Chiral topological superconductivity in twisted bilayer and double bilayer graphene

TL;DR

The paper develops a Bogoliubov–de Gennes framework for twisted bilayer graphene and twisted double bilayer graphene with chiral $p_{x}+i p_{y}$ pairing, revealing a rich landscape of topological superconducting phases characterized by Chern numbers that depend on twist angle, chemical potential, and pairing strength. By analyzing the direct band gap and tracking gap closings in the moiré Brillouin zone, it identifies multiple topological transitions and demonstrates how trigonal warping reshapes phase boundaries and enables higher-Chern-number phases. The results establish tBLG and tDBLG as tunable platforms for engineering chiral topological superconductivity and potential Majorana modes, with a practical route suggested to realize effective $p+ip$ pairing from conventional $s$-wave superconductivity under appropriate spin–orbit and magnetic conditions. Overall, the work highlights the interplay between moiré band structure and unconventional pairing as a route to exotic, tunable topological superconducting states in graphene-based moiré materials.

Abstract

We present a theoretical investigation of the emergence of chiral topological superconductivity in small-angle twisted bilayer graphene (tBLG) and twisted double bilayer graphene (tDBLG). Using the low-energy continuum model and incorporating spin-triplet $p_{x}+i p_{y}$ pairing in each graphene layer, we construct the effective models for both tBLG and tDBLG with superconductivity. By varying the chemical potential, superconducting order parameter, and twist angle, we explore the emergence of topological superconducting phases via the calculation of Chern numbers. Our phase diagrams for tBLG and tDBLG (both AB-AB and AB-BA stackings) reveal distinct topological transitions, which are consistently marked by bulk gap-closing points. To gain further insight, we analyze the evolution of Chern numbers by tracking the number and location of gap closings within the moiré Brillouin zone. Additionally, we illustrate representative squared amplitude of Bloch states corresponding to different topological phases. In the later part of our study, the effect of trigonal warping on the topological superconducting properties is also discussed. Beyond the quantitative results, our study highlights how the interplay between moiré band structure and unconventional pairing symmetries enriches the landscape of possible superconducting states in twisted graphene systems. The framework developed here may also be extended to other multilayer moiré materials, offering a route towards engineering exotic topological superconductivity with tunable parameters.

Figures (13)

• Figure 1: (a) Schematic of a single layer graphene where $A$, $B$-sublattices are denoted by black and red color dots respectively. Here, $\delta_{1}$, $\delta_{2}$, $\delta_{3}$ denote the nearest neighbor vectors and $\mathbf{a}_{1}$ and $\mathbf{a}_{2}$ represent the lattice vectors. (b) Schematic representation of the double bilayer graphene with AB-AB and AB-BA stackings. Here, $L_{1}$, $L_{2}$, $L_{3}$, $L_{4}$ respectively represent the layer-1, 2 belonging to the first bilayer (shown in blue color) and layer-3, 4 belonging to the second bilayer (shown in red color). (c) The red and cyan hexagons denote the Brillouin zone of the two rotated layers centering each other by $+\theta$/2 and $-\theta$/2. The smaller hexagonal tiling is the folded Brillouin zone i.e., the mBZ. One of the smaller hexagons (the moiré Brillouin zone) is zoomed in for clarity and shown with the high symmetry path along which the band structure is calculated.
• Figure 2: The electronic band dispersion of tBLG near valley-$K$ along the high symmetry path is demonstrated in presence of different values of chemical potential ($\mu$) and superconducting order parameter ($\Delta_{\mathrm{sc}}$) at twist angle $\theta = 1.05^{o}$. We choose the other model parameters as following, in panel (a) $\Delta_{\mathrm{sc}}$ = 0 meV, $\mu$ = 0 meV, panel (b) $\Delta_{\mathrm{sc}}$ = 5 meV, $\mu$ = 0 meV, panel (c) $\Delta_{\mathrm{sc}}$ = 0 meV, $\mu$ = 1 meV, and panel (d) $\Delta_{\mathrm{sc}}$ = 5 meV, $\mu$ = 1 meV.
• Figure 3: The electronic band dispersion of tDBLG near valley-$K$ along the high symmetry path is demonstrated in presence of different values of chemical potential ($\mu$) and superconducting pairing ($\Delta_{\mathrm{sc}}$) at the twist angle $\theta = 1.3^{o}$. In the upper panel (i.e., (a)-(d)) results are shown for AB-AB tDBLG and in the lower panel (i.e., (e)-(h)) for AB-BA tDBLG respectively. We choose the model parameters as following, in panels (a) and (e) $\Delta_{\mathrm{sc}}$ = 0 meV, $\mu$ = 0 meV, panels (b) and (f) $\Delta_{\mathrm{sc}}$ = 10 meV, $\mu$ = 0 meV, panels (c) and (g) $\Delta_{\mathrm{sc}}$ = 0 meV, $\mu$ = 5 meV, and panels (d) and (h) $\Delta_{\mathrm{sc}}$ = 15 meV, $\mu$ = 5 meV.
• Figure 4: Density plots and line plots respectively for the topological phase diagram and direct band gap in tBLG with chiral $p_{x} + ip_{y}$ superconductivity around valley-$K$ and spin-up are depicted. In panel (a), Chern number ($\mathcal{C}^{K}_{\uparrow}$) is shown in the plane of twist angle ($\theta$) and chemical potential ($\mu$) for superconducting order parameter ($\Delta_{\mathrm{sc}} = 5$ meV). In panel (b), the same is shown in the chemical potential ($\mu$) and superconducting order parameter ($\Delta_{\mathrm{sc}}$) plane for the twist angle $\theta = 1.05^{o}$. Direct band gap ($\delta_{\text{dir}}$ in meV) is displayed as a function of twist angle ($\theta$) and chemical potential ($\mu$) respectively in panels (c) and (d) for the same system with various values of twist angle ($\theta$), chemical potential ($\mu$) and superconducting order parameter ($\Delta_{\mathrm{sc}}$), as mentioned in the figures.
• Figure 5: Density plots for the band gap (in meV) are shown in the $k_{x} - k_{y}$ plane within the mBZ. Panels (a) - (c) correspond to $\mu = 3$ meV, $\mu = 14.5$ meV and $\mu = 20$ meV in case of $p_{x} + ip_{y}$ superconducting tBLG with $\Delta_{\mathrm{sc}} = 5$ meV and twist angle $\theta = 1.5^{o}$ near valley-$K$ and spin-up. On the other hand, panels (d) - (f) correspond to $\mu = 3$ meV, $\mu = 6.3$ meV and $\mu = 15$ meV with $\Delta_{\mathrm{sc}} = 20$ meV and twist angle $\theta = 1.05^{o}$ in the same system.