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

We present a method to study the effects of isotopic composition on the Raman spectra of crystals, in which disorder is treated exactly without resorting to any kind of mean-field approximation. The Raman cross section is expressed in terms of a suitable diagonal element of the vibrational Green’s function, which is accurately and efficiently calculated using the recursion technique. This method can be used in conjunction with both semiempirical lattice-dynamical models and with first-principles interatomic force constants. We have applied our technique to diamond and germanium using the most accurate interatomic force constants presently available, obtained from density-functional perturbation theory. Our method correctly reproduces the light scattering in diamond—where isotopic effects dominates over the anharmonic ones—as well as in germanium, where anharmonic effects are larger.