Anisotropic Andreev reflection and Josephson effect in ballistic phosphorene

We study Andreev reflection and the Josephson effect in a ballistic monolayer of black phosphorous, known as phosphorene. Due to the anisotropic band structure of this system, the supercurrent changes with an order of magnitude when comparing tunneling along two perpendicular directions in the monolayer. We show that the main reason for this effect is a large difference in the number of transverse modes in Andreev bound states. The oscillatory behavior of the supercurrent as a function of the length and chemical potential of the junction also differs substantially depending on the orientation of the superconducting electrodes deposited on the phosphorene sheet. For Andreev reflection, we show that gate voltaging controls the probability of this process and that the anisotropic behavior found in the supercurrent case is also present for conductance spectra.

Figure 1

Top view of the proposed experimental setup for transport measurements. (a) Supercurrent measurements: Two superconducting electrodes are deposited on top of a phosphorene sheet. A supercurrent flows between them upon current biasing the system. The magnitude of the supercurrent depends on how the separation vector between the electrodes is oriented on the phosphorene sheet. This orientation is quantified by the angle $\delta$. (b) For conductance spectroscopy, one of the superconducting electrodes is replaced with a normal-metal electrode. A gate voltage between the electrodes can be used to tune the local chemical potential.

Figure 2

(a) Contour plot of the upper band ${\epsilon }_{+}$ in the long-wavelength limit $(k_j{a}_j\ll 1)$. Band structure for ${\epsilon }_{+}$ for fixed (b) $k_y=0$ and (c) $k_x=0$, demonstrating the anisotropy of the dispersion relation.

Figure 3

Supercurrent-phase relation for ${\mu }_N=4t_4+1.05m$ for propagation directions $\delta =0$ and $\delta =\pi /2$. The magnitude of the critical current changes by nearly an order of magnitude.

Figure 4

Left panel: Critical current vs the chemical potential for $L/W=0.03$. Right panel: Critical current vs the length $L$ of the junction for ${\mu }_N=4t_4+1.05m$.

Figure 5

Normalized conductance for ${\mu }_N=4t_4+1.01m$ for (a) $\delta =0$ and (b) $\delta =\pi /2$. The lines in the panels correspond to: a. ${\mu }_{N^{\prime }}=4t_4+1.1m$, b. ${\mu }_{N^{\prime }}=4t_4+1.3m$, and c. ${\mu }_{N^{\prime }}=4t_4+1.6m$.

Figure 6

Normalized conductance for ${\mu }_N=4t_4+1.1m$ for (a) $\delta =0$ and (b) $\delta =\pi /2$. The lines in the panels correspond to: a. ${\mu }_{N^{\prime }}=4t_4+1.1m$, b. ${\mu }_{N^{\prime }}=4t_4+1.3m$, and c. ${\mu }_{N^{\prime }}=4t_4+1.6m$.

Figure 7

Normalized conductance at zero bias as a function of ${\mu }_{N^{\prime }}$ for (a) ${\mu }_N=4t_4+1.01m$ and for (b) ${\mu }_N=4t_4+1.1m$.