Spin-Orbit Interaction in Symmetric Wells with Two Subbands

We investigate the spin-orbit (SO) interaction in two-dimensional electron gases in quantum wells with two subbands. From the $8\times 8$ Kane model, we derive a new intersubband-induced SO term which resembles the functional form of the Rashba SO but is nonzero even in symmetric structures. This follows from the distinct parity of the confined states (even or odd) which obliterates the need for asymmetric potentials. We self-consistently calculate the new SO coupling strength for realistic wells and find it comparable to the usual Rashba constant. Our new SO term gives rise to a nonzero ballistic spin-Hall conductivity, which changes sign as a function of the Fermi energy (${\epsilon }_F$) and can induce an unusual Zitterbewegung with cycloidal trajectories without magnetic fields.

Figure 1

Square well with its ground-state ${\phi }_e(z)$ and first excited-state ${\phi }_o(z)$ wave functions. The new intersubband-induced SO coupling $\eta$ in Eq. (4) is nonzero even in symmetric wells due to the distinct parities of ${\phi }_e(z)$ (even) and ${\phi }_o(z)$ (odd), which yield a nonvanishing matrix element for the derivative of the symmetric potential.

Figure 2

Calculated SO coupling strengths as a function of the external gate $V_b$ for realistic wells. (a) For the single GaInAs [16, 20] well studied, the intersubband coupling $\eta$ is larger than the Dresselhaus ${\beta }_i$ and the Rashba ${\alpha }_i$ constants ($i=e,o$). Note that $|{\alpha }_e|\ge |{\alpha }_o|$ and both change sign across $V_b=0$ (in contrast to ${\beta }_i$ and $\eta$). (b) For the InSb double well considered, $\eta$ shows a “resonant behavior” about $V_b=0$ [symmetric configuration, lower-left inset in (b)]. This occurs because the subband splitting ${\epsilon }_o-{\epsilon }_e$ reaches a minimum at $V_b=0$ and the double-well wave functions are very similar (though of distinct parities) for $V_b\sim 0$. This also makes ${\alpha }_e\sim -{\alpha }_o$ around $V_b=0$. Upper-right inset in (b): Energy dispersions ${\epsilon }_{\pm }(\overrightarrow{k})$ [Eq. (10)] of the symmetric double well.

Figure 3

Zitterbewegung due to the SO coupling $\eta$ for distinct ratios $\mu ={ϵ}_{-}/2{\epsilon }_{\mathrm{SO}}$. Note the peculiar trajectories with the forward injected electrons moving backward (I) and even in a closed path (II). This follows from the SO induced change in the curvature of the bands which renormalizes the effective masses. Here we use ${\lambda }_{\mathrm{SO}}{k}_{0y}=\mu /10$, ${\lambda }_{\mathrm{SO}}^{-1}=m^{*}\eta /\hslash$.