Interaction driven transverse thermal resistivity in a phonon gas

Abstract

The amplitude of the Hall response of electrons can be understood without invoking interactions. Most theories of the phonon thermal Hall effect have likewise opted for a non-interacting picture. Here, we challenge this approach. Our study of WS$_2$, a transition metal dichalcogenide (TMD) insulator, finds that longitudinal, $κ_{xx}$, and transverse, $κ_{xy}$, thermal conductivities peak at almost the same temperature. Their ratio obeys an upper bound, as in other insulators. We then compare transverse thermal transport in a phonon gas and in a molecular gas. In the latter, the Senftleben-Beenakker effect is driven by the competition between molecular collisions and applied magnetic field in setting the distribution of molecular angular momenta. An off-diagonal transport response arises thanks to interactions between non-spherical particles, which do not need to be chiral. By analogy, we argue that in a phonon gas, magnetic field will influence phonon-phonon interactions, and generates a transverse thermal \emph{resistivity}, whose order of magnitude can be accounted for by invoking a Berry force on the drift velocity of the nuclei in the presence of a finite heat. This simple picture gives a reasonable account of the experimentally measured transverse thermal resistivity of seven different crystalline insulators.

Figures (9)

• Figure 1: Longitudinal transport in WS$_2$. (a) The crystal structure of WS$_2$ is a stack of two-dimensional layers each consisting of covalently bonded S-W-S sandwiches. (b) Schematic setup for measuring longitudinal and transverse thermal conductivities. (c) Temperature dependence of the longitudinal thermal conductivity $\kappa_{xx}$ at zero field (black) and in a magnetic field of 9 T (blue). Also shown are previous experimental data pisoni2016anisotropic and theoretical calculations Lindroth2016. The inset shows a comparison of $\kappa_{xx}$ and $L_0T\sigma_{xx}$, implying that the electronic contribution to heat transport is negligible. (d) Temperature dependence of lattice specific heat according to experiment (black symbols) and theoretical calculations (blue line). (e) Temperature dependence of phonon mean-free-path $\ell_p$, extracted from specific heat, thermal conductivity and sound velocity compared with the sample thickness.
• Figure 2: Thermal Hall effect in WS$_2$. (a, b) The field dependence of the thermal Hall angle ($\nabla_y T /\nabla_x T$) at 23.3 K and 30.5 K. Red and black symbols correspond to two sets of data obtained for opposite directions of field sweep. The thermal Hall angle is linear in the magnetic field. (c) Temperature dependence of $\kappa_{xy}$ multiplied by 650, compared to $\kappa_{xx}$. (d) Comparison of the transverse $\kappa_{ij} / B$ and the longitudinal thermal conductivity $\kappa_{ii}$ in different insulators Ideue2017Li2020Grissonnanche2020Boulanger2020Akazawa2020Chen2022Uehara2022Lefran2022Li2023Meng2024lishi2025Boulanger2022.
• Figure 3: Transverse thermal conductance in a molecular gas and in a phonon gas. (a) Magnetic field exerts a torque on a diatomic molecule (in red) with an angular momentum of $\vec{L}$ and a magnetic moment of $\vec{J}$. This generates a precession of $\vec{J}$. (b) The precession does not affect the amplitude of $\vec{J}$, but gives traveling molecules a handedness depending on the mutual orientation of velocity and magnetic field. (c) The collision cross section becomes skewed, because the angular momentum of the binary system composed of the colliding molecules have opposite signs for molecules coming from opposite lateral directions. (d) Left: An atom inside an anharmonic potential. This leads to permanent fluctuation in the phonon number. As shown on the right, this occurs thanks to emission and absorption events. The combination of two phonons generates a third phonon which in turn disintegrates to two other phonons. In thermal equilibrium, with no net flow of phonons ($\vec{J}_n=0$), emission and absorption events compensate each other. (e) When phonons flow ($\vec{J}_n\neq 0$), there is an excess of emission events. In the absence of time reversal symmetry, collisions from left and right are no more equivalent. The pseudo-momentum sum rule, which states that the sum of initial, $\vec{q}_i$, and the final phonon wave-vectors,$\vec{q}_f$ , are equal, does not hold any more.
• Figure 4: Transverse thermal resistivity in different materials. (a) Thermal Hall resistivity $W_{\perp} = \frac{\nabla_yT}{J_q^x}$ normalized by magnetic field, $B$, as a function of temperature $T$ in WS$_2$. Other panels show the same quantity for black P Li2023 (b), Si lishi2025 (c), Ge lishi2025 (d), SrTiO$_3$Li2020 (e) and Nd$_{2}$CuO$_4$Boulanger2020 (f). Light red stripes cover the data points in the explored temperature range. The dashed vertical lines show the position of the peak in the thermal Hall angle.
• Figure 5: Transverse thermal Hall angle in different materials. (a) Thermal Hall angle $\frac{\kappa_{xy}}{\kappa_{xx}}$ normalized by magnetic field, $B$, as a function of temperature $T$ in WS$_2$. Other panels show the same quantity for black P Li2023 (b), Si lishi2025 (c), Gelishi2025 (d), SrTiO$_3$Li2020 (e) and Nd$_{2}$CuO$_4$Boulanger2020 (f). The dashed vertical lines show the position of the peak in the thermal Hall angle.