Activation energies and localization in the fractional quantum Hall effect

This paper summarizes an extensive study of the temperature dependence of magnetotransport in the fractional quantum Hall effect in GaAs-${\mathrm{Al}}_{\mathrm{x}}$ ${\mathrm{Ga}}_{1\mathrm{-}\mathrm{x}}$As heterostructure devices of varying mobility and density. For devices with electron mobility 400 000≲μ≲1 000 000 ${\mathrm{cm}}^2$/V s, we find a single activation energy, ${}_{}{}^3{}_{}{}^{}/2\mathrm{}$, in the longitudinal transport coefficients, ${\sigma }_{\mathrm{xx}}$ and ${\rho }_{\mathrm{xx}}$, for Landau-level filling factors ν=(1/3, (2/3, (4/3, and (5/3, with a magnetic field dependence which is vanishingly small for B≲5.5 T and increases to ${}_{}{}^3{}_{}{}^{}6$.8 K at B=30 T. The observed ${}_{}{}^3{}_{}{}^{}\mathrm{\Delta }$ is smaller by more than a factor of 3 than either the unbound quasiparticle-quasihole pair-creation energy gap or the magneto-roton energy gap, calculated for an ideal two-dimensional electron system. Observations for devices of mobility ${\mu }_0$≊300 000 ${\mathrm{cm}}^2$/V s yield even smaller ${}_{}{}^3{}_{}{}^{}\mathrm{\Delta }$. Adequate fitting of all our results requires inclusion of finite electron layer thickness and disorder, with the effect of decreasing the energy gaps and providing a finite magnetic field threshold. At low temperatures and high magnetic fields, deviations from activated conduction are observed. These deviations are attributed to two-dimensional hopping conduction in a magnetic field. Samples of sufficiently low mobility, ${\mu }_0$≲150 000 ${\mathrm{cm}}^2$/V s exhibit no evidence of activated conduction. Rather, the transport is qualitatively consistent with two-dimensional hopping alone. Studies at Landau-level filling factors ν=(2/5 and (3/5 also yield a single activation energy, ${}_{}{}^5{}_{}{}^{}/2\mathrm{}$, with a weak magnetic field dependence. Experimentally, we find ${}_{}{}^5{}_{}{}^{}\mathrm{}_{}^3{}_{}{}^{}$Δ∼0.4, compared with an expected ratio of 0.28 from simple theoretical considerations.