Probing unconventional superconductivity in inversion-symmetric doped Weyl semimetal

Unconventional superconductivity has been predicted to arise in the topologically nontrivial Fermi surface of doped inversion-symmetric Weyl semimetals (WSMs). In particular, Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) and nodal BCS states are theoretically predicted to be possible superconductor pairing states in inversion-symmetric doped WSMs. In an effort to resolve the preferred pairing state, we theoretically study two separate four-terminal quantum transport methods that each exhibit a unique electrical signature in the presence of FFLO and nodal BCS states in doped WSMs. We first introduce a Josephson junction that consists of a doped WSM and an $s$-wave superconductor in which we show that the application of a transverse uniform current in $s$-wave superconductors effectively cancels the momentum carried by FFLO states in doped WSMs. From our numerical analysis, we find a peak in Josephson current amplitude at finite uniform current in $s$-wave superconductors that serves as an indicator of FFLO states in doped WSMs. Furthermore, we show using a four-terminal measurement configuration that the nodal points may be shifted by an application of transverse uniform current in doped WSMs. We analyze the topological phase transitions induced by nodal pair annihilation in nonequilibrium by constructing the phase diagram and we find a characteristic decrease in the density of states that serves as a signature of the quantum critical point in the topological phase transition, thereby identifying nodal BCS states in doped WSMs.

Figure 1

A schematic of the system. $H_L$ is a WSM and $H_R$ is an ordinary metal superconductor. A weak coupling between $H_L$ and $H_R$ is assumed. A Josephson current flows in the $\widehat{x}$ direction (blue dashed arrow) and a uniform supercurrent in the $\widehat{z}$ direction (red solid arrow) gives center-of-mass momentum $q$ to the $H_R$ system.

Figure 2

Plot of Josephson current maximum $I_{J,max}$ in Eq. (14) as a function of momentum, $q$, in the BCS superconductor as described by $H_R$. There are two clear peaks when the $q$ matches with the $\pm Q$ in the doped WSM described by $H_L$. The parameters $t_m=1,{\mu }_R=0$ are used for $H_R$ and $t_x=0.5,t_y=0.5,t_z=1.0,\lambda =0.5,{\mu }_L/t=0.2$, and $Q=0.1\pi$ are used for $H_L$. The pairing potentials ${\mathrm{\Delta }}_L/t={\mathrm{\Delta }}_R/t=0.2$ are used and the number of points along the longitudinal direction ($\widehat{x}$), $N_x=10$, is fixed for both $H_L$ and $H_R$. In order to see the finite-size effect of the Josephson junction, we plot $N_z=20$ to $N_z=50$.

Figure 3

(a) A schematic of the system. Uniform supercurrent $J_S$ is driven to the Weyl superconductor system $H_w$. Differential conductance is read from a current measured in perpendicular direction ($I$). (b) Phase diagram of the number of nodal point pairs from Hamiltonian Eq. (15) at $k_x=k_y=0$. A DOS is obtained in the particular direction indicated by the red vertical arrow and plotted in Fig. 4. For the WSM Hamiltonian, the same parameters used in Fig. 2 are adopted. The range of $q$ presented here is $0\le q\le \pi /2$ due to the fact that a relevant range of total Cooper pair momentum is $\left|2q\right|\le \pi$.

Figure 4

A DOS plot along the vertical red arrow in Fig. 3. (a) DOS is plotted as a function of $q$ within a superconducting gap. Pairing potential is set to be ${\mathrm{\Delta }}_0/t_z=0.2$ with (a) $N_y=50$ and (c) $N_y=5$. A chemical potential is $\mu /t_z=0.2$. (b) DOS plot at $E=0$ as a function of $q$ for $N_y=50$ [white dotted line in (a)]. (d) DOS plot at $E=0$ as a function of $q$ for $N_y=5$ [white dotted line in (c)].