Spin polarization decay in spin-$\frac{1}{2}$ and spin-$\frac{3}{2}$ systems

We present a general unifying theory for spin polarization decay due to the interplay of spin precession and momentum scattering that is applicable to both spin-$\frac{1}{2}$ electrons and spin-$\frac{3}{2}$ holes. Our theory allows us to identify and characterize a wide range of qualitatively different regimes. For strong momentum scattering or slow spin precession, we recover the D’yakonov-Perel result [Sov. Phys. Solid State 13, 3023 (1972)], according to which the spin relaxation time is inversely proportional to the momentum relaxation time. On the other hand, we find that in the ballistic regime, the carrier spin polarization shows a very different qualitative behavior. In systems with isotropic spin splitting, the spin polarization can oscillate indefinitely, while in systems with anisotropic spin splitting, the spin polarization is reduced by spin dephasing, which is nonexponential and may result in an incomplete decay of the spin polarization. For weak momentum scattering or fast spin precession, the oscillations or nonexponential spin dephasing is modulated by an exponential envelope proportional to the momentum relaxation time. Nevertheless, even in this case, in certain systems a fraction of the spin polarization may survive at long times. Finally, it is shown that despite the qualitatively different nature of spin precession in the valence band, spin polarization decay in spin-$\frac{3}{2}$ hole systems has many similarities to its counterpart in spin-$\frac{1}{2}$ electron systems.

Figure 1

Incomplete spin dephasing of (a) electron spins in the $k^3$-Dresselhaus model and (b) heavy-hole spins (solid lines) and light-hole spins (dashed lines) in bulk GaAs in the ballistic limit. The vertical axis shows the normalized spin polarization $\mathit{s}\left(t\right)\bullet {\widehat{\mathit{s}}}_0∕\mid {\mathit{s}}_0\mid$. The initial spin polarization ${\mathit{s}}_0$ is assumed to point along [001]. The Fermi wave vector is $k_F={10}^8{\mathrm{m}}^{-1}$. Note the different time scales in (a) and (b).