Theory of thermal conductivity in extended-$s$ state superconductors: Application to ferropnictides

Within a two-band model for the recently discovered ferropnictide materials, we calculate the thermal conductivity assuming general superconducting states of $A_{1g}$ (“$s$-wave”) symmetry, considering both currently popular isotropic “sign-changing” $s$ states and states with strong anisotropy, including those which manifest nodes or deep minima of the order parameter. We consider both intraband and interband disorder scattering effects, and show that in situations where a low-temperature linear-$T$ term exists in the thermal conductivity, it is not always “universal” as in $d$-wave superconductors. We discuss the conditions under which such a term can disappear, as well as how it can be induced by a magnetic field. We compare our results to several recent experiments.

Figure 1

(Color online) (a) Density of states and (b) thermal conductivity for an isotropic $s_{\pm }$ state with ${\Delta }_1=-{\Delta }_2$, shown for $U_d=U_{11}=U_{22}$ (intraband scattering) and scattering-rate parameters $c=0.07$ and $\Gamma =0.3T_{c0}$ in cases (i) weak intraband scattering only, $U_{12}/U_d=0$ (solid line); (ii) pair-breaking scattering with midgap impurity band, $U_{12}/U_d=0.98$ (dashed line); and (iii) pair-breaking scattering with impurity band overlapping Fermi level, $U_{12}/U_d=1.0$ (dotted line).

Figure 2

(Color online) Density of states (inset) and thermal conductivity for an isotropic $s_{\pm }$ state with scattering parameters as Fig. 1 but $U_{12}/U_d=1.0$ and $\Gamma =0.2,0.25$, and $0.3T_{c0}$.

Figure 3

(Color online) Density of states $N\left(w\right)/N_{Total}$ (top) and normalized thermal conductivity $\left[\kappa \left(T\right)/T\right]/\left({\kappa }_n/T_c\right)$ vs $T/T_c$ for two-band anisotropic model with isotropic order parameter on sheet 1 and anisotropic order parameter ${\Delta }_1=-1.1T_{c0}$ on sheet 1 and ${\Delta }_{iso}=1.3T_{c0}$, $r=1.3$ on sheet 2 [Eq. (6)]. Results shown for various values of intraband-scattering rate $\Gamma /T_{c0}$ and $c=0.07$, and no intraband scattering, $U_{12}=0$.

Figure 4

(Color online) Magnitude of linear-$T$ term in thermal conductivity in limit $T\to 0$ for case with nodes $r=1.3$, plotted as a function of intraband-scattering rate $\Gamma /T_{c0}$ for $c=0.07$. Solid line: exact numerical result. Dashed line: analytical estimate from text, Eq. (14). Here ${\kappa }_{00}$ is the thermal conductivity for a $d$-wave superconductor with order parameter $\Delta \left(\varphi \right)={\Delta }_{ani}\text{cos}2\varphi$. ${\Delta }_{ani}$ is same as anisotropic component in clean system for the model specified by Eq. (6).

Figure 5

(Color online) (Top) normalized density of states $N\left(w\right)/N_0$ vs $\omega /T_{c0}$ for two-band anisotropic model with isotropic gap on sheet 1 and $r=0.9$ (deep gap minima) on sheet 2 [Eq. (6)]. Results are shown for various values of intraband-scattering rate $\Gamma /T_{c0}$ and $c=0.07$. Bottom: normalized thermal conductivity $\left[\kappa \left(T\right)/T\right]/\left({\kappa }_n/T_c\right)$ vs $T/T_c$.

Figure 6

(Color online) Suppression of critical temperature, $T_c/T_{c0}$, by impurities, $\Gamma /T_{c0}$, for the cases of only intraband scattering ($U_{12}=0$, solid curve), and uniform scattering ($U_{12}=U_d$, dashed curve).

Figure 7

(Color online) Density of states $N\left(w\right)/N_0$ (top) and normalized thermal conductivity $\kappa \left(T\right)T_c/\left({\kappa }_{n}T\right)$ vs $T/T_c$ for two-band anisotropic model with isotropic order parameter on sheet 1 and $r=0.9$ (deep gap minima) on sheet 2 [Eq. (6)]. Results are shown for equal intraband and interband scattering $U_{12}=U_d$.

Figure 8

(Color online) $\kappa /T$ in $T\to 0$ limit, for the state with deep gap minima. ${\kappa }_{00}$ is thermal conductivity for pure $d$-wave state. Band 1 has isotropic order parameter $\Delta =-1.7T_{c0}$, and band 2 has $r=0.9$ and ${\Delta }_{iso}=1.5T_{c0}$.

Figure 9

(Color online) The field dependence of the thermal conductivity at $T=0.02T_{c0}$, for the order parameter with nodes, $r=1.3$, near unitary limit ($c=0.07$). (a) $\kappa (H)$ for different impurity concentrations, no interband scattering; (b, c) evolution of the field dependence with increased interband scattering for two characteristic $\Gamma /T_{c0}=0.31,0.94$. The field unit, $H_0={\varphi }_{0/2\pi }{\xi }_0\sim H_{c2}$, where ${\varphi }_0$ is the magnetic flux quantum, and ${\xi }_0=v_{F/2\pi }{T}_{c0}$ is the coherence length.

Figure 10

(Color online) The field dependence of the thermal conductivity at $T=0.02T_{c0}$ for the state with deep minima in the gap, $r=0.9$, for several impurity concentrations in the limits of weak (a) and strong (b) interband scattering.

Figure 11

(Color online) The low temperature field dependence of the thermal conductivity for the $s_{\pm }$ state, unitary limit ($c=0.07$). (a) $\kappa (H)$ without pairbreaking interband scattering; dependence of $\kappa (H)$ on the interband scattering for small (b) and large (c) impurity concentrations, corresponding to $\Gamma /T_{c0}=0.13$ and $\Gamma /T_{c0}=0.94$.

Figure 12

(Color online) These are the impurity averaged diagrams, which contribute to the self-energy of the first-band Green’s function. Here the interband contribution comes through processes, which involve even number of interband scatterings. The diagrams also take into account the order of interband and intraband scatterings. $U_{ij}$ is the impurity potential strength, where $i,j$ are the band indexes. $i=j$ denotes the intraband and $i\ne j$ denotes the interband potential strength with $U_{ij}=U_{ji}$ and $G_i$ is the bare Green’s function.