  We use a fully relativistic layer Green's functions approach to investigate
spin-dependent tunneling through a symmetric indirect band gap barrier like
GaAs/AlAs/GaAs heterostructure along [100] direction. The method is based on
Linear Muffin Tin Orbitals and it is within the Density Functional Theory (DFT)
in the Local Density Approximation (LDA). We find that the results of our {\it
ab-initio} calculations are in good agreement with the predictions of our
previous empirical tight binding model [Phys. Rev. {\bf B}, 075313 (2006)]. In
addition we show the $k_{||}$-dependence of the spin polarization which we did
not previously include in the model. The {\it ab-initio} calculations indicate
a strong $k_{||}$-dependence of the transmission and the spin polarization due
to band non-parabolicity. A large window of 25-50 % spin polarization was found
for a barrier of 8 AlAs monolayers at $k_{||}$ = 0.03 $2\pi/a$. Our
calculations show clearly that the appearance of energy windows with
significant spin polarization depends mostly on the location of transmission
resonances and their corresponding zeros and not on the magnitude of the spin
splitting in the barrier.
