RKKY interaction in Mn-doped 4 times 4 Luttinger systems

We consider Mn-doped bulk zinc-blende semiconductors described by the 4 $ \times$ 4 Luttinger Hamiltonian. In these semiconductors, Mn atom acts as an acceptor providing the system a mobile hole, and also acts like a magnetic impurity of spin $S=5/2$. We obtain exact analytical expressions of the hole mediated Ruderman-Kittel-Kasuya-Yoshida (RKKY) exchange interaction between two Mn${}^{2+}$ ions. The RKKY interaction of the Luttinger system consists of collinear Heisenberg-like and Ising-like interactions. The characteristic beating patterns appear in the range functions of the RKKY interaction owing to the presence of multiple Fermi wave-vectors of the underlying $j=3/2$ states. As an application of the analytical form of the range function, from the finite temperature evaluation of the correlation functions, we calculate the contribution of RKKY interaction to the Curie-Weiss temperatures of a particular dilute magnetic semiconductor ZnMnTe where $4\times4$ Luttinger Hamiltonian is valid.