Condensation of electron-hole pairs in a two-layer graphene system: Correlation effects

Condensation of pairs formed by spatially separated electrons and holes in a system of two isolated graphene layers is studied beyond the mean-field approximation. Suppression of the screening of the pairing interaction at large distances, caused by the appearance of the gap, is considered self-consistently. A mutual positive feedback between the appearance of the gap and the enlargement of the interaction leads to a sharp transition to a correlated state with a greatly increased gap above some critical value of the coupling strength. At a coupling strength below the critical value, this correlation effect increases the gap approximately by a factor of 2. The maximal coupling strength achievable in experiments is close to the critical value. This indicates the importance of correlation effects in closely spaced graphene bilayers at weak substrate dielectric screening. Another effect beyond the mean-field approximation considered is the influence of vertex corrections on the pairing, which is shown to be very weak.

Figure 1

The effective static polarizability $\Pi$ (in units of the density of states $\mathcal{N}$) in a graphene bilayer with the pairing gap as a function of momentum $q$ at different values of $\Delta /\mu$, indicated above corresponding curves (solid lines). Dotted line: the intrinsic static polarizability ${\Pi }_0$ at $\Delta =0$.

Figure 2

Solid lines: the right-hand side $I$ of the gap equation (6) $I\left(\Delta \right)=1$ as a function of $\Delta$ (in logarithmic scale) at different values of $r_{\mathrm{s}}$, indicated on the right.

Figure 3

Solid line: the largest value of the gap $\Delta$ (in logarithmic scale), found from Eq. (6) at different values of $r_{\mathrm{s}}$. At $r_{\mathrm{s}}\approx 2.35$, the gap jumps (dashed line) to a much larger value due to the appearance of three solutions of Eq. (6). Dotted line: the gap, calculated in the mean-field approximation.

Figure 4

(a) The bare vertex of the electron-electron Coulomb interaction and (b) the simplest correction to it.

Figure 5

The dimensionless second-order correction ${\lambda }_{++}^{\left(1\right)}$ (dotted line) to the coupling constant as a function of $r_{\mathrm{s}}$ in comparison with the first-order intraband ${\lambda }_{\gamma \gamma }^{\left(0\right)}$ (solid line) and interband ${\lambda }_{\gamma ,-\gamma }^{\left(0\right)}$ (dashed line) coupling constants.