Magnetic polarons and electronic structure in semimagnetic superlattices: Application to CdTe/${\mathrm{Cd}}_{1\mathrm{-}\mathrm{x}}$${\mathrm{Mn}}_{\mathrm{x}}$Te

We report a theoretical study of the magnetic polaron in semimagnetic superlattices, based on the model previously developed for isolated quantum wells, which reduces the master equation to an effective one-dimensional nonlinear Schrödinger equation. This equation has been solved rigorously by a numerical procedure and explicit computations are reported for the CdTe/${\mathrm{Cd}}_{1\mathrm{-}\mathrm{x}}$ ${\mathrm{Mn}}_{\mathrm{x}}$Te system. For superlattices of infinite size, we find two new localized magnetic-polaron states. One is centered in the middle of the semimagnetic layers and is always the ground state. The other one is centered in the middle of the layers made of the linear (nonmagnetic) medium. Both bound-magnetic-polaron states have binding energies which are decreasing functions of the temperature T and magnetic field B. The case of superlattices of finite size has also been investigated. The most drastic size effects are predicted for a superlattice made of a few bilayers bounded by semi-infinite layers of the nonmagnetic medium. Here the bound magnetic polaron is stable only at high magnetic fields, and the binding energy for the ground state decreases with B, but increases with T. A first-order transition between localized and delocalized states is predicted for the quasiparticle ground state. In the region of the phase diagram where the Schrödinger equation has no bound state, we find the existence of virtual bound-magnetic-polaron states. The lifetime of such resonances, which is finite owing to a tunneling effect through the superlattice, is also determined. The conditions for observation of those magnetic-polaron effects specific to the superlattice geometry are discussed, together with the related magnetooptical properties.