  We demonstrate first measurements of successful spin generation in
crystalline Co$_2$FeSi/MgO/GaAs hybrid structures grown by molecular-beam
epitaxy (MBE), with different MgO interlayer thicknesses. Using non-local spin
valve and non-local Hanle measurement configurations, we determine spin
lifetimes of ${\tau \approx 100}$~ns and spin diffusion lengths of ${\lambda
\approx 5.6}$~$\mu$m for different MgO layer thicknesses proving the high
quality of the GaAs transport channel. For an optimized MgO layer thickness,
the bias dependence of the spin valve signals indicates the verification of the
half-metallic gap (upper edge) of Co$_2$FeSi in accordance with first principle
calculations. In addition to that, spin generation efficiencies up to 18$\%$
reveal the high potential of MgO interlayers at the Co$_2$FeSi/GaAs interface
for further device applications.
