Spin trajectory along an evanescent loop in zinc-blende semiconductors

Due to the absence of inversion symmetry, spin-orbit interaction leads to a very particular topology of the evanescent states in zinc-blende semiconductors, which may consist of loops connecting different spin sub-bands at the zone center. The spin-vector motion along such loops is analytically or numerically studied. A surprising picture emerges from this detailed analysis. Namely, the two spin sub-bands do not correspond to opposite spin states near the Brillouin-zone center and merge with identical spins at larger wave vector. This determines the spin-filtering capabilities of the semiconductor barrier.

Figure 1

(Color online) Evanescent loop in the wave vector—energy plane. The energy origin is set at the bottom of the first conduction band. The wave-vector $\mathbf{k}$ lies along the $\left[\text{tan}\theta ,0,i\right]$ direction, at $\eta =\xi /K=0.4$ $\left(\theta =21.8°\right)$. The spin vector, in the $xOy$ plane, is represented, after a numerical calculation, for several pairs of points (identical symbols) related to the two sub-bands at a given wave vector. For small wave vectors—in the D’yakonov-Perel limit (out of the representation domain)—the spin vector is given by $n_{-}$ (respectively, $-n_{+}$) for the lower-(respectively, upper-) energy band. Lower right inset: top view of the intercepts of two evanescent loops with a constant energy plane determining the evanescent components of the wave-vector $K$ and $K^{\prime }$ at constant $\xi$. To the first order, the wave-vector change $\delta K$ can be directly measured on the loop in the main figure. Lower left inset: the unit sphere allows a simple visualization of the spin-direction trajectory along the loop (take care that the sphere is pointing down to the north hemisphere, the axis being along $Ox$, the spin-filter axis). The imaginary (in-plane) component of the wave vector, $K$ $\left(\xi \right)$ lies along $Oz$ $\left(Ox\right)$. The calculated path, with the symbols referring to the points on the loop, connects two symmetrical spots where the D’yakonov-Perel’ sub-bands collapse ($DP$ symbols), located in the north (shaded area) and south hemispheres. The departure location in the north hemisphere lies at the colatitude ${\Theta }_{DP}$.