Zero-field spin splitting and spin lifetime in $n\text{−}\mathrm{In}\mathrm{Sb}∕{\mathrm{In}}_{1-x}{\mathrm{Al}}_x\mathrm{Sb}$ asymmetric quantum well heterostructures

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 $\delta$-doped $n$-type $\mathrm{In}\mathrm{Sb}∕{\mathrm{In}}_{1-x}{\mathrm{Al}}_x\mathrm{Sb}$ quantum wells using the established eight-band $\mathit{k}\cdot \mathit{p}$ formalism [J. Deng et al., Phys. Rev. B 59, R5312 (1999)]. We present calculations for conditions of zero bias at $10\mathrm{K}$. 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 ${\tau }_{s\left[1\overline{1}0\right]}$ for spins oriented along $\left[1\overline{1}0\right]$ 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. ${\tau }_{s\left[1\overline{1}0\right]}$ 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 ${\tau }_{s\left[1\overline{1}0\right]}$, 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 $\left[1\overline{1}0\right]$ direction.

Figure 1

Conduction band profiles $V\left(z\right)$ of a typical $\delta$-doped ${\mathrm{In}}_{0.8}{\mathrm{Al}}_{0.2}\mathrm{Sb}∕\mathrm{In}\mathrm{Sb}∕{\mathrm{In}}_{0.85}{\mathrm{Al}}_{0.15}\mathrm{Sb}$ $20\mathrm{nm}$ QW with spacer thickness $S=20\mathrm{nm}$ (left axis), and corresponding normalized wave function $\Psi \left(z\right)$ of the first subband (right axis) at $10\mathrm{K}$. The hatched region represents a schematic of the doping profile used. The doping density $N_d$ and resulting carrier density $n_s$ for this solution are given. The axis is normalized to $V_{\mathit{QW}}\left(0\right)=0$.

Figure 2

Calculated results for the SIA SO coupling parameter $\alpha$ as a function of $n_s$ for (a) the various well widths considered (open symbols) along with the $k$-linear BIA term for comparison (closed symbols), and (b) for a $20\mathrm{nm}$ QW showing the contributions from the field and interface terms. The inset of (a) shows the two components of the field term $F$, which result in the observed dependence on $n_s$.

Figure 3

(a) Calculated results for the BIA SO coupling parameters $\beta$ and $\gamma$ as a function of QW width $W$ for fixed Fermi energy $E_F=40\mathrm{meV}$, which show $\beta$ calculated from $\gamma ⟨k_z^2⟩$ (dashed line) and ${\gamma }_{\mathit{\text{bulk}}}{(\pi ∕W)}^2$ (dot-dashed line) for comparison. (b) Calculated parameters $\alpha$ and $-\beta$ as a function of barrier asymmetry for a $20\mathrm{nm}$ QW with fixed carrier density $n_s=2.6\times {10}^{11}{\mathrm{cm}}^{-2}$; the solid lines are a guide to the eyes.

Figure 4

(a) Calculated spin-orbit splitting at the Fermi energy as a function of $k_{\parallel }=(k_x,k_y)$ for three $15\mathrm{nm}$ QW heterostructures with $k_F=0.81\times {10}^8{\mathrm{m}}^{-1}$ (dotted line), $k_F=1.26\times {10}^8{\mathrm{m}}^{-1}$ (solid line), and $k_F=1.81\times {10}^8{\mathrm{m}}^{-1}$ (dashed line), which show strong anisotropy in the [110] and $\left[1\overline{1}0\right]$ directions due to the presence of both SIA and BIA. $2\beta ∕\gamma k_F^2>1$ for the data shown. (b) Fraction of the total averaged spin splitting contributed from BIA as a function of carrier density for the different well widths. The dashed line indicates the point when both SIA and BIA equally contribute to $⟨\Delta E_{\mathit{tot}}⟩\left(k_F\right)$.

Figure 5

Calculated spin lifetimes for the spin component directed along $\left[1\overline{1}0\right]$ as a function of the ratio of BIA and SIA $k$-linear parameters for a $15\mathrm{nm}$ QW (dashed line) and $30\mathrm{nm}$ QW (dotted-dashed line) with $k_F=1.26\times {10}^8{\mathrm{m}}^{-1}$. Data for a $15\mathrm{nm}$ QW structure with $k_F=0.8\times {10}^8{\mathrm{m}}^{-1}$ (crossed and dashed line) and the result of taking $\gamma =0$ (solid line) are plotted for comparison. Data are presented on a ${\log }_{10}$ scale for ease of comparison.

Figure 6

(a) Results for the calculation of ${\tau }_{s\left[1\overline{1}0\right]}\mu$ as a function of $n_s$ for all $\mathrm{In}\mathrm{Sb}∕{\mathrm{In}}_{1-x}{\mathrm{Al}}_x\mathrm{Sb}$ structures. Data are presented on a ${\log }_{10}$ scale. The solid lines are guide to the eyes. (b) Spin relaxation rate ${\left({\tau }_{s\left[1\overline{1}0\right]}\mu \right)}^{-1}$ as a function of $n_s$ for the $15\mathrm{nm}$ QW data showing the separate contributions from terms proportional to $n_s$, $n_s^2$, and $n_s^3$ in Eq. (13). The inset shows the contributions for the $25\mathrm{nm}$ QW data for comparison where a turnover is observed at lower $n_s$ (axes are in the same units).

Figure 7

Squared effective magnetic field perpendicular to spins oriented in the $\left[1\overline{1}0\right]$ direction averaged over the Fermi circle $⟨{\Omega }_{\left[110\right]}^2⟩\left({\mathit{k}}_F\right)$ as a function of $n_s$ for all structures of varying well widths. Data are presented on a ${\log }_{10}$ scale for comparison to Fig. 6a. The solid lines are a guide to the eyes.