Non-Markovian spin transfer dynamics in magnetic semiconductors despite short memory times

A quantum kinetic theory of the spin transfer between carriers and Mn atoms in a Mn-doped diluted magnetic semiconductor is presented. It turns out that the typical memory time associated with these processes is orders of magnitude shorter than the time scale of the spin transfer. Nevertheless, Markovian rate equations, which are obtained by neglecting the memory, work well only for bulk systems. For quantum wells and wires the quantum kinetic results qualitatively deviate from the Markovian limit under certain conditions. Instead of a monotonic decay of an initially prepared excess electron spin, an overshoot or even coherent oscillations are found. It is demonstrated that these features are caused by energetic redistributions of the carriers due to the energy-time uncertainty.

Figure 1

Time evolution of the total electron spin ${\Sigma }_e^z$ in Zn${}_{0.93}$Mn${}_{0.07}$Se for (a) a bulk system; (b) a quantum well; (c) a quantum wire. The solid red lines have been calculated with the full quantum kinetic Eqs. (4), the blue lines with bullets have been obtained by neglecting $Q_{\overline{K}}$ and $\overline{C}$ in Eqs. (4), and the green lines with squares represent the results of the Markovian rate Eqs. (14). The pink curve with triangles in (b) has been obtained from Eqs. (19), where the energetic redistribution of electrons is neglected.

Figure 2

Spin memory function $G_k^2\left(\tau \right)$ of a ZnMnSe quantum well for different values of $k$.

Figure 3

Time evolution of the total electron spin ${\Sigma }_e^z$ for different values of the rescaled electron Mn coupling $F$ [cf. Eq. (17)] at fixed mass ratio $m_0/m_e$. $F_0$ corresponds to the parameters used in Fig. 1b.

Figure 4

Time evolution of the electron distribution in energy space.

Figure 5

Time evolution of the electron-spin distribution in energy space. The electron spin is given in multiples of $\frac{\hslash }{2}$.

Figure 6

Distribution of (a) the electron occupation and (b) the electron spin as a function of the kinetic energy for different times $t$. The insets show magnified sections of the main figures.

Figure 7

(a) Time evolution of the total electron spin ${\Sigma }_e^z$ for different central energies $E_0$ of the initial Gaussian distribution; (b) distribution of the electron occupation as a function of the kinetic energy for the central energy $E_0=3$ meV and different times $t$; (c) the corresponding distribution of the electron spin.