  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.
