The exchange interaction between magnetic ions and charge carriers in semiconductors is considered as prime tool for spin control. Here, we solve a long-standing problem by uniquely determining the magnitude of the long-range $p-d$ exchange interaction in a ferromagnet-semiconductor (FM-SC) hybrid structure where a 10~nm thick CdTe quantum well is separated from the FM Co layer by a CdMgTe barrier with a thickness on the order of 10~nm. The exchange interaction is manifested by the spin splitting of acceptor bound holes in the effective magnetic field induced by the FM. The exchange splitting is directly evaluated using spin-flip Raman scattering by analyzing the dependence of the Stokes shift $Delta_S$ on the external magnetic field $B$. We show that in strong magnetic field $Delta_S$ is a linear function of $B$ with an offset of $Delta_{pd} = 50-100~mu$eV at zero field from the FM induced effective exchange field. On the other hand, the $s-d$ exchange interaction between conduction band electrons and FM, as well as the $p-d$ contribution for free valence band holes, are negligible. The results are well described by the model of indirect exchange interaction between acceptor bound holes in the CdTe quantum well and the FM layer mediated by elliptically polarized phonons in the hybrid structure.
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