Figure 1

Low-temperature transport properties of CeRu${}_2$Si${}_2$ as a function of magnetic field $B$. (a) The $A$ coefficient of the resistivity $\rho$ peaks at $B_{\mathrm{MMT}}=7.80\pm 0.05\mathrm{T}$. For comparison, similar data are included for the sample of Ref. 5 and also published specific-heat data on another sample (Ref. 17). (b) A peak in $\rho \left(B\right)$ appears at higher temperatures (lighter curves). The resistivity measured at 100 mK is close to the residual resistivity ${\rho }_0$, however, and shows no such peak at the MMT. (c) The derivative of ${\rho }_0$ with respect to magnetic field, compared to the sample of Ref. 5. The data of Ref. 5 presented in (a) and (c) have been slightly scaled so that $B_{\mathrm{MMT}}$ matches, accounting for a misalignment of the sample. The MMT appears much broader in $\rho$ of the present sample. (d) The thermal conductivity. Field sweeps (filled circles) and temperature sweeps (open squares) are in excellent agreement. The inset shows the Lorenz ratio at 100 mK. (e) The thermopower divided by temperature shows multiple sign changes and a strong negative peak near the MMT. There is a sharp kink in the thermopower precisely at $B_{\mathrm{MMT}}$, which is more clearly seen in Fig. 2. An additional peak at 1 T becomes sharper as the temperature is reduced.Reuse & Permissions

Figure 2

Comparing thermopower to thermodynamics. We calculate the electronic specific-heat coefficient $\gamma$ in two ways. On the left scale we plot $\gamma$ obtained from the Kadowaki-Woods ratio, by using the $A$ coefficient of the electrical resistivity in the formula $A/{\gamma }^2=R_{KW}$. On the right scale we use the $q$ ratio (see text) to scale the thermopower data into units of $\gamma$. We also invert the sign and offset it by 0.45 J mol${}^{-1}$ K${}^{-2}$ to match the left scale at zero field. An offset is justified by the presence of other bands, while the sign inversion is appropriate for heavy fermion materials. Clearly, the structure in the thermopower near the MMT is more complex than the thermodynamic signature. The inset shows an enlargement at $B_{\mathrm{MMT}}$ showing the sharp kink in the thermopower-derived data, which persists to higher temperatures.Reuse & Permissions

Figure 3

Model calculations of transport and thermodynamic properties at zero temperature. In the Lifshitz model (dark red/gray), the spherical Fermi surface is spin split until one band becomes completely unoccupied at $B_{\mathrm{MMT}}$. The total number of occupied states is conserved. In the peak model (light blue/gray), the Fermi energy passes through a peak in the density of states at $B_{\mathrm{MMT}}$. A combination of these models (see text for details, dashed green/gray) can qualitatively reproduce key features of both transport and thermodynamic properties. (a) The specific heat $C/T$. The observed specific heat can be well reproduced by the peak model. (b) The resistivity and Hall effect only have sharp features in the Lifshitz model calculation. (c) The thermopower of the Lifshitz model has an asymmetric structure and a sharp kink at $B_{\mathrm{MMT}}$.Reuse & Permissions

Figure 4

Lifshitz model calculation with experimental field dependence. The Zeeman energy $E$ is plotted in the top panel. Dashed lines show the results for $E\propto B$ and solid lines for $E\propto M$. The peak in the measured thermopower (dark red/gray, right axis) is asymmetric with respect to the critical field $B_{\mathrm{MMT}}$, with a greater apparent weight for $B<B_{\mathrm{MMT}}$. If the Zeeman splitting is instead taken to be proportional to the experimental magnetization (from Ref. 12, dashed lines) the features in the calculation appear much narrower.Reuse & Permissions