TIGHT-BINDING MOLECULAR-DYNAMICS STUDIES OF SILICON SURFACES

Abstract

In this work, we have successfully coded a program to stimulate tight-binding molecular-dynamics (MD) for silicon. Our work is based on that by Khan and Broughton with the total-energy proposed by Tomanek and Schluter, modified with smooth cut-off functions for MD simulation. This total-energy of the electron-ion system is more complete than that by Chadi and hence more suitable for MD. A band-structure energy, EBS which is the one-electron energy calculated within the tight-binding approximation. A repulsive energy E2 which is a semi-empirical correction to the double counting of electron-electron interactions in EBS, it includes the exchange correlation and ion-ion repulsion energies. E3 corrects for energy changes that arise from the different number of bonds in different geometries of the same cluster size. It ensures that the system will not always favor maximum coordination and metallic structures. E4 is similar to the atomic charge neutrality scheme, it corrects the unrealistic large charge, transfer between atoms intrinsic within the tight-binding model. For simulation of surface, "hydrogen" atoms are included to mimic bulk silicon. For MD simulation, the total energy of the system must be conserved, hence the electronic structure problem must be solved with great accuracy. Furthermore, instead of a set of simultaneous equations, we have to solve a set of non-linear simultaneos equations because of E4. We use Broyden's mixing scheme for this iterative self-consistent diagonalization. This is known as quenching the electrons to the Borm-Oppenheimer surface; i.e., forcing the electrons to the ground state for that configuration. We have also implemented Car and Parrinello's "fictitious Lagrange" procedure where the electronic degrees of freedom (i.e., coefficients of the wave-functions) are treated as "position" variables of classical particles with fictitious mass and the trajectories of the ionic and electronic coordinates are predicted via MD with the forces on the electrons being calculated from the "fictitious Lagrange". The outline for the simulation is as follows. For a given positions of the ions, the electrons are quenched to the Born-Oppenheimer surface. During the MD step, the positions of the ionic and electronic coordinates are updated simultaneously and is repeated for NBorn times. The process repeats by quenching the electrons. We implement a constraint MD technique known as SHAKE to ensure that the electronic trajectories will not drift from its orthonormality constraints during simulation. We employ the scheme to progressively complex problems. Predicted optimal geometry of several Si clusters are similar to ab initio results. Hence this scheme is reliable in an area where most classical potentials have failed. Next we determine that the relaxed surface of Si(001) consists of asymmetric dimers. Relaxed Si(001) c(4 x 2) and p(2 x 2) surfaces are also obtained and their surface states calculated. Occupied states are quite good but the unoccupied states in the conduction band are not as good, a shortfall of the tight-binding model. Finally, vacancy in bulk and on Si(001) c( 4 x 2) surface are studied; monovacancy on this surface is unstable and degenerates into a divacancy. By relaxing several saddle-point configurations we have also determine an anisotropy in the diffusion of vacancy on this surface.