Spin tunneling through an indirect barrier

Spin-dependent tunneling through an indirect bandgap barrier like the GaAs/AlAs/GaAs heterostructure along [001] direction is studied by the tight-binding method. The tunneling is characterized by the proportionality of the Dresselhaus Hamiltonians at $\Gamma $ and $X$ points in the barrier and by Fano resonances. The present results suggest that large spin polarization can be obtained for energy windows that exceed significantly the spin splitting. We also formulate two conditions that are necessary for the existence of energy windows with large polarization.