Wiedemann-Franz violation and thermal Hall effect in kagome metal TbCr6Ge6

Abstract

The thermal Hall effect has emerged as a powerful probe of exotic excitations in correlated quantum materials, providing access to charge-neutral heat carriers that remain invisible to electrical transport. To directly examine how heat and charge respond in relation within a kagome metal, we investigate the ferrimagnetic rare-earth 1-6-6 compound TbCr6Ge6 using the Wiedemann-Franz (WF) framework. We observe a dramatic breakdown of the WF law across the ferrimagnetic transition, where both longitudinal and transverse Lorenz ratios, L_{xx,xy} = κ_{xx,xy} / (T σ_{xx,xy}), deviate strongly from the Sommerfeld value L_0. After a partial recovery toward L_0 near 5-7 K, the Lorenz ratios are sharply suppressed well below L_0 despite a metallic charge response. We further find a pronounced low-temperature suppression of both L_{xx} and L_{xy} and a sign-changing transverse Lorenz ratio, indicating a clear decoupling between heat and charge transport and signaling substantial contributions from charge-neutral excitations whose Berry-curvature-driven transverse response evolves with temperature and magnetic field. TbCr6Ge6 thus provides a tunable metallic platform in which exchange-driven ferrimagnetism governs both longitudinal and transverse thermal responses, enabling controlled departures from Wiedemann-Franz behavior over an experimentally accessible temperature and field range.

Figures (4)

• Figure 1: Crystal structure and phase diagram of TbCr6Ge6.a. Crystal and magnetic structure of TbCr$_6$Ge$_6$ (HfFe$_6$Ge$_6$-type, $P6/mmm$) showing Cr kagome (blue) and Tb triangular (purple) layers stacked along $c$. The canted ferrimagnetic arrangement with Tb and Cr moments tilted by $\approx 25^{\circ}$ and $143^{\circ}$ with respect to the $c$ axis is adapted from neutron data schobinger1997atomic. b. Actual image of the TbCr$_6$Ge$_6$ crystal. c. Top view of the Cr kagome network, the interleaved Tb triangular sublattice, and the surrounding Ge coordination completing the HfFe$_6$Ge$_6$ structural motif. d. Temperature-dependent susceptibility $\chi(T)$ for $\mu_0 H \parallel c$ (0.03–4 T). A pronounced maximum appears near 10 K, progressively broadening and shifting to lower temperatures as the magnetic field increases. e. For low magnetic fields ($\mu_0 H < H_c$), a sharp anomaly signals the onset of long-range magnetic order (LRO). As the field approaches $H_c$, this feature broadens and shifts to lower temperatures, evolving into a crossover scale associated with short-range order (SRO). At higher temperatures, the SRO regime connects smoothly to the paramagnetic (PM) phase. The points shown correspond to fits of Eq. \ref{['eq:sigmoid']} to the temperature-dependent susceptibility data, following the procedure described in Ref. gokhfeld2025new. Smooth curve is a guide to the eye.
• Figure 2: Field-dependent electronic transport and magnetization of TbCr$_6$Ge$_6$.a. Longitudinal electronic resistivity $\rho_{xx}(H)$ from 2--10 K showing suppression of spin-dependent scattering with increasing field. b. Background-subtracted longitudinal resistivity $\rho_{xx}(H)$ revealing a central low-field peak consistent with spin-dependent scattering. c. Temperature dependence of the parameters $M_{0}$ (black squares) and $\rho_{S}$ (red circles) obtained from fits to $\rho_{xx}(H,M)=\rho_{xx}^{(2c)}(H)+A_M M^{2}+\rho_{S}\tanh[k(M^{2}-M_{0}^{2})]$. Here, $\rho_{xx}^{(2c)}(H)$ represent the classical two-carrier (electron--hole) background that depends on $H$ but not on $M$Yao2025YbFe6Ge6Zhu2018BismuthValley. The term $A_M M^{2}$ accounts for the continuous reduction of spin-disorder scattering with increasing magnetization in $\rho_{xx}$, while the heuristic sigmoidal contribution $\rho_{S}\tanh[k(M^{2}-M_{0}^{2})]$ phenomenologically captures a crossover between scattering regimes associated with different magnetic ordered regions with distinct resistivities, where $M_{0}$ sets the characteristic magnetization scale for the crossover and $k$ controls its sharpness (see SI for more details). The progressive reduction of $M_{0}$ toward the magnetic transition near 8--10 K is consistent with the decreasing magnetization scale associated with the crossover between ferrimagnetic and paramagnetic regimes. The pronounced extremum in $\rho_{S}$ marks the temperature range in which the two-regime contribution is strongest; above the transition, $\rho_{S}$ decreases rapidly and becomes negligible. Arrows denote the corresponding axes. d. Normalized magnetization $m(H)$ displaying strong anisotropy and rapid saturation above 0.5 T. e.$M^{2}(H)$ curves. f. Temperature dependence of $\rho_{xx}$ at 0 T highlighting two regimes: below $\sim 9$ K, $\rho_{xx}\propto T^{3}$ consistent with single-magnon scattering in the ferrimagnetically ordered state, and above $\sim 9$ K, $\rho_{xx}\propto T^{2}$ reflecting coherent electronic scattering in the paramagnetic metallic regime. Smooth curves are guides to the eye. Error bars represent one standard deviation ($1\sigma$).
• Figure 3: Thermal transport and Wiedemann--Franz analysis of TbCr$_6$Ge$_6$.a. Sketch of the steady-state thermal transport experiment. Three thermometers are connected to the sample via Au wires to measure longitudinal ($\Delta T_{xx}$) and transverse ($\Delta T_{xy}$) temperature gradients. The inset photograph shows the single-crystal device mounted with heater and thermometers inside the dilution refrigerator. b. Field dependence of the longitudinal thermal conductivity $\kappa_{xx}(H)$ at selected temperatures from 0.5 to 20 K. A broad enhancement centered at zero field is most pronounced near $T_c$ and is reduced with increasing field. c. Temperature dependence of the normalized longitudinal Lorenz ratio $L_{xx}/L_0 = \kappa_{xx}/(L_0 T \sigma_{xx})$ for $\mu_0 H = 0$, 3, 6, and 9 T. The shaded regions I–IV indicate distinct regimes discussed in the text: I. a low-temperature regime ($T \lesssim 4$–5 K) where $L_{xx}/L_0$ is strongly reduced; II. an intermediate regime ($\sim$5–7 K) where $L_{xx}/L_0$ approaches unity; III. a peak region around the ferrimagnetic transition temperature $T_c$ where $L_{xx}/L_0$ is strongly enhanced; and IV. a higher-temperature regime ($T \gtrsim 10$ K) in which $L_{xx}/L_0$ is positive and varies only weakly with temperature. The smooth curves are guides to the eye. d. Magnified view of normalized Lorenz ratio $L_{xx}/L_0$ in I and II regions, emphasizing the continuous evolution from the low-temperature regime with strongly suppressed normalized Lorenz ratio toward a higher-temperature regime that can be viewed as an extension of region II. The shaded background qualitatively indicates these limiting regimes. The smooth curves are guides to the eye. Error bars represent one standard deviation ($1\sigma$).
• Figure 4: Transverse thermal transport and normalized Lorenz analysis of TbCr$_6$Ge$_6$.(a, b) Field-dependent thermal Hall conductivity $\kappa_{xy}(H)$ from 0.5--20 K, with a 3--4 K magnified view showing a smooth evolution from negative to positive slopes and a nonlinear regime near 8 K. (c) Temperature--field map of $\kappa_{xy}(T,H)$ displaying a strongly antisymmetric response with sign reversals. (d) Temperature dependence of the low-field $\kappa_{xy}(H)$ slope, peaking near the ferrimagnetic transition $T_c$ (inset: low-$T$ sign change). (e) Normalized transverse Lorenz ratio $L_{xy}/L_0 = \kappa_{xy}/(L_0 T \sigma_{xy})$ for $\mu_0 H = 3$--11 T. Regimes I--IV correspond to: I, a suppressed and negative ratio ($T\lesssim4$ K); II, a crossover toward unity (4--7 K); III, an enhanced response near $T_c$; IV, a weakly $T$-dependent, positive ratio above $\sim10$ K. (f) Low-temperature enlargement of (e). Smooth curves are guides to the eye. Error bars represent one standard deviation ($1\sigma$).