Effects of isotopic disorder on the Raman spectra of crystals: Theory and ab initio calculations for diamond and germanium

Isotopic disorder is possibly the simplest phenomenon that breaks translational invariance in a crystal, thus lifting the selection rules due to the conservation of crystal momentum. As a consequence of this symmetry lowering, the position, width, and shape of the Raman line in elemental semiconductors depends on isotopic composition. In this work the effects of isotopic disorder on the Raman spectra of elemental semiconductors are studied 'exactly' using a newly developed theoretical method which does not rely on any kind of mean-field approximation. We first determine the interatomic force constants using density-functional perturbation theory; disorder effects are then simulated using very large super-cells of up to 60 000 atoms; the spectral function which determines the Raman intensity is finally calculated via a reciprocal-space variant of the recursion method. We present theoretical results obtained for diamond and germanium, and compare them with existing theoretical data.