When are the q/3 fractional quantum Hall states stable?

Magneto-transport measurements in a wide GaAs quantum well in which we can tune the Fermi energy ($E_F$) to lie in different Landau levels of the two occupied electric subbands reveal a remarkable pattern for the appearance and disappearance of fractional quantum Hall states at $\nu$ = 10/3, 11/3, 13/3, 14/3, 16/3, and 17/3. The data provide direct evidence that the $q/3$ states are stable and strong even at such high fillings as long as $E_F$ lies in a ground-state (N=0) Landau level of either of the two electric subbands, regardless of whether that level belongs to the symmetric or the anti-symmetric subband. Evidently, the node in the out-of-plane direction of the anti-symmetric subband does not destabilize the $q/3$ fractional states. On the other hand, when $E_F$ lies in an excited ($N>0$) Landau level of either subband, the wavefunction node(s) in the in-plane direction weaken or completely destabilize the $q/3$ fractional quantum Hall states. Our data also show that the $q/3$ states remain stable very near the crossing of two Landau levels belonging to the two subbands, especially if the levels have parallel spins.