Spontaneous spin momentum locking and anomalous Hall effect in ${\mathrm{BiFeO}}_3$

The studies for the discovery of new nontrivial degrees of freedom of Bloch electrons are at the forefront of condensed matter physics. The focus remains on the spin-orbit coupling (SOC)-induced pseudospin valley state and its interplay with time inversion asymmetry. Here we propose a distinct magnetic order-protected, momentum-dependent spin-splitting valley (SSV) state in polar bulk ${\mathrm{BiFeO}}_3$. The octahedral distortion-induced spatial asymmetry of the superexchange interaction path breaks spin degeneracy and antiferromagnetic time-reversal symmetry. The SSV polarization is caused by crystal symmetry-related intrinsic SOC and leads to nonzero anomalous Hall conductivity (AHC) and magneto-optical effects. The anomalous Hall conductivity and magneto-optical effects are expected to be most prominent if magnetic and electric dipoles are orthogonal. Due to the ferroelectric nature of ${\mathrm{BiFeO}}_3$, such unconventional magneto-optical effect and AHC in fully compensated magnetic order offers straightforward tunability under an electric field.

Figure 1

The atomic and magnetic structure and its relationship with antiferromagnetic time-reversal symmetry ($T_{\mathrm{AF}}$). (a) Atomic structure of high-symmetry cubic ($Pm\text{−}3m$) phase and magnetic exchange interactions. The transparent pink planes represent the equivalent mirror symmetry ($m$). (b) Schematics of octahedral rotations of the oxygen frame (upper panel) and polar distortion (lower panel). (c) Schematic of asymmetry of magnetic density. (d) Distorted rhombohedral structure ($R3c$) ground-state structure. The upper and lower panels show side and top views; the dashed line indicates glide plane symmetry $c$. (e) Schematics of the superexchange mechanism between two oppositely spin-polarized iron atoms, mediated by oxygen atoms. The upper panel represents the $T_{\mathrm{AF}}$ symmetry case ($Pm\text{−}3m$), the second panel is for broken $T_{\mathrm{AF}}$ symmetry due to the octahedral rotations ($R\text{−}3c$), and the lowest panel is for the case of complete polar distortion ($R3c$). The red, yellow, and purple balls represent iron, oxygen, and bismuth atoms, respectively. The blue and green colors represent up and down spin, respectively.

Figure 2

The magnetic exchange-induced SSVs of a ground-state rhombohedral ($R3c$) structure. (a) Spin splitting eigenvalues $\mathrm{\Delta }E\left(k\right)=E\uparrow \left(k\right)\text{–}E\downarrow \left(k\right)$ (in the unit of meV) projected on the sphere with a radius of one third of the reciprocal lattice vector. Here, one can only see three pairs of valleys as the [111] direction is out of the paper. (b) The Fermi surface for the chemical potential (–10 meV) at the valence band maxima. (c) The two-dimensional band structure along the a-b plane, where $a$ and $b$ are reciprocal lattice vectors; the SSV is represented in ΔE($k$) (in the unit of meV). (d) and (e) Band structures along the path of $H_2\text{−}\mathrm{\Gamma }\text{−}{H}_2^{\prime }$ and $H_6\text{−}\mathrm{\Gamma }\text{−}{H}_0$, respectively. The red and blue colors represent the opposite spin.

Figure 3

The SSV polarization in the ($R3c$) structure and magnetic alignments. (a) Schematic of the valley polarization with SOC. (b) The band structure along the path $H_0\text{−}\mathrm{\Gamma }\text{−}{H}_6$ with magnetic alignment along the $x$-axis. (c) Spin splitting eigenvalue $\mathrm{\Delta }E\left(k\right)=E\uparrow \left(k\right)\text{–}E\downarrow \left(k\right)$ in the unit of meV, projected on the sphere with a radius of 0.333 in the unit of reciprocal lattice vector. The upper and lower panels are for $z$ and $x$ spin alignment, respectively. The black arrows and dots denote the [111] direction.

Figure 4

The nature of Berry curvature and AHC. (a) The reciprocal space spin splitting (left) and the Berry curvature (right) in the (111) plane. (b) The AHC for in-plane $x$ spin alignment. (c) The AHC for out-of-plane $z$ spin alignment.

Figure 5

The magneto-optical effect showing the frequency-dependent imaginary and real part of the antisymmetric optical conductivity tenser for (a) in-plane ($x$) spin alignment and (b) out-of-plane ($z$) spin alignment.