Gapfull and gapless $1$D Topological Superconductivity in Spin-Orbit Coupled Bilayer Graphene

TL;DR

This work proposes a TMD-proximitized Bernal-stacked bilayer graphene platform to realize a one-dimensional topological superconductor without magnetic fields by exploiting velocity mismatch and Ising spin-orbit coupling in a double Josephson junction. A Brillouin-Wigner reduced low-energy Hamiltonian and a Bogoliubov–de Gennes framework show that Ising SOC yields a gapless topological phase, while Rashba SOC opens a gap via Andreev-band inversion; topological transitions occur as the two junction phases $(\theta,\phi)$ and the orientation angle $\beta$ are tuned. The authors introduce two topological principles for gapless and gapped phases, analyze disorder and perturbation effects, and demonstrate domain-wall Majorana modes in knee geometries where $\beta$ varies spatially. They further reveal reentrant topological superconductivity as a function of chemical potential, indicating robust, tunable Majorana platforms in 2D heterostructures. All results are presented with a focus on experimental feasibility and device design in spin-orbit coupled graphene–TMD heterostructures.

Abstract

We propose a way to generate a one-dimensional topological superconductor from a monolayer of a transition metal dichalcogenide coupled to a Bernal-stacked bilayer of graphene under a displacement field. With proper gating, this structure may be tuned to form three parallel pads of superconductors creating two planar Josephson junctions in series, in which normal regions separate the superconductors. Two characteristics of the system which are essential for our discussion are spin orbit coupling induced by the transition metal dichalcogenides and the variation of the Fermi velocities along the Fermi surface. We demonstrate that these two characteristics lead to one-dimensional topological superconductivity occupying large parts in the parameter space defined by the two phase differences across the two junctions and the relative angle between the junctions and the lattice. An angle-shaped device in which this angle varies in space, combined with proper phase tuning, can lead to the formation of domain walls between topological and trivial phases, supporting a zero-energy Majorana mode, within the bulk of carefully designed devices. We derive the spectrum of the Andreev bound states and show that Ising spin-orbit coupling leaves the topological superconductor gapless, and the Rashba spin-orbit coupling opens a gap in its spectrum. Our analysis shows that the transition to a gapped topological state is a result of the band inversion of Andreev states.

Figures (3)

• Figure 1: (A) A schematic of the system we consider, that consists of Bernal stacked bi-layer of graphene in proximity to TMD (which is not shown), on which patterned gates define superconducting (light blue) and normal (white) regions. The angle $\beta$ between the superconducting areas and the armchair orientation of the graphene bi-layer determines the velocity difference between the two Fermi branches. With an adjustment of the phase difference between the superconductors that form two Josephson junctions, we can bring the system into a topological state supporting Majorana zero modes. (B) Fermi surface of TMD/BBG heterostructure ($\beta=\pi/2$) in a perpendicular displacement field $D=1$ V$/$nm and $\mu=-12.653$ meV and $\mu=-12.2$ meV. For Fermi energy smaller than $-12.652$ meV we have three Fermi pockets (FP) per valley and for larger -- one Fermi surface (FS) per valley. The $k_{\tilde{x}}$ and $k_{\tilde{y}}$ are the coordinates in momentum space aligned with zig-zag and armchair directions, respectively.
• Figure 2: (A) Fermi velocities (expressed in units of energy $v^{\prime}_{\rm f}=\hbar v_{\rm f} / a$, where $v_{\rm f}$ is Fermi velocity and $a=2.46 \text{ \r{A}}$. The energy scale $300 \text{meV}$ corresponds to $\approx 10^5 \text{m} /\text{s}$) in different valleys ($K_{+}$ and $K_{-}$) as a function of angle $\beta$$(k_{y}=0)$, for $\mu=-12.652$ meV and $\mu=-12.5$ meV. (B) and (C) The spectrum of $2$JJ for $\beta=0$ near the upper part (B) of the phase diagram and the lower part (C) for $\mu = -12.5$, $a=0.1$, $|d|=1$, ${\bf B}=0$ and $\Delta=0.17$. In panel (B), the conduction and the valence band of the lowest Andreev levels have opposite spins and the topological area is a consequence of band inversion of the Andreev spectrum. The value of $s_z$ of the negative energy band close to $E=0$ differs between neighboring topological triangles. Panel (C) shows that when Rashba coupling is added, left and right Fermi pockets can undergo a band inversion earlier than the middle Fermi pocket. (D) A deviation from high symmetry point $\beta=0$ to $\beta\neq0$, at which a threshold of Rashba coupling is required for the spectrum to be gapped. Panels (E) and (F) represent the minimal gap $E_{M}=Q\min_{k_{x},b}|E^{A}_{b}/\Delta|$ of Andreev state in $2$JJ over the phase space without (E) with (F) Rashba SO coupling, where $Q=-1$ in the topological area and $Q=1$ in trivial. White lines correspond to the maximal topological area according to the DV condition. $B$ and $B^{\prime}$ label topological gapless phases and $A$ and $A^{\prime}$ are gapped trivial phases related by spin flip.
• Figure 3: (A) $\beta - \alpha_R$ qualitative phase diagram of a $2$JJ. The phases of the superconductors are fixed in the middle of a topological region. The diagram shows two gapped regions (Trivial and Topological) and a gapless region. The gapless region connects between gapped trivial and topological superconductors. The angle $\beta_t$ represents the alignment angle of the device with respect to the tangential direction of the Fermi pockets that separates the trivial and the topological states. (B) A possible configuration for a $2$JJ device oriented on the lattice of BBG in such a way that, for some value of $\theta$ and $\phi$, it acquires a trivial region in one half of the device and topological region at another, with Majorana zero modes between them. The $\beta_{1}=0$ and $\beta_{2}$ are corresponding angles, and color crosses are phase-controlling fluxes. (C) Phase diagram for the case of four velocity branches, for $\mu=-15$ meV and $\beta=0$. The colored regions represent gapped topological phases. The diagram shows that the topological region can expand when the system has two bands per valley instead of one. (D) Phase diagram for $\beta_{1}=0$ (the shaded blue region is topological) and $\beta_{2}=\pi/24$ (the shaded gold region is topological). The chemical potential is $\mu=-12$ meV. In a knee geometry with angle $\pi/24$, when both parts are topological (the blue and gold regions overlap), there is no MZM at the knee. When only one of them is topological, the knee hosts an MZM. (E) $\mu$–$\phi$ phase diagram for different values of $\theta$. One can clearly observe the strong dependence of the phase diagram on the chemical potential, as well as the reentrance of the topological phase as $\mu$ is lowered. (F) $\mu$–$\phi$ phase diagram for fixed $\theta=0$ and different values of $\beta$.