Band topology and dynamic multiferroicity induced from dynamical Dzyaloshinskii-Moriya interactions in centrosymmetric lattices

Abstract

We develop a theory of a dynamical Dzyaloshinskii-Moriya interaction (dDMI) in centrosymmetric crystals by generally considering the vibration of both cations and anions. It gives rise to an antisymmetric spin-lattice coupling, inducing magnon-phonon hybridized topological excitations. Moreover, we find that this dDMI naturally exhibits a magnetoelectric feature, leading to the presence of dynamic multiferroicity with finite toroidal moment distribution in the momentum space. By comparing toroidal moments with band skyrmion structure, we reveal the intrinsic connection between band topology and dynamic multiferroicity through the dDMI.

Figures (7)

• Figure 1: (a) Static four-atom diamond cell with $ij$ (blue) atoms magnetic but $lm$ (red) atom non-magnetic. (b) The schematic for dynamical DMI from lattice dynamics. $k_{u} (k_{uv})$ is the vibrational force constant between magnetic ions and nearest-neighbor (non-)magnetic ions. Purple and orange arrows indicate the DM vectors from purple and orange triangles, respectively. (c) A ferromagnet in a diatomic square lattice, where blue (red) spheres stand for (non-)magnetic ions. Blue arrows indicate magnetic moments along $z$-direction. The inset shows the Brillouin zone for the square lattice with a high-symmetry path $\Gamma XM\Gamma$. (d) The propagating spin-wave (blue arrows) produces varying polarizations (green arrows) $\mathbf{P}_{ij}^s$ that induce dynamical distortion of the lattice with electrical polarization $\mathbf{P}_{iljm}\propto(u_i^z+u_j^z-v_l^z-v_m^z)\hat{\mathbf{z}}$.
• Figure 2: Band dispersion (left) and Berry curvature distribution (right) in log scale $L(\Omega_n^z)\equiv\text{sgn}(\Omega_n^z)\ln(1+|\Omega_n^z|)$ for (a) $B_\text{eff}=2$ meV, (b) $B_\text{eff}=5$ meV, (c) $B_\text{eff}=8$ meV. Red (Blue) dashed lines in the dispersion are uncoupled phonon (magnon) bands. $C_B, C_M,\text{ and }C_T$ is the Chern number for the bottom, middle, and top band, respectively.
• Figure 3: The dependence of thermal Hall effects on $B_\text{eff}$ for $T=15\text{ K},20\text{ K},25\text{ K},30\text{ K},35\text{ K, and }40\text{ K, respectively}$. The topological gap in Fig. \ref{['fig:Band_topo']} closes at $B_{c1}\approx 3.60$ meV and $B_{c2}\approx 6.37$ meV.
• Figure 4: (a) Non-zero toroidal moment distribution (in the unit of $\hbar\gamma Q^*\sqrt{\frac{S}{2}}$) for the middle band for $B_\text{eff}=8$ meV when $D_0\neq 0$, characterizing dynamic type-II multiferroicity. (b) Toroidal moments become zero everywhere in the momentum space when $D_0=0$. (c) The rotated toroidal moment $(T^y_\mathbf{k},T^x_\mathbf{k})$, which is approximately proportional to $(\hat{d}^x_\mathbf{k},\hat{d}^y_\mathbf{k})$ as shown in Fig. \ref{['fig:TM']}(d). (d) The skyrmion structure of the $\boldsymbol{d}$-vector for the effective magnon-phonon Hamiltonian.
• Figure B1: (a) The dependence of longitudinal thermal conductivity $\kappa_{xx}$ on $B_\text{eff}$ for $T = 15$ K, $20$ K, $25$ K, $30$ K, $35$ K, and $40$ K, respectively. (b) The dependence of the thermal Hall angle on temperature $T$ for $B_{\text{eff}} = 2$ meV, $4$ meV, $6$ meV, $8$ meV, respectively.