Planar thermal Hall effect from phonons in a Kitaev candidate material

Abstract

The thermal Hall effect has emerged as a potential probe of exotic excitations in spin liquids. In the Kitaev magnet $\alpha$ -RuCl₃, the thermal Hall conductivity ${\kappa }_{xy}$ has been attributed to Majorana fermions, chiral magnons, or phonons. Theoretically, the former two types of heat carriers can generate a "planar" ${\kappa }_{xy}$ , whereby the magnetic field is parallel to the heat current, but it is unknown whether phonons also could. Here we show that a planar ${\kappa }_{xy}$ is present in another Kitaev candidate material, Na₂Co₂TeO₆. Based on the striking similarity between ${\kappa }_{xy}$ and the phonon-dominated thermal conductivity ${\kappa }_{xx}$ , we attribute the effect to phonons. We observe a large difference in ${\kappa }_{xy}$ between different configurations of heat current and magnetic field, which reveals that the direction of heat current matters in determining the planar ${\kappa }_{xy}$ . Our observation calls for a re-evaluation of the planar ${\kappa }_{xy}$ observed in $\alpha$ -RuCl₃.

Fig. 1. Crystal structure of Na₂Co₂TeO₆ and experimental setup.

a The crystal structure of Na₂Co₂TeO₆. Honeycomb layers of edge-sharing CoO₆ octahedra sandwiched between Na layers and stacked along the c direction in an ABAB format. The two types of inequivalent environments result in two different Co²⁺ sites which are labeled as Co (1) and Co (2). b The honeycomb layer viewed along the crystal c axis. The Co²⁺ ions are surrounded by oxygen octahedra. a denotes the zigzag direction (perpendicular to the Co-Co bond), a* denotes the armchair direction (parallel to the Co-Co bond). Schematic of the thermal transport measurement setup with (c) H // J and (d) H $\perp$ J (see Methods). Directions of both thermal current J and external magnetic field H are shown with colored arrows.

Fig. 2. Thermal transport in Na₂Co₂TeO₆ with H // J // a and H // J // a*.

Thermal conductivity ${\kappa }_{xx}$ vs temperature T in Na₂Co₂TeO₆ (a) sample A measured with H // J // a and (c) sample B measured with H // J // a* at H = 0 T, 5 T, 10 T, and 15 T, with J the thermal current and H the external magnetic field. Thermal Hall conductivity ${\kappa }_{xy}$ vs T in Na₂Co₂TeO₆ (b) sample A and (d) sample B measured at H = 5 T, 10 T, and 15 T. In panel (d), the inset shows the full range of data. a denotes the zigzag direction (perpendicular to the Co-Co bond), a* denotes the armchair direction (parallel to the Co-Co bond).

Fig. 3. Ratio of κ_xy/κ_xx for candidate Kitaev magnets.

a Ratio between the thermal Hall conductivity and thermal conductivity ${\kappa }_{xy}/{\kappa }_{xx}$ vs temperature T of the planar thermal Hall response in Na₂Co₂TeO₆ at a magnetic field H = 15 T of the four measured samples. Although the ratio clearly shows a sample dependence, the order of magnitude (0.1% at H = 15 T and T = 20 K) is typical of the phonon thermal Hall effect in various insulators^–. b Ratio ${\kappa }_{xy}/{\kappa }_{xx}$ vs T of the planar thermal Hall response in candidate Kitaev magnets Na₂Co₂TeO₆ and $\alpha$-RuCl₃. The purple curve is obtained from data in Fig. 3.28 of ref. . a denotes the zigzag direction (perpendicular to the Co-Co bond), a* denotes the armchair direction (parallel to the Co-Co bond).

Fig. 4. Thermal transport data in Na₂Co₂TeO₆ for H // J and H ⊥ J.

a Thermal conductivity ${\kappa }_{xx}$ and (b) thermal Hall conductivity ${\kappa }_{xy}$ vs temperature T in Na₂Co₂TeO₆, measured at a magnetic field H = 15 T for H // a: on sample C with J // a (red) and sample D with J // a* (blue). c, d Corresponding data for H // a*. In both field directions, ${\kappa }_{xy}$ measured with H $\perp$ J is much smaller than that measured with H // J. a denotes the zigzag direction (perpendicular to the Co-Co bond), a* denotes the armchair direction (parallel to the Co-Co bond).