Figure shows the longitudinal magnetoresistivity measurements on Sample A as a function of B
at various temperatures. It can be seen that at a crossing field B
c = 0.9 T, ρxx
is approximately T
-independent. For B
decreases with increasing temperature, characteristics of an insulating regime [16
]. For B
increases with increasing temperature, and therefore the 2DES is in the quantum Hall regime. As the 2DES enters the ν = 4 QH state from the insulating regime, a direct 0-4 transition where the symbol 0 corresponds to the insulator has been observed. It is worth pointing out that before the 2DES enters the ν = 4 QH state, resistance oscillations due to Landau quantization in the insulating regime have already been observed [15
]. Therefore, the experimental results of this study clearly demonstrate that the crossover from localization from Landau quantization actually covers a wide range of magnetic field, in sharp contrast to Huckestein's argument [19
ρxx(B) at various temperatures ranging from 0.25 to 2.85 K (Sample A).
As mentioned earlier, a GaN-based electron system can be affected by nanoscaled dislocation and impurities. It is therefore interesting to study such a system. Figure shows magnetoresistance measurements on Sample B as a function of magnetic field at different temperatures. The data deviate slightly from the expected symmetric behavior, i.e., R
) = R
). The reason for this could be due to slight misalignment of the voltage probes. Nevertheless, it can be seen that at Bc
= 11 Tand -Bc
= -11 T, the measured resistances are approximately temperature independent. The corresponding Landau level filling factor is about 50 in this case. Therefore, a direct 0-50 transition has been observed. Note that even at the highest attainable field of approximately 15 T
, there is no sign of resistance oscillations due to the moderate mobility of our GaN system. Therefore, the experimental results of this study clearly demonstrate that the observed direct I-QH transition is irrelevant to Landau quantization. Therefore, the onset of Landau quantization does not necessarily accompany the direct I-QH transition, inconsistent with Huckestein's model [20
ρxx(B) at various temperatures ranging from 0.28 to 20 K (Sample B).
Figure shows magnetoresistance measurements on Sample C as a function of magnetic field at various temperatures. It can be seen that the 2DES undergoes a 0-8 transition characterized by an approximately temperature-independent point in ρ xx
at the crossing field Bc
. Near the crossing field, ρxx
is very close to ρxy
shows a weak T
dependence. For B
, no resistance oscillation is observed. At first glance, our experimental results are consistent with Huckestein's model. However, it is noted that Landau quantization should be linked with quantum mobility, not classical Drude mobility [36
]. Moreover, the observed oscillations for B
do not always correspond to formation of quantum Hall states. As mentioned in our previous study [36
], the observed oscillations can be well approximated by conventional Shubnikov-de Haas (SdH) formalism. It is noted that the SdH formula is derived without considering quantum localization effects which give rise to formation of quantum Hall state. Therefore, quantum localization effects are not significant in the system under this study. Actually, as shown in Figure , the crossing point in σxy
at around 7.9 Tmay correspond to the extended states due to the onset of the strong localization effects. Therefore, in this study, the onset of strong localization actually occurs at a magnetic field approximately 4 Thigher than the crossing point.
ρxx(B) at various temperatures ranging from 0.3 to 4 K (Sample C). ρxx at T = 0.3 K and T = 4 K are shown.
Converted σxx(B) and σxy(B) at various temperatures ranging from 0.3 to 4 K (Sample C).
It has been suggested that by converting the measured resistivities into longitudinal and Hall conductivities, it is possible to shed more light on the observed I-QH transition [5
]. Figure shows such results at various temperatures. Interestingly, for B
< 5 T, σxy
is nominally T
independent. Such data are consistent with electron-electron interaction effects. Over the whole measurement range, σxx
decreases with increasing T
, consistent with electron-electron interaction effects. Unlike σxy
shows a significant Tdependence.
By inspecting the conductivies, previously the authors have studied the renormalized mobility [43
] of a GaN-based 2DES at high temperatures (Sample B) [44
]. It is therefore interesting to study such a mobility for both Sample A and Sample C. It has been suggested the electron-electron interaction effects can renormalize the mobility μ' given by
Figure and the inset to Figure show σxy
, together with fits to Equations 1 and 2 over limited ranges for Sample C, respectively. From the fits, it is possible to determine the respective renormalized mobilites as a function of temperature as shown in Figure for Sample C and in Figure for Sample A. The renormalized mobility calculated using Equation 1 is only slightly larger than that using Equation 2. It may be possible that different mobilities should be taken into account to understand the direct I-QH transition [37
σxy(B) and the fit to Equation 1 for 0 <B < 3.5 T. The inset shows σxx(B) and the fit to Equation (2) for 1 T <B < 3.5 T.
Calculated renormalized mobilities due to electron-electron interaction effects using Equations (1) and (2) for (a) Sample C and (b) Sample A, respectively.