Tight-binding study of bilayer graphene Josephson junctions

Using highly efficient simulations of the tight-binding Bogoliubov–de-Gennes model, we solved self-consistently for the pair correlation and the Josephson current in a superconducting-bilayer graphene–superconducting Josephson junction. Different doping levels for the non-superconducting link are considered in the short- and long-junction regimes. Self-consistent results for the pair correlation and superconducting current resemble those reported previously for single-layer graphene except at the Dirac point, where remarkable differences in the proximity effect are found, as well as a suppression of the superconducting current in the long-junction regime. Inversion symmetry is broken by considering a potential difference between the layers and we found that the supercurrent can be switched if the junction length is larger than the Fermi length.

Figure 1

Layout of the SNS-BLG Josephson junction where the superconducting leads are modeled by assuming on-site attractive pairing potential $U_S$ and a heavy doping ${\mu }_S$ in the regions under the contacts labeled by $S$. For the non-superconducting link, with length $L$ and labeled by $N$, we took the pairing potential $U_N=0$ and a varying chemical potential ${\mu }_N$. Phases ${\varphi }_L$ and ${\varphi }_R$ are kept fixed during the self-consistent calculation.

Figure 2

Absolute value of the pair correlation in a long (a) and short (b) BLG Josephson junction as a function of the position along the $\widehat{y}$ direction perpendicular to the $\mathit{SN}$ interface. Both inequivalent sites, dimer at $A$ and nondimer at $B$, are plotted separately for different doping levels considered for the non-superconducting region: ${\mu }_N=0$, $0.3t$, and $0.6t$. The first one corresponds to the case when the Fermi level is pinned at the Dirac point while the last one corresponds to no FLM at the interface. (c) Pair correlation at $L/2$ in the $N$ region as a function of the junction length $L$ for different values of ${\mu }_N$. SLG self-consistent results are shown for comparison with the BLG undoped case.

Figure 3

LDOS for A (red) and B (black) sublattice sites in the $N$ region with (a) zero doping (at the Dirac point) and (b) heavy doping (or with no-FLM between the superconducting and normal regions). (c) LDOS in the highly doped $S$ region showing the superconducting gap. The inset in panel (b) emphasizes the difference in the LDOS between A and B sublattice: a vanishing and finite density of states around the Dirac point. A magnification of (a) around the Dirac point is shown in panel (d), as well the LDOS of atomic sites at the interface for both sublattices. A close up view of the panels (b) and (c) around the zero energy is shown in (e) where the superconducting gap in the $S$ region and the Andreev states in the highly doped $N$ are discernible. SLG analogous cases (blue) have been included for comparison purposes. Note that the results presented here correspond to the long-junction limit.

Figure 4

Self-consistent Josephson current as a function of position along the $\widehat{y}$ direction for top (a) and bottom (b) layer as well as the average current (c) in the system for different values of the pairing potential, that is, different coherence lengths. The interlayer current between the $A$-$\tilde{B}$ dimer sites is plotted in (d).

Figure 5

Current-phase relation normalized to the critical current found in (e) for SLG- and BLG-based junctions considering different doping levels in the $N$ region: (e), (f) undoped ${\mu }_N=0$; (c), (d) slightly doped ${\mu }_N=0.1t$; and (a), (b) no-FLM ${\mu }_N={\mu }_S=0.6t$ for short (a), (c), (e) and long (b), (d), (f) junctions. The short- and long-junction lengths are $L=10$ and $L=50$, respectively.

Figure 6

Critical density current in a short BLG junction, with $L=40a$ and $U_S=-1.2t$, as a function of the gate voltage $V_g$ at the Dirac point (a) and the chemical potential ${\mu }_N$ (b). In the left panel two different regimes are considered, $L<{\lambda }_F$ and $L>{\lambda }_F$, showing an enhancement and suppression of the superconducting current, respectively. In the right panel $\mu$ dependence is plotted for bias and unbias cases. SLG is included for comparison.