Perturbation theory of impurity diffusion.

Abstract

A derived generalized Langevin equation for classical impurity diffusion in a lattice is studied in perturbation theory. Temperature dependence of the activation barrier to impurity motion is absent for an interaction Hamiltonian linear in phonon amplitudes, but appears for an interaction Hamiltonian including quadratic or higher orders in phonon amplitudes. The time integral of the memory function, known as the friction constant, separates into one-phonon and multiphonon contributions to dissipation. At low temperatures the friction constant becomes a power series in temperature. The coefficients of this power series are evaluated for a Gaussian impurity-lattice interaction potential. When the potential is short ranged, one-phonon acoustic umklapp processes dominate and the friction constant goes to a constant value as temperature vanishes; otherwise, umklapp processes can be ignored, and the leading term in the power series is the linear temperature coefficient which, in lowest order, is dominated by two-phonon normal quasielastic scattering process. Transverse modes in the two-phonon processes contribute significantly to the latter, but only when the interaction Hamiltonian contains terms of at least quadratic order in the phonon amplitudes. Due to the power series form of the friction constant and the temperature dependence of the activation energy, the diffusion constant obtained using Kramers's formulas assumes a generalized Arrhenius form.