Multiband tight--binding approach to tunneling in semiconductor heterostructures: Application to $\Gamma X$ transfer in GaAs

We study tunneling in semiconductor heterostructures where the constituent materials can have a direct or indirect bandgap. In order to have a good description of the lowest conduction band, we have used the nearest-- neighbour $sp^3s^*$ tight--binding model put forward by P. Vogl {\em et al.}. A recursive Green--function method yields transmission coefficients from which an expression for the current density may be written down. The method is applied to GaAs/AlAs heterostructures. Electrons may traverse the AlAs barriers via different tunneling states $\psi_{\Gamma}$ and $\psi_X$ ($\Gamma X$ mixing). With an applied bias $V>0.5$ V electrons may enter the GaAs collector contact in both the $\Gamma$ and the $X$ valley ($\Gamma X$ transfer). We have studied a number of GaAs/AlAs structures. For very narrow barriers there is little $\Gamma X$ transfer, but AlAs barriers wider than about 25 \AA act as ``$\Gamma X$ filters'', i.e., most transmitted electrons have been transfered to the $X$ valley.