It may be possible that as our 2DEG density is

considerably higher than those reported signaling pathway in the seminal work of Tutuc, Melinte, and Shayegan. Therefore we do not see such a trend in our system. Figure 5 Local Fermi energy E and the corresponding 2D carrier density n 2D . The local Fermi energy E and the corresponding 2D carrier density n 2D for n = 1↓ and n = 1↑, Landau levels as a function of B for ARRY-438162 chemical structure Sample C at T = 0.3 K. Let us now turn our attention to the activation energy measurements. Figure 6 shows ln (ρ xx) as a function of 1/T for eight different carrier densities while maintaining the filling factor at ν = 3 for sample C. The resistivity shows activated behavior . Figure 6 shows the activation energy Δs determined from a least-square fit to the experimental data shown in Figure 5. We can see that the spin gaps Δs drops approximately linearly to zero at a critical magnetic field B c ~ 3.47 T. The spin gap is expected to have the form Δ s = g 0 μ B B + E ex = g * μ B B[12], where E ex is the many-body exchange energy which lifts the g-factor from its bare value (0.44 in GaAs) to its enhanced value g *. Figure 7 shows that the measured Δs is greatly enhanced over the single particle Zeeman energy (shown in

the dotted line), yielding g * = 4.64 ± 0.30. Moreover, the exchange energy shows a roughly linear B dependence. The disorder broadening Γs can be estimated from the critical magnetic B c [12]. From this we obtain a quantum lifetime of Γs = 0.71 ps, in qualitative agreement with the value 0.40 ps obtained from the Dingle plot. For the low-field regime where Δs < see more Γs, the many-body interactions are destroyed by the disorder, and there is no spin-splitting for the magnetic field less than B c. As shown in Figure 7, the ‘spin gap’ measured by the conventional activation energy studies is very different from that measured by the direct measurements (shown in the dashed line). This is consistent with the fact that activation energy studies yield a mobility gap which is smaller than the real spin gap in the spectrum. Moreover, the measured by studying the slopes of the n = 1 L-gulonolactone oxidase spin-split Landau levels is approximately 2.4 times

larger than that determined from the activation energy studies. Our data shows that both the spin gaps and g * measured by the activation energy studies are very different from those determined from direct measurements. A possible reason for this is that there exists disorder within 2D system which is indispensable to the observation of the IQHE. The direct measurements are performed in the zero disorder limit. On the other hand, in the activation energy studies, the disorder within the quantum Hall system must be considered. As shown in the inset of Figure 7, the spin gap in the zero disorder limit is the energy difference between neighboring peaks in the density of states N(E) which is larger than the energy spacing between the edges of the localized states given the finite extended states.