Thermodynamics of Ferrotoroidic Materials

The three primary ferroics, namely ferromagnets, ferroelectrics and ferroelastics exhibit corresponding large (or even giant) magnetocaloric, electrocaloric and elastocaloric effects when a phase transition is induced by the application of an appropriate external field ($E$, $H$ and stress). Recently the suite of primary ferroics has been extended to include ferrotoroidic materials in which there is an ordering of toroidic moments in the form of magnetic vortex-like structures, examples being LiCo(PO$_4$)$_3$ and Ba$_2$CoGe$_2$O$_7$. We formulate the thermodynamics of ferrotoroidic materials. Within a Landau free energy framework we calculate the toroidocaloric effect by quantifying isothermal entropy change (or adiabatic temperature change) in the presence of an applied toroidic field ($G=E\times H$) when usual magnetization and polarization may also be present simultaneously.