We present a theoretical study of the effect of reduced dimensionality on the Zeeman splittings of the electron and hole states. It is based on the effective mass approximation for the confined states and the wave vector dependence of the $\mathrm{sp}-d$ exchange interaction parameters in the bulk semimagnetic semiconductor. We find that the ratio of the exciton Zeeman splitting to the magnetization in a nanostructure is reduced with respect to the bulk by a factor $\rho$ that depends on the degree of confinement. The effect is relatively small (a few percent) in quantum wells, but $\rho$ as a function of the well width shows a minimum, in qualitative agreement with experimental data in ${\mathrm{Cd}}_{1-x}{\mathrm{Mn}}_x{\mathrm{T}\mathrm{e}/\mathrm{C}\mathrm{d}}_{1-x-y}{\mathrm{Mn}}_x{\mathrm{Mg}}_y\mathrm{Te}$ heterostructures. In small-size quantum dots of ${\mathrm{Cd}}_{1-x}{\mathrm{Mn}}_x\mathrm{Te}$ the predicted reduction is drastic: $\rho \approx 0.5$ for the dot radius $R=10\mathrm{}\hspace{0.5em}\mathrm{\AA };$ it increases rapidly with R and then saturates to about 0.84. We account for the available magnetoabsorption data in the wurtzite-structure nanocrystals of ${\mathrm{Cd}}_{1-x}{\mathrm{Mn}}_x\mathrm{Se},$ where uniaxial anisotropy also contributes to a reduction of the average Zeeman splitting.

• Received 4 February 1998

DOI:https://doi.org/10.1103/PhysRevB.58.15660