Fig. 1: Ferromagnetism and valley polarisation at ν < 2.

a Schematic of hBN-encapsulated TBG-WSe₂ heterostructure on SiO₂/Si substrate. The carrier density in the system is tuned by applying top gate voltage V_G. The longitudinal V_xx and transverse V_xy voltages are measured by driving current I through the channel of the Hall bar device in the presence of an out-of-plane magnetic field B. The black arrow indicates the direction of the magnetic field. b Four-probe longitudinal resistance R_xx as a function of filling ν measured at T = 0.3 K and B = 0 T. c Hall resistance R_xy for three perpendicular magnetic fields B = 0.05, 0.4 and 1.2 T for density being swept back and forth. The red and blue arrows indicate directions of density sweep. The horizontal black dashed lines at R_xy = 0 are drawn for a better illustration of the zero crossings in R_xy that disappear with increasing B-field. d R_xy at three different fillings for B swept back and forth, as indicated by the arrows. A reversal of hysteresis is seen at ν = 1.86. The coercive fields are indicated as B⁺ and B⁻. e Colour plot of ΔR_xy, defined as the difference between the values of R_xy for the opposite field sweeps, as a function of ν and B. The change in colour along the vertical dashed line shows the reversal of magnetisation that accompanies the occupation of electrons in different K and \({K}^{{\prime} }\) valleys (see the inset schematics). The red and blue circles around two valleys represent the electronic orbital, and the difference in thickness of these circles indicates a valley imbalance. In the bottom panel, the ratio of the magnitude of negative and positive coercive fields α is plotted as a function of ν. Coercive fields are asymmetric in positive and negative B. The asymmetry flips exactly when the magnetisation is reversed as evident from the data points above (α > 1) and below (α < 1) the horizontal black dashed line at α = 1. The error bars in determining the coercive fields are negligible.