Non-reciprocal directional dichroism in the AFM phase of BiFeO$_3$ at THz frequencies*

We did THz absorption spectroscopy of BiFeO$_3$ single crystals in the AFM phase, where the spin cycloid is destroyed in magnetic fields between 18\,T and 32\,T in Voigt geometry at 1.6\,K. If $\mathbf{B}_0 \parallel [1\bar{1}0]$, we see strong directional dichroism (DD) of absorption of the magnon mode with light propagating along the direction of the ferroelectric polarization $\mathbf{k} \parallel \mathbf{P} \parallel [111]$ and $\mathbf{e}^\omega \parallel [1\bar{1}0]$, $\mathbf{b}^\omega \parallel [\bar{1}\bar{1}2]$. The sign of DD can be reversed (\i) by reversing the direction of $\mathbf{B}_0$ or (ii) by flipping the sample, thus reversing the propagation direction of light. The observed effect is caused by the strong magneto-electric coupling in the collinear AFM phase.