Exciton magnetic polaron in semimagnetic semiconductor nanocrystals

Abstract

We present a theoretical study of the magnetic polaron associated with an electron-hole pair in a diluted magnetic semiconductor quantum dot. It is based on the effective-mass approximation in the strong confinement regime, which incorporates the coupling between the light- and heavy-hole bands. The magnetic polaron, arising from the sp-d exchange interaction between the confined carriers and the magnetic ions, is treated in a self-consistent mean-field approach that leads to coupled nonlinear Schr\"odinger equations for the electron and the hole. The local response to the effective field is modeled by the experimental high-field magnetization curve in the bulk. The electron-hole Coulomb interaction is taken into account. An exact numerical solution of the three coupled equations is used to calculate the equilibrium polaron size, binding energy (${\mathrm{E}}_{\mathrm{p}}$), and spin (${\mathrm{S}}_{\mathrm{p}}$). Results are first presented for ${\mathrm{Cd}}_{1\mathrm{\ensuremath{-}}\mathrm{x}}$${\mathrm{Mn}}_{\mathrm{x}}$Te nanocrystals with x=0.11. ${\mathrm{E}}_{\mathrm{p}}$ decreases and the orbital contraction increases with an increasing quantum dot radius (a). In small dots, approaching saturation in the core region, ${\mathrm{E}}_{\mathrm{p}}$ decreases slowly as the temperature (T) increases. In large dots ${\mathrm{E}}_{\mathrm{p}}$(T) decreases rapidly towards the fluctuation regime, where ${\mathrm{E}}_{\mathrm{p}}$\ensuremath{\propto}${\mathrm{a}}^{\mathrm{\ensuremath{-}}3}$. A similar temperature dependence is obtained for ${\mathrm{S}}_{\mathrm{p}}$; the fluctuation-regime value is, however, size independent. The light-induced magnetization enhancement due to polaron formation is considered and an optimal quantum dot radius is predicted to be \ensuremath{\sim}30 \AA{}. We have also calculated ${\mathrm{E}}_{\mathrm{p}}$ as a function of an applied magnetic field, which shows a decreasing behavior that depends on a and T. Theoretical results for ${\mathrm{Cd}}_{1\mathrm{\ensuremath{-}}\mathrm{x}}$${\mathrm{Mn}}_{\mathrm{x}}$Se nanocrystals show a good agreement with recently reported experimental data on the photoluminescence Stokes shift versus magnetic field.