Theory of thermal and charge transport in diffusive normal metal/superconductor junctions

Thermal and charge transport in diffusive normal metal (DN)/insulator/$s$-, $d$-, and $p$-wave superconductor junctions are studied based on the Usadel equation with the Nazarov’s generalized boundary condition. We derive a general expression of the thermal conductance in unconventional superconducting junctions. Thermal conductance, electric conductance of junctions and their Lorentz ratio are calculated as a function of resistance in DN, the Thouless energy, magnetic scattering rate in DN and transparency of the insulating barrier. We also discuss transport properties for various orientation angles between the normal to the interface and the crystal axis of superconductors. It is demonstrated that the proximity effect does not influence the thermal conductance while the midgap Andreev resonant states suppress it. Dependencies of the electrical and thermal conductance on temperature are sensitive to pairing symmetries and orientation angles. The results imply a possibility to distinguish one pairing symmetry from another based on the results of experimental observations.

Figure 1

Schematic illustration of trajectories of incoming and outgoing quasiparticles at a DN/S interface with pair potential ${\Delta }_{\pm }\left(\varphi \right)$. The pair potentials ${\Delta }_{\pm }$ are given by ${\Delta }_{\pm }=\Delta \left(T\right)\cos \left[2(\varphi \mp \alpha )\right]$ for $d$-wave superconductors ($D$-wave) where $\alpha$ denotes the angle between the normal to the interface and the crystal axis of $d$-wave superconductors. For $p$-wave superconductors ($P$-wave) the pair potentials ${\Delta }_{\pm }$ are given by ${\Delta }_{\pm }=\pm \Delta \left(T\right)\cos \left[(\varphi \mp \alpha )\right]$, where $\alpha$ denotes the angle between the normal to the interface and the lobe direction of the $p$-wave pair potential. In the above, $\varphi$ denotes the injection angle of the quasiparticle measured from the $x$ axis and $\Delta \left(T\right)$ is the maximum amplitude of the pair potential at a temperature $T$.

Figure 2

Normalized tunneling conductance for $s$-wave superconductor with $Z=3$. (a) $R_d∕R_b=1$ and $E_{\mathrm{Th}}∕\Delta \left(0\right)=0.1$. (b) $R_d∕R_b=0.1$ and $E_{\mathrm{Th}}∕\Delta \left(0\right)=0.1$. (c) $R_d∕R_b=1$ and $E_{\mathrm{Th}}∕\Delta \left(0\right)=0.01$. (d) $R_d∕R_b=0.1$ and $E_{\mathrm{Th}}∕\Delta \left(0\right)=0.01$.

Figure 3

Normalized tunneling conductance for $s$-wave superconductor with $Z=0$. (a) $R_d∕R_b=1$ and $E_{\mathrm{Th}}∕\Delta \left(0\right)=0.1$. (b) $R_d∕R_b=0.1$ and $E_{\mathrm{Th}}∕\Delta \left(0\right)=0.1$. (c) $R_d∕R_b=1$ and $E_{\mathrm{Th}}∕\Delta \left(0\right)=0.01$. (d) $R_d∕R_b=0.1$ and $E_{\mathrm{Th}}∕\Delta \left(0\right)=0.01$.

Figure 4

(Color online) Normalized thermal conductance for $s$-wave superconductor with $Z=3$ and $R_d∕R_b=1$. (a) $\gamma ∕\Delta \left(0\right)=0$ and (b) $E_{\mathrm{Th}}∕\Delta \left(0\right)=0.1$.

Figure 5

Normalized thermal conductance for $s$-wave superconductor with $E_{\mathrm{Th}}∕\Delta \left(0\right)=0.1$ and $\gamma ∕\Delta \left(0\right)=0$. (a) $Z=3$ and (b) $Z=0$.

Figure 6

Normalized Lorentz ratio for $s$-wave superconductor with $E_{\mathrm{Th}}∕\Delta \left(0\right)=0.1$ and $\gamma ∕\Delta \left(0\right)=0$. (a) $Z=3$ and (b) $Z=0$.

Figure 7

Normalized tunneling conductance (a) and thermal conductance (b) for $s$-wave superconductor with $T∕T_C=0.8$, $E_{\mathrm{Th}}∕\Delta \left(0\right)=0.1$, and $\gamma ∕\Delta \left(0\right)=0$.

Figure 8

Normalized tunneling conductance for $d$-wave superconductor with $R_d∕R_b=0.1$, $E_{\mathrm{Th}}∕\Delta \left(0\right)=0.1$, and $\gamma ∕\Delta \left(0\right)=0$. (a) $Z=3$ and (b) $Z=0$.

Figure 9

Normalized thermal conductance for $d$-wave superconductor with $R_d∕R_b=0.1$, $E_{\mathrm{Th}}∕\Delta \left(0\right)=0.1$, and $\gamma ∕\Delta \left(0\right)=0$. (a) $Z=3$ and (b) $Z=0$.

Figure 10

Normalized Lorenz ratio for $d$-wave superconductor with $R_d∕R_b=0.1$, $E_{\mathrm{Th}}∕\Delta \left(0\right)=0.1$, and $\gamma ∕\Delta \left(0\right)=0$. (a) $Z=3$ and (b) $Z=0$.

Figure 11

Normalized tunneling conductance (a) and thermal conductance (b) for $d$-wave superconductor with $T∕T_C=0.8$, $Z=3$, $E_{\mathrm{Th}}∕\Delta \left(0\right)=0.1$, and $\gamma ∕\Delta \left(0\right)=0$.

Figure 12

Normalized tunneling conductance (a) and thermal conductance (b) for $d$-wave superconductor with $T∕T_C=0.8$, $Z=0$, $E_{\mathrm{Th}}∕\Delta \left(0\right)=0.1$, and $\gamma ∕\Delta \left(0\right)=0$.

Figure 13

Normalized tunneling conductance for $p$-wave superconductor with $R_d∕R_b=0.1$, $E_{\mathrm{Th}}∕\Delta \left(0\right)=0.1$, and $\gamma ∕\Delta \left(0\right)=0$. (a) $Z=3$ and (b) $Z=0$.

Figure 14

Normalized thermal conductance for $p$-wave superconductor with $R_d∕R_b=0.1$, $E_{\mathrm{Th}}∕\Delta \left(0\right)=0.1$, and $\gamma ∕\Delta \left(0\right)=0$. (a) $Z=3$ and (b) $Z=0$.

Figure 15

Normalized Lorenz ratio for $p$-wave superconductor with $R_d∕R_b=0.1$, $E_{\mathrm{Th}}∕\Delta \left(0\right)=0.1$, and $\gamma ∕\Delta \left(0\right)=0$. (a) $Z=3$ and (b) $Z=0$.

Figure 16

Normalized tunneling conductance (a) and thermal conductance (b) for $p$-wave superconductor with $T∕T_C=0.8$, $Z=3$, $E_{\mathrm{Th}}∕\Delta \left(0\right)=0.1$, and $\gamma ∕\Delta \left(0\right)=0$.

Figure 17

Normalized tunneling conductance (a) and thermal conductance (b) for $p$-wave superconductor with $T∕T_C=0.8$, $Z=0$, $E_{\mathrm{Th}}∕\Delta \left(0\right)=0.1$, and $\gamma ∕\Delta \left(0\right)=0$.

Figure 18

Chart for distinguishing $s$-, $d$-, and $p$-wave superconductors.

Figure 19

Magnetic field dependence of normalized tunneling conductance with $Z=3$ and $E_{\mathrm{Th}}∕\Delta \left(0\right)=0.1$ for (a) $d$-wave superconductor with $\alpha ∕\pi =0$ and (b) $p$-wave superconductor with $\alpha ∕\pi =0.25$.

Figure 20

The conductance measurable by STS at $x=3L∕4$ with $Z=3$, $R_d∕R_b=0.1$, $E_{\mathrm{Th}}∕\Delta \left(0\right)=0.1$, and $\gamma ∕\Delta \left(0\right)=0$ for $d$-wave superconductor with $\alpha ∕\pi =0.125$ and for $p$-wave superconductor with $\alpha ∕\pi =0.25$.