Self-consistent solution for proximity effect and Josephson current in ballistic graphene SNS Josephson junctions

We use a tight-binding Bogoliubov–de Gennes (BdG) formalism to self-consistently calculate the proximity effect, Josephson current, and local density of states in ballistic graphene superconductor-normal conductor-superconductor (SNS) Josephson junctions. Both short and long junctions, with respect to the superconducting coherence length, are considered, as well as different doping levels of the graphene. We show that self-consistency does not notably change the current-phase relationship derived earlier for short junctions using the non-self-consistent Dirac–BdG formalism [M. Titov and C. W. J. Beenakker, Phys. Rev. B 74, 041401(R) (2006)] but predict a significantly increased critical current with a stronger junction-length dependence. In addition, we show that in junctions with no Fermi-level mismatch between the N and S regions, superconductivity persists even in the longest junctions we can investigate, indicating a diverging Ginzburg-Landau superconducting coherence length in the normal region.

Figure 1

Schematic of (a) an experimental SNS graphene Josephson junction and (b) the model setup with input parameters: pairing potential $U$, effective chemical potential $\tilde{\mu }$, length of normal region $L$, and region where the phase of the order parameter will be kept fixed $X$.

Figure 2

(Color online) The graphene honeycomb lattice with the two different atomic sites $f$ and $g$, the unit cell vectors $\left\{{\mathbf{c}}_1,{\mathbf{c}}_2\right\}$, the three nearest-neighbor directions $\left\{{\mathbf{a}}_1,{\mathbf{a}}_2,{\mathbf{a}}_3\right\}$, the zigzag interface with its $T=a=2.46\text{Å}$ long unit cell in the $\hat{\mathbf{y}}$ direction, and the $\left(n,m\right)$ notation for labeling each unit cell.

Figure 3

(Color online) Normalized pair amplitude $F_U$ profiles for a $\text{S}=50$, $\text{N}=50$ unit cells junction with three different doping levels in N: at Dirac point 0 eV (black), moderate doping 0.7 eV (red), no FLM 1.5 eV (green).

Figure 4

(Color online) $I\left(\varphi \right)$ relationship, normalized with respect to the critical current in (e) for short $L<\xi$ (a), (c), (e) and long $L>\xi$ (b), (d), (f) junctions when the N region is doped at the Dirac point $\tilde{\mu }=0\text{eV}$ (e), (f), moderately doped $\tilde{\mu }=0.7\text{eV}$ (c), (d), and with no FLM $\tilde{\mu }=1.5\text{eV}$ (a), (b). Self-consistent numerical data are in black with the lines being guides to the eye. Red curves are least-squares fits (of the overall constant) to the DBdG results whereas green curves are fits to the functional form $\text{sgn}\left[\text{cos}\left(\varphi /2\right)\right]\text{sin}\left(\varphi /2\right)$.

Figure 5

(Color online) Normalized critical Josephson current $I_c$ as function of length $L$ of the junction at (a) the Dirac point $\tilde{\mu }=0\text{eV}$ and (b) moderately heavy doping $\tilde{\mu }=0.7\text{eV}$. Self-consistent results (crosses), Eqs. (19, 20) (black), and least-square fits to the functional forms $C{L}^b$ (red) and ${\text{Ce}}^{-L/{\xi }_N}$ (green). The current is normalized such that the critical current in Eq. (20) is equal to 1.

Figure 6

(Color online) Typical LDOS for SN junction as function of the normalized energy defined with reference to the Dirac point. Upper plots (a), (b): N region is undoped, at the Dirac point. Lower plots (c), (d): N is moderately doped. Right-hand plots are magnifications of the gap features of the left-hand plots. Different colors represent different positions inside the junction: middle of S (black), middle of N (red), interface (green), interface $\pm 2$ unit cells (magenta, cyan), interface $-20$ unit cells (blue).