After a brief introduction to the quantum Hall effects and the composite fermion model, I will discuss recent results in the second Landau level (LL). The data were obtained at very low temperatures (sample temperatures as low as 9 mK) in a very high mobility 2D electron system. We observe well-quantized FQHE states at the LL filling nu= 2+1/2, 2+1/3, and nu= 2+2/3 in coexistence with the reentrant integer quantum Hall states, and a new FQHE state at nu= 2+2/5. The origin of the 2+2/5 state is not clear and the numerical results strongly suggest that it is not a conventional FQHE state but a parafermionic state. There is also evidence for a second even-denominator FQHE state at nu= 2+3/8.
Most importantly, we discovered an unexpected quantization of the diagonal resistance, Rxx, at the edges of several quantum Hall states. Each quantized Rxx value is close to the difference between the two adjacent Hall plateaus in the off-diagonal resistance, Rxy. Surprisingly, we can trace this observation back to a small density gradient, about 1%/cm, in our sample. Under this scenario almost all Rxx features can be explained quantitatively by an electron density gradient. These findings have very important implications for any Rxx data taken on two-dimensional electron systems, since Rxx seems now to be solely determined by Rxy, while its relationship to the diagonal resistivity, rxx, is unclear. These findings are further corroborated by data from a different sample at a temperature of 1.2K. There, Rxx shows a strictly linear dependence on the magnetic field, except for sharp spikes at B-fields where the IQHE develops. Interestingly, this linear magnetoresistance can also be explained by the density gradient model.