Chiral anomaly induced oscillations in the Josephson current in Weyl semimetals

Weyl semimetals are a three-dimensional topological phase of matter with linearly dispersed Weyl points which appear in pairs and carry opposite chirality. The separation of paired Weyl points allows charge transfer between them in the presence of parallel electric and magnetic fields, which is known as the chiral anomaly. In this paper, we theoretically study the influence of the chiral anomaly induced chiral charge imbalance on the Josephson current in a Weyl superconductor–Weyl semimetal–Weyl superconductor junction. In Weyl superconductors, two types of pairings are considered, namely, zero momentum BCS-like pairing and finite momentum Fulde-Ferrell-Larkin-Ovchinnikov (FFLO)-like pairing. For BCS-like pairing, we find that the Josephson current exhibits $0\text{−}\pi$ transitions and oscillates as a function of ${\lambda }_{0}L$, where ${\lambda }_0$ is the chirality imbalance induced by parallel electric and magnetic fields and $L$ is the length of the Weyl semimetal. The amplitude of the Josephson current also depends on the angle $\beta$ between the line connecting two paired Weyl points and the transport direction along the junction. For FFLO-like pairing, chirality imbalance induced periodic oscillations are absent and the Josephson current is also independent of the angle $\beta$. These findings are useful in detecting the chiral anomaly and distinguishing the superconducting pairing mechanism of Weyl semimetals.

Figure 1

Left: Schematic diagram of the WSC-WSM-WSC Josephson junction: A WSM at $0<z<L$ sandwiched between two WSCs at $z<0$ and $z>L$. The left (right) WSC is characterized by the phase ${\phi }_L$ (${\phi }_R$). Right: Schematic of the momentum space with two WPs located at $\pm {\mathbf{k}}_0$ in the $k_x\text{−}{k}_z$ plane; the angle between the line joining the two WPs and the $k_z$ axis is $\beta$. There exists a chirality imbalance between two WPs.

Figure 2

Energy dispersion $E\left(k_z\right)$ of the middle WSM region: (a) for BCS-like pairing and (b) for FFLO-like pairing with $\mu =0.3$. Solid (dashed) lines are for electrons (holes). Red (blue) lines are for with (without) the chirality imbalance.

Figure 3

Josephson current as a function of the superconducting phase difference $\phi$. (a) and (b) are for BCS-like pairing, and (c) and (d) are for FFLO-like pairing. (a) and (c) are for different values of the length $L$ with fixed ${\lambda }_0=0.3$ and (b) and (d) are for different values of ${\lambda }_0$ with fixed $L=6.6\pi$. The other parameters are $\beta =\pi /4$, $\mu =0.3$, and $T=0.5T_c$.

Figure 4

Energy levels $E_b$ of the Andreev bound states [solutions of Eq. (7)] for BCS-like pairing. In (b)–(f), the black curves represent bound levels for the pair $H_B^{+}$ and the red curves represent the bound levels for the pair $H_B^{-}$. The other parameters are the same as in Fig. 3.

Figure 5

(a) The Josephson current $J(\pi /2)$ for BCS-like pairing as a function of $L$ for various ${\lambda }_0$; the other parameters are the same as those in Fig. 3. (b) The Josephson current $J(\pi /2)$ as a function of the angle $\beta$, for BCS-like pairing (black solid line) and FFLO-like pairing (blue dotted line); the other parameters are $L=0$, ${\lambda }_0=0.3$, $\mu =0.3$, and $T=0.5T_c$.

Figure 6

Left: The critical current $J_c$ as a function of the length $L$ with fixed ${\lambda }_0=0.3$ for (a) BCS-like and (c) FFLO-like pairing. Right: The critical current $J_c$ as a function of ${\lambda }_0$ with fixed length $L=6.6\pi$ for (b) BCS-like and (d) FFLO-like pairing. The other parameters are $\beta =\pi /4$, $\mu =0.3$, and $T=0.5T_c$.