Mesoscopic model of a system possessing both relaxor ferroelectric and relaxor ferromagnetic properties

A pseudospin model of a multiferroic system which exhibits both relaxor ferroelectric and relaxor ferromagnetic behavior is presented. The electric and magnetic degrees of freedom associated with the simultaneous presence of polar nanoregions and magnetic nanoregions are described by two sets of pseudospin variables, which satisfy separate spherical conditions. The spin-glass-like random interactions within each subset are assumed to be infinitely ranged. In addition, the polar nanoregions are subject to random electric fields. By introducing strain modulation of the corresponding random interaction parameters, a fourth-order interaction between polar and magnetic degrees of freedom is derived whose strength can be estimated from the phenomenological electrostriction and magnetostriction coefficients. Dynamic dielectric susceptibility in the presence of a static magnetic field H is calculated from the Langevin equations of motion. The value of the critical magnetic field at which long-range ferroelectric order appears is determined. By considering the corresponding free-energy density functional, the local electric field inside the polar nanoregions is derived and it is shown that the mechanism of growth and percolation of polar nanoregions is also affected by the magnetic field. Thus the Vogel-Fulcher relaxation time is predicted to diverge on a line of percolation critical points in the H,T plane, in agreement with recent experiments.