Phonon-induced superconductivity at high temperatures in electrical graphene superlattices

We discuss the BCS theory for electrons in graphene with a superimposed electrical unidirectional superlattice (SL) potential. New Dirac points emerge together with van Hove singularities (VHSs) linking them. We obtain a superconducting transition temperature $T_c$ for chemical potentials close to the VHSs assuming that acoustic phonon coupling should be the dominant mechanism. Pairing of two onsite electrons with one electron close to the $\mathbf{K}$ and the other close to the $-\mathbf{K}$ point is the most stable pair formation. The resulting order parameter is almost constant over the entire SL.

Figure 1

Left panel: Energy spectrum ${\epsilon }_{+}^0$ [Eq. (12)] at ${\tilde{k}}_x=0$ and $\tilde{\mu }=0$, $s=1$ for various SL potentials $\tilde{V}$. Dotted curves show the corresponding exact spectrum obtained by evaluating the transfer-matrix eigenvalue equation (10) numerically. Inset shows $D_c(1,0)$ [Eq. (21)] as a function of ${\tilde{k}}_y/\tilde{V}$ for ${\epsilon }^0=0$ and SL potentials $\tilde{V}=4,6.66,4\pi$. The curves are calculated by using the outer valley VHS chemical potentials $\tilde{\mu }=0.185,0.5$ in the case $\tilde{V}=4,6.66$, and by using the average chemical potential of the three existent VHSs $\tilde{\mu }=0.415$ being of similar absolute energy value for $\tilde{V}=4\pi$. Right panel: $D_c(1,0)$ and $D_c(0,1)$ for ${\epsilon }^0=0$, $k_y=k_y^n$ and chemical potentials $\tilde{\mu }$ at the VHS $n=1,...,3$ where $n=1$ corresponds to the outermost VHS. The dotted curves show $D_c(1,0)$ by going one order higher taking into account (13) up to order ${(\tilde{E}\tilde{V}/{\alpha }_0)}^2$. $D_c(0,1)$ is not changed within this approximation. Inset shows the energy spectrum ${\tilde{\epsilon }}_{+}$ (20) for $d_1^c=0$ as a function of ${\tilde{d}}_1^s$ for ${\epsilon }^0=0$ and also one further value ${\epsilon }^0\ne 0$. The specific value can be read off from the intersection of the spectral curve with the $y$ axis. The corresponding dashed curves are calculated by a numerical diagonalization of (2) using a transfer-matrix method similar to (5)–(10).

Figure 2

We show the condensate quantities ${\tilde{d}}_1^c$ (upper solid curves) and ${\tilde{d}}_1^s$ (lower dotted curves) for various SL potentials $\tilde{V}$ as a function of temperature by minimizing the free energy (23). The chemical potentials are chosen to lie at the outer valley VHSs for $\tilde{V}=4,6.66$ and at the average of the VHS energies for $\tilde{V}=4\pi$ (see caption to Fig. 1).