Pairing in graphene: A quantum Monte Carlo study

To address the issue of electron correlation driven superconductivity in graphene, we perform a systematic quantum Monte Carlo study of the pairing correlation in the $t-U-V$ Hubbard model on a honeycomb lattice. For $V=0$ and close to half filling, we find that pairing with $d+id$ ($d_{x^2-y^2}+i{d}_{xy}^{\prime }$ in its specific form) symmetry dominates pairing with extended-$s$ symmetry. However, as the system size or the on-site Coulomb interaction increases, the long-range part of the $d+id$ pairing correlation decreases and tends to vanish in the thermodynamic limit. An inclusion of nearest-neighbor interaction $V$, either repulsive or attractive, has a small effect on the extended-$s$ pairing correlation, but strongly suppresses the $d+id$ pairing correlation.

Figure 1

(a) Sketch of graphene with double-48 sites, (b) Phases of the $d+id$ and ES pairing symmetries on the honeycomb lattice. For clarity, the phase for the ES symmetry is omitted, which equals 0 for all ${\mathit{\delta }}_l$.

Figure 2

(a) Spin structure factor $S\left(\mathbf{q}\right)$ on the double-48 lattice along the high symmetry lines in the first Brillouin zone (BZ) at electron fillings $\langle n\rangle =1.00$, 1.10, and $1.20$. (b) NN spin correlation $S_{\langle i,j\rangle }^{zz}$ vs $\langle n\rangle$ on different lattices. The inset of (a) shows the first BZ, with $a$ denoting the distance between NN lattice sites. The results are presented for temperature $T=\left|t\right|/6$.

Figure 3

(a) Pairing susceptibility $P_{\alpha }$ as a function of temperature $T$ for different pairing symmetries and different fillings. (b) ${\tilde{P}}_{d+id}$ and $P_{d+id}$ as a function of temperature at $\langle n\rangle =1.05$ and $\langle n\rangle =1.10$. Dashed and dashed-dotted lines in (a) represent the $d+id$ pairing susceptibility at $U=0$. The inset of (b) shows the effective pairing interaction $P_{d+id}-{\tilde{P}}_{d+id}$ as a function of temperature.

Figure 4

Pairing correlation $C_{\alpha }$ as a function of distance for the $d+id$ and ES pairing symmetries on the double-75 lattice with $\langle n\rangle =1.107$ (a) and double-108 lattice with $\langle n\rangle =1.13$ (b). The inset of (b) shows the average of the long-range $d+id$ pairing correlation ${\overline{C}}_{d+id}$ vs $\frac{1}{\sqrt{N_s}}$ for different $U$’s at a filling $\langle n\rangle \approx 1.1$.

Figure 5

Pairing correlation as a function of distance for the $d+id$ (filled symbols) and ES (open symbols) pairing symmetries on the double-75 lattice with $\langle n\rangle =1.107$. The value of NN interaction $V$ is indicated by the shape of the symbol, which is also applied for the ES symmetry.