Energy and vibrational spectrum of the Si(111) (7×7) surface from empirical potentials

The stability and vibrational properties of the Si (111) (77) Takayanagi, or dimer-adatom-stacking-fault structure (DAS) surface are investigated using the Stillinger-Weber and modified Tersoff interatomic potentials for silicon. The diamond-cubic bulk and surface-terminated bulk and (3 3) structures are treated for comparison. The modified Tersoff potential underestimates the bulk transverse-acoustic branch frequencies while the Stillinger-Weber does the opposite. Both potentials produce z expansions away from the idealized (constant-bond-length) structures. The Stillinger-Weber potential produces surface energies in the following order: terminated bulk (33) (77) DAS, whereas the modified Tersoff potential reverses this order in accord with local-density-functional calculations and experiment. The two potentials produce surprisingly similar features in the local densities of vibrational states on surface atoms. In the case of adatoms, a high-frequency state split off from the bulk continuum and a mid-frequency state are observed. The high-frequency state has significant weight in the z direction on the adatom, (x, y) weight (only) on the three first-layer atoms to which the adatom is attached, and z weight again on the second- and third-layer atoms directly beneath; the mid-frequency state corresponds to the z motion of the adatom and the two atoms underneath in phase with each other, while out of phase with the three first-layer atoms attached which symmetrically vibrate in the (x, y) plane as well. Atoms with dangling bonds (first-layer atoms not directly attached to adatoms and atoms in the open site at the corners of the unit cell) have small force constants in the z direction and their local spectra only exhibit weight at low frequencies. The z-polarized adatom vibrations agree nicely with recent inelastic electron-energy-loss spectra by Daum, Ibach, and M\"uller.