Confinement engineering of s-d exchange interactions in GaMnAs quantum wells

Recent measurements of coherent electron spin dynamics have observed antiferromagnetic $s$-$d$ exchange coupling between conduction band electrons and electrons localized on Mn$^{2+}$ impurities in GaMnAs quantum wells \footnote{R. C. Myers, M. Poggio, N. P. Stern, A. C. Gossard, and D. D. Awschalom, \textit {Phys. Rev. Lett.} \textbf{95}, 017204 (2005).}. Here we discuss systematic measurements of the $s$-$d$ exchange interaction in Ga$_{1-x}$Mn$_{x}$As/Al$_{y}$Ga$_{1-y}$As quantum wells with different confinement potentials using time- resolved Kerr rotation spectroscopy. Extending previous investigations of the well width dependence of the $s$-$d$ exchange, $N_{0} \alpha$, we find that the magnitude of the exchange parameter, $N_{0} \alpha$, varies as a function of both well width and well depth ($y$). Both phenomena reduce to a general dependence on confinement energy, which is well-fit to a model taking into account the effect of kinetic exchange and band mixing on the exchange parameters.