Spin transport in a graphene normal-superconductor junction in the quantum Hall regime

The quantum Hall regime of graphene has many unusual properties. In particular, the presence of a Zeeman field opens up a region of energy within the zeroth Landau level, where the spin-up and spin-down states localized at a single edge propagate in opposite directions. We show that when these edge states are coupled to an $s$-wave superconductor, the transport of charge carriers is spin-filtered. This spin-filtering effect can be traced back to the interplay of specular Andreev reflections and Andreev retro-reflections in the presence of a Zeeman field.

Figure 1

Electron band structures of the few lowest Landau levels [12]. (a) Band structure of a zigzag ribbon and (b) of an armchair ribbon. The ribbons are shown in the insets. (c), (d) The corresponding band structures in the presence of a Zeeman field that splits the energies for spin-up (red) and spin-down (blue) electrons.

Figure 2

A graphene ribbon in the normal state (gray region N) with two leads $L_0$ and $L_1$ and a superconducting lead $L_2$ (green region S) attached to the top edge. The width of the lead $L_i$ is denoted by $W_i$. An external magnetic field smaller than the critical field of the superconductor is applied such that the normal region is in the quantum Hall regime. The superconductor couples electron and hole edge states propagating along the upper edge. We assume that a bias voltage $V$ is applied between leads $L_0$ and $L_1$.

Figure 3

On the horizontal axis, we plot in (a)–(c) the transport coefficients $T_{ee}$, $T_{he}$, and $R_{he}$ for spin-up (red) and spin-down (blue) particles as a function of energy $E$ (vertical axis). Similarly, (d) shows the band structure of the spin-split zeroth Landau level for electrons (full lines) and holes (dashed lines) of the normal lead $L_0$, (e) the spin polarization, both as a function of $E$. (f) (differential) charge conductance as a function of $E=eV$, where $V$ is the bias voltage applied between leads $L_0$ and $L_1$. The thin horizontal dashed lines mark the energies where the edge states change the direction of propagation, while the thick ones correspond to $\left|E\right|=\mathrm{\Delta }$. Here, the edge terminations of $L_0$ and $L_1$ are zigzag while the edge termination of $L_2$ is armchair. The parameters are $\mathrm{\Delta }=10\text{meV}$, $E_F=0.3\mathrm{\Delta }$, ${\mathrm{\Delta }}_Z=0.5\mathrm{\Delta }$, $B=10\text{T}$, $W=600a$, $W_2=510a$, and $y_0=300a$.

Figure 4

(a) Band structure of electrons (full lines) and holes (dashed lines) of the spin-split zeroth Landau level for spin up (red) and spin down (blue) in a graphene zigzag ribbon. (b) Electron and hole edge states in lead $L_0$ and their propagation direction indicated by arrows shown for the three energy regions I–III. While there are four edge states at each edge, we show only the states relevant for the transport in our geometry (see Fig. 2). The representative electron (full circle) and hole state (empty circle) for spin up (red) and spin down (blue) in each energy region is marked in (a) as well as in (b).

Figure 5

(a) Spin polarization and (b) (differential) charge conductance for three different interface lengths (values of $W_2$) that correspond to three different valley polarizations. The charge conductance depends on the angle between the valley isospins, while the spin polarization does not (up to the region where $(e\downarrow )$ leaks to $L_1$ due to the smaller induced gap.) Here, $\mathrm{\Delta }=20\text{meV}$, $E_F=0.3\mathrm{\Delta }$, ${\mathrm{\Delta }}_Z=0.5\mathrm{\Delta }$, $B=10\text{T}$, $W=600a$, and $y_0=300a$.

Figure 6

(a), (b) Band structure of excitations along the graphene-superconductor interface in a quantizing magnetic field for $\mathrm{\Delta }/t=0.05$ and 0.1, respectively. Dispersing states at the interface for $\left|E\right|<\mathrm{\Delta }$ evolve into the flat zeroth Landau level (ZLL) away from the interface. The bulk ZLL is split into four: electrons and holes are coupled via the superconductor, while the spin degeneracy is lifted due to the Zeeman field. The interface states are valley degenerate since the interface is smooth on the scale of the lattice constant. When $E_F<{\mathrm{\Delta }}_Z$, the ZLL edge states develop an effective band gap ${\mathrm{\Delta }}^{*}$ (red arrows) due to the coupling to a superconductor. ${\mathrm{\Delta }}^{*}$ increases from (a) to (b) with increasing superconducting pair potential $\mathrm{\Delta }$. Here, $E_F=0.3\mathrm{\Delta }$, ${\mathrm{\Delta }}_Z=0.5\mathrm{\Delta }$, and the interface is along the zigzag direction. The parameters $\mathrm{\Delta }$ and magnetic field are chosen to be larger than their realistic values to obtain better visibility.

Figure 7

Same as Fig. 3 but with armchair edge terminations for $L_0$ and $L_1$ and zigzag edge termination for $L_2$.