Helical states in curved bilayer graphene

We study spin effects of quantum wires formed in bilayer graphene by electrostatic confinement. With a proper choice of the confinement direction, we show that in the presence of magnetic field, spin orbit interaction induced by curvature, and intervalley scattering, bound states emerge that are helical. The localization length of these helical states can be modulated by the gate voltage which enables the control of the tunnel coupling between two parallel wires. Allowing for proximity effect via an $s$-wave superconductor, we show that the helical modes give rise to Majorana fermions in bilayer graphene.

Figure 1

A bilayer graphene (BLG) sheet with a fold at $x=0$ along the $z$ axis is placed between two pairs of gates that are set to opposite polarities $\pm V_0/2$, inducing the bulk gap. There are midgap bound states, localized in transverse $x$ direction around $x=0$. At the same time, they freely propagate along the $z$ direction, forming an effective quantum wire.[15] An externally applied magnetic field $\mathbf{B}=B{\mathbf{e}}_{x^{\prime }}$ breaks time-reversal symmetry. The spin orbit interaction $\beta$ is induced by the curvature of the wire, which is characterized by the radius $R$. In the insets we show the BLG structure in momentum (left) and real (right) space for a chosen chirality $\theta =0$. The edges of the BLG sheet can be arbitrary.

Figure 2

The spectrum of the BLG structure for $V_0/2=50$ meV and chirality $\theta =0$. The green area in the inset corresponds to the bulk spectrum. The midgap bound states for valleys $K$ (full line) and $K^{\prime }$ (dashed line) have opposite velocities. The main figure shows the details of $K$-$K^{\prime }$ crossing region (shaded region in the inset). The curvature induced SOI shifts spin-up and spin-down levels in opposite directions by the SOI parameter $\beta$. A magnetic field $\mathbf{B}$ assisted by intervalley scattering ${\Delta }_{K{K}^{\prime }}$ results in the anticrossing gap $2{\Delta }_g$ of two Kramers partners at $k_z=0$. If the chemical potential $\mu$ is tuned inside the gap [shaded area with $\mu \approx (V_0/2\sqrt{2})\pm \beta$], the system is equivalent to three Dirac cones (only one is shown in the main figure), resulting in the helical mode regime. Here $\beta \approx 60$ $\mu \text{eV}$ ($R=5\mathrm{nm}$), ${\Delta }_{K{K}^{\prime }}=30$ $\mu \text{eV}$, and ${\Delta }_Z=30$ $\mu \text{eV}$, so the opened gap is $2{\Delta }_g\approx 30$ $\mu \text{eV}$$\approx 300$ mK.

Figure 3

(a) The profile of the gate potential $V\left(x\right)$ along the curved BLG. The reversed polarity at the two ends gives rise to midgap states, localized in $x$ direction. The density profile of three right-moving states, whose energy $E$ is inside the gap ${\Delta }_g$, allows us to estimate the localization lengths $\xi$: (b) $V_0=100$ meV, $E=29.13$ meV, $d=R$ (dashed white line in Fig. 2), $\xi \approx 9$ and 8 nm, and (c) $V_0=10$ meV, $E=3.49$ meV, $d=R$, $\xi \approx 30$ and 24 nm. We note that states with larger momenta have shorter localization lengths. (d) The localization length follows approximately $\tilde{\xi }=\xi -{\xi }_0\propto 1/\sqrt{V_0}$. The circles are extracted from our numerical calculations $\xi =\langle x^2\rangle$ for energies in the middle of the gap ${\Delta }_g$, equivalent to the white dashed line in Fig. 2. The lines are fits $\tilde{\xi }\propto 1/V_0^p$ for the states at large $k_z$ ($p=0.59$), shown as dashed lines, and for the states near $k_z=0$ ($p=0.52$), shown as full lines.