Phase Boundary of Diluted Hexaferrites Near the Magnetic Percolation Transition

The magnetic phase boundary of diluted hexagonal ferrites, specifically, PbF_12−xGa_xO₁₉ has attracted attention recently because of its unusual shape [Phys. Rev. 96, 020407 (2017)]. The critical temperature as a function of dilution varies as T_c ∝ (1 − x/x_c)^2/3 over the entire concentration range from x = 0 to x_c, where x_c is very close to the percolation threshold of the lattice. To explain this behavior we employ a classical percolation scenario. We perform large-scale Monte Carlo simulations to determine the magnetic phase boundary of diluted XY and Heisenberg models on simple cubic and hexaferrite lattices. Our findings are twofold. (i)In the asymptotic critical region close to x_c, the phase boundary follows T_c ∼ (1 − x/x_c )^φ with φ = 1.09(2) in agreement with the percolation theory. (ii) For smaller dilutions, φ takes smaller values but the simulation does not reproduce the 2/3 power law. We also discuss the effects of additional weak couplings which frustrate the ferrimagnetic order on the shape of the phase boundary.