THz spectroscopy and directional dichroism in the high magnetic field AFM phase of bulk BiFeO$_3$*

Magnetic ordering can induce local electric polarization via three mechanisms in BiFeO$_3$: spin-current, exchange-striction, and single-ion anisotropy. Only the spin-current mechanism causes the observed large non-reciprocal directional dichroism (NDD) at THz frequencies in the cycloidal state [PRL, 115:127203 (2015)]. However, the static magneto-electric effect persists even in the canted AFM state [Nat. Comm., 6:5878, (2015)] where the cycloid is destroyed by high magnetic field. Single crystal samples were studied in fields up to 35\,T with $\mathbf{B}\parallel \mathbf{a}$, $\mathbf{B}\perp \mathbf{a}$ and $\mathbf{B}\parallel \mathbf{c}$. NDD was found in the canted AFM phase above 18\,T. The number of observed spin wave modes is not consistent with a current model of two AFM-coupled spin sublattices in the canted AFM phase.