Tensor properties of the magnetoelectric coupling in hexaferrites

Recent data reported on magnetoelectric coupling parameter α for hexaferrite materials are very high; hence, these materials are expected to potentially impact future technologies in a major way. At this juncture, it is imperative to address the fundamental question, “What does α depend on?” in order to advance the design of new hexaferrite materials with even higher α. A complete mathematical formulation of a microscopic model would be rather complicated, especially for complex crystal structures such as the $Z$-type hexaferrites. Herein we present a simple, yet elegant thermodynamic argument by which we derive a functional relationship between α and the material parameters. We show that α is best described as a tensor proportional to the product of magnetostriction and piezoelectric strain tensors. Using this relationship, quantitative values for α are estimated and compared to experimentally measured values for various hexaferrites, composites, and ${\mathrm{Cr}}_2{\mathrm{O}}_3$.