Low-Temperature Thermal Conductivity of Superconductors with Gap Nodes

We report a detailed analytic and numerical study of electronic thermal conductivity in $d$-wave superconductors. We compare theory of the crossover at low temperatures from $T$ dependence to $T^3$ dependence for increasing temperature with recent experiments on ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_7$ in zero magnetic field for $T\in [0.04\mathrm{K},0.4\mathrm{K}]$ by Hill et al. [Phys. Rev. Lett. 92, 027001 (2004)]. Transport theory, including impurity scattering and inelastic scattering within strong-coupling superconductivity, can consistently fit the temperature dependence of the data in the lower half of the temperature regime. We discuss the conditions under which we expect power-law dependences over wide temperature intervals.

Figure 1

Temperature dependence of the electronic contribution to the thermal conductivity for elastic scattering rates as indicated for a phase shift ${\delta }_0=89.5°$. Inset (a) the two electron-boson coupling spectra used in the calculations. Inset (b) single-particle density of states in the superconducting state at $T=0$.

Figure 2

(a)–(c) The low-temperature parameters in Eqs. (1, 3) for three values of the $T^2$ coefficient in Eq. (2) as a function of scattering phase shift. (d) Three sets of parameters that represent our best fits to the data of Ref. 12. The shaded region indicates the temperature range where a good fit can be made. The dashed curves are from a strong-coupling calculation while the underlying solid lines are generated by the effective theory. Curve (II) is the low-$T$ part of the black solid line in Fig. 1. In all cases $k=19.2{\mathrm{K}}^{-2}$. Note that the material appears cleaner in the effective theory as ${\Gamma }_0^{*}={\Gamma }_0/(1+{\lambda }_{\mathsf{i}\mathsf{n}})$ with ${\Gamma }_0=n_{\mathsf{i}\mathsf{m}\mathsf{p}}/\pi {\mathcal{N}}_f$. We normalized ${\kappa }_{\mathrm{el}}(T)/T$ to the $T\to 0$ extrapolated value $0.16\mathrm{mW}/({\mathrm{K}}^2\mathrm{cm})$. The energy scale is $T_c=90\mathrm{K}$.

Figure 3

(a)–(b) The impurity scattering self-energy and (c) the density of states for ${\Gamma }_0=0.01T_c$ and phase shifts ranging from 90° (dashed line) to 80° (solid line) in steps of 2°. (d) The kernel $\tilde{\kappa }(ϵ)$ of the thermal conductivity contains structures at low energies for phase shifts below 90° because of the split impurity bands seen in (a)–(c). Also shown is the derivative of the Fermi function that has a peak at the Fermi level of width $\sim T$. Here is $T=0.005T_c=0.45\mathrm{K}$ for $T_c=90\mathrm{K}$.