Nonexponential spin decay in a quantum kinetic description of the D'yakonov-Perel' mechanism mediated by impurity scattering

The electron spin dynamics in an optically excited narrow quantum well is studied, where the electron spins precess in a $\mathbf{k}$-dependent magnetic field, while the electrons scatter at localized impurities. For the resulting spin decay, which is commonly known as the D'yakonov-Perel' mechansim, analytical expressions in the strong- and weak-scattering limits are available. It is found by the numerical solution of quantum kinetic equations in a broad range of parameters that, in situations that are typically relevant for ultrafast optical experiments, the dynamics of the total spin polarization significantly deviates from the pertinent analytical results. This is attributed to the broad spectral width of the optically excited spin-polarized electron distribution, which gives rise to a spin dephasing due to inhomogeneous broadening. Furthermore, it is found that the decay of the spin polarization need not be exponential. The notion of a spin decay time becomes ambiguous and different definitions of spin decay times can lead to different outcomes. The long-term dynamics of the decay of the spin polarization is even dominated by an algebraically decaying component. These findings highlight the importance of the effects of the broad spectral distribution of optically excited carriers in ultrashort magneto-optical experiments.

Figure 1

(a) Time evolution of the optically induced electron spin calculated using the Markovian equation (16) for different impurity concentrations $x$ and different initial electron distributions (Gaussian with central energy $E_c$ above the band edge and standard deviation $E_s$) shown in (b).

Figure 2

(a) Spin decay time ${\tau }_s$ as a function of the momentum scattering rate ${\tau }_p^{-1}$. The spin decay times are determined either by fitting a stretched exponential to the time evolution of the total spin polarization (${\tau }_s^f$) or by extracting the time after which the spin polarization is decayed to $\frac{1}{e}$ of its initial value (${\tau }_s^e$). The black pluses represent the results of calculations using the full quantum kinetic equations, while the remaining results are based on the Markovian equations of motion. The initial electron distributions used for the calculations are Gaussians with standard deviation $E_s=1$ meV ($E_s\to 0$ for the results depicted as purple squares) centered at $E_{\mathrm{c}}=0\mathrm{meV}$ [cf. Fig. 1]. The analytic expressions in the strong- and weak-scattering limits are depicted as lines. For comparison, a linear fit through the last five points of the Markovian calculation of ${\tau }_s^f$ is shown. (b) Precession frequency of the total spin polarization obtained from the fitting procedure.

Figure 3

Spin decay time ${\tau }_s^f$ as a function of the momentum scattering rate ${\tau }_p^{-1}$ for Gaussian initial electron distributions centered at (a) $E_c=3$ and (b) 7 meV above the band edge and with standard deviation $E_s=1$ meV.

Figure 4

Spin dynamics in a quantum well without momentum scattering ($x=0$) subject to Dresselhaus (D), Rashba (R), or both (R+D) fields. The initial electron distribution is chosen to be a Gaussian centered at $E_c=0$ with standard deviation $E_s=1$ meV and $E_s\to 0$, respectively.

Figure 5

Double-logarithmic plot of the long-time behavior of the spin dynamics calculated using the parameters $E_c=0,E_s=1$ meV, ${\alpha }_R=0,{\beta }_D=1$ meV nm, and $x=1.5\%$.

Figure 6

(a) Comparison of the spin dynamics according to the quantum kinetic theory (QKT) and the Markovian approach for $x=10\%, d=4\mathrm{nm},{\beta }_D=7\mathrm{meV}\mathrm{nm}, E_c=0\mathrm{meV}$, and $E_s=0.4\mathrm{meV}$ yielding a momentum scattering rate ${\tau }_p^{-1}=1.12{\mathrm{ps}}^{-1}$. (Inset) The spectral electron distribution for the quantum kinetic and Markovian calculations at $t=0$ and 20 ps. (b) Dynamics of the average correlation energy as defined in Eq. (26) and the total energy per electron in the quantum kinetic theory.