Resonating valence bonds and mean-field $d$-wave superconductivity in graphite

We investigate the possibility of inducing superconductivity in a graphite layer by electronic correlation effects. We use a phenomenological microscopic Hamiltonian which includes nearest-neighbor hopping and an interaction term which explicitly favors nearest-neighbor spin singlets through the well-known resonance valence bond (RVB) character of planar organic molecules. Treating this Hamiltonian in mean-field theory, allowing for bond-dependent variation of the RVB order parameter, we show that both $s$- and $d$-wave superconducting states are possible. The $d$-wave solution belongs to a two-dimensional representation and breaks time-reversal symmetry. At zero doping there exists a quantum critical point at the dimensionless coupling $J∕t=1.91$ and the $s$- and $d$-wave solutions are degenerate for low temperatures. At finite doping the $d$-wave solution has a significantly higher $T_c$ than the $s$-wave solution. By using density functional theory we show that the doping induced from sulfur absorption on a graphite layer is enough to cause an electronically driven $d$-wave superconductivity at graphite-sulfur interfaces. We also discuss applying our results to the case of the intercalated graphites, as well as the validity of a mean-field approach.

Figure 1

(Color online) (a) shows the 2D honeycomb lattice with lattice vectors ${\mathbf{c}}_1$ and ${\mathbf{c}}_2$ and nearest-neighbor vectors ${\mathbf{a}}_1$, ${\mathbf{a}}_2$, and ${\mathbf{a}}_3$ indicated as well as the two different lattice sites $f$ (blue) and $g$ (red). (b) Band structure of graphene, $\pm t{ϵ}_{\mathbf{k}}$, in the first Brillouin zone. The band structure close to the corners of the Brillouin zone has a linear dispersion with a pointlike Fermi surface at zero doping.

Figure 2

(Color online) (a) Dimensionless transition temperature $k_B{T}_c∕t$ as function of $J∕t$ for doping levels $\delta =0$, 0.001, 0.01, 0.05, 0.1, and 0.2. Big (green) arrow indicates direction of increasing doping. The dimensionless temperature scale spans approximately $0.03–3000\mathrm{K}$ for $t=2.5\mathrm{eV}$. (b) Transition temperature $T_c$ calculated using $t=2.5\mathrm{eV}$ as a function of doping level $\delta$ for $J∕t=0.8$, 0.9, 1, 1.1, and 1.2. Big (green) arrow indicates direction of increasing $J∕t$.

Figure 3

(Color online) (a) Band structure for ${\mathrm{C}}_2$ (black) and ${\mathrm{C}}_2\mathrm{S}$ in the hollow configuration [red (gray)]. The Fermi level is indicated with a dotted line. Inset shows an enlargement around the $K$ point. (b) Band structure for ${\mathrm{C}}_{18}$ (black) and ${\mathrm{C}}_{18}\mathrm{S}$ in the top configuration [red (gray)]. Inset shows an enlargement around the $\Gamma$ point.