Spin tunneling through an indirect barrier: Tight-binding calculations

Abstract

Spin-dependent tunneling through an indirect-band-gap barrier like the $\mathrm{GaAs}∕\mathrm{AlAs}∕\mathrm{GaAs}$ heterostructure along the $[001]$ direction is studied by the tight-binding method. The tunneling is characterized by the proportionality of the Dresselhaus Hamiltonians at the $\ensuremath{\Gamma}$ and $X$ points in the barrier and by Fano resonances (i.e., pairs of resonances and antiresonances or zeros in transmission). The present results suggest that large spin polarization can be obtained for energy windows that significantly exceed the spin splitting. The widths of these energy windows are mainly determined by the energy difference between the resonance and its associated zero, which in turn increases with the decrease of barrier transmissibility at direct tunneling. We formulate two conditions that are necessary for the existence of energy windows with large polarization. First, the resonances must be well separated such that their corresponding zeros are not pushed away from the real axis by mutual interaction. Second, the relative energy order of the resonances in the two spin channels must be the same as the order of their corresponding zeros. The degree to which the first condition is satisfied is determined by the barrier width and the longitudinal effective mass at the $X$ point. In contrast, the second condition can be satisfied by choosing an appropriate combination of spin splitting strength at the $X$ point and transmissibility through the direct barrier.