Anisotropic lattice anomalies and the pressure effect on the ferroelectricity in multiferroic HoMn$_{2}$O$_{5}$

Distinctive anomalies in the linear thermal expansivities along all principal axes were observed in multiferroic HoMn$_{2}$O$_{5}$ at the magnetic (T$_{N1}$ and T$_{N2})$ and ferroelectric (T$_{C1}$ and T$_{C2})$ transitions with a notable negative thermal expansivity along the $c$-axis for T$<$25K. These measurements provide a clear evidence of significant coupling between the magnetic and lattice orders in this material. Two main anomalies were observed, one at the onset of ferroelectricity (T$_{C1})$ and the other at T$_{C2}$ where another sharp change in the dielectric order was seen. The lattice anomalies were highly anisotropic where the $a $and $ b$-axes expand as the $c$-axis shrinks upon cooling through T$_{C1}$ and T$_{C2}$. The incommensurate AFM ordering of the Mn$^{3+}$ spins has been associated with T$_{N1}$. This is characterized by a high degree of frustration that subsequently drives the ferroelectric displacements at the lock-in transition to a commensurate magnetic structure. The second magnetic anomaly at T$_{N2}$ was shown via neutron diffraction to be due to a spin-reorientation phase transition. Dielectric measurements under isotropic pressure revealed that the two ferroelectric transitions are correlated and that the pressure stabilizes the ferroelectricity below T$_{C2.}$