Interatomic potential for Si–O systems using Tersoff parameterization

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Abstract

A parameter set for Tersoff potential has been developed to investigate the structural properties of Si–O systems. The potential parameters have been determined based on ab initio calculations of small molecules and the experimental data of α-quartz. The structural properties of various silica polymorphs calculated by using the new potential were in good agreement with their experimental data and ab initio calculation results. Furthermore, we have prepared SiO2 glass using molecular dynamics (MD) simulations by rapid quenching of melted SiO2. The radial distribution function and phonon density of states of SiO2 glass generated by MD simulation were in excellent agreement with those of SiO2 glass obtained experimentally.

Introduction

Silicon (Si) and oxygen (O) are key elements in modern microelectronics technologies. In particular, O is an important impurity in Czochralski Si crystals to obtain the gettering effects. Moreover, the thermal oxidation of Si crystals leads to the formation of SiO2 layers, and the Si/SiO2 interface is considered to be the essential element in metal–oxide–semiconductor devices. As the design rule of electronic devices decreases, it becomes important to understand the Si–O system at the atomic level.

Molecular dynamics (MD) simulation using an empirical potential is a powerful tool for the analysis of a large system composed more than 10,000 atoms. For Si, several empirical potentials [1], [2], [3] has been used extensively in MD simulations. Stillinger and Weber proposed an empirical potential composed two- and three-body interactions [1]. Stillinger–Weber (SW) potential is widely used in MD or MC simulations of silicon, since the melting point of silicon and other properties of liquid Si are well reproduced. Recently, Sastry and Angell successfully observed a liquid–liquid first-order phase transition deep into the supercooled liquid Si by using SW potential [4]. In 1989, Tersoff potential incorporating the dependence of bond order was proposed. The melting point of Tersoff Si are much higher than the real liquid Si. The Tersoff potential is, however, applied in the recent simulations, since it can reasonably well describe the properties of the liquid and amorphous phases of Si [5], [6], [7] in addition to the crystal phase. Balamane et al. reported that SW potential reproduce the structure of liquid Si and other properties are in good agreement with experimental value [8]. They also reported that Tersoff potential seems to favor fourfold coordination in the liquid in disagreement with experiment. In 1998, Justo et al. developed a new empirical potential for silicon that incorporates several coordination-dependent functions to adapt the interactions for different coordinations [9]. It predicts the correct reconstruction for both the 90°- and 30°-partial dislocations, and the reconstruction energies are in agreement with ab initio data. The Justo potential predicts a quench directly from the liquid into the amorphous structure without the artificial procedure. Overall, the properties of the amorphous Si is in good agreement with ab initio results.

Various interatomic potentials have been proposed for the Si–O system so far. Tsuneyuki [10], BKS [11] and Vashishta [12] potentials were employed to analyze the SiO2 structures. More recently, Yasukawa [13] proposed a set of modified potential parameters based on the Tersoff potential including a charge transfer function. These empirical potentials, however, are difficult to apply to large-scale calculations because they include long-range functions to describe the Coulomb interactions. Although long-range Coulomb interactions can be usually handled by using the Ewald method, there are various studies to replace the Coulomb interactions with short-range interactions by using a cut-off length [14], [15], [16], [17]. They succeeded in representing properties of ionic materials. Pettifor et al. proposed an analytic bond-order potential (BOP) and parametrized for some materials [18], [19], [20]. The potential addresses σ and π bonding and the valence-dependent character of heteroatomic bonding. The structural and binding energy trends of this potential excellently match experimental observations and ab initio calculations.

In this study, although less accurate than BOP or ab initio calculation, we choose the Tersoff potential among the many potentials for performance of large system. Our parameter set is applicable to large Si–O systems by excluding the Coulomb interaction terms. In spite of such drastic simplification, it can well reproduce structural and dynamical properties of SiO2.

Section snippets

Parameterization

Tersoff potential, which has been widely used for atomic simulations [7], [21], [22], [23], is an empirical function composed of two-body terms depending on the local environments. The potential energy E is taken to beE=iEi=12ijVij;Vij=fc(rij)[fR(rij)+bijfA(rij)];fR(rij)=Aijexp(-λijrij);fA(rij)=Bijexp(-μijrij);fc(rij)=1,rij<Rij12+12cosπrij-RijSij-Rij,Rij<rij<Sij0,rij>Sij;bij=χij(1+βiniζijni)-1/2ni;ζij=ki,jfc(rik)ωikg(θijk);g(θijk)=1+ci2/di2-ci2/[di2+(hi-cosθijk)2];λij=(λi+λj)/2,μij=(μi+μj)/

Results and discussion

Table 1 shows the determined parameters and these were applied to molecular calculations of O2. The obtained bond energy, 1.38 eV, is in good agreement with the experimental value of 1.43 eV [27], but the bond length, 1.50 Å, is appreciably larger than the experimental value of 1.2 Å [28]. Table 2 shows the structural properties of silica polymorphs calculated by using the Tersoff potential described above. The results are generally in good agreement with both experimental data [29], [30], [25],

Conclusions

We have proposed a new parameter set of Tersoff potential applicable to Si–O systems. In spite of complete neglect of the Coulomb interactions, it can reproduce structural and dynamical properties of SiO2 systems. One of the most important advantages using our proposed potential is the ability to perform large-scale simulations of systems composed of Si, SiO2 and SiOx. Further calculations are required to determine the feasibility of those parameters developed in this study. More extensive

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