Pairing symmetry and spontaneous vortex-antivortex lattice in superconducting twisted-bilayer graphene: Bogoliubov-de Gennes approach

We study the superconducting pairing symmetry in twisted bilayer graphene by solving the Bogoliubov-de Gennes equation for all electrons in moiré supercells. With increasing the pairing potential, the system evolves from the mixed nontopological $d+id$ and $p+ip$ phase to the $s+p+d$ phase via the first-order phase transition. In the time-reversal symmetry breaking $d+id$ and $p+ip$ phase, vortex and antivortex lattices accompanying spontaneous supercurrent are induced by the twist. The superconducting order parameter is nonuniform in the moiré unit cell. Nevertheless, the superconducting gap in the local density of states is identical in the unit cell. The twist-induced vortices and nontopological nature of the mixed $d+id$ and $p+ip$ phase are not captured by the existing effective models. Our results suggest the importance of long-range pairing interaction for effective models.

Figure 1

(a) Rescaled TBLG lattice. The black parallelogram encloses a MUC. AA, AB, and BA mark three different stacking patterns in a MUC. (b) The schematic BZs of the two graphene layers (red and blue hexagons) and the MBZ of the TBLG (black hexagon). (c) The band structure of the TBLG. The green points are from the unrescaled model, while the curves are from the rescaled model. The red curves denote the four flat bands. (d) The DOS of the TBLG. (e), (f) Fermi surface and LDOS of the TBLG at the filling $\nu =(N-1.8)/N$, where $N$ is the total number of energy bands.

Figure 2

Superconducting order parameter distribution in the TBLG for the pairing strength $V=0.8\mathrm{eV}$ (a) and 1.6eV (b).

Figure 3

Averaged order parameter components $s$, $p_x+d_{xy}$, and $p_y+d_{x^2-y^2}$ in the two graphene layers. The amplitude of different components are shown in (a) and (c), and the phase difference between different components are shown in (b) and (d). The insets of (a) and (c) show the averaged $s$ component around the phase transition point. The inset of (d) shows the phase difference between the $p_x+d_{xy}$ components of the two layers.

Figure 4

(a), (b) Profiles of different superconducting order parameter components for the two graphene layers in a MUC with $V=0.8\mathrm{eV}$. The upper panels show the amplitude distribution and the lower panels show the phase distribution. The numbers in the lower panels denote the winding numbers of the vortices.

Figure 5

(a), (b) Profiles of different superconducting order parameter components for the two graphene layers in a MUC with $V=1.6\mathrm{eV}$. The upper panels show the amplitude distribution and the lower panels show the phase distribution.

Figure 6

Profile of the superconducting order parameter $\mathrm{\Delta }\left(k\right)$ in the MBZ with $V=0.8\mathrm{eV}$. (a), (b) Amplitude and phase of the even part of order parameter ${\mathrm{\Delta }}_g\left(k\right)$. (c), (d) Amplitude and phase of the odd part of order parameter ${\mathrm{\Delta }}_u\left(k\right)$. The numbers in (b) and (d) mark the winding numbers of the vortices. There are six vortices at the ${\mathbf{M}}_M$ points of the MBZ boundary in (d). These vortices are shared by two adjacent MBZs and contribute half winding number to the total vorticity in a MBZ which is zero.

Figure 7

Profile of the superconducting order parameter $\mathrm{\Delta }\left(k\right)$ in the MBZ with $V=1.6\mathrm{eV}$. (a), (b) Amplitude and phase of the even part of order parameter ${\mathrm{\Delta }}_g\left(k\right)$. (c), (d) Amplitude and phase of the odd part of order parameter ${\mathrm{\Delta }}_u\left(k\right)$.

Figure 8

Distribution of spontaneous supercurrent for $V=0.8\mathrm{eV}$ in a MUC with the PBC (a) and OBC (b), respectively. The black arrows denote the direction of the supercurrent and the background color represents the magnitude of the supercurrent.

Figure 9

LDOS at four different points in a MUC for $V=0.8\mathrm{eV}$ (a) and $1.6\mathrm{eV}$ (b), respectively. The four different points marked as a, b, c, d are shown in the inset of (b), where the black parallelogram denotes the MUC.