Magnetoelastic phase diagrams in magnetoelectric TbPO₄: a neutron and X-ray scattering study

The interplay between elastic and magnetic properties is particularly important in the case of magnetoelectric materials, where bulk magnetization and electrical polarization are coupled. Magnetoelectric materials are poised to play a role in the next generation of information technology, enabling low-power transistors or spintronic devices. Understanding the complex coupling mechanisms of magnetoelectric materials is interesting in its own right, but can also guide future discovery of functional materials.
TbPO₄ exhibits one of the strongest magnetoelectric couplings among bulk materials. Since the interaction between electric and magnetic properties is mediated by strain, the significant magnetoelectric effect in TbPO₄ arises from the spin-lattice coupling characteristic of Jahn-Teller compounds like TbPO₄. In this work, the magnetoelastic phase diagrams of TbPO₄ have been extensively studied using various experimental techniques such as magnetization, heat capacity measurements, X-ray diffraction, neutron diffraction, and inelastic neutron scattering.
The experimental phase diagram for the field applied along the c axis is presented. For applied magnetic fields in the interval 0.6 T-1.1 T, a first-order field-induced transition is detected. This phase is shown to be characterized by the absence of magnetic moments, a linear behavior of the heat capacity as a function of temperature, and a massive orthorhombic distortion in the ab plane. The order of this phase is identified as a ferroquadrupolar order, in which the spin-lattice coupling induces a collective effect that causes the magnetic quadrupoles to interact. 
A second experimental phase diagram is studied for the magnetic field applied along the hard axis a. A new boundary, almost linear with temperature and field, has been determined up to 7 T. The unconventional behavior shown can characterize a very sharp cross-over or a broad phase transition. This boundary separates the paramagnetic phase, with only a field-induced structural distortion, from the distorted ferromagnetic regime. This distorted phase shows three strains, the moments are almost aligned along the field direction, but the presence of a tilting component is suggested by one of the detected strains.
In the absence of an external magnetic field, the structural distortion associated with the collinear tilting of the magnetic moment is found to have two strain components. 
Empirical inelastic neutron data are presented to complement the two phase diagrams. The measurements show a complex dynamical behavior in the magnetoelastic phases, characterized by multiple dispersive excitations that avoid crossing each other in different cases. A continuum excitation is revealed in the orthorhombic ferroquadrupolar phase. There are several non-negligible magneto-elastic terms giving rise to unconventional magneto-structural phase diagrams, where several distinct strain-induced lattice distortions  accompany changes in the magnetic structure. Here, a comprehensive and detailed analysis of the different phases is introduced.