Microscopic theory of the proximity effect in superconductor-graphene nanostructures

We present a theoretical analysis of the proximity effect at a graphene-superconductor interface. We use a tight-binding model for the electronic states in this system, which allows us to describe the interface at the microscopic level. Two different interface models are proposed: one in which the superconductor induces a finite pairing in the graphene regions underneath, thus maintaining the honeycomb structure at the interface and one that assumes that the graphene layer is directly coupled to a bulk superconducting electrode. We show that properties such as the Andreev reflection probability and its channel decomposition critically depend on the model used to describe the interface. We also study the proximity effect on the local density of states on the graphene. For finite layers, we analyze the induced minigap and how it is reduced when the length of the layer increases. The results for the local density of states profiles for finite and semi-infinite layers are presented.

Figure 1

The honeycomb structure of a graphene sheet is formed by combining two triangular sublattices, which are denoted by A (dots) and B (open dots). The unit cell on each horizontal line includes one atom from each sublattice. The axis selection used in this work is indicated.

Figure 2

(Color online) Schematic picture of an Andreev reflection in the bulk-BCS and in the HDSC models. In the first case, the sites on each sublattice of the graphene edge are coupled to the superconductor through independent channels, thus breaking the coherence of the honeycomb structure. In the other case, the coupling with the superconducting region maintains this structure. The hopping element $t_c$ in the bulk-BCS model controls the normal transparency of the interface. The vertical arrows represent the coupling between electrons and holes in Nambu space given by the gap parameter $\Delta$.

Figure 3

(Color online) Andreev reflection probability on the two eigenchannels as a function of the parallel momentum $q$ for fixed energy $\omega =0.2\Delta$. The left panel corresponds to the bulk-BCS model (with $\beta =1$) and the right one to the HDSC model. The different curves correspond to different values of the doping level $ϵ$: 0 (full lines), 0.1 (dashed lines), 0.3 (dotted lines), and $0.4\Delta$ (dashed-dotted lines).

Figure 4

(Color online) Total differential conductance per unit length due to Andreev reflection $G_{AR}$ normalized to the conductance per unit length of a ballistic graphene layer $g_0\left(V\right)=\frac{4e^2}{h}\left(eV+ϵ\right)/\left(\pi \hslash v_F\right)$. The results for the HDCS model (full lines) and for the bulk-BCS model (dashed lines) are compared with increasing doping level $ϵ=0$, 0.2, 0.4, and $0.6\Delta$.

Figure 5

(Color online) Evolution of the lowest energy level $E_g^{\ast }$ of a finite graphene layer coupled to a superconductor as a function of $\beta$ within the bulk-BCS model for $L=40\xi$ (left panel) and with the length of the layer $L$ within both the bulk-BCS (at $\beta =1$) and the HDSC models (right panel). In the case of the bulk-BCS model for arbitrary $\beta$, three different behaviors are found depending on the $N\text{mod}3$ value. On the contrary, a universal behavior of $E_g^{\ast }$ is found within the HDSC model regardless of the $N$ value (see text).

Figure 6

(Color online) Spatial variation of the LDOS on a finite layer $\left(L=9\xi \right)$ in the HDSC model. The plots on the top panels correspond to the lines inside the layer with $n\text{mod}3=1,2$ and those on lower ones correspond to $n\text{mod}3=0$. The uncoupled layer has a metallic behavior $\left(N\text{mod}3=2\right)$. The LDOS for this case, at the edges of the layer (top left panel) and at the center of the layer (lower left panel), is plotted with dashed lines. When coupled to the superconductor, the LDOS is modified by the appearance of a minigap (denoted as $m_0$ in the pictures) and with the breaking of the uncoupled bands into a pair of sub-bands ($m_1$ and $m_2$ in the pictures). The evolution of the LDOS along the layer is shown in the right panels. The results are normalized to the LDOS of a bulk graphene layer with zero doping at $\omega =\Delta$, denoted by ${\rho }_0$.

Figure 7

(Color online) LDOS on lines of type $n\text{mod}3=1$ for a semi-infinite graphene layer coupled to a superconductor within the bulk-BCS model (lower panels). The upper panels show the corresponding results for the uncoupled case. The plots on the left correspond to the undoped case while those on the right correspond to $ϵ=\Delta$.

Figure 8

(Color online) Oscillation pattern on the LDOS of a semi-infinite graphene layer at zero doping within the bulk-BCS model (full line), the HDCS model (dashed line), and in the uncoupled case (dotted line). The energy is fixed at $\omega =2\Delta$.