Semiclassical kinetic theory of electron spin relaxation in semiconductors

We develop a semiclassical kinetic theory for electron spin relaxation in semiconductors. Our approach accounts for elastic as well as inelastic scattering and treats Elliott-Yafet and motional-narrowing processes, such as D’yakonov-Perel’ and variable $g$-factor processes, on an equal footing. Focusing on small spin polarizations and small momentum transfer scattering, we derive, starting from the full quantum kinetic equations, a Fokker-Planck equation for the electron spin polarization. We then construct, using a rigorous multiple time scale approach, a Bloch equation for the macroscopic ($\vec{k}$-averaged) spin polarization on the long time scale, where the spin polarization decays. Spin-conserving energy relaxation and diffusion, which occur on a fast time scale, after the initial spin polarization has been injected, are incorporated and shown to give rise to a weight function that defines the energy averages required for the calculation of the spin relaxation tensor in the Bloch equation. Our approach provides an intuitive way to conceptualize the dynamics of the spin polarization in terms of a “test” spin polarization that scatters off “field” particles (electrons, impurities, phonons). To illustrate our approach, we calculate for a quantum well the spin lifetime at temperatures and densities where electron-electron and electron-impurity scattering dominate. The spin lifetimes are nonmonotonic functions of temperature and density. Our results show that at electron densities and temperatures where the crossover from the nondegenerate to the degenerate regime occurs, spin lifetimes are particularly long.

Figure 1

Diagrammatic representation of self-energies in the Born approximation for electron-electron (a)–(d), electron-impurity (e), and electron-phonon scattering (f). Diagram (g) denotes the self-energy due to spin off-diagonal Hamiltonian matrix elements.

Figure 2

Graphical illustration of the collision terms within the diffusion approximation: A “test” spin polarization scatters off a generalized bath of equilibrated “field” particles (electrons, impurities, and/or phonons). Whereas spin-flip scattering is elastic (on-shell) within our approximation, spin-conserving scattering can be on- or off-shell, i.e. the “test” spin polarization can lose or gain energy, because the “internal degrees” of the bath can absorb or emit energy.

Figure 3

Distribution function (for the equilibrated state at $t\to \infty$), dynamical friction coefficient, dynamical diffusion coefficient, and angle randomization coefficient (from top to bottom) for a modulation doped quantum well at an electron density $n=4\times {10}^9{\mathrm{cm}}^{-2}$ and $T=10\mathrm{K}$. For the calculation of the coefficients, only electron-electron scattering is taken into account. The unit of $\epsilon$ is $14.93\mathrm{meV}$. The units of the functions $v$, $w$, and $u$, are $22.68\mathrm{meV}∕\mathrm{ps}$, $338.44{\left(\mathrm{meV}\right)}^2∕\mathrm{ps}$, and $1.521∕\mathrm{ps}$, respectively.

Figure 4

Distribution function (for the equilibrated state at $t\to \infty$), dynamical friction coefficient, dynamical diffusion coefficient, and angle randomization coefficient (from top to bottom) for a modulation-doped quantum well at an electron density $n=4\times {10}^{11}{\mathrm{cm}}^{-2}$ and $T=10\mathrm{K}$. As in Fig. 3, for the calculation of the coefficients, only electron-electron scattering is considered. The units of $\epsilon$, $v$, $w$, and $u$ are the same as in Fig. 3.

Figure 5

Weight function for a modulation-doped quantum well at $T=10\mathrm{K}$ and three electron densities $n=4\times {10}^9{\mathrm{cm}}^{-2}$ (solid line), $n=4\times {10}^{11}{\mathrm{cm}}^{-2}$ (dotted line), $n={10}^{12}{\mathrm{cm}}^{-2}$ (dashed line), taking only electron-electron scattering into account. The inset shows the corresponding electron distribution functions in the equilibrated state at $t\to \infty$.

Figure 6

The upper panel shows $P(k_{\mathrm{typ}};q)$ for $T=10\mathrm{K}$ and two electron densities: $n=4\times {10}^9{\mathrm{cm}}^{-2}$ (nondegenerate electrons) and $n=4\times {10}^{11}{\mathrm{cm}}^{-2}$ (degenerate electrons). For nondegenerate electrons, the typical momentum $k_{\mathrm{typ}}$ corresponds to the thermal energy (vertical solid line), whereas for degenerate electrons, $k_{\mathrm{typ}}$ is the Fermi wave number (vertical dashed line). In the lower panel we present the cumulant $C(k_{\mathrm{typ}};q)$, defined in Eq. (109), which is a measure of the relative importance of the scattering processes with a momentum transfer less than $q$.

Figure 7

D’yakonov-Perel’ spin lifetime due to electron-electron scattering for a $25\text{−}\mathrm{nm}$ modulation-doped quantum well as a function of electron density at three temperatures $T=10\mathrm{K}$, $T=20\mathrm{K}$, and $T=30\mathrm{K}$. The solid dots are experimental data at $T=10\mathrm{K}$ from Ref. 30.

Figure 8

D’yakonov-Perel’ spin lifetime for a $25\text{−}\mathrm{nm}$ quantum well at $T=40\mathrm{K}$ and three values of ionized (donor and acceptor) impurity concentrations: no impurities $(x=0)$, ionized impurity concentration equal to the electron density $(x=1)$, and ionized impurity concentration equal to four times the electron density $(x=4)$. For $x=0$, only electron-electron scattering contributes to the spin lifetime, whereas for $x\ne 0$ both electron-electron and electron-impurity scattering determine the spin lifetime.

Figure 9

D’yakonov-Perel’ spin lifetime for a $25\text{−}\mathrm{nm}$ quantum well as a function of temperature at an electron density $n=3\times {10}^{10}{\mathrm{cm}}^{-2}$ and three values of ionized (donor and acceptor) impurity concentrations: no impurities $(x=0)$, ionized impurity concentration equal to the electron density $(x=1)$, and ionized impurity concentration equal to four times the electron density $(x=4)$. As in Fig. 8, for $x=0$, only electron-electron scattering contributes to the spin lifetime, whereas for $x\ne 0$ both electron-electron and electron-impurity scattering determine the spin lifetime.

Figure 10

D’yakonov-Perel’ spin lifetime for a $10\text{−}\mathrm{nm}$ modulation-doped GaAs quantum well as a function of temperature at an electron density $n=1.86\times {10}^{11}{\mathrm{cm}}^{-2}$ (solid line). The solid boxes are experimental data from Ref. 47.