Unconventional thermal magnon Hall effect in a ferromagnetic topological insulator

We present theoretically the thermal Hall effect of magnons in a ferromagnetic lattice with a Kekule-O coupling (KOC) modulation and a Dzyaloshinskii-Moriya interaction (DMI). Through a strain-based mechanism for inducing the KOC modulation, we identify four topological phases in terms of the KOC parameter and DMI strength. We calculate the thermal magnon Hall conductivity ${\kappa }^{xy}$ at low temperature in each of these phases. We predict an unconventional conductivity due to a nonzero Berry curvature emerging from band proximity effects in the topologically trivial phase. We find sign changes of ${\kappa }^{xy}$ as a function of the model parameters, associated with the local Berry curvature and occupation probability of the bulk bands. Throughout, ${\kappa }^{xy}$ can be easily tuned with external parameters such as the magnetic field and temperature.

Figure 1

KOC modulation of the honeycomb lattice for different values of the KOC parameter $\mathrm{\Delta }$. The intracell (intercell) couplings are shown as black (red) lines of widths proportional to their strength. The isotropic, unstrained case with $\mathrm{\Delta }=0$ is shown in (b), with the unstrained NN vectors $\mathit{\rho }$, NNN vectors $\mathit{\mu }$, and lattice parameter $a$ indicated.

Figure 2

(a) The KOC-modulated honeycomb lattice $(\mathrm{\Delta }>0)$ with DMI complete with the indexed site basis in the unit cell shown as a dotted green hexagon. The strengths of the various couplings are indicated and described in the main text. The arrows give the directions in which ${\nu }_{ij}=1$. (b) The vertices of the unstrained Brillouin zone (dashed hexagon) are folded into the center of the strained Brillouin zone (shaded hexagon) due to the KOC modulation. The path $\mathrm{\Gamma }MK\mathrm{\Gamma }$ in $\mathit{k}$ space between the high-symmetry points shown as black circles used in later figures is shown in (b).

Figure 3

The thermal Hall conductivity ${\kappa }^{xy}$ of a KOC-modulated ${\mathrm{CrI}}_3$ monolayer against the KOC parameter $\mathrm{\Delta }$ and DMI strength $D$ at temperature $T=30\mathrm{K}$ with magnetic field $B=0^{+}\mathrm{T}(\xi =1)$. Each topological phase is distinguished by its set of Chern numbers $(C_1,C_2,\cdots ,C_6)$. Gap-closing transitions separating phases are depicted as dotted black lines. The four points in each phase we use in later figures lying on the line $D=0.31\mathrm{meV}$ with $\mathrm{\Delta }=-0.27$ (I), $-0.136$ (II), $0.136$ (III), and $0.27$ (IV) are shown as black dots. The two color bars apply to region I and regions II–IV, respectively, and have been added for clarity as the magnitude of ${\kappa }^{xy}$ is much smaller in region I.

Figure 4

The thermal Hall conductivity (${\kappa }^{xy}$, top) and its band contributions (${\kappa }_{\lambda }^{xy}$, bottom) in a KOC modulated honeycomb lattice with DMI as a function of temperature $T$ for the points in each phase given in Fig. 3.

Figure 5

The band structure along the path $\mathrm{\Gamma }MK\mathrm{\Gamma }$ shown in Fig. 2 (left), the isoenergy surfaces of the Chern numbers $\mathcal{D}{C}_{\lambda }$ (middle), and the isoenergy surfaces of the contributions of each band to the thermal Hall conductivity $\mathcal{D}{\kappa }_{\lambda }^{xy}$ (right). The parameters are given in Fig. 3 with (a) $\mathrm{\Delta }=-0.27$ and $D=0.31\mathrm{meV}$ and (b) $\mathrm{\Delta }=-0.2$ and $D=0.05\mathrm{meV}$. The energies of the bands at the $\mathrm{\Gamma }, M$, and $K$ points are highlighted in each plot by gray horizontal lines. The Chern numbers $C_{\lambda }$ and contribution to the conductivity ${\kappa }_{\lambda }^{xy}$ in units of ${10}^{-13}\mathrm{W}{\mathrm{K}}^{-1}$ are given for each band as text of the corresponding color in each subfigure. The color scheme of the energy bands and its contributions are the same as in Fig. 4.

Figure 6

The total thermal conductivity ${\kappa }^{xy}$ (blue, solid line) and its current ${\kappa }_E^{xy}$ (yellow, dashed line) and orbital ${\kappa }_O^{xy}$ (green, dotted-dashed line) contributions against (a) the temperature $T$ at $\mathrm{\Delta }=0.136$ and (b) the KOC parameter $\mathrm{\Delta }$ at $T=30\mathrm{K}$ at the point in phase III from Fig. 3.

Figure 7

The thermal Hall conductivity ${\kappa }^{xy}$ against magnetic field $B$ at various temperatures $T$ at the point in phase III in Fig. 3.