Extended Data Fig. 2: Estimation of carrier densities.

(a). B_⊥/eR_xy versus V_f in a 50 μm × 50 μm Hall-bar at 300 mK under B_⊥= 1 T. Inset shows a schematic of the asymmetric Hall-bar, and the dashed box region is covered by the front gate. The yellow dashed line is obtained from the integration of measured capacitance over V_f, representing the net-carrier density |n_e − n_p| (see ref. ¹²). In the linear regime of B_⊥/eR_xy, n_p is negligible with |n_e − n_p| ≈ n_e, and the yellow line represents n_e (blue arrow). B_⊥/eR_xy deviates from the yellow line due to the increased n_p in the two-carrier transport. In this regime, V_f would tune both n_e and n_p. Since the geometry capacitance remains constant and dominates the measured capacitance, |n_e − n_p| would still change linearly with V_f and follow the yellow line. The green dashed line represents the V_f dependence of n_e (blue arrow) in the two-carrier transport regime¹². The difference between the green and yellow lines corresponds to n_p (red arrow). At V_f = −2 V, B_⊥/eR_xy drops because the topological edge occurs and contributes a large R_xx component on the R_xy due to the asymmetrical Hall-bar geometry¹². The edge state persists in a voltage range from −2 V to −3 V (topological EI regime). At V_f = −2 V, n_e ≈1.1 × 10¹¹ cm⁻² and n_p ≈4.5 × 10¹⁰ cm⁻². From −2 V to −2.5 V, the e-h density imbalance regime with electrons as the majority carrier overlaps the topological EI regime. Symmetrically, a similar case occurs from −2.5 V to −3 V but with holes as the major carrier. (b). Calculated n_e and n_p using a two-carrier model (Methods). The calculated results agree well with those obtained in (a).