Interplane and intraplane heat transport in quasi-two-dimensional nodal superconductors

We analyze the behavior of the thermal conductivity in quasi-two-dimensional superconductors with line nodes. Motivated by measurements of the anisotropy between the interplane and intraplane thermal transport in $\mathrm{Ce}\mathrm{Ir}{\mathrm{In}}_5$ we show that a simple model of the open Fermi surface with vertical line nodes is insufficient to describe the data. We propose two possible extensions of the model taking into account (a) additional modulation of the gap along the axial direction of the open Fermi surface and (b) dependence of the interplane tunneling on the direction of the in-plane momentum. We discuss the temperature dependence of the thermal conductivity anisotropy and its low $T$ limit in these two models and compare the results with a model with a horizontal line of nodes (“hybrid gap”). We discuss possible relevance of each model for the symmetry of the order parameter in $\mathrm{Ce}\mathrm{Ir}{\mathrm{In}}_5$, and suggest further experiments aimed at clarifying the shape of the superconducting gap.

Figure 1

(Color online) Fermi surfaces for models considered in the paper. Left panel: Models A, B, and D. Right panel: Model C. Notice the similarity between the Fermi surface shown in that panel and the Fermi surface suggested for $\mathrm{Ce}\mathrm{Ir}{\mathrm{In}}_5$ in Ref. 23.

Figure 2

(Color online) Temperatures dependence of the gap parameters in model B. Left panel: temperature evolution of the gaps in two channels for typical coupling values. Note the independence of the result on the cutoff frequency ${\Omega }_0$: for different cutoffs ${\Omega }_0∕T_c=10,100,1000$ the curves lie on top of one another. Right panel: Temperature dependence of the gap anisotropy $\delta$ for different coupling constants. Note the temperature variation of the anisotropy in the regime of moderately large ${\lambda }_2=1.6$ and small ${\lambda }_1=0.14$, where we present the results for ${\Omega }_0∕T_c=10,100,1000$.

Figure 3

(Color online) Gap amplitude as a function of the gap anisotropy for model B at low temperature. The BCS value ${\Delta }_{00}=2.14k_B{T}_c$. Note the smooth evolution of the gap amplitude across the value $\delta =1$, where a horizontal line of nodes appears.

Figure 4

(Color online) In-plane thermal conductivity for the modulated state, model B.

Figure 5

(Color online) Interplane thermal conductivity for the modulated state, model B.

Figure 6

(Color online) Anisotropy of the thermal conductivity for model B.

Figure 7

(Color online) Temperature dependence of the in-plane and the interplane thermal conductivity for the model with modulated hopping. The impurities are assumed to be in the unitarity limit, with $\Gamma$ the normal state scattering rate.

Figure 8

(Color online) Temperature dependence of the anisotropy of the interplane to the intraplane thermal conductivity. Note that the residual value is increasing with increasing $t_0∕t_1$.

Figure 9

(Color online) Temperature dependence of the in-plane and the interplane thermal conductivity for the hybrid gap. The impurities are assumed to be in the unitarity limit, with $\Gamma$ the normal state scattering rate.

Figure 10

(Color online) Temperature dependence of the anisotropy of the interplane to the intraplane thermal conductivity for the hybrid gap. For comparison, we include results with deviations from the unitarity scattering limit.

Figure 11

(Color online) Comparison of the temperature dependence of the anisotropy between the in-plane and $c$-axis conductivities for all the models considered. We show a strong modulation for model B, $\delta =-0.8$, and a moderately strong tunneling anisotropy for model C, $t_1=4t_0$. Notice the overlap of the results for models C (anisotropic tunneling, vertical lines of nodes) and D (hybrid gap, horizontal line of nodes).

Figure 12

(Color online) Increase of the residual thermal conductivity with impurity scattering rate.