Carrier-impurity spin transfer dynamics in paramagnetic II-VI diluted magnetic semiconductors in the presence of a wave-vector-dependent magnetic field

Quantum kinetic equations of motion for carrier and impurity spins in paramagnetic II-VI diluted magnetic semiconductors in a $\mathbf{k}$-dependent effective magnetic field are derived, where the carrier-impurity correlations are taken into account. In the Markov limit, rates for the electron-impurity spin transfer can be derived for electron spins parallel and perpendicular to the impurity spins corresponding to measurable decay rates in Kerr experiments in Faraday and Voigt geometry. Our rigorous microscopic quantum kinetic treatment automatically accounts for the fact that, in an individual spin flip-flop scattering process, a spin flip of an electron is necessarily accompanied by a flop of an impurity spin in the opposite direction and the corresponding change of the impurity Zeeman energy influences the final energy of the electron after the scattering event. This shift in the electron energies after a spin flip-flop scattering process, which usually has been overlooked in the literature, turns out to be especially important in the case of extremely diluted magnetic semiconductors in an external magnetic field. As a specific example for a $\mathbf{k}$-dependent effective magnetic field the effects of a Rashba field on the dynamics of the carrier-impurity correlations in a ${\mathrm{Hg}}_{1-x-y}{\mathrm{Cd}}_y{\mathrm{Mn}}_x\mathrm{Te}$ quantum well are described. It is found that, although accounting for the Rashba interaction in the dynamics of the correlations leads to a modified $\mathbf{k}$-space dynamics, the time evolution of the total carrier spin is not significantly influenced. Furthermore, a connection between the present theory and the description of collective carrier-impurity precession modes is presented.

Figure 1

Magnetic field dependence of the parallel $\left(i=1\right)$ and perpendicular $\left(i=2\right)$ spin transfer rates normalized with respect to $B=0$ in a 8-nm-wide ${\mathrm{Cd}}_{0.983}{\mathrm{Mn}}_{0.017}\mathrm{Te}$ quantum well at temperature $T=4$ K. Red and blue lines (PESC) represent rates according to the theory of the present article [Eqs. (45)] and red and blue crosses show the rates calculated by the projection operator method (proj.) of Ref. [32]. Furthermore, cyan and orange triangles and lines show the results of Eqs. (45), when the energetic shifts due to the Zeeman impurity splittings in spin-flip-flop processes $\left({\omega }_{Mn}=0\right)$ or additionally the spin splittings $\left({\omega }_e={\omega }_{Mn}=0\right)$ are neglected. $T_{\uparrow }$ and $T_{\downarrow }$ are the relaxation rates of spin-up and spin-down occupations, respectively.

Figure 2

Magnetic field dependence of the electron (red solid line) and impurity Zeeman energy (green dashed line) as well as their difference (blue dotted line) for a DMS quantum well (same parameters as for Fig. 1).

Figure 3

Magnetic field dependence of spin transfer rates for a ${\mathrm{Cd}}_{0.9983}{\mathrm{Mn}}_{0.0017}\mathrm{Te}$ quantum well (cf. Fig. 1).

Figure 4

Magnetic field dependence of the Zeeman energies for a ${\mathrm{Cd}}_{0.9983}{\mathrm{Mn}}_{0.0017}\mathrm{Te}$ quantum well (cf. Fig. 2).

Figure 5

Time evolution of the total electron-spin polarization after spin-polarized optical excitation in a magnetic field perpendicular to the quantum well plane (cf. text for parameters). The red solid line describes the results according to Eqs. (10b) and (10c) with the Markovian expression for the correlations from Eq. (34). The green dashed line corresponds to a calculation without Rashba coupling, where only the $s\text{−}d$ interaction is considered. The blue dotted line presents the results of the case in which only the Rashba interaction is present. The mean-field calculation, which is obtained by dropping the correlations completely, is shown as the purple circles. The cyan crosses describe the results where the effects due to the Rashba interaction on the dynamics of the carrier-impurity correlations are neglected, so that in addition to the mean-field terms, the time derivative of the carrier variables obtains the correlation induced contribution of Eqs. (40).

Figure 6

Time evolution of the total electron-spin polarization after spin-polarized optical excitation in a magnetic field parallel to the quantum-well plane (cf. Fig. 5).

Figure 7

Kinetic-energy dependence of the occupation of carrier states at times $t=0$ ps and $t=50$ ps for the calculations shown in Fig. 5.