Rashba spin splitting in biased semiconductor quantum wells

Rashba spin splitting (RSS) in biased semiconductor quantum wells is investigated theoretically based on eight-band $\mathbf{k}\bullet \mathbf{p}$ theory. We find that at large wave vectors, RSS is both nonmonotonic and anisotropic as a function of in-plane wave vector, in contrast to the widely used isotropic linear model. We derive an analytical expression for RSS, which can qualitatively reproduce such nonmonotonic behavior at large wave vectors. We also investigate numerically the dependence of RSS on the various band parameters and find that RSS increases with decreasing band gap and subband index, increasing valence band offset, external electric field, and well width. All these dependences can be qualitatively described by our analytical model.

Figure 1

RSS of the lowest subband vs $k_x$ for $20\text{−}\mathrm{nm}$-wide (a) ${\mathrm{Ga}}_{0.47}{\mathrm{In}}_{0.53}\mathrm{As}∕{\mathrm{Al}}_{0.48}{\mathrm{In}}_{0.52}\mathrm{As}$ and (b) ${\mathrm{Hg}}_{0.74}{\mathrm{Cd}}_{0.26}\mathrm{Te}∕\mathrm{Cd}\mathrm{Te}$ quantum wells with the electric field $100\mathrm{kV}∕\mathrm{cm}$. Solid lines denote the full numerical solutions to the eight-band Hamiltonian. Dashed [dotted] lines are obtained from Eqs. (2, 4) [Eq. (5)]. The two contributions $\Delta E^{\left(1\right)}$ and $\Delta E^{\left(2\right)}$ are denoted by dash-dotted lines.

Figure 2

RSS of the lowest three subbands (the subband indices are indicated in the figure) vs $k_{\Vert }$ along [100] (solid lines) and [110] (dashed lines) for $20\text{−}\mathrm{nm}$-wide (a) ${\mathrm{Ga}}_{0.47}{\mathrm{In}}_{0.53}\mathrm{As}∕{\mathrm{Al}}_{0.48}{\mathrm{In}}_{0.52}\mathrm{As}$ and (b) ${\mathrm{Hg}}_{0.74}{\mathrm{Cd}}_{0.26}\mathrm{Te}∕\mathrm{Cd}\mathrm{Te}$ quantum wells with fixed electric field $50\mathrm{kV}∕\mathrm{cm}$ and different valence band offsets $0.5\Delta E_g$ and $0.1\Delta E_g$, respectively.

Figure 3

RSS for the lowest subband for a $20\text{−}\mathrm{nm}$-wide ${\mathrm{Hg}}_{0.74}{\mathrm{Cd}}_{0.26}\mathrm{Te}∕\mathrm{Cd}\mathrm{Te}$ quantum well with electric field $50\mathrm{kV}∕\mathrm{cm}$.

Figure 4

Rashba coefficient for the lowest subband of a ${\mathrm{Hg}}_{0.74}{\mathrm{Cd}}_{0.26}\mathrm{Te}∕\mathrm{Cd}\mathrm{Te}$ quantum well as a function of electric field and well width.