The trajectory of an electron bends as it moves through a magnetic field. This Hall effect has been thought to not appear in insulators because of the apparent absence of mobile electrons. But recent reports of a thermal version of this effect, known as the thermal Hall effect, in numerous insulators has led to broad attention from researchers to understand its origin as well as potential applications for thermal current control.

Thermal current in an insulator is carried by spins and phonons. This leads us to ask how these charge neutral carriers can be bent by magnetic fields and how one can separate these two effects. To clarify the issue, we investigate the magnetic insulator Cd-kapellasite CdCu₃(OH)₆(NO₃)₂, a kagomé antiferromagnet [1]. In Cd-kapellasite, Cu²⁺ ions form a kagomé structure with a dominant nearest-neighbor antiferromagnetic interaction J/k_B = 45 K. The geometrical frustration effect of the kagomé structure suppresses the magnetic order down to T_N = 4 K, realizing a spin liquid state in a wide temperature range T_N < T < J/k_B.

We find clear thermal Hall signals in the spin liquid phase in all samples. The thermal Hall conductivity (κ_xy) is larger in a higher-quality sample with a larger longitudinal thermal conductivity (κ_xx), whereas the temperature dependence of κ_xy is similar (Fig. 1(a)). At 15 T, κ_xy and κ_xx are found to show a peak at almost the same temperature, a telltale sign of a phonon contribution κ^ph_xy in κ_xy at high fields. In addition, we find that the field dependence of κ_xy turns to be nonlinear at low temperatures and at low fields (Fig. 1(b)), concomitantly with the appearance of the field suppression of κ_xx, indicating the presence of a spin contribution κ^sp_xy in κ_xy at low fields. This is the first observation of both κ^ph_xy and κ^sp_xy in the same magnetic insulator.

Fig. 1. (a) The temperature dependence of the thermal Hall conductivity of Cd-kapellasite of different samples at 15 T [1]. All the samples show a peak at the same temperature, but with different magnitudes. (b) The field dependence of κ_xy at different temperatures. The non-linear part (colored region) shows the spin contribution.

Remarkably, by assembling the κ_xx dependence of κ^sp_xy data of other kagome antiferromagnets, we find that, whereas κ^sp_xy stays a constant in the low-κ_xx region (the “intrinsic” line in Fig. 2), κ^sp_xy starts to increase as κ_xx does in the high-κ_xx region (the “extrinsic” line in Fig. 2). This κ_xx dependence bears similarity to that of the anomalous Hall effect in ferromagnetic metals; the intrinsic mechanism by the Berry curvature is dominant in a moderate dirty metal, whereas the extrinsic mechanism by skew scatterings is dominant for a superclean metal. This good analogy indicates the presence of a similar duality of intrinsic-extrinsic mechanisms for the spin thermal Hall effect of an insulator.

Fig. 2. The dependence of the spin thermal Hall conductivity per the 2D kagomé layer (κ^sp,2D_xy) on the longitudinal thermal conductivity (κ_xx) of Cd-kapellasite [1], Volborthite [2], and Ca-kapellasite [3]. The blue and pink dashed lines are guides to the eye for the intrinsic and extrinsic contributions, respectively.

Furthermore, we find that both κ^ph_xy and κ^sp_xy disappear in the antiferromagnetic ordered phase at low fields, showing that phonons alone do not exhibit the thermal Hall effect. A high field above approximately 7 T induces κ^ph_xy, concomitantly with a field-induced increase of κ_xx and the specific heat, suggesting a coupling of the phonons to the field-induced spin excitations as the origin of κ^ph_xy.

References

• [1] M. Akazawa et al., Phys. Rev. X 10, 041059 (2020).
• [2] D. Watanabe et al., Proc. Natl. Acad. Sci. USA 113, 8653 (2016).
• [3] H. Doki, M. Akazawa, H.-Y. Lee et al., Phys. Rev. Lett. 121, 097203 (2018).

Authors

• M. Akazawa, M. Shimozawa^a, S. Kittaka^b, T. Sakakibara, R. Okuma, Z. Hiroi, H.-Y. Lee^c, N. Kawashima, J. H. Han^d, M. Yamashita
• ^aOsaka University
• ^bChuo University
• ^cKorea University
• ^dSungkyunkwan University