Josephson effect in graphene Corbino disks

Abstract

Peculiar features of the Josephson effect in graphene were described theoretically by Titov and Beenakker [Phys. Rev. B 74, 041401(R) (2006)], who solved the Dirac-Bogoliubov-de-Gennes equation for a superconductor-graphene-superconductor junction with rectangular geometry. Here, we adopt the analysis for graphene Corbino disks, finding out that -- for the outer to inner radii ratio $r_2/r_1\gtrsim{}5$ -- such systems may demonstrate, when varying the electrochemical potential and the spatial profile of the electrostatic barrier, crossover from standard Josephson tunneling (SJT), via graphene-specific multimode Dirac-Josephson tunneling (MDJT), towards the ballistic Josephson effect (BJE). Signatures of SJT appear only near the Dirac point when the barrier shape is close to rectangular, MDJT appears in the tripolar range and is very robust against varying the barrier shape, and BJE is restored in the unipolar range when smoothing the barrier shape. A comparison with the results of a numerical simulation of quantum transport on the honeycomb lattice is also given.

Figures (7)

• Figure 1: Left: Schematic of a graphene disk with inner radius $r_1$ and outer radius $r_2$, contacted by two circular superconducting electrodes (dark areas). A current source drives a dissipationless supercurrent through the annular region (white). A separate gate electrode (not shown) allows one to tune the carrier concentration around the neutrality point. Right: Electrostatic potential profiles given by Eq. (\ref{['vrmpot']}) with $m=2$, $8$ and $m=\infty$ (i.e., the rectangular barrier). The Fermi energy $E$ is defined with respect to the top of a barrier. $E > 0$ corresponds to unipolar n-n-n doping in the device; for $E<0$, circular n-p-n (tripolar) structure is formed. Arcs (dashed lines) mark the interfaces between the disk area [$r_1<r<r_2$] and contact regions [$r<r_1$ or $r>r_2$].
• Figure 2: (a)--(d) Current-phase relation for the Corbino-Josephson setup in graphene in the case of rectangular potential barrier and infinitely-doped leads, corresponding to $m\rightarrow\infty$ and $V_0\rightarrow\infty$ in Eq. (\ref{['vrmpot']}). (a,b) The high-doping limit ($|\mu|\gg{}\hbar{}v_F/r_1$), (c,d) the Dirac point ($\mu=0$). The radii ratio $r_2/r_1$ is specified at each panel. Results obtained from Eq. (\ref{['ithsubsh']}) are displayed with blue thick lines in (a,b); red thick line visualizes Eq. (\ref{['ijotunn']}) in (c) or Eq. (\ref{['ithpdiff']}) in (d). Remaining color thick lines in (d) visualize Eq. (\ref{['ijoth']}) with the probabilities $T_j(0)$ given by Eq. (\ref{['tjzero']}) calculated for $r_2/r_1=2$ (orange), $r_2/r_1=5$ (green), and $r_2/r_1=10$ (cyan). Thin black lines in all panels visualize the tunneling limit, see Eq. (\ref{['ijotunn']}) (dashed lines), and the ballistic limit, see Eq. (\ref{['ijoball']}) (solid lines).
• Figure 3: Product of critical current and normal-state resistance $I_cR_N$ displayed versus skewness of the current-phase relation (datapoints). Two datasets represent the results of Table \ref{['icrntable']}, for the high-doping limit ($|\mu|\gg{}\hbar{}v_F/r_1$) [circles], and for the Dirac point ($\mu=0$) [diamonds]. Double arrow indicates the multimode Dirac-Josephson tunneling (MDJT) regime, bounded by the $(I_cR_N,S)$ values given in Eqs. (\ref{['icrnpdiff']}) and (\ref{['icrnsubsh']}). Black solid line presents the results following from the optimization of Eq. (\ref{['ijoth']}) for a single nonzero eigenvalue $0<T\leqslant{}1$; black dashed line depicts the results obtained within the multimode toy model defined by Eq. (\ref{['tttoy']}).
• Figure 4: Normal-state conductance $1/R_N$ (top) and critical current $I_c$ (bottom) for the system of Fig. \ref{['diskmpot']} as functions of the chemical potential ($\mu=E$). The parameters are: $r_2=5\,r_1=250\,$nm, $V_0=t_0/2=1.35\,$eV. The exponent $m$ in Eq. (\ref{['vrmpot']}) is specified for each dataset (solid lines). Insets (top) depict the potential profiles for $m=2$ and $m=\infty$. Dashed line depicts the sub-Sharvin conductance and the corresponding critical current for $r_2/r_1\rightarrow\infty$, see Eq. (\ref{['icrnsubshwide']}).
• Figure 5: Left: Product $I_cR_N$ for the data shown in Fig. \ref{['gicR50L200vsEab']}. Right: Skewness of the current-phase relation $S$ as a function of the chemical potential for the same system parameters. The exponent $m$ in Eq. (\ref{['vrmpot']}) is varied between the rows. Horizontal lines bordering the yellow areas mark the MDJT range, defined by the values for rectangular barrier of an infinite height ($m\rightarrow\infty$, $V_0\rightarrow\infty$), corresponding to $\mu=0$ and $|\mu|\gg{}\hbar{}v_F/r_1$ for the narrow-disk limit, $r_2/r_1\rightarrow{}1$, see Eqs. (\ref{['icrnpdiff']}) and (\ref{['icrnsubsh']}); the values for $r_2/r_1=5$ are also marked in the bottom rows.