Suppression of carrier-induced ferromagnetism by composition and spin fluctuations in diluted magnetic semiconductors

We suggest an approach to account for spatial (composition) and thermal fluctuations in “disordered” magnetic models with given spatial dependences of the magnetic spin-spin interaction. Our approach is based on the introduction of a fluctuating molecular field (rather than mean field) acting between the spins. The distribution function of the above field is derived self-consistently. Our approach permits one to derive the equation for a critical temperature $T_c$ of a ferromagnetic phase transition with respect to the above fluctuations. On the base of the latter equation we analyze the influence of the fluctuations on $T_c$ in diluted magnetic semiconductors with RKKY spin-spin interaction. Our calculations predict the deviation of $T_c$ from its mean-field value $T_c^{\mathrm{MF}}$ by the factor $(1\mathrm{}-\nu /{\nu }_c{)}^{\lambda },$ where $\nu {=n}_e{/n}_i$ is a ratio of carriers ${(n}_e)$ and magnetic ion ${(n}_i)$ concentrations; ${\nu }_c$ is a critical ratio such that at $\nu >{\nu }_c$ any ferromagnetic ordering is impossible. In the case of Mn ions with spins $S=5/2,$ we find ${\nu }_c\simeq 0.264$ and $\lambda \simeq 0.45,$ that constitute an optimal ratio $\nu \simeq 0.11$ to obtain maximal $T_c$ as a function of carrier concentration.