Odd-momentum pairing and superconductivity in vertical graphene heterostructures

Vertical graphene heterostructures made up of graphene layers separated by boron nitride spacers allow for novel ways of tuning the interactions between electrons. We study the possibility of electron pairing mediated by modified repulsive interactions. Long-range intravalley and short-range intervalley interactions give rise to different anisotropic phases. We show that a superconducting state with gaps of opposite signs in different valleys, an odd momentum pairing state, can exist above carrier densities of $5–10\times {10}^{13}{\mathrm{cm}}^{-2}$. The dependence of the transition temperature on the different parameters of the devices is studied in detail.

Figure 1

Sketch of the heterostructure considered in the text. Two graphene mono- or bilayers are embedded into three dielectric media. The three dielectrics need not have the same dielectric constant. The device is characterized by the carrier densities in the graphene layers, ${\rho }_t$ and ${\rho }_b$, the distance between layers, $d$, and the three dielectric constants, ${\epsilon }_1={\epsilon }_t,{\epsilon }_2={\epsilon }_m$, and ${\epsilon }_3={\epsilon }_b$.

Figure 2

Sketch of the scattering processes of Cooper pairs considered in the text: (a) scattering within a given valley and (b) scattering between valleys.

Figure 3

Diagram included in the calculation of the effective interaction, $V\left(q\right)$. Red and blue bubbles stand for the polarizabilities in the two layers in the heterostructure. The dashed lines describe the Coulomb interaction, which can be intra- or interlayer.

Figure 4

Imaginary part of the effective potential for a heterostructure with ${\rho }_t=4\times {10}^{13}{\mathrm{cm}}^{-2},{\rho }_b={10}^{13}{\mathrm{cm}}^{-2},{\epsilon }_1={\epsilon }_2={\epsilon }_3=4$, and $d=10$ nm. Left: top layer. Right: bottom layer.

Figure 5

As in Fig. 4 but with ${\epsilon }_t=10$. Left: top layer. Right: bottom layer.

Figure 6

Effective interaction $A\left(\theta \right)=(2/\pi )\rho \left({\epsilon }_F^t\right)V\left[2k_F^t\sin (\theta /2)\right]$ as a function of $\theta$ for a heterostructure made of two single layer graphene sheets with densities ${\rho }_t={\rho }_b={10}^{13}{\mathrm{cm}}^{-2}$ and $d=10,40,70,...,280$ Å (from bottom to top).

Figure 7

Effective interaction $A\left(\theta \right)$ for a graphene heterostructure made of single layer graphene and bilayer graphene with densities ${\rho }_{slg}={\rho }_{blg}={10}^{13}$ cm${}^{-2}$ and $d=10$ Å.

Figure 8

Values of ${\lambda }_n$ as a function of $k_F^{t}d$ for the same parameters as in Fig. 6.

Figure 9

Sketch of the gap with alternating signs in different valleys induced by intervalley scattering. The odd momentum pairing state $\Delta \left(\vec{\mathbf{k}}\right)=-\Delta (-\vec{\mathbf{k}})$ breaks time reversal and inversion symmetry.

Figure 10

Dependence of the parameter ${\lambda }_0$ in Eq. (16) on distance for a heterojunction with ${\epsilon }_1={\epsilon }_2={\epsilon }_3=4,n_t={10}^{14}{\mathrm{cm}}^{-2}$, and $n_b=2\times {10}^{14}{\mathrm{cm}}^{-2}$.

Figure 11

Dependence of the critical temperature on carrier density for a heterostructure made of two single layers with the same concentration, separated by a distance $d=10$ Å, and ${\epsilon }_1={\epsilon }_2={\epsilon }_3=4$.

Figure 12

Dependence of $T_c$ on interlayer distance and dielectric constant. Top: Dependence on distance, for $U=10$ eV and ${\epsilon }_1={\epsilon }_2={\epsilon }_3=4$. Bottom: Dependence on ${\epsilon }_1$, for $U=10$ eV, and $d=10$ Å.

Figure 13

Dependence of $T_c$ on the electron-phonon coupling $g_{ph}$. The remaining parameters are $U=10$ eV, d $=$ 10 Å, and ${\epsilon }_1={\epsilon }_2={\epsilon }_3=4$.

Figure 14

Subbands of a graphene ribbon with a superconducting gap which has opposite signs in the two valleys. In order to better illustrate the main features, the Fermi energy is ${\epsilon }_F=0.5$ eV and the gap is $\Delta =0.2$ eV. Top: armchair edge. Bottom: zigzag edge.