Magnetic-field-induced critical dynamics in magnetoelectric TbPO${}_4$

We report on the magnetoelectric dynamics in the linear magnetoelectric antiferromagnet TbPO${}_4$ studied by broadband dielectric spectroscopy. For the phase transition into the magnetoelectric antiferromagnetic phase at $T_N\approx 2.3$ K, a finite magnetic field $H$ induces critical behavior in the quasi-static permittivity ${\epsilon }^{\mathrm{\prime }}$. Plotting the corresponding anomaly as function of $T\mathrm{/}{T}_N(H)$, we observe the scaling behavior $\mathrm{\Delta }{\epsilon }^{\mathrm{\prime }}\propto H^2$, a clear fingerprint of linear magnetoelectric antiferromagnets. Above the phase transition, we find a critical slowing down of the ferroic fluctuations in finite magnetic field. This behaviour can be understood via a magnetic-field-induced relaxational response that resembles the soft-mode behaviour in canonical ferroelectrics and multiferroics.