Interlayer excitonic superfluidity in graphene

We discuss the conditions under which the predicted (but not yet observed) zero-field inter-layer excitonic condensation in double layer graphene has a critical temperature high enough to allow detection. Crucially, disorder arising from charged impurities and corrugation in the lattice structure—invariably present in all real samples—affects the formation of the condensate via the induced charge inhomogeneity. In the former case, we use a numerical Thomas-Fermi-Dirac theory to describe the local fluctuations in the electronic density in double layer graphene devices and estimate the effect these realistic fluctuations have on the formation of the condensate. To make this estimate, we calculate the critical temperature for the interlayer excitonic superfluid transition within the mean-field BCS theory for both optimistic (unscreened) and conservative (statically screened) approximations for the screening of the interlayer Coulomb interaction. We also estimate the effect of allowing dynamic contributions to the interlayer screening. We then conduct similar calculations for double quadratic bilayer graphene, showing that the quadratic nature of the low-energy bands produces pairing with critical temperature of the same order of magnitude as the linear bands of double monolayer graphene.

Figure 1

$T_c$ for the unscreened interaction in dielectric environment $\epsilon =3.9$ (a) and (b) show color plots as a function of $\overline{\mu }$ and $\delta \mu$ for $d=1$ and $5\mathrm{nm}$, respectively. (c) $T_c$ as a function of $\overline{\mu }$ for various $\delta \mu$ and $d=1\mathrm{nm}$. (d) $T_c$ as a function of $\delta \mu$ for various $\overline{\mu }$ and $d=1\mathrm{nm}$.

Figure 2

Static interlayer screening for DMG with ${\Delta }_{\mathrm{k}}={\Delta }_{k_F}$. (a) For various $\delta \mu$ as a function of $\overline{\mu }$ and (b) for various $\overline{\mu }$ as a function of $\delta \mu$. Note that the scale on the vertical axis is micro-Kelvin.

Figure 3

Comparison of (a) the statically screened and (b) the real part of the dynamically screened interlayer interaction potentials to the unscreened interaction $V_q=2{\pi }^2{e}^{-qd}/\left(\epsilon q\right)$ for DMG with $\epsilon =3.9$. The inset to (b) shows the imaginary part of the interaction.

Figure 4

Spatial plots of $\delta \mu$ calculated via the TFDT. The left column is for $d=1\mathrm{nm}$, $d_B=1\mathrm{nm}$, and $\overline{\mu }=50\mathrm{meV}$, which roughly approximates the experimental situation in Ref. 7. The right column is for $d=5\mathrm{nm}$, $d_B=20\mathrm{nm}$, and $\overline{\mu }=200\mathrm{meV}$, which roughly approximates the experimental situation in Ref. 6. The color bar at the bottom of each column applies to all three plots in each column. The first row is for $n_{\mathrm{imp}}={10}^{11}{\mathrm{cm}}^{-2}$, the second row is $n_{\mathrm{imp}}={10}^{10}{\mathrm{cm}}^{-2}$, the third row is $n_{\mathrm{imp}}={10}^9{\mathrm{cm}}^{-2}$.

Figure 5

Root-mean-square of the distribution of the local $\delta \mu$ as a function of the global chemical potential for two experimentally relevant geometries. (a) and (b) $d_B=1\mathrm{nm}$, $\epsilon =3.9$ corresponding to double layer graphene placed straight onto a SiO${}_2$ substrate per the experiments in Ref. 7. (c) and (d) $d_B=20\mathrm{nm}$, $\epsilon =3.9$ corresponding to double layer graphene placed onto a 20$\mathrm{nm}$ slab of hBN per the experiments in Ref. 6.

Figure 6

(a) $T_c$ as a function of $\overline{\mu }$ with $\delta \mu =0$ for double bilayer graphene with the unscreened interlayer interaction. (b) $T_c$ as a function of $\delta \mu$ with $\overline{\mu }=30\mathrm{meV}$. (c) The statically screened interlayer interaction for DQBG. (d) The dynamically screened interlayer interaction for DQBG in the high density regime ($k_F=0.2{\mathrm{nm}}^{-1}$).

Figure 7

(a) The diagram for $\Pi (\mathrm{q},\omega )$ in our calculation. (b) The diagram for $\Pi (\mathrm{q},\omega )$ in the calculation of Refs. 3 and 13. (c) The first-order diagram absent from (b). (d) The second-order diagrams absent from (b). The dashed lines represent interlayer electron-electron interactions.