Phase-dependent heat transport in Josephson junctions with $p$-wave superconductors and superfluids

We investigate phase-coherent thermal transport in Josephson junctions made from unconventional superconductors or superfluids. The thermal conductance is evaluated for one- and two-dimensional junctions within the Bogoliubov–de Gennes formalism. We analyze three different scenarios of junctions between two triplet superconductors with (i) helical pairing, (ii) unitary chiral pairing, and (iii) nonunitary chiral pairing. We find that the phase-dependent thermal conductance allows us to distinguish the different pairings and provides insight into the formation of topologically nontrivial Andreev bound states in the junction.

Figure 1

(a) Short Josephson junction consisting of two superconducting reservoirs with order parameters ${\widehat{\mathrm{\Delta }}}_L$ and ${\widehat{\mathrm{\Delta }}}_R$ connected by a normal region $N$. The two superconductors are at the same chemical potential $\mu$ while a temperature bias $\mathrm{\Delta }T$ and phase bias $\varphi$ are applied across the junction. (b) Scattering states: an incoming electronlike state ${\psi }_{e,i}$ (dashed blue line) impinging on the junction under an angle $\alpha$ gives rise to transmitted electronlike quasiparticles $t_{e,j}{\tilde{\psi }}_{e,j}$ (blue solid line) and holelike quasiparticles $t_{h,j}{\tilde{\psi }}_{h,j}$ (red solid line) as well as to reflected electronlike quasiparticles $r_{e,j}{\psi }_{e,j}$ (blue solid line) and holelike quasiparticles $r_{h,j}{\psi }_{h,j}$ (red solid line). Note that the group velocity ${\mathbf{v}}_{\mathbf{g}}\propto {\partial }_{\mathbf{k}}\varepsilon$ is parallel to the momentum vector for electronlike modes and antiparallel for holelike modes.

Figure 2

Thermal conductance $\kappa$ in units of the heat flux quantum $G_Q={\pi }^2{k}_B^{2}T/\left(3h\right)$ for a one-dimensional SNS junction with helical pairing. (a) Comparison of $\kappa \left(\varphi \right)$ between the $s$- and helical $p$-wave cases for varying transmission $\tau$. (b) Comparison of $\kappa \left(\tau \right)$ between the $s$- and $p$-wave cases for $\varphi =0$ and $\varphi =\pi$. In contrast to the trivial $s$-wave pairing there is always a minimum of $\kappa$ at $\varphi =\pi$ for the $p$-wave case independent of $\tau$. The temperature is set to $k_{\text{B}}T=\left|{\mathrm{\Delta }}_0\right|/2$.

Figure 3

Transmission function $\mathcal{T}(\varepsilon ,\varphi )$ of a one-dimensional SNS junction with helical pairing for different transmissions $\tau$. The presence of a topologically protected Majorana bound state at $\varphi =\pi$ is connected to a minimum of the transmission above the superconducting gap.

Figure 4

(a) Phase-dependent thermal conductance for a unitary chiral one-dimensional SNS junction for different angles $\vartheta$ between the left and right $d$ vectors and transmission $\tau =0.8$. (b) Transmission function $\mathcal{T}(\varepsilon ,\varphi )$ for $\vartheta =\pi /2$ and $\tau =0.8$. Parameters are chosen as in Fig. 2.

Figure 5

(a) Thermal conductance $\kappa \left(\varphi \right)$ of a one-dimensional SNS junction with nonunitary chiral pairing for different angles $\vartheta$ between the $d$ vectors on the left and right sides for a transmission $\tau =0.8$. (b) Transmission function $\mathcal{T}(\varepsilon ,\varphi )$ for $\vartheta =\pi /2$ and $\tau =0.8$. We remark that $\mathcal{T}(\varepsilon ,\varphi )$ accounts for transmission of electron- and holelike quasiparticles of both spins and hence is bounded by 4. Parameters are chosen as in Fig. 2.