Bandwidth-tuned Mott transition and superconductivity in moiré WSe2

The Hubbard model, describing electrons hopping on a lattice with a hopping amplitude t between neighbouring sites and an on-site electron–electron repulsion U, provides a simplified representation of high-temperature superconductors^1,2,3,4,5. The electron hopping leads to a finite bandwidth W (=8t and 9t, respectively, for square and triangular lattices). In cuprates, W and U are comparable^3,5, indicating that these materials are in the moderate correlation regime and are near a Mott transition from an insulator to a metal^3,15. The idea of doping a Mott insulator for superconductivity has been extensively studied for decades^1,2,3,4,5. Although the nature of the superconducting state itself has been qualitatively understood, a full understanding of the rich phase diagram remains unknown because accurately solving the Hubbard model in the moderate correlation regime, in which different orders intricately compete in the ground state, is difficult⁶. Studying high-transition-temperature (T_c) phenomenology in a tunable quantum system^16,17 may, therefore, shed light on how superconductivity emerges and pave the way for designing new high-T_c materials.

Transition metal dichalcogenide (TMD) moiré heterobilayers, such as WSe₂/WS₂, are established quantum simulators of the two-dimensional (2D) triangular lattice Hubbard model^18,19. For small twist angles, the moiré period (consequently U/W) is largely determined by the lattice mismatch in heterobilayers. Studies so far have focused on the strong correlation limit, and superconductivity has not yet been realized. The recent demonstration of robust superconductivity in twisted WSe₂ (tWSe₂) homobilayers^7,8, for which U/W can be readily tuned by the twist angle^20,21,22, provides the possibility of exploring the superconducting phase diagram in the moderate correlation regime^23,24,25. Here, we report the phase diagram of tWSe₂, focusing on twist angle around 4.6°, for which U and W are comparable. The transport characteristics, supplemented by magnetic susceptibility, reveal rich high-T_c phenomenology, including an antiferromagnetic (AF) insulator, superconducting domes and strange metals. The phase diagrams for both electron and hole doping of the AF insulator and their continuous evolution as the system undergoes a band-structure-tuned Mott transition can be obtained using a single dual-gated device (Fig. 1a).

a, Schematic side view of dual-gated transport devices. The tWSe₂ sample is contacted by platinum (Pt) electrodes and controlled by voltages V_tg and V_bg applied on the hBN/Gr gates. The Pd contact and split gates turn on the Pt contacts and turn off the parallel channels, respectively. b, Topmost moiré valence bands for the K-valley state (blue solid line) and K′-valley state (orange dashed line) at E = 0 mV nm⁻¹ (top) and 100 mV nm⁻¹ (bottom). Arrows mark the vHS. c, Electronic DOS as a function of ν and E. The vHS with high DOS shifts towards higher ν with increasing E. Results in b and c are from the continuum model calculations for 4.6° tWSe₂. d, Resistance as a function of ν and E at 50 mK in tWSe₂ with twist angle varying from 2.1° to 4.7°. e, θ–E phase diagram at ν = 1 constructed from experiment (symbols). Red, correlated insulator; dark blue, superconductor (SC); yellow, AF metal; light blue, metal. f, Superconducting transitions (for the highest T_c) for different twist angles. g, Highest T_BKT (Berezinskii–Kosterlitz–Thouless transition temperature) versus θ. The blue triangule is from ref. ⁸. Dashed lines in c and e denote the boundary between the layer-hybridized and layer-polarized regions.

Source data

Monolayer WSe₂ is a triangular lattice semiconductor with its valence band maxima located at the K and K′ points of the hexagonal Brillouin zone (BZ)¹⁹. A triangular moiré lattice with period \({a}_{{\rm{M}}}=\frac{a}{\sqrt{2(1-\cos \theta )}}\) is formed when two layers are stacked with a relative twist angle θ, where a = 0.33 nm is the monolayer lattice constant. Figure 1b,c shows, respectively, the band structure and density of states (DOS) as a function of hole filling factor (ν) and vertical electric field (E) for 4.6° tWSe₂ from the continuum model calculations (Methods). The K-valley states of the top and bottom layers fold, respectively, onto the κ and κ′ valleys of the moiré BZ, which are swapped for the K′-valley states^{9,10,11,26,27}. The bands from the two layers hybridize and generate a saddle point at which they intersect. The hybridized bands remain spin degenerate for small θ because of spin–valley locking in monolayer TMDs¹⁹ (Fig. 1b, top). Application of a finite E-field places the two layers at different potentials and lifts the spin degeneracy¹⁰ (Fig. 1b, bottom); it also shifts the van Hove singularity (vHS) with a diverging DOS continuously from ν < 1 to ν > 1 (Fig. 1c). Sufficiently large fields eventually polarize the holes to one of the layers. Theoretical studies showed that the topmost moiré valence band, if nontopological, can be described by the triangular lattice Hubbard model with an additional E-dependent spin–orbit coupling term^9,10,11. This applies to samples with θ ≥ 4° (ref. ²⁸).

Figure 1d shows resistance R as a function of E and ν at temperature T = 50 mK in a series of dual-gated tWSe₂ devices with twist angle increasing from 2.1° to 4.7° (see Methods for details on the device fabrication and characterizations; see Extended Data Fig. 1 for a device image; see Extended Data Fig. 2 for the determination of the moiré density). The resistance maps qualitatively agree with the DOS map in Fig. 1c, including a layer-hybridized region centred at E = 0 and a vHS with enhanced resistance due to the large DOS. Not captured by the band theory are the insulating states at fractional fillings of the first moiré band, ν = 1, 1/3 and 1/4, which exhibit strong electron correlations. As the twist angle increases, the correlated insulators gradually melt and are indiscernible at 4.7°. This is expected because the correlation effects weaken as the moiré period decreases. The E-field dependence of the ν = 1 state is further enriched by the presence of the vHS, which enhances the correlation effects^7,8,10,20,29 (see Methods for further discussions).

Superconductivity starts to emerge near θ = 3.5–3.6° and persists over a range of twist angle up to about 5° (beyond which superconductivity is expected to fade away together with the moiré effect). Rather than following the vHS, the most robust superconducting state always appears right next to the ‘melting point’ of the ν = 1 insulator on the low E-field end in the layer-hybridized region. The optimal T_c increases with θ (Fig. 1f,g). At ν = 1, superconductors occupy a narrow strip (dark blue) in the (θ–E) phase diagram (Fig. 1e). Both θ and E can induce an insulator-to-superconductor transition. Theoretical studies examined the phase diagram in the small and large twist angle limits (large and small U/W)^{30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45}. The moderate U/