  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.
