Theory of bond-length variations in relaxed, strained, and amorphous silicon-germanium alloys

We present a theoretical study of bond-length and angle variations in relaxed, epitaxially strained, and amorphous Si1-xGex alloys. Our approach is based on Monte Carlo simulations, within the semigrand-canonical ensemble utilizing Ising-like identity flips, and in conjuction with energies calculated using the empirical potential of Tersoff [Phys. Rev. B 39, 5566 (1989)]. The method offers great statistical precision enabling us to extract clear variations through the whole composition range and for all types of bonds. Our simulations show that in relaxed crystalline alloys, where the lattice constant takes its natural value, bond lengths depend on composition x and that these variations are type specific, in agreement with recent experimental studies. Similar type-specific variations are found for the angles and the second-nearest-neighbor distances. This analysis also reveals that the negative deviation of the lattice constant from Vegard's law is mainly due to radial, and not angular, relaxations. In the epitaxially strained alloys, bond lengths decrease with x due to the two-dimensional confinement in the growth layers, in good agreement with predictions based on the macroscopic theory of elasticity. The dimer bond lengths at the (100)-(2x1)-reconstructed alloy surface remain nearly constant, and they are elongated with respect to the bulk values. In the amorphous alloys, we unravel a remarkable behavior of bond lengths at the dilute low-x alloy limit, characterized by strong relaxations and elongation. Furthermore, the bond lengths decrease with increasing Ge content. We offer an explanation of this effect based on the analysis of the enthalpy of formation of the amorphous alloy.