Doping fingerprints of spin and lattice fluctuations in moiré superlattice systems

Twisted Van der Waals systems offer the unprecedented possibility to tune different states of correlated quantum matter with external noninvasive electrostatic doping. The nature of the superconducting order presents a recurring open question in this context. In this work, we assess quantitatively the case of spin-fluctuation-mediated pairing for $\mathrm{\Gamma }$-valley twisted transition metal dichalcogenide homobilayers. We calculate self-consistently and dynamically the doping-dependent superconducting transition temperature $T_{\mathrm{c}}$ revealing a superconducting dome with a maximal $T_{\mathrm{c}}\approx 0.1–1\mathrm{K}$ depending on twist angle. We compare our results with conventional phonon-mediated superconductivity, and we identify clear fingerprints in the doping dependence of $T_{\mathrm{c}}$, which enable experiments to distinguish between different pairing mechanisms.

Figure 1

$\mathrm{\Gamma }$-valley twisted TMDCs. (a) Moiré pattern of twisted TMDCs. AA (gray shaded), AB (blue shaded), and BA (red shaded) regions in the moiré pattern correspond to different stackings of the two layers, as shown on the right side. Dashed black lines serve as a guide to the eye to identify the honeycomb superlattice. The most relevant hopping processes are sketched with black arrows. (b) Continuum model for ${\mathrm{WS}}_2$ at a twist angle of $\theta =3.5^{\circ }$ (black solid line) with a third-nearest-neighbor hopping tight-binding model (red dashed line) of the highest valence bands. The effective honeycomb lattice is formed by the AB and BA moiré sites. The right panel shows a zoom of the flat Dirac bands. (c) Twist-angle dependence of the hopping parameters for different $\mathrm{\Gamma }$-valley twisted TMDCs, ${\mathrm{WS}}_2, {\mathrm{MoS}}_2$, and ${\mathrm{MoSe}}_2$ obtained via Wannier projection.

Figure 2

Spin fluctuation characteristics of $\mathrm{\Gamma }$-valley twisted TMDCs at $T/t=0.003$. (a) Leading Stoner enhancement factor ${\alpha }_{\mathrm{S}}={\max }_{\mathbf{q}}\left\{U{\chi }^0\left(\mathbf{q}\right)\right\}$ for different Coulomb interaction strengths $U/t$ and dopings $\delta$ with respect to the Dirac point as obtained from FLEX. A transition to a quasiordered magnetic state is assumed for ${\alpha }_{\mathrm{S}}\ge 0.99$. (b) Real-space components of the static spin susceptibility ${\chi }^{\mathrm{s}}\left(\mathbf{r}\right)$ for $U/t=6$. Up to eighth-nearest-neighbor components are shown, with $\left|\mathbf{r}\right|$ denoting the distance between two spins in terms of the moiré unit length ${\lambda }^{\mathrm{M}}$. Solid (dashed) lines correspond to the AA (AB) components of ${\chi }^{\mathrm{s}}$, i.e., correlations between same (different) sublattice sites. The area around the Van Hove singularities (VHS) is not accessible because of too strong fluctuations; it is marked by a gray shaded area [cf. panel (a)].

Figure 3

Doping dependence of the superconducting transition temperature $T_{\text{c}}$ in $\mathrm{\Gamma }$-valley twisted TMDCs. (a) Phase diagram for spin-fluctuation-mediated pairing for different local Coulomb interaction strengths $U/t$ from FLEX calculations. The critical temperature $T_{\text{c}}/t$ belongs to the dominant singlet $d$-wave pairing symmetry. (b) Phonon-mediated $T_{\mathrm{c}}$ for an Einstein-Holstein phonon mode ${\omega }_0$ and for different Coulomb pseudopotentials ${\mu }^{*}=0.0$ (blue), 0.1 (gray), and 0.2 (red). From left to right, the effective attractive interaction $U_{\mathrm{eff}}$ from electron-phonon coupling is increased. The trend of the density of states (DOS) is indicated by black dashed lines. (c) Comparison of the doping-dependent phase diagram of the maximum $T_{\text{c}}$ obtained for spin fluctuations and phonons. This phase diagram holds qualitatively for all $\mathrm{\Gamma }$-valley twisted TMDCs.