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.
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