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The spin polarization of Sb overlayers on the semi-Heusler alloy NiMnSb is investigated in terms of the Landau-Ginzburg approach. The half-metallic semi-Heusler alloy NiMnSb acts as a ferromagnetic perturbation and induces a spin polarization in the semimetallic Sb overlayer. Using a Gaussian approximation, the propagation of the spin perturbation in the overlayer is calculated. The results are compared with spin-polarized inverse photoemission spectroscopy (SPIPES) results and with recent spin-dependent envelope-function approximation (SDEFA) predictions. The Landau-Ginzburg parameters are both band-structure and temperature dependent, and it is argued that thermal spin excitations lead to an injection depth decreasing as 1//T law at high temperatures.