Thermal bias induced charge current in a Josephson junction: From ballistic to disordered

It is known that Josephson junction (JJ) hosting scattering centers with energy-dependent scattering amplitudes, which breaks the $\omega \to -\omega$ symmetry (where $\omega$ is the excitation energy of electron about the Fermi level) exhibits finite thermoelectric response. In contrast, here we show that even in a ballistic JJ this symmetry is broken and it leads to a nonzero thermal-bias induced charge current above the gap, when the junction length is of the order of coherence length of the superconductor and the corresponding response confirms to the universal sinusoidal dependence on ${\varphi }_{12}$, where ${\varphi }_{12}$ is the superconducting phase bias. In presence of multiple scatterers in the junction region, we have numerically shown that the sign of the even-in-${\varphi }_{12}$ part of the this response fluctuates violently from one disorder configuration to another hence averaging to vanishingly small values while the odd part tends towards the universal sinusoidal dependence on ${\varphi }_{12}$ as we approach the large disorder limit under disorder averaging.

Figure 1

Schematic of a finite-length Josephson junction set-up based on helical edge state of a 2D quantum spin Hall insulator. Superconducting leads ($S_1$ and $S_2$) are kept at different temperatures. Also there is a superconducting phase bias between $S_1$ and $S_2$. The up and down arrows indicate the direction of spin-polarization of the corresponding edge states.

Figure 2

(a) Pictorial representation of below the gap $(\omega <{\mathrm{\Delta }}_0)$ tunneling of Cooper pairs (CP) form left (right) to right (left) via two different Andreev bound states ${\omega }_0^{21}$ (indicated by blue lines) and ${\omega }_0^{12}$ (indicated by orange lines). The dotted lines represent the fact that tunneling of the quasielectrons (quasiholes) above the gap $(\omega >{\mathrm{\Delta }}_0)$ across the junction are in correspondence with distinct bound states, as can be noted from the poles of the quasielectron (quasihole) transmission probabilities ${\mathcal{T}}_{ee}^{21}$ and ${\mathcal{T}}_{ee}^{12}$ (or ${\mathcal{T}}_{hh}^{12}$ and ${\mathcal{T}}_{hh}^{21}$). (b) Two types of Andreev bound states as a function of superconducting phase difference ${\varphi }_{12}$ are plotted for different values of junction lengths where $\xi =\hslash v_F/{\mathrm{\Delta }}_0$ is the superconducting coherence length. (c) Density plot for the thermocurrent coefficient ${\kappa }_e^{21}$ of a ballistic JJ based on the edge states of a quantum spin Hall insulator in proximity to a $s$-wave superconductor, as a function of superconducting phase bias ${\varphi }_{12}$ and junction length $L$. The average temperature of the junction is considered to be $k_B{T}_{\text{avg}}=0.5{\mathrm{\Delta }}_0$ where the effective superconducting gap at this temperature is taken to be ${\mathrm{\Delta }}_{T_{\text{avg}}}\sim 0.56{\mathrm{\Delta }}_0$. Both the plots (b) and (c) are valid for any value of the chemical potential $\mu$.

Figure 3

Different transmission probabilities of the quasiparticles through a ballistic Josephson junction based on the helical edge state of a quantum spin Hall insulator, in the space of energy ($\omega$) and superconducting phase difference (${\varphi }_{12}$) for different values of junction length ($L$). A clear asymmetry between the electron and hole transmission probability from left (right) to right (left) develops as we increase the length of the junction. The plot in energy window $\left(\right|\omega |<{\mathrm{\Delta }}_0)$ signifies the evolution of the pole (location of Andreev bound states) of the transmission amplitude as a function of ${\varphi }_{12}$. The plots are valid for any value of the chemical potential $\mu$.

Figure 4

Thermocurrent coefficient of a Josephson junction based on helical edge state of quantum spin Hall insulator in presence of four random scattering centers. (a) Total thermocurrent coefficient (b) the part of thermal conductance that is even in ${\varphi }_{12}$ (c) the part of conductance that is odd in ${\varphi }_{12}$, for single random configuration of scattering centers and after averaging over different numbers of random configurations. The average temperature of the junction is considered to be $k_B{T}_{\text{avg}}=0.5{\mathrm{\Delta }}_0$, where the effective superconducting gap at this temperature is taken to be ${\mathrm{\Delta }}_{T_{\text{avg}}}\sim 0.56{\mathrm{\Delta }}_0$, and the overall chemical potential to be $\mu =10{\mathrm{\Delta }}_0$. Length of the junction is considered to be $L=0.87\xi$ where $\xi$ is the superconducting coherence length.

Figure 5

Disordered averaged mean value (left) and the variance (right) of the thermocurrent coefficient ${\kappa }_e^{21}$ of a Josephson junction based on the helical edge states of a quantum spin Hall insulator with proximity to a $s$-wave superconductor, are plotted as a function of superconducting phase difference ${\varphi }_{12}$. The average temperature of the junction is considered to be $k_B{T}_{\text{avg}}=0.5{\mathrm{\Delta }}_0$, where the effective superconducting gap at this temperature is taken to be ${\mathrm{\Delta }}_{T_{\text{avg}}}\sim 0.56{\mathrm{\Delta }}_0$, and the overall chemical potential to be $\mu =10{\mathrm{\Delta }}_0$. Length of the junction is considered to be $L=0.87\xi$ where $\xi$ is the superconducting coherence length. Average is done over 500 disorder configurations. The middle plot shows that, as we increase the number of scatterers, the curves for thermocurrent coefficients tend to a $sin\left({\varphi }_{12}\right)$ curves (solid lines) with an amplitude ($\max ({\kappa }_e^{12}$)-$\min ({\kappa }_e^{21}$))/2.

Figure 6

The maximum possible thermocurrent coefficient of a JJ with (left) $s$-wave and (right) $p$-wave superconductivity for a given junction length and normal state reflection probability $r$ (as calculated with the analytic approximation $\mu \gg {\mathrm{\Delta }}_0,k_{B}T$). Note that the scatterer is assumed to be energy independent and is placed at the middle of the junction. The parameters are assumed to be $\mu =100{\mathrm{\Delta }}_0$ and $k_B{T}_{\text{avg}}=0.5{\mathrm{\Delta }}_0$ where the effective superconducting gap at this temperature is taken to be ${\mathrm{\Delta }}_{T_{\text{avg}}}\sim 0.56{\mathrm{\Delta }}_0$.