Zero-field spin splitting and spin lifetime inn−InSb∕In1−xAlxSbasymmetric quantum well heterostructures

Abstract

The spin-orbit (SO) coupling parameters for the lowest conduction subband due to structural inversion asymmetry (SIA) and bulk inversion asymmetry (BIA) are calculated for a range of carrier densities in [001]-grown $\ensuremath{\delta}$-doped $n$-type $\mathrm{In}\mathrm{Sb}∕{\mathrm{In}}_{1\ensuremath{-}x}{\mathrm{Al}}_{x}\mathrm{Sb}$ quantum wells using the established eight-band $\mathbit{k}\ensuremath{\cdot}\mathbit{p}$ formalism [J. Deng et al., Phys. Rev. B 59, R5312 (1999)]. We present calculations for conditions of zero bias at $10\phantom{\rule{0.3em}{0ex}}\mathrm{K}$. It is shown that both the SIA and BIA parameters scale approximately linearly with carrier density, and exhibit a marked dependence on well width when alloy composition is adjusted to allow maximum upper barrier height for a given well width. In contrast to other material systems, the BIA contribution to spin splitting is found to be of significant and comparable value to the SIA mechanism in these structures. We calculate the spin lifetime ${\ensuremath{\tau}}_{s[1\overline{1}0]}$ for spins oriented along $[1\overline{1}0]$ based on D'yakonov--Perel' mechanism using both the theory of Averkiev et al. [J. Phys.: Condens. Matter 14, R271 (2002)] and also directly the rate of precession of spins about the effective magnetic field, taking into account all three SO couplings, which show good agreement. ${\ensuremath{\tau}}_{s[1\overline{1}0]}$ is largest in the narrowest wells over the range of moderate carrier densities considered, which is attributed to the reduced magnitude of the $k$-cubic BIA parameter in narrow wells. The inherently large BIA induced SO coupling in these systems is shown to have considerable effect on ${\ensuremath{\tau}}_{s[1\overline{1}0]}$, which exhibits significant reduction in the maximum spin lifetime compared to previous studies that consider systems with relatively weak BIA induced SO coupling. The relaxation rate of spins oriented in the [001] direction is found to be dominated by the $k$-linear SIA and BIA coupling parameters and at least an order of magnitude greater than in the $[1\overline{1}0]$ direction.