Detection of Valley Polarization in Graphene by a Superconducting Contact

Because the valleys in the band structure of graphene are related by time-reversal symmetry, electrons from one valley are reflected as holes from the other valley at the junction with a superconductor. We show how this Andreev reflection can be used to detect the valley polarization of edge states produced by a magnetic field. In the absence of intervalley relaxation, the conductance $G_{\mathrm{NS}}=(2e^2/h)(1-\cos \Theta )$ of the junction on the lowest quantum Hall plateau is entirely determined by the angle $\Theta$ between the valley isospins of the edge states approaching and leaving the superconductor. If the superconductor covers a single edge, $\Theta =0$ and no current can enter the superconductor. A measurement of $G_{\mathrm{NS}}$ then determines the intervalley relaxation time.

Figure 1

Three diagrams of a graphene sheet contacted by one normal-metal ($N$) and one superconducting ($S$) electrode. Edge states approaching and leaving the superconductor are indicated by arrows. The solid line represents an electron state (green, approaching superconductor: isospin ${\mathit{\nu }}_1$; blue, leaving superconductor: isospin ${\mathit{\nu }}_2$), and the dashed line represents a hole state (red: isospin $-{\mathit{\nu }}_2$).

Figure 2

Dispersion relation of edge states in graphene along the normal-superconducting interface, calculated from Eq. (15) for $|\epsilon |\ll {\Delta }_0$. The dotted lines are for $\mu =0$, the solid lines for $\mu =0.4\hslash v/l_m$.

Figure 3

Cyclotron orbits of Andreev reflected electrons and holes.

Figure 4

Dispersion relation of states along the insulating edge, calculated from Eqs. (16, 17) for $\mu =0.4\hslash v/l_m$ and $\theta =\pi /2$. The solid lines are the electron states (blue ${\epsilon }_e^{+}$, red ${\epsilon }_e^{-}$), the dashed lines are the hole states (blue ${\epsilon }_h^{+}$, red ${\epsilon }_h^{-}$).