Tunneling conductance in $s$- and $d$-wave superconductor-graphene junctions: Extended Blonder-Tinkham-Klapwijk formalism

We investigate the conductance spectra of a normal-superconductor graphene junction using the extended Blonder-Tinkham-Klapwijk formalism, considering pairing potentials that are both conventional (isotropic $s$-wave) and unconventional (anisotropic $d$-wave). In particular, we study the full crossover from normal to specular Andreev reflection without restricting ourselves to special limits and approximations, thus expanding results obtained in previous work. In addition, we investigate in detail how the conductance spectra are affected if it is possible to induce an unconventional pairing symmetry in graphene—for instance, a $d$-wave order parameter. We also discuss the recently reported conductance oscillations that take place in normal-superconductor graphene junctions, providing both analytical and numerical results.

Figure 1

(Color online) (a) Sketch of a real-space lattice of graphene, consisting of two hexagonal sublattices $A$ and $B$. The interatomic distance $a$ is equal between all lattice points. (b) $\mathbf{k}$-space (momentum-space) lattice of graphene, including the hexagonal Brillouin zone. Only two inequivalent points exist on the BZ boundary, termed $K$ and $K^{\prime }$.

Figure 2

(Color online) Sketch of the Fermi surface and anisotropic $d$-wave gap as believed to be present in high-$T_c$ cuprate superconductors. The nodal fermions reside at the intersection of the nodal lines of the gap and the Fermi surface.

Figure 3

(Color online) (a) The energy dispersion for graphene in the Brillouin zone. The upper band is the antibonding $\pi$ orbital, while the lower band is the bonding $\pi$ orbital. It is seen that the bands touch at Fermi level $(E_{\mathrm{F}}=0)$ at six discrete points, which constitutes the effective Fermi surface. (b) Contour plot of the dispersion relation, clearly showing the hexagonal structure of the Fermi points. The center of each red droplike structure represents either $K$ or $K^{\prime }$.

Figure 4

(Color online) Graphical illustration of the different scattering processes that may take place at a (i) metallic and (ii) graphene N-S junction. While the Andreev reflected hole in case (i) retraces the trajectory of the incoming electron, the hole in case (ii) may be specularly reflected. This peculiar property is a result of the existence of two bands (conduction and valence) close to the Fermi energy in graphene. In the low-energy transport regime—i.e., quasiparticle energies $E$ of order $\mathcal{O}\left(\Delta \right)$—retroreflection dominates if $E_{\mathrm{F}}\gg \Delta$, while specular reflection dominates if $E_{\mathrm{F}}\ll \Delta$.

Figure 5

(Color online) The effective impedance caused by FVM illustrated schematically for all possible cases of smaller, equal, and larger Fermi velocity in the normal part of the system. Except for the case when the Fermi velocities are identical in the normal and superconducting parts of the system $(\kappa =1)$, parts of the Fermi surface does not participate in the scattering processes, resulting in a reduction in conductance. For instance, when $\kappa <1$, total reflection occurs at angles ${\theta }_N>\mathrm{asin}\left(\kappa \right)$, such that FVM effectively acts as a source of normal reflection (Ref. 23).

Figure 6

(Color online) The scattering processes taking place at a N-S or N-I-S graphene junction. In the former case, the insulating region is completely absent, and only the six depicted processes take place. Note that only normal Andreev reflection or specular Andreev reflection takes place at any given energy $E$, never both. For a N-I-S graphene junction, there are transmitted and reflected electrons and holes in the insulating region corresponding to ${\tilde{\psi }}_{\mathrm{I}}$, not shown in the above figure.

Figure 7

(Color online) Conductance spectra for graphene in (a) with $E_{\mathrm{F}}^{\prime }∕\Delta ={10}^3$ and in (b) with $E_{\mathrm{F}}^{\prime }=E_{\mathrm{F}}$. The spectra in (a) are identical to the result of Ref. 2. In (b), the subgap conductance is always close to $2G_{\mathrm{N}}$, but becomes more constant for increasing $E_{\mathrm{F}}$ since ${\theta }_{\mathrm{c}}\to \pi ∕2$. We have plotted the ratios $\{\mathrm{50,100},1000\}$ of $E_{\mathrm{F}}∕\Delta$ in (b), from bottom to top.

Figure 8

(Color online) Sketch of a 2D N-S graphene junction with an anisotropic superconductor. The orientation of the gap in $\mathbf{k}$ space is modeled by the angle $\alpha$.

Figure 9

(Color online) Conductance spectra for doped graphene in with $E_{\mathrm{F}}^{\prime }={10}^4\Delta$ for a $d$-wave order parameter with orientation angle $\alpha =\pi ∕4$. A ZBCP is present for large FVM and becomes unobservable narrow for $E_{\mathrm{F}}∕\Delta <10$. For $\alpha =0$, the conductance spectra are essentially identical to those in Fig. 7.

Figure 10

(Color online) Conductance spectra for (a) doped graphene in with $E_{\mathrm{F}}^{\prime }={10}^4\Delta$ for a $d$-wave order parameter and (b) undoped graphene with $E_{\mathrm{F}}=E_{\mathrm{F}}^{\prime }$. In both cases, we have set $E_{\mathrm{F}}∕\Delta =100$ and investigate how the conductance spectra evolves upon rotating $\alpha$ from $0$ to $\pi ∕4$ in steps of $\pi ∕20$. In (a), it is seen that the peak of the conductance shifts from $eV=\Delta$ to progressively lower values as $\alpha$ increases. This is in contrast to metallic N-S junctions. In (b), we plot the conductance for $\alpha =\pi ∕4$. We find virtually no difference between various orientations of the gap in this case, and the spectra are quite similar to the metallic N-S case with zero barrier.

Figure 11

(Color online) Conductance spectra for a N-I-S graphene junction with $E_{\mathrm{F}}^{\prime }∕\Delta =E_{\mathrm{F}}∕\Delta =100$, using $s$-wave pairing. We reproduce the same results as Ref. 12 with a $\pi$ periodicity in the parameter $\chi$. However, we obtain a phase shift of $\pi ∕2$ in $\chi$ compared to their results. We believe that this difference pertains to a minor sign error in the wave functions used in Ref. 12.

Figure 12

(Color online) Plot of the zero-bias conductance as a function of $\chi$ with $E_{\mathrm{F}}∕\Delta =100$ for a N-I-S graphene junctions with $s$-wave pairing.

Figure 13

(Color online) Conductance spectra for a N-I-S graphene junction with $E_{\mathrm{F}}^{\prime }∕\Delta =E_{\mathrm{F}}∕\Delta =100$, using $d$-wave pairing.

Figure 14

(Color online) Conductance spectra for a N-I-S graphene junction with $E_{\mathrm{F}}^{\prime }∕E_{\mathrm{F}}=10$ and $E_{\mathrm{F}}∕\Delta =100$, using $d$-wave pairing.

Figure 15

(Color online) Tunneling conductance of a N-I-S graphene junction for $s$-wave pairing in the undoped and doped cases (see main text for parameter values). We have fixed $V_0∕\Delta =500$ and $E_{\mathrm{F}}^{\prime }∕\Delta =100$. It is seen that for increasing $w$, a novel oscillatory behavior of the conductance as a function of voltage is present in all cases.

Figure 16

(Color online) Same as Fig. 15, but now for $d$-wave pairing with $\alpha =\pi ∕4$. We have fixed $V_0∕\Delta =500$ and $E_{\mathrm{F}}^{\prime }∕\Delta =100$.