Resonant tunneling through superconducting double barrier structures in graphene

We study resonant tunneling through a superconducting double barrier structure in graphene as a function of the system parameters. At each barrier, due to the proximity effect, an incident electron can either reflect as an electron or a hole (specular as well as retro Andreev reflection in graphene). Similarly, transport across the barriers can occur via electrons as well as via the crossed (specular and/or retro) Andreev channel, where a hole is transmitted nonlocally to the other lead. In this geometry, in the subgap regime, we find resonant suppression of Andreev reflection at certain energies due to the formation of Andreev bound levels between the two superconducting barriers, where the transmission probability $T$ for electrons incident on the double barrier structure becomes unity. The evolution of the transport through the superconducting double barrier geometry as a function of the incident energy for various angles of incidence shows the damping of the resonance as normal reflection between the barriers increases.

Figure 1

(Color online) Cartoon of the SDB structure in a graphene sheet. Two patches at the two places on the graphene sheet depict superconducting material deposited on top of it. The schematic of the potential profile seen by an incident electron is shown below.

Figure 2

(Color online) The electron and hole paths contributing to the formation of Andreev bound levels between the two superconductors. The electrons have been shown as red lines, the retro AR, CAR holes as blue lines and the SAR, SCAR holes as green lines. The bound levels formed by multiple retro AR have been shown as thick red and blue lines. However, not all possible paths have been shown in the figure.

Figure 3

(Color online) The bottom graph shows the behavior of angle-resolved conductance, obtained numerically, in units of $4e^2/h$ as a function of energy in the subgapped regime $\left(ϵ⪡\Delta \right)$ for $k_y=0.75$. Here, $\Delta /E_F=0.05$ and $U_0/E_F=10.0$. The top graph depicts the denominator of Eq. (20) for the same parameter values.

Figure 4

(Color online) The behavior of all possible quantum-mechanical scattering probabilities $\left({\left|r_c\right|}^2,{\left|t_c\right|}^2,{\left|r_{Ac}\right|}^2,{\left|t_{Ac}\right|}^2\right)$ through the graphene SDB structure are plotted as a function of energy in the subgapped regime $\left(ϵ⪡\Delta \right)$ for three different values of $k_y$. In (a)–(c) solid red, green, blue, and dashed black lines correspond to the ${\left|r_{Ac}\right|}^2$, ${\left|r_c\right|}^2$, ${\left|t_{Ac}\right|}^2$, and ${\left|t_c\right|}^2$ respectively. Here, $\Delta /E_F=0.05$ and $U_0/E_F=10.0$.

Figure 5

(Color online) (a) The behavior of the transmission resonances $\left(T={\left|t_c\right|}^2\right)$ is plotted as a function of energy in the subgapped regime $\left(ϵ⪡\Delta \right)$ for three different values of $a/L$ ratio. The red, blue, and green lines correspond to the three different values of $a/L$ which are 0.012, 0.017, and 0.022, respectively. (b) The distance between consecutive resonances for the same three different values of $a/L$. For both the figures $k_y=0.125$.

Figure 6

(Color online) The behavior of the angle-resolved differential conductance in units of $4e^2/h$ as a function of energy in the subgapped regime $\left(ϵ⪡\Delta \right)$ for three different values of $k_y$. Here, $\Delta /E_F=0.05$ and $U_0/E_F=10.0$.