Characteristic energies, transition temperatures, and switching effects in clean S$|$N$|$S graphene nanostructures

We study proximity effects in clean nanoscale superconductor–normal-metal–superconductor (S$\mid$N$\mid$S) graphene heterostructures using a self-consistent numerical solution to the continuum Dirac Bogoliubov–de Gennes (DBdG) equations. We obtain results for the pair amplitude and the local density of states (DOS) as a function of doping and of the geometrical parameters determining the width of the structures. The superconducting correlations are found to penetrate the normal graphene layers even when there is extreme mismatch in the normal and superconducting doping levels, where specular Andreev reflection dominates. The local DOS exhibits peculiar features, which we discuss, arising from the Dirac cone dispersion relation and from the interplay between the superconducting and Thouless energy scales. The corresponding characteristic energies emerge in the form of resonant peaks in the local DOS, which depend strongly on the doping level, as does the energy gap, which declines sharply as the relative difference in doping between the S and N regions is reduced. We also linearize the DBdG equations and develop an essentially analytical method that determines the critical temperature $T_c$ of a S$\mid$N$\mid$S nanostructure self-consistently. We find that for S regions that occupy a fraction of the coherence length, $T_c$ can undergo substantial variations as a function of the relative doping. At finite temperatures and by manipulating the doping levels, the self-consistent pair amplitudes reveal dramatic transitions between a superconducting and resistive normal state of the structure. Such behavior suggests the possibility of using the proposed system as a carbon-based superconducting switch, turning superconductivity on or off by tuning the relative doping levels.

Figure 1

Schematic of the S$\mid$N$\mid$S graphene nanostructure studied. The sample is in the $x-y$ plane. The widths of the superconducting and normal segments are $d_S$ and $d_N$, respectively. The electrostatic potential in the normal region might be controlled via a gate voltage.

Figure 2

Normalized pair amplitude vs $X$ (see text) at $T=0$ for an S$\mid$N$\mid$S heterostructure. The S portions have a dimensionless width (see text) of $D_S=150$ each. For clarity, the outermost parts of the sample are not shown. Each panel corresponds to a different normalized normal graphene width of (a) $D_N=20$, (b) $D_N=50$, (c) $D_N=100$, and (d) $D_N=300$. For each case, four curves representing different doping levels are shown. From top to bottom in the central N region (green, blue, black, red), these curves correspond to ${\tilde{\mu }}_N=0.5,0.2,0$, and 10.

Figure 3

Normalized pair amplitude versus $X$, as in Fig. 2. The dimensionless geometrical quantities $D_N$ and $D_S$ are now $D_N=300$ and $D_S=300$, so that the S$\mid$N boundaries are at $\left|X\right|=150$. The top set of panels depicts most of the S$\mid$N$\mid$S structure and the bottom set of panels shows a magnified part of the N region near the interface for the panel above it. Panels (a) and (c) are for electron-doped N with curves shown, from top to bottom at $X=0$, for ${\tilde{\mu }}_N=0.5$, 0.2, and $0.1$ (red, green, and blue), while panels (b) and (d) are for hole-doped N with ${\tilde{\mu }}_N=-0.1,-0.2$, and $-0.5$ with a similar scheme. In both cases results for ${\tilde{\mu }}_N=0$, (black) lowest curve at $X=0$, are shown for reference.

Figure 4

Local DOS for a normal, finite width, graphene layer that has (a) zero doping (${\tilde{\mu }}_N=0$), and (c) moderate doping (${\tilde{\mu }}_N=0.2$). The normalizations of DOS and energies are chosen (see text) so that for an infinite width layer the plots would be straight lines of slope $\pm 1$. Four different widths are illustrated, in order of decreasing height of the main peaks: $D_N=20$ (black), $D_N=100$ (blue), $D_N=300$ (red), and $D_N=600$ (cyan). Each bottom panel is a magnification of the results above it, over a narrower energy range. The main peaks are related to the Thouless scale; see Eq. (22) and discussion below it.

Figure 5

Local DOS (normalized as explained in the text) for S$\mid$N$\mid$S systems with both vanishing and very low doping in the N layers. Panels (a)–(d) correspond to ${\tilde{\mu }}_N=0$, while panels (e) and (f) correspond to ${\tilde{\mu }}_N=0.005$ (or equivalently ${\mu }_N/{\Delta }_0=0.5$). Results are shown for the S region [panels (a) and (b)] and in the N region [(c)–(f)]. In (a), (c), and (e) we have $D_N={\Xi }_0/5$ (solid black curves) and $D_N={\Xi }_0$ (blue dashed curves), while in (b), (d), and (f), the N layers are larger: $D_N=3{\Xi }_0$ (solid red curves) and $D_N=6{\Xi }_0$ (dashed green curves).

Figure 6

Local DOS (plotted and normalized as in Fig. 5) for an S$\mid$N$\mid$S system with moderately doped S layers (${\tilde{\mu }}_N=0.2$). The panel arrangement, curve (color and) structure, and all other parameters are the same as that in Fig. 5. See text for discussion and comparison

Figure 7

The excitation gap, $E_{\mathrm{gap}}$ as defined in the text, as a function of the relative doping parameter, ${\tilde{\mu }}_N$. Four different normal graphene widths are considered: $D_N=20$, 50, 100, and 300 (red circles, blue squares, green diamonds, and cyan triangles respectively). Lines are straight segments joining points.

Figure 8

The critical temperature $T_c$ (normalized by $T_0$) for a S$\mid$N$\mid$S system as a function of (a) ${\tilde{\mu }}_N$, and (b) the S width, $D_S$. In (a) the doping dependence is shown for two values of $D_S=50$ (red circles) and 100 (blue squares). In (b), results are shown as a function of $D_S$ for three doping levels: ${\tilde{\mu }}_N=0$ (red circles), $0.2$ (blue squares), and $0.4$ (green diamonds). The doping level in these cases has a dramatic effect on $T_c$ for small $D_S$, but is less detrimental for larger $D_S$, where the curves tend to coalesce in the limit of bulk S widths. In both (a) and (b), the normal graphene layer has $D_N=100$.

Figure 9

The normalized pair amplitude as a function of position. In (a), the temperature is set at $T=0.87T_0$, and the normal and superconductor widths are $D_N=100$ and $D_S=50$, respectively. In (b), we have $T=0.92T_0$, while $D_N=100$ and $D_S=100$. In both cases, the doping parameter, ${\tilde{\mu }}_N$, is varied from 0 to $0.4$ in increments of $0.1$. The arrow depicts the direction of increasing ${\tilde{\mu }}_N$, which eventually leads to the vanishing of the pairing correlations.