Title:First-Order Spin-Reorientation Transition and Incomplete Softening of the Antiferromagnetic Resonance Mode in Multiferroic GdFe$_3$(BO$_3$)$_4$

Abstract:The multiferroic ferroborate GdFe$_3$(BO$_3$)$_4$ with huntite-type structure exhibits magnetic ordering below T$_N$ = 38 K and contains two magnetic subsystems associated with Gd and Fe ions. Competing anisotropies of these subsystems drive a spin reorientation transition at T$_{SR}$ = 10.7 K, switching the ground state from easy-axis to easy-plane. Using antiferromagnetic resonance, we investigate the spin dynamics across this transition. The observed incomplete softening of a magnon mode during both field- and temperature-induced spin-reorientation transitions indicates the first-order nature of the phase transition, which is accompanied by a discontinuous jump in the effective anisotropy field. We reproduce this behavior using a simple model that attributes the jump in the anisotropy field to the presence of an effective fourth-order anisotropy constant, responsible for the discontinuous character of the transition. Remarkably, for in-plane magnetic fields, we identify a new AFMR mode that persists from 12 K up to T$_N$. This mode likely corresponds to the dynamics of a long period incommensurate state, previously detected by resonant elastic X-ray scattering.