Interaction between band electrons and transition-metal ions in diluted magnetic semiconductors

We present a theoretical study of the effective exchange interaction arising from the hybridization between the valence-band states and the localized d orbitals of a transition-metal impurity in a II-VI semiconductor. The irreducible tensor method is used to deduce the effective Hamiltonian in the manifold of the ground multiplet of a 3${\mathit{d}}^{\mathit{n}}$ ion in a tetrahedral crystal field. There is no coupling in the cases of ${\mathrm{Sc}}^{2+}$ and ${\mathrm{Ti}}^{2+}$ ions. For ${\mathrm{Mn}}^{2+}$, ${\mathrm{Fe}}^{2+}$, and ${\mathrm{Co}}^{2+}$ the coupling reduces to the usual spin-exchange Kondo Hamiltonian, in agreement with experiments; the observed increase of the exchange parameter ‖${\mathit{N}}_0$β‖ from Mn to Fe to Co in a given host is also explained. In the cases of ${\mathrm{V}}^{2+}$, ${\mathrm{Cr}}^{2+}$, ${\mathrm{Ni}}^{2+}$, and ${\mathrm{Cu}}^{2+}$ additional coupling terms involving the orbital degrees of freedom are obtained; these predict drastic modifications of the valence-band splitting in a magnetic field.