Device geometry and fabrication
The tWSe 2 field-effect devices use contact gates and split gates (Fig. 1a and Extended Data Fig. 1) to achieve low contact resistances (typically about 4 kΩ) and to turn off unwanted parallel conduction channels, respectively. The device channel is defined by gates, including the top, bottom, contact and split gates. Details of the device geometry and device fabrication have been described in ref. 7. In brief, 2D flakes, including few-layer graphite, hBN and monolayer WSe 2 , were mechanically exfoliated from bulk crystals onto silicon substrates. Large WSe 2 monolayers were cut into half by an atomic force microscope tip. These flakes were sequentially picked up using a polycarbonate stamp. From top to bottom, the heterostructures consist of a hBN capping layer, a graphite and hBN (3.4 nm) top gate, the first and second half of a WSe 2 monolayer with a small twist angle, and a hBN (12.0 nm) and graphite bottom gate. The heterostructures were released onto silicon substrates with pre-patterned titanium/platinum (5 nm/25 nm) electrodes. Titanium/palladium (5 nm/35 nm) contact gates and split gates were deposited using the standard electron-beam lithography and evaporation techniques.
Electrical measurements
The electrical transport measurements were performed in a dilution refrigerator (Bluefors LD250) equipped with a 12 T superconducting magnet. Silver-epoxy filters (Basel Precision Instruments MFT25) and additional resistor–capacitor filters were installed on the mixing chamber plate to ensure sufficient thermalization and efficient filtering of the high-frequency radiation, achieving a base electron temperature of less than 100 mK. The longitudinal and transverse resistances were measured using the standard low-frequency (5.777 Hz) lock-in technique. The excitation current was kept at 1 nA to minimize sample heating. The sample resistivity, ρ xx = R xx /(L/W), was calculated from the longitudinal resistance R xx , where L (≈1.6 μm) is the centre-to-centre distance between the voltage probes and W (≈1.3 μm) is the channel width. The Hall resistivity ρ xy is the anti-symmetrized transverse resistance R xy under a positive and negative B-field.
Magneto-optical measurements
Magneto-optical measurements were performed in a closed-cycle cryostat (attocube, attoDRY 2100) down to 1.6 K. Details of the measurements have been reported in ref. 22. In short, a linear polarizer and a quarter-wave plate were used to generate circularly polarized light (σ+ and σ–) from a light-emitting diode. The polarized light beam was focused onto samples by a low-temperature objective lens (numerical aperture 0.8) with a spot size of about 1 µm. The excitation intensity was kept below 50 nW µm−2 to minimize sample heating (no measurable changes in the MCD were observed on further reduction of the excitation power by an order of magnitude). The reflected light was collected by the same objective and directed to a spectrometer equipped with a liquid-nitrogen-cooled charge-coupled device for spectral acquisition. The MCD spectrum is defined as (I− − I+)/(I− + I+), where I− and I+ are the reflection spectra for the σ− and σ+ incident light, respectively. To obtain the data shown in the figures, we integrated the MCD signal over a narrow spectral window (735–740 nm) covering the moiré exciton resonance of tWSe 2 . The magnetic susceptibility was extracted from the slope of the B-field dependence of the MCD signal at zero field. The reflection contrast spectrum is defined as (I – I 0 )/I 0 , where I is the raw reflection spectrum and I 0 is the reference spectrum at a high doping density with no distinct excitonic resonances. To obtain the reflection contrast map in Fig. 2d, we used the mean value of the reflection contrast over the spectral window 736–741 nm that covers the moiré exciton resonance.
Band structure calculations
The low-energy electronic band structure of small twist angle tWSe 2 was calculated using the continuum model26,27. Monolayer WSe 2 is a direct-gap semiconductor with the bandgap located at the two inequivalent corners of the hexagonal BZ, K and K′. The electron valley degree of freedom is locked to the spin degree of freedom because of the broken inversion symmetry and large spin–orbit coupling51. The K and K′ states, carrying opposite spin polarizations, are related by time-reversal symmetry. In tWSe 2 , the valley pocket K t and K b , originated from the top and bottom monolayers, respectively, are slightly displaced in momentum; they define the corners of the moiré BZ for each spin flavour. The low-energy physics of hole-doped tWSe 2 is captured by a two-band k · p model within an effective mass description.
The effective moiré Hamiltonian for the K-valley (spin-up) states can be written as
$${H}_{\uparrow }=\left(\begin{array}{cc}-\frac{{{\hbar }}^{2}{({\bf{k}}-{{\boldsymbol{\kappa }}}_{+})}^{2}}{2{m}^{\ast }}+{\varDelta }_{t}({\bf{r}}) & {\varDelta }_{T}({\bf{r}})\\ {\varDelta }_{T}^{\dagger }({\bf{r}}) & -\frac{{{\hbar }}^{2}{({\bf{k}}-{{\boldsymbol{\kappa }}}_{-})}^{2}}{2{m}^{\ast }}+{\varDelta }_{b}({\bf{r}})\end{array}\right).$$ (1)
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