Synthesizing Coulombic superconductivity in van der Waals bilayers

Synthesizing a polarizable environment surrounding a low-dimensional metal to generate superconductivity is a simple theoretical idea that still awaits a convincing experimental realization. The challenging requirements are satisfied in a metallic bilayer when the ratio between the Fermi velocities is small and both metals have a similar, low carrier density. In this case, the slower electron gas acts as a retarded polarizable medium (a “dielectric” environment) for the faster metal. Here we show that this concept is naturally optimized for the case of an atomically thin bilayer consisting of a Dirac semimetal (e.g., graphene) placed in atomic-scale proximity to a doped semiconducting transition metal dichalcogenide (e.g., ${\mathrm{WSe}}_2$). The superconducting transition temperature that arises from the dynamically screened Coulomb repulsion is computed using the linearized Eliashberg equation. In the case of graphene on ${\mathrm{WSe}}_2$, we find that $T_c$ can exceed 100 mK, and it increases further when the Dirac valley degeneracy is reduced. Thus, we argue that suspended van der Waals bilayers are in a unique position to realize experimentally this long-anticipated theoretical concept.

Figure 1

The proposed setup for synthethesizing Coulombic superconductivity. (a) A high velocity Dirac semimetal (DSM) is placed in proximity to a semiconducting transition metal dichalcogenide (TMD). The density of the two layers can be controlled independently using distant gate electrodes [17]. (b) As the density on both layers is tuned to zero, the ratio of their Fermi velocities also trends to zero (i.e., $v_s/v_d\ll 1$), where $v_d$ is the Dirac velocity and $v_s$ is the Fermi velocity of the TMD. (c) The optical plasmon is a collective mode of two propagating charge density waves, one on each layer, which are “in-phase” with each other. (d) The acoustic plasmon is a collective mode for which the two charge density waves are “out-of-phase” with each other. As such, this mode is charge neutral overall, resulting in an acoustic mode. (e) The schematic dispersion of the optical plasmon (solid) and the acoustic plasmon (dashed) superimposed on the particle-hole continuum of the Dirac semimetal (blue) and the TMD (red). The velocity of the acoustic mode scales as the geometric mean of the two Fermi velocities $\sim \sqrt{v_s{v}_d}$; thus it becomes very slow for low densities and can mediate superconductivity efficiently.

Figure 2

Results of the estimation of $T_c$ using only the attraction from the acoustic plasmon. Top: the screened Coulomb repulsion ${\mu }^{*}$ Eq. (7) (dot-dashed lines) and the acoustic plasmon attraction $\lambda$ Eq. (8) (solid lines) vs density $n_s=n_d$ for different DSM degeneracies, indicated on by color on the lines in the lower panel. Bottom: The corresponding transition temperature Eq. (9) vs density for the different DSM cases. An additional attraction coming from the phonons ${\lambda }_{\text{ph}}=0.05$ [40] is included in the graphene case, and we also show its impact on the single-valley DSM.

Figure 3

The vertex function for $s$-wave pairing, Eq. (11), as a function of Matsubara frequency for $k_d=k_s, k=0.9k_d, k^{\prime }=1.1k_d, Q_d/k_d=8.75$, and $m_s=0.5m_e$. The interaction decreases in two steps marked by grey arrows, corresponding to the dynamical contribution of the optical and acoustic plasma modes. The red dashed curve denotes the vertex function without the semiconducting layer. The difference between these curves is the contribution of the acoustic plasmon, which leads to superconductivity. The dashed orange curve corresponds to the acoustic plasmon approximation, Eq. (4), which does not account for the optical plasmon.

Figure 4

The transition temperature obtained from the numerical solution of Eq. (10) as a function of the density of the semiconducting layer $n_s$ for different values of the density of the Dirac semimetal $n_d$ (indicated in the legend). Solid lines correspond to a single-valley DSM, $g_v^d=1$ and the dashed lines to double valley (i.e., graphene). All data for $g_{\sigma }^s{g}_{\sigma }^d=2$. See Appendix pp2 for technical details.

Figure 5

(a) The eigenvector obtained from diagonalizing the kernel in Eq. (10) for $T=180$ mK, $n=n_d=n_s={10}^{11}{\mathrm{cm}}^{-2}, g_v^d=1, a=0.5, N=30, {\beta }_k=0.45, {\beta }_{\omega }=4, \mathrm{\Omega }={\epsilon }_d/2, \mathrm{\Lambda }=\mathrm{\Omega }/2v_d, v_d={10}^6$ m/s, and $m_s=0.5m_e$. (b) The frequency dependance of the eigenvector for $k=k_d$. (c) The momentum dependance for $\omega =\pi T$ (red) and $\omega =\mathrm{\Omega }$ (black).

Figure 6

The dependence of $T_c$ on the frequency cutoff $\mathrm{\Omega }$.

Figure 7

The dependence of $T_c$ on the number of grid $1/N_k$ for $N_{\omega }=20, {\beta }_k=0.55, {\beta }_{\omega }=4, \mathrm{\Omega }=0.5{\epsilon }_d, \mathrm{\Lambda }=0.2k_d$. The dashed black line is the interpolation to the continuum limit $N_k\to \infty$.

Figure 8

The transition temperature vs the dielectric constant $\varepsilon$ calculated numerically from Eq. (10) for three different cases: double-valley DSM—black, single-valley DSM—red, and for surface states of a topological insulator—orange. The frequency cutoff $\mathrm{\Omega }$ for these three cases are ${\epsilon }_d, 0.5{\epsilon }_d$, and $0.32{\epsilon }_d$, respectively. All other parameters are the same as in Fig. 4.

Figure 9

(a) $T_c$ vs density for the case of $n=n_d=n_s$ for the case of a single-valley DSM $g_v=1$. Solid lines correspond to $a=0.5$ nm and dashed lines to $a=0.1$ nm. (b) $T_c$ as a function of the layer separation $a$. Red curves correspond to single-valley DSMs and black to double-valley DSMs.

Figure 10

$T_c/T_{c0}$ as a function of $k_{d}l$ for monolayer graphene at different experimentally relevant carrier densities, assuming $T_{c0}=100\text{mK}$ in all cases for simplicity.