Thermal Hall Effect of Spin Excitations in a Kagome Magnet

At low temperatures, the thermal conductivity of spin excitations in a magnetic insulator can exceed that of phonons. However, because they are charge neutral, the spin waves are not expected to display a thermal Hall effect. However, in the kagome lattice, theory predicts that the Berry curvature leads to a thermal Hall conductivity ${\kappa }_{xy}$. Here we report observation of a large ${\kappa }_{xy}$ in the kagome magnet Cu(1-3, bdc) which orders magnetically at 1.8 K. The observed ${\kappa }_{xy}$ undergoes a remarkable sign reversal with changes in temperature or magnetic field, associated with sign alternation of the Chern flux between magnon bands. The close correlation between ${\kappa }_{xy}$ and ${\kappa }_{xx}$ firmly precludes a phonon origin for the thermal Hall effect.

Figure 1

The in-plane thermal conductivity $\kappa$ (in zero $B$) measured in the kagome magnet Cu(1,3-bdc). At 40–50 K, $\kappa$ displays a broad peak followed by a steep decrease reflecting the freezing out of phonons [panel (a)]. The spin excitation contribution becomes apparent below 2 K. The inset is a schematic of the kagome lattice with the LRO chiral state [1]. The arrows on the bonds indicate the direction of advancing phase $\varphi ={\tan }^{-1}D/J$. Panel (b) plots $\kappa$ (black symbols) and $\kappa /T$ (red) for $T<4.5\mathrm{K}$. Below the ordering temperature $T_C=1.8\mathrm{K}$, the magnon contribution to $\kappa$ appears as a prominent peak that is very $B$ dependent. Values of $\kappa$ and $\kappa /T$ at large $B$ (identified with the phonon background) are shown as open symbols.

Figure 2

The thermal Hall conductivity ${\kappa }_{xy}$ measured in Cu(1,3-bdc). In panel (a), we plot the strongly nonmonotonic profiles of ${\kappa }_{xy}$ vs $B$ in sample 2. The dispersionlike profile changes sign below $\sim 1.7\mathrm{K}$. The right scale gives ${\kappa }^{2D}/(k_B^2/\hslash )$ (per plane) obtained by multiplying ${\kappa }_{xy}$ by $d\hslash /k_B^2=443.2$ (SI units). Panels (b) and (c) show corresponding curves in sample 3 (now plotted as ${\kappa }_{xy}/T$). Above $T_C$ [panel (b)], ${\kappa }_{xy}/T$ is $p$ type. The behavior below 1.90 K is shown in panel (c). At 1.09 K, the $n$-type contribution appears in weak $B$, and eventually changes ${\kappa }_{xy}/T$ to $n$-type at all $B$. Right scale in (c) reports ${\kappa }_{xy}^{2D}/(T{k}_B^2/\hslash )$. In panel (d), we plot the $T$ dependence of the quantity $[{\kappa }_{xy}/TB{]}_0$ which measures the thermal Hall response in the limit $B\to 0$. The $T$ dependence of $[{\kappa }_{xy}/TB{]}_0$ closely correlates with ${\kappa }_{xx}^S$ vs $T$ (aside from the sign change).

Figure 3

The effect of field $B$ on ${\kappa }_{xx}$ and scaling behavior at low $T$, for sample 3. The curves in panel (a) show that the $B$ dependence of ${\kappa }_{xx}$ is resolved (in the range $|B|<14\mathrm{T}$) only at $T<\sim 6.5\mathrm{K}$. The expanded scale in panel (b) shows that, near $T_C$ (1.8 K), ${\kappa }_{xx}$ has a nonmonotonic profile with a $\mathsf{V}$-shaped minimum at $B=0$ (identified with stiffening of the magnon bands by the field). Below 1 K, however, ${\kappa }_{xx}$ has a strictly monotonic profile that terminates in a sharp cusp peak as $B\to 0$. At each $T<T_C$, the constant “floor” profile at large $B$ is identified with ${\kappa }_{\text{ph}}$. The pattern in panel (b) simplifies when plotted as ${\kappa }_{xx}^s/T$ vs $B/T$ [panel (c)]. Multiplying by a scaling factor $s(T)$ collapses all the curves below 1 K to a “universal” curve, shown on log scale in panel (d). The slope at large $B$ gives a Zeeman gap with $g=1.6$. The Hall curve $-{\kappa }_{xy}/T$ has a similar slope at large $B$.