Theory of the spin EPR shift and application to Pb$_{1-x}$Mn$_{x}$Te

We consider a system with a periodic potential, spin-orbit interaction, conduction electron-local moment interaction and an applied magnetic field. We derive a theory for the spin-contribution to the electron-paramagnetic resonance shift (P$_{s})$ by considering an effective equation of motion of the Green's function in a representation defined by the periodic part of the Bloch function. The spin-EPR shift is expressed as a function of the matrix elements of the momentum, Pauli spin-operators, and conduction electron-local moment interactions. We apply the theory to calculate P$_{s}$ at Mn$^{2+}$ ion in the diluted magnetic semiconductor Pb$_{1-x}$Mn$_{x}$Te, as a function of the carrier concentration. Contributions from band-edge interactions as well as from far bands are included and their relative strengths are analyzed. P$_{s}$ is found to be anisotropic arising mainly due to spin-orbit interactions. Our results of P$_{s}$ for two typical hole densities agree fairly well with the recent experimental results for p-Pb$_{1-x}$Mn$_{x }$Te.