This paper introduces a phase-field crystal (PFC) approach that couples the atomic-scale PFC density field to order parameters describing ferromagnetic and ferroelectric ordering, as well to a solute impurity field. This model extends the magnetic PFC model introduced by Faghihi et al. [N. Faghihi, Ph.D. Thesis, The University of Western Ontario, 2012; N. Faghihi, N. Provatas, K. R. Elder, M. Grant, and M. Karttunen, Phys. Rev. E 88, 032407 (2013)] to incorporate polarization and concentration fields, as well as anisotropic ordering of the magnetization and polarization fields as determined by the local crystalline orientation. Magnetoelectric coupling is incorporated through the elastic coupling. Analytic calculations for a body centered-cubic (BCC) system are presented to illustrate that the model reduces to the standard multiferroic phase-field models when only a single crystal is considered. Two special cases of the model are then studied, the first focusing on magnetocrystalline interactions in a system described by the two-point correlation function of the XPFC model developed by Greenwood et al. [M. Greenwood, N. Provatas, and J. Rottler, Phys. Rev. Lett. 105, 045702 (2010); M. Greenwood, J. Rottler, and N. Provatas, Phys. Rev. E 83, 031601 (2011)], and the second focusing on electrocrystalline interactions in a system described by the original PFC kernel developed by Elder et al. K. R. Elder, M. Katakowski, M. Haataja, and M. Grant, Phys. Rev. Lett. 88, 245701 (2002); K. R. Elder and M. Grant, Phys. Rev. E 70, 051605 (2004)]. We examine the small deformation properties of these two realizations of the model . Numerical simulations are performed to illustrate how magnetocrystalline coupling can be exploited to design a preferential grain texture and how defects and grain boundaries influence the ferroelectric coercivity.

• Received 13 July 2015

DOI:https://doi.org/10.1103/PhysRevB.92.184109