Comparison between a quantum kinetic theory of spin transfer dynamics in Mn-doped bulk semiconductors and its Markov limit for nonzero Mn magnetization

We investigate the transfer between carrier and Mn spins due to the $s$-$d$-exchange interaction in a Mn-doped bulk semiconductor within a microscopic quantum kinetic theory. We demonstrate that the spin transfer dynamics is qualitatively different for components of the carrier spin parallel and perpendicular to the Mn magnetization. From our quantum kinetic equations we have worked out the corresponding Markov limit, which is equivalent to rate equations based on Fermi's golden rule. The resulting equations resemble the widely used Landau-Lifshitz-Gilbert equations, but also describe genuine spin transfer due to quantum corrections. Although it is known that the Markovian rate description works well for bulk systems when the initial Mn magnetization is zero, we find large qualitative deviations from the full quantum kinetic theory for finite initial Mn magnetizations. These deviations mainly reflect corrections of higher than leading order in the interaction, which are not accounted for in golden-rule-type rates.

Figure 1

Time evolution of the total electron spin polarization (a) and its components parallel (b) and perpendicular (c) to the Mn spin assuming the electrons to be initially spin polarized along a direction at an angle of ${45}^{\circ }$ relative to the Mn magnetization. The solid red line describes the spin dynamics according to the full quantum kinetic theory, the dashed green line shows its Markov limit (analytic solutions, cf. Appendix). Blue circles and purple squares correspond to approximate quantum kinetic calculations where only a subset of source terms for the correlations (as indicated in the key of the figure) has been accounted for.

Figure 2

Dynamics of the electron spin polarization for initially unpolarized electron spins. Line styles and symbols have the same meaning as in Fig. 1.