current of the tip electronic structure was approached by Tersoff-Hamann by assuming a simple, s-wavc tip model with wave functions centered at a point Fq in the tip. In the limit of low-bias voltages, the total tunnel current can then be expressed as follows ... 

Fig. 2. STM image (78 A x 76 A) of nitrogen atom adsorbates on an Fe(l 00) surface. Because the nitrogen adsorbates deplete the LDOS at the Fermi level, the nitrogen atoms are imaged as depressions in accord with the Tersoff-Hamann model. From the STM image it is concluded that nitrogen atoms adsorb in fourfold hollow sites on Fe(l 0 0). This is just one of many examples illustrating how the STM contrast may depend on the details of the LDOS around an adsorbate and produce a somewhat counterintuitive picture. Adapted from Reference (dd).

Therefore, the interpretation of such structures should be performed carefully and preferably in combination with other experimental or theoretical techniques. Nevertheless, the large database of STM results, which in combination with other experimental and theoretical surface science investigations have solved many problems correctly, often rely on the Tersoff Hamann interpretation (62-65). In this sense, the Tersoff-Hamann theory has proved itself very successful, but when applying it to STM characterizations of more complicated samples, one has to be aware of the limitations of the model. The sample wave functions may be distorted by the close proximity of the tip to the surface, and the forces between the tip and sample may lead to geometric relaxations of the atoms in the surface layer beneath the tip (66). Moreover, the tip is in this model represented by a simple x-wave, so that the chemical composition of the tip is neglected. In reality the nature of the tip may, however, differ significantly from this situation because of adsorbates or other contaminants present on the tip apex (53). 

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