Figure 1

(a) $k(T)$ for $\mathrm{U}({\mathrm{Ru}}_{1-x},{\mathrm{Rh}}_x{)}_2{\mathrm{Si}}_2$. A dramatic increase in $\kappa$ is observed at the hidden order transition, sharply suppressed with Rh doping. AF order is realized at $x=3\%$, with only a slight feature in $\kappa$. No ordering was observed at $x=4\%$. Inset: The pure sample displays a low-$T$ $T^2$ power law, indicating phonon conduction in the presence of $e$-ph scattering. (b) $\kappa (T)$ of pure ${\mathrm{URu}}_2{\mathrm{Si}}_2$ with heat ($Q$) applied along the $a$ axis (dashed line) and the $c$ axis (closed squares). The $H$ dependence of the $c$ axis $\kappa (T)$ is shown (closed symbols) up to 14 T, with $H\parallel c$. Inset: $\delta ={\gamma }_{\mathrm{LT}}/({\gamma }_{\mathrm{HT}}-g_{\mathrm{LT}})$ as a function of Rh doping (%). ${\gamma }_{\mathrm{LT}}$ and ${\gamma }_{\mathrm{HT}}$ were extracted from the published $C(T)$ data.Reuse & Permissions

Figure 2

(a) The electronic contribution to $\kappa (T)$ calculated from the thermal Hall effect in pure ${\mathrm{URu}}_2{\mathrm{Si}}_2$ measured in $H$ up to 6 T (closed symbols). The zero field ${\kappa }_E(T)$ was extrapolated from the high-field data (open circles). At low $T$, the error bars are about the size of the symbols. Inset: Comparison of the total ${\kappa }_C(T)$, the electronic contribution calculated using the Weidemann-Franz law ($L_0{\sigma }_{C}T$), and the zero field data in the main panel (${\kappa }_E$). (b) Raw thermal Hall signal as a function of $H$ at fixed $T$. Inset: Electrical Hall angle as a function of $T$ at various fixed $H$. (c) ${\kappa }_{XY}$ as a function of $H$ at various fixed $T$. Inset: Hall Lorenz ratio, $L_{XY}$, as a function of $T$ and normalized to $L_0$, For clarity, only selected data sets are shown for panels (b) and  (c). Lines are guides for the eye.Reuse & Permissions

Figure 3

(a) The BRT model (open symbols, described in text) describes the observed increase in $\kappa (T)$ for ${\mathrm{URu}}_2{\mathrm{Si}}_2$ (solid line) very well. A Debye-Calloway approximation (dashed line) was fit to the experimental $\kappa (T)$ for the $x=4\%$ sample (open up-triangles). The energy gap, $2\Delta (T)/k_B{T}_{\mathrm{HO}}$ determined by Rodrigo et al. (inset) was directly inserted into the BRT model. The parameter $\delta$ (defined in Fig. 1) was added in varying amounts (described in text) in parallel to the BRT model, in order to simulate a partially gapped Fermi surface. (b) An energy gap model using the $\Delta (T)$ data of Rodrigo et al. also describes the heat capacity measurements rather well, shown in the lower right inset, for which a value of $\delta =0.5$ (open squares) best accounts for the ungapped amount of DOS. A mean-field solution for $\Delta (T)$ was inserted into the BRT equation for the Rh doped samples, using a $\delta$ value determined from the inset of Fig. 1b. The ratio $2\Delta (0)/k_B{T}_C$ was progressively increased in order to match the experimental results. The red symbols indicate the best fit to the data for both panels.Reuse & Permissions