Sub-Sharvin conductance and Josephson effect in graphene

Abstract

Titov and Beenakker [Phys. Rev. B 74, 041401(R) (2006)] found, by solving the Dirac-Bogoliubov-De-Gennes equation, that the product of critical current and normal-state resistance for superconductor-graphene-superconductor (S-g-S) Josephson junction takes values (for a short junction and zero temperature) between $I_cR_N\approx{}2.1$ and $I_cR_N\approx{}2.4$ in units of $e/Δ_0$, where $Δ_0$ is the superconducting gap. These values are notably higher than the tunnelling bound ($π/2$), but lower than the ballistic bound ($π$). Here we analyze numerically the tunneling of Cooper pairs through S-g-S junctions in which the longitudinal electrostatic potential profile is tuned, within gates electrodes, from a rectangular to a parabolic one. In the unipolar regime (i.e., when the chemical potential is above the top of a barrier, $μ>0$), it is found that $I_cR_N$ gradually evolves from the graphene-specific to the ballistic value. At the same time, the normal-state conductance increases from the sub-Sharvin value of $1/R_N\approx(π/4)\,G_{\rm Sharvin}$ towards to the Sharvin value $G_{\rm Sharvin}=g_0|μ|W/(π\hbar{}v_F)$, with the conductance quantum $g_0=4e^2/h$, the junction width $W$, and the Fermi velocity in graphene $v_F$. In contrast, in the tripolar regime ($μ<0$), both normal-state conductance and the critical current are suppressed when smoothing the potential; however, $I_c{}R_N$ remains close to the graphene-specific range, even for a parabolic potential. The skewness of the current-phase relation is also discussed.

Figures (6)

• Figure 1: Top: Schematic of a graphene strip of width $W$, contacted by two superconducting electrodes (dark areas) at a distance $L$. A current source drives a dissipationless supercurrent through the central region. A separate gate electrode (not shown) allows us to tune the carrier concentration around the neutrality point. The lattice parameter $a=0.246\,$nm is also shown. Middle: Electrostatic potential profiles given by Eq. (\ref{['vxmpot']}) with $m=2$, $8$, $32$, 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 $E<0$, a tripolar n-p-n structure is formed. Bottom: Absolute value of the pair potential with the superconducting phases $\theta/2$ for $x<-L/2$ (left electrode) and $-\theta/2$ for $x>L/2$ (right electrode).
• Figure 2: Current-phase relation for S-g-S Josephson junction in the case of rectangular potential barrier and infinitely-doped leads, corresponding to $m\rightarrow\infty$ and $V_0\rightarrow\infty$ in Eq. (\ref{['vxmpot']}). (a) The Dirac point ($\mu=0$), (b) the high-doping limit ($|\mu|\gg{}\hbar{}v_F/L$). Results obtained from Eqs. (\ref{['ithpdiff']}) and (\ref{['ithsubsh']}) are displayed with color thick lines. Thin black lines visualize the tunneling limit, see Eq. (\ref{['ijotunn']}) (dashed lines), and the ballistic limit, see Eq. (\ref{['ijoball']}) (solid lines).
• Figure 3: Normal-state conductance $1/R_N$ (top) and critical current $I_c$ (bottom) for the system of Fig. \ref{['setupjos']} as functions of the chemical potential ($\mu=E$). The parameters are: $W=5\,L=1000\,$nm, $V_0=t_0/2=1.35\,$eV. The exponent $m$ in Eq. (\ref{['vxmpot']}) 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 given by Eq. (\ref{['rnsubsh']}) and the corresponding critical current, see Eq. (\ref{['ithsubsh']}). (The values of $G_{\rm Sharvin}$ are not shown, as they closely follow the numerical results for $m=2$ and $\mu>0$.) Additional insets (bottom) visualize the tripolar (n-p-n) and the unipolar (n-n'-n) doping for $\mu<0$ and $\mu>0$ (respectively).
• Figure 4: Product $I_cR_N$ for the data shown in Fig. \ref{['gicV0t0ov2ab']}. The exponent $m$ in Eq. (\ref{['vxmpot']}) is varied between the rows. The potential profile for each $m$ is displayed on the left. Horizonal lines bordering the yellow areas on the right mark the values for rectangular barrier of an infinite height ($m\rightarrow\infty$, $V_0\rightarrow\infty$), corresponding to $\mu=0$ and $|\mu|\gg{}\hbar{}v_F/L$, see Eqs. (\ref{['icrnpdiff']}) and (\ref{['icrnsubsh']}).
• Figure 5: Skewness of the current-phase relation $S$ displayed versus the chemical potential for the same system parameters as in Fig. \ref{['gicV0t0ov2ab']} (solid lines). The exponent $m$ in Eq. (\ref{['vxmpot']}) is specified for each panel. Horizonal lines mark graphene-specific values of $S$, see Eqs. (\ref{['icrnpdiff']}) and (\ref{['icrnsubsh']}).