Josephson current in graphene: Role of unconventional pairing symmetries

We investigate the Josephson current in a graphene superconductor/normal/superconductor junction, where superconductivity is induced by means of the proximity effect from external contacts. We take into account the possibility of anisotropic pairing by also including singlet nearest-neighbor interactions, and investigate how the transport properties are affected by the symmetry of the superconducting order parameter. This corresponds to an extension of the usual on-site interaction assumption, which yields an isotropic $s$-wave order parameter near the Dirac points. Here, we employ a full numerical solution as well as an analytical treatment, and show how the proximity effect may induce exotic types of superconducting states near the Dirac points, e.g., $p_x$- and $p_y$-wave pairing or a combination of $s$- and $p+\text{i}p$-wave pairing. We find that the Josephson current exhibits a weakly damped, oscillatory dependence on the length of the junction when the graphene sheet is strongly doped. The analytical and numerical treatments are found to agree well with each other in the $s$-wave case when calculating the critical current and current-phase relationship. For the scenarios with anisotropic superconducting pairing, there is a deviation between the two treatments, especially for the effective $p_x$-wave order parameter near the Dirac cones which features zero-energy states at the interfaces. This indicates that a numerical, self-consistent approach becomes necessary when treating anisotropic superconducting pairing in graphene.

Figure 1

(Color online) The experimental setup proposed in this paper. Two superconducting electrodes in close proximity to a graphene sheet induces superconductivity. The Josephson current flows through the graphene sheet indicated by the red arrow. Top and bottom gates contacted to the graphene sheet permit local control over the chemical potential.

Figure 2

(Color online) The graphene honeycomb lattice with the two different atomic sites $A$ and $B$, the three nearest-neighbor directions $\left\{{\mathit{a}}_1,{\mathit{a}}_2,{\mathit{a}}_3\right\}$, and the zigzag and armchair interfaces marked.

Figure 3

(Color online) Josephson current for the $s$-wave symmetry with no FVM between the $S$ and $N$ regions $\left({\mu }_{\text{S}}={\mu }_{\text{N}}=\mu \right)$. (a) Plot of the length dependence of the critical current. (b) Plot of the current-phase relationship for $\mu /{\Delta }_0=50$.

Figure 4

(Color online) Josephson current for the $s$-wave symmetry with a strong FVM between the $S$ and $N$ regions $\left({\mu }_N=\mu \right)$. (a) Plot of the length dependence of the critical current. (b) Plot of the current-phase relationship for $L/\xi =0.1$.

Figure 5

(Color online) Plot of the $\mathit{q}$ dependence of the two $d$-wave symmetry order parameters $\left(C_{\mathit{q}}\right)$. Red color represents positive sign, blue color negative sign, with zero being white. The first Brillouin zone is marked with a black line. Near the Dirac points, $\left(q_x,q_y\right)=\left(0,\pm \frac{4\pi }{3\sqrt{3}a}\right)$, at the zone corners, effective $p_y$- or $p_x$-wave symmetries emerge in the low-energy regime.

Figure 6

(Color online) Josephson current for the $d_{xy}$-wave symmetry (giving rise to an effective $p_x$-wave symmetry at the Dirac points) with no FVM between the $S$ and $N$ regions. (a) Plot of the length dependence of the critical current. (b) Plot of the current-phase relationship for $\mu /{\Delta }_0=50$.

Figure 7

(Color online) Josephson current for the $d_{xy}$-wave symmetry (giving rise to an effective $p_x$-wave symmetry at the Dirac points) with a strong FVM between the $S$ and $N$ regions. (a) Plot of the length dependence of the critical current. (b) Plot of the current-phase relationship for $L/\xi =0.1$.

Figure 8

(Color online) Proximity effect in terms of normalized pair amplitudes when $\mu \left(\text{N}\right)=0\text{eV}$ (a), $\mu \left(\text{N}\right)=0.7\text{eV}$ (b), and $\mu \left(\text{N}\right)=1.5\text{eV}$ (no FVM) (c). Conventional $s$-wave contacts (solid black), $d_{x^2-y^2}$-wave contacts with zigzag interfaces (dashed red), $d_{xy}$-wave contacts with zigzag interfaces (dotted blue), and $d_{x^2-y^2}$-wave contacts with armchair interfaces (dash-dotted green). The width of the $N$ region is $L=0.42\xi$ and the interfaces are marked with vertical black lines.

Figure 9

LDOS plots for conventional $s$-wave contacts (a, d), $d_{x^2-y^2}$-wave contacts with zigzag interfaces (b, e) and armchair interfaces (c, f). The doping level in the $N$ region is zero (a, b, c) and moderately doped at 0.7 eV (d, e, f). The energy scale has been normalized by the order parameter ${\Delta }_U$. Black color represent 2 states/eV/unit cell.

Figure 10

(Color online) $I$ vs $\Delta \varphi$ for ${\mu }_{\text{N}}=0\text{eV}$ (a), 0.7 eV (b), and 1.5 eV (no FVM) (c) for conventional $s$-wave contacts (black, $\times$), $d_{x^2-y^2}$-wave contacts with zigzag interfaces (red, ○) and armchair interfaces (green, $+$). The current is given in units of $I_0=eW/\left(\hslash \xi \right)$ when $v_F$ is set to 1. Analytical results in (a, b) are shown with dashed lines and calculated for the width $W=30\xi$, which approximates the infinite width of the numerical results.

Figure 11

(Color online) $I_c$ vs $L$ for ${\mu }_{\text{N}}=0\text{eV}$ (a) and 0.7 eV (b) for conventional $s$-wave contacts (black, $\times$), $d_{x^2-y^2}$-wave contacts with zigzag interfaces (red, ○) and armchair interfaces (green, $+$). The current is given in units of $I_0=eW/\left(\hslash \xi \right)$ when $v_F$ is set to 1. Analytical results are shown with dashed lines and calculated for the width $W=30\xi$, which approximates the infinite width of the numerical results.

Figure 12

(Color online) $I_c$ (black, left axis) and $\Delta {\varphi }_c$ (red, right axis) vs $L$ for ${\mu }_{\text{N}}=0\text{eV}$ (a) and 0.7 eV (b) for the $d_{x^2-y^2}$-wave solution on the armchair interface.