Fig. 7: Non-equilibrium excited-state FQHE occur at the intersection of branches arising from equilibrium FQHEs.
Each illustration (a–j) shows the source (blue), a barrier, the Landau level (LL) (gray), another barrier, and the drain (blue). The source (drain) electrochemical potential is μS (μD). Orange and brown bands denote FQHE mobility gaps at two distinct LL fillings. a, b With IDC = 0, an equilibrium (`orange') FQHE occurs at magnetic field B = B2, where EF coincides with the orange gap, μS, and μD. Note the forbidden entry and forbidden exit at μS and μD. c, d Similarly, with IDC = 0, another equilibrium (`brown') FQHE occurs at B = B6. e At B3 > B2, the LL filling (green bar) falls below the orange band. However, with IDC = +I0, μD comes into coincidence with the orange gap, and carrier entry at μD becomes forbidden. f IDC = −I0 brings μS into coincidence with the orange gap, and carrier entry at μS becomes forbidden. Thus, (e, f) indicate a resistance minimum at B = B3. g At B5 < B6, the brown gap falls below the LL-Fermi level. However, IDC = +I0 serves to bring μS into coincidence with the brown gap, so that carrier exit becomes forbidden. h IDC = −I0 brings μD into coincidence with the brown gap. Both (g) and (h) result in a resistance minimum at B = B5. i, j At B = B4, the source-drain bias equals the spacing between the two mobility gaps. In (i), the orange gap prevents carrier entry at μD, and the brown gap prevents carrier exit at μS. In (j), carrier entry at μS and carrier exit at μD are forbidden. Thus, these two gaps work together to prevent carrier entry and exit, similar to the canonical FQHE cases (a, b) or (c, d). k A IDC vs. B graph of the coincidence conditions exhibits the diamond shapes observed in Fig. 3. And, non-equilibrium, excited-state FQHEs, such as the 15/11 in Fig. 4g, occur at the diamond vertices. Orange (brown) dots signify coincidence between orange (brown) gap and μS and/or μD. Orange-brown dots represent superposition. Magenta dots represent cases not examined.
