$d$-wave superconductivity on the honeycomb bilayer

We introduce a microscopic model on the honeycomb bilayer, which in the small-momentum limit captures the usual (quadratic dispersion in the kinetic term) description of bilayer graphene. In the limit of strong interlayer hopping it reduces to an effective honeycomb monolayer model with also third-neighbor hopping. We study interaction effects in this effective model, focusing on possible superconducting instabilities. We find $d_{x^2-y^2}$ superconductivity in the strong-coupling limit of an effective $tJ$-model-like description that gradually transforms into $d+id$ time-reversal symmetry-breaking superconductivity at weak couplings. In this limit the small-momentum order-parameter expansion is ${(k_x+i{k}_y)}^2$ [or ${(k_x-i{k}_y)}^2$] in both valleys of the effective low-energy description. The relevance of our model and investigation for the physics of bilayer graphene is also discussed.

Figure 1

(a) View of Bernal stacked honeycomb lattices 1 and 2 with corresponding sublattice sites A1, B1 and A2, B2, respectively. (b) Model reduced to a monolayer model with the third-neighbor hopping $\tilde{t}\equiv t^2/t_{\perp }$ and the nearest-neighbor hopping $2\tilde{t}$ (see the text).

Figure 2

(a) Noninteracting dispersion and (b) density of states of the projected monolayer model. (c) Linear dispersion in the vicinity of the K points in the graphene monolayer in comparison to (d) quadratic dispersion in our model. We use $\tilde{t}=t^2/t_{\perp }$ for the unit of energy.

Figure 3

$C_k$ in the first Brillouin zone calculated for three possible symmetries on monolayer and projected bilayer lattices.

Figure 4

(a) Order-parameter amplitude $\Delta$ in the $(\mu ,J)$ parameter space, obtained by a minimization of the free energy, (b) single-particle excitation gap, (c) contribution of $i{d}_{xy}$, and (d) $s$-wave component in the ground-state order parameter. The green dashed line marks where $\Delta$ drops below ${10}^{-4}$. Below this line, our numerics is not reliable. We use $\tilde{t}=t^2/t_{\perp }$ for the unit of energy.

Figure 5

(a) Order-parameter amplitude $\Delta$ and (b) single-particle excitation gap as a function of $J$, for $\mu =0.04,0.55,1$. (c–e) The contributions of three relevant symmetry components. The $d_{x^2-y^2}$ component is the dominant one for large $J$. The contribution of $i{d}_{xy}$ increases with decreasing $J$ until the two contributions are equal and we find a pure $d+id$-wave symmetry. We use $\tilde{t}=t^2/t_{\perp }$ for the unit of energy. The data are plotted only above the value for the coupling $J$ which is numerically significant, as mentioned in the text (see also the dashed green line in Fig. 4).