Enhanced magnetoelastic coupling in hexagonal multiferroic HoMnO3

We have investigated the magnetoelastic coupling in HoMnO${}_{3}$ by measuring the temperature and magnetic field dependence of two elastic moduli ${C}_{11}$ and ${C}_{33}$ with an ultrasonic velocity technique. In comparison with similar data in YMnO${}_{3}$, we have identified an enhanced elastic softening, predominantly on ${C}_{11}$, that scales with the spin susceptibility over a wide temperature range. Spin fluctuations induced by Ho-Mn interactions are responsible for this softening: when the softening is quenched by an in-plane magnetic field with a concomitant stabilization of the Mn order, it is rather enhanced for a field oriented along the $c$ axis. The H-T phase diagrams deduced from elastic anomalies appearing at the different phase transitions are consistent with previous microwave data. The origin of the Ho-Mn coupling is discussed in terms of a fourth-order trigonal anisotropy term in the spin Hamiltonian.