Evolution of pair correlation symmetries and supercurrent reversal in tilted Weyl semimetals

We study the effective symmetry profiles of superconducting pair correlations and the flow of charge supercurrent in ballistic Weyl semimetal systems with a tilted dispersion relation. Utilizing a microscopic method in the ballistic regime and starting from both opposite-pseudospin (the band degree of freedom) and equal-pseudospin phonon-mediated opposite-spin electron-electron couplings, we calculate the anomalous Green's function to study various superconducting pair correlations that Weyl semimetal systems may develop. The momentum-space profile reveals that by properly manipulating the parameters of Weyl semimetal systems, including the tilting parameter, the effective symmetry class of even-parity $s$-wave (odd-parity $p$-wave) superconducting correlations can be converted into a $d$-wave ($f$-wave) symmetry class that consists of equal-pseudospin and opposite-pseudospin channels. We also find that the supercurrent in a ballistic Weyl Josephson junction can be made to vanish or switch directions, depending on the tilt of the Weyl cones, in addition to the relevant parameters characterizing the Weyl semimetal and junction. We show that inversion-symmetry-breaking terms introduce transitions that result in the appearance of self-biased current at zero difference between the macroscopic phases of the superconducting segments, creating a ${\phi }_0$ Josephson state. Weyl semimetal systems are shown to offer several experimentally tunable parameters to control the induction of higher harmonics into the current phase relations.

Figure 1

Real and imaginary parts of the superconducting correlations for the opposite-pseudospin phonon-mediated electron-electron interaction plotted as a function of momenta $k_x$ and $k_z$. For illustrative purposes, the momentum component $k_y$ is set to zero. The following pair correlations are shown: $f_{\uparrow \downarrow }^{AA}$ and $f_{\uparrow \downarrow }^{AB}$. (a), (b) ($\gamma >\beta$) We use the representative parameters $\gamma =1.3$, $\beta =0.1$, ${\alpha }_z={\alpha }_{xy}=0.2$, $\mu =0.5$, $\eta =1$, and $Q=0.4\pi$. (c), (d) ($\gamma <\beta$) We use the representative parameters $\gamma =0.1$, $\beta =1.3$, ${\alpha }_z={\alpha }_{xy}=0.2$, $\mu =0.5$, $\eta =1$, and $Q=0.4\pi$.

Figure 2

Real and imaginary parts of the superconducting correlations for the equal-pseudospin phonon mediated electron-electron interaction plotted as a function of momenta $k_x$ and $k_z$. The following pair correlations are shown: $f_{\uparrow \downarrow }^{AA}$ and $f_{\uparrow \downarrow }^{AB}$. The parameters here are the same as in Fig. 1.

Figure 3

Real and imaginary parts of the superconducting correlations $f_{\uparrow \downarrow }^{AA}$ and $f_{\uparrow \downarrow }^{AB}$ plotted as a function of momenta $k_x$ and $k_z$. (a), (b) The opposite-pseudospin phonon-mediated electron-electron interaction is considered and parameters are identical to those of Figs. 1 and 1  except we now set $\gamma =-1.3$. (c), (d) The equal-pseudospin phonon mediated electron-electron interaction is set and parameters are identical to those of Figs. 2 and 2, except now we flip the sign of $\gamma$ to minus $\gamma =-1.3$.

Figure 4

Schematic of a Weyl semimetal Josephson junction. The junction interfaces lie in the $xy$ plane, with the normal to the interfaces along the $z$ direction. The interfaces between the normal-state Weyl semimetal and the superconductors are located at $z=\pm d/2$. The width of the junction is $W$ and the macroscopic phase of the left and right superconductors are labeled by ${\phi }_l$ and ${\phi }_r$, respectively.

Figure 5

Normalized current flowing in the $z$ direction, $J_z\left(\phi \right)$, as a function of the superconducting phase difference $\phi$ applied across the junction for various values of the tilting parameter $\beta$ and separation of Weyl nodes $Q$ (see legends). In (a), the orbital term ${\alpha }_z$ is set zero. In (b), the orbital term ${\alpha }_z$ possesses a representative finite value of 0.2. In (c), we set ${\alpha }_z$ zero and vary $Q$. The other parameters are set as follows: $\gamma =0.5$, $\beta =0.2$, ${\alpha }_{x,y}=1$, and $\eta =1$.

Figure 6

Current normalized charge current $J_{\text{max}}$ flowing across the Josephson junction shown in Fig. 5 as a function of junction thickness $d$. In (a), we set $Q=0.4\pi$ and (b) $Q=0.7\pi$ and show the influence of the tilting parameter $\beta$ on the critical supercurrent. The remaining junction parameters are set as follows: $\gamma =0.5$, ${\alpha }_{x,y}=1$, ${\alpha }_z=0$, and $\eta =1$.