The fourth ferroic order: Current status on ferrotoroidic materials

Abstract

We review the topic of ferrotoroidicty and the inorganic solids proposed to exhibit this most elusive category of primary ferroic orders. We first define how ferrotoroidic order fits in with the other well known ferroic orders: ferromagnetism, ferroelectricity, and ferroelasticity. Since the order parameter is a toroidal moment, which is a moment of magnetization, these materials have been difficult to characterize. We present some of the symmetry constraints for an extended solid to exhibit ferrotoroidic order, and the most prominent characterization tools to measure the order parameter, which include magnetoelectric measurements, spherical neutron polarimetry, and second-harmonic generation optical spectroscopy and microscopy. In addition to being primary ferroics, ferrotoroidic materials also display the secondary effect known as magnetoelectricity. The magnetically ordered state must display long-range order with a k=0 structure, that is a magnetic unit cell that is the same size as the chemical unit cell. Of the 122 known Heesch-Shubnikov (or colored) point groups, only 31 allow ferrotoroidicity. With these preliminaries covered, we then review the leading candidates for ferrotoroidicity including the metal orthophosphates with the triphylite-type structure such as LiCoPO4, the pyroxene germanates and silicates such as LiFeSi2O6, and a few other systems. Common to the various solid state compounds is a first-row transition metal with some orbital degree of freedom that may not be completely quenched. From a crystal chemistry point of view, many include transition metals in an octahedral crystal field arranged in motifs separated by oxoanions such as phosphate or silicate groups.