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Theory of Half-Metallic Double Perovskites II: Effective Spin Hamiltonian and Disorder Effects

Double perovskites like Sr$_2$FeMoO$_6$ are materials with half-metallic ground states and ferrimagnetic T$_{\rm{c}}$'s well above room temperature. This paper is the second of our comprehensive theory for half metallic double perovskites. Here we derive an effective Hamiltonian for the Fe core spins by "integrating out" the itinerant Mo electrons and obtain an unusual double square-root form of the spin-spin interaction. We validate the classical spin Hamiltonian by comparing its results with those of the full quantum treatment presented in the companion paper "Theory of Half-Metallic Double Perovskites I: Double Exchange Mechanism". We then use the effective Hamiltonian to compute magnetic properties as a function of temperature and disorder and discuss the effect of excess Mo, excess Fe, and anti-site disorder on the magnetization and T$_{\rm{c}}$. We conclude with a proposal to increase T$_{\rm{c}}$ without sacrificing carrier polarization.