Abstract
We have performed magnetotransport measurements on a multilayer graphene flake. At the crossing magnetic field B_{c}, an approximately temperatureindependent point in the measured longitudinal resistivity ρ_{xx}, which is ascribed to the direct insulatorquantum Hall (IQH) transition, is observed. By analyzing the amplitudes of the magnetoresistivity oscillations, we are able to measure the quantum mobility μ_{q} of our device. It is found that at the direct IQH transition, μ_{q}B_{c} ≈ 0.37 which is considerably smaller than 1. In contrast, at B_{c}, ρ_{xx} is close to the Hall resistivity ρ_{xy}, i.e., the classical mobility μB_{c} is ≈ 1. Therefore, our results suggest that different mobilities need to be introduced for the direct IQH transition observed in multilayered graphene. Combined with existing experimental results obtained in various material systems, our data obtained on graphene suggest that the direct IQH transition is a universal effect in 2D.
Keywords:
Insulatorquantum Hall transition; Graphene flake; Multilayer grapheneBackground
Graphene, which is an ideal twodimensional system [1], has attracted a great deal of worldwide interest. Interesting effects such as Berry's phase [2,3] and fractional quantum Hall effect [46] have been observed in mechanically exfoliated graphene flakes [1]. In addition to its extraordinary electrical properties, graphene possesses great mechanical [7], optical [8], and thermal [9] characteristics.
The insulatorquantum Hall (IQH) transition [1013] is a fascinating physical phenomenon in the field of twodimensional (2D) physics. In particular, a direct transition from an insulator to a high Landaulevel filling factor ν > 2 QH state which is normally dubbed as the direct IQH transition continues to attract interest [14]. The direct IQH transition has been observed in various systems such as SiGe hole gas [14], GaAs multiple quantum well devices [15], GaAs twodimensional electron gases (2DEGs) containing InAs quantum dots [1618], a deltadoped GaAs quantum well with additional modulation doping [19,20], GaNbased 2DEGs grown on sapphire [21] and on Si [22], InAsbased 2DEGs [23], and even some conventional GaAsbased 2DEGs [24], suggesting that it is a universal effect. Although some quantum phase transitions, such as plateauplateau transitions [25] and metaltoinsulator transitions [2629], have been observed in singlelayer graphene and insulating behavior has been observed in disordered graphene such as hydrogenated graphene [3033], graphene exposed to ozone [34], reduced graphene oxide [35], and fluorinated graphene [36,37], the direct IQH transition has not been observed in a graphenebased system. It is worth mentioning that the Anderson localization effect, an important signature of strong localization which may be affected by a magnetic field applied perpendicular to the graphene plane, was observed in a doublelayer graphene heterostructure [38], but not in singlelayer pristine graphene. Moreover, the disorder of single graphene is normally lower than those of multilayer graphene devices. Since one needs sufficient disorder in order to see the IQH transition [11], multilayer graphene seems to be a suitable choice for studying such a transition in a pristine graphenebased system. Besides, the top and bottom layers may isolate the environmental impurities [3942], making multilayer graphene a stable and suitable system for observing the IQH transition.
In this paper, we report magnetotransport measurements on a multilayer graphene flake. We observe an approximately temperatureindependent point in the measured longitudinal resistivity ρ_{xx} which can be ascribed to experimental evidence for the direct IQH transition. At the crossing field B_{c} in which ρ_{xx} is approximately Tindependent, ρ_{xx} is close to ρ_{xy}. In contrast, the product of the quantum mobility determined from the oscillations in ρ_{xx} and B_{c} is ≈ 0.37 which is considerably smaller than 1. Thus, our experimental results suggest that different mobilities need to be introduced when considering the direct IQH transition in graphenebased devices.
Methods
A multilayer graphene flake, mechanically exfoliated from natural graphite, was deposited onto a 300nmthick SiO_{2}/Si substrate. Optical microscopy was used to locate the graphene flakes, and the thickness of multilayer graphene is 3.5 nm, checked by atomic force microscopy. Therefore, the layer number of our graphene device is around ten according to the 3.4 Å graphene interlayer distance [1,43]. Ti/Au contacts were deposited on the multilayer graphene flake by electronbeam lithography and liftoff process. The multilayer graphene flake was made into a Hall bar pattern with a lengthtowidth ratio of 2.5 by oxygen plasma etching process [44]. Similar to the work done using disordered graphene, our graphene flakes did not undergo a postexfoliation annealing treatment [45,46]. The magnetoresistivity of the graphene device was measured using standard AC lockin technique at 19 Hz with a constant current I = 20 nA in a He^{3} cryostat equipped with a superconducting magnet.
Results and discussion
Figure 1 shows the curves of longitudinal and Hall resistivity ρ_{xx}(B) and ρ_{xy}(B) at T = 0.28 K. Features of magnetoresistivity oscillations accompanied by quantum Hall steps are observed at high fields. In order to further study these results, we analyze the positions of the extrema of the magnetoresistivity oscillations in B as well as the heights of the QH steps. Although the steps in the converted Hall conductivity ρ_{xy} are not well quantized in units of 4e^{2}/h, they allow us to determine the Landaulevel filling factor as indicated in the inset of Figure 1. The carrier density of our device is calculated to be 9.4 × 10^{16} m^{−2} following the procedure described in [47,48].
Figure 1. Longitudinal and Hall resistivity ρ_{xx}(B) and ρ_{xy}(B) at T = 0.28 K. The inset shows the converted ρ_{xy} (in units of 4e^{2}/h ) and ρ_{xx} as a function of B.
We now turn to our main experimental finding. Figure 2 shows the curves of ρ_{xx}(B) and ρ_{xy}(B) as a function of magnetic field at various temperatures T. An approximately Tindependent point in the measured ρ_{xx} at B_{c} = 3.1 T is observed. In the vicinity of B_{c}, for B < B_{c}, the sample behaves as a weak insulator in the sense that ρ_{xx} decreases with increasing T. For B > B_{c}, ρ_{xx} increases with increasing T, characteristic of a quantum Hall state. At B_{c}, the corresponding Landaulevel filling factor is about 125 which is much bigger than 1. Therefore, we have observed evidence for a direct insulatorquantum Hall transition in our multilayer graphene. The crossing points for B > 5.43 T can be ascribed to approximately Tindependent points near half filling factors in the conventional Shubnikovde Haas (SdH) model [17].
Figure 2. Longitudinal and Hall resistivity ρ_{xx}(B) and ρ_{xy}(B) at various temperatures T. An approximately Tindependent point in ρ_{xx} is indicated by a crossing field B_{c}.
By analyzing the amplitudes of the observed SdH oscillations at various magnetic fields and temperatures, we are able to determine the effective mass m^{*} of our device which is an important physical quantity. The amplitudes of the SdH oscillations ρ_{xx} is given by [49]:
where , ρ_{0}, k_{B}, h, and e are a constant, the Boltzmann constant, Plank's constant, and electron charge, respectively. When , we have
where C_{1} is a constant. Figure 3 shows the amplitudes of the SdH oscillations at a fixed magnetic field of 5.437 T. We can see that the experimental data can be well fitted to Equation 2. The measured effective mass ranges from 0.06m_{0} to 0.07m_{0} where m_{0} is the rest mass of an electron. Interestingly, the measured effective mass is quite close to that in GaAs (0.067m_{0}).
Figure 3. Amplitudes of the observed oscillations Δρ_{xx}at B = 5.437 T at different temperatures. The curve corresponds to the best fit to Equation 2.
In our system, for the direct IQH transition near the crossing field, ρ_{xx} is close to ρ_{xy}. In this case, the classical Drude mobility is approximately the inverse of the crossing field 1/B_{c}. Therefore, the onset of Landau quantization is expected to take place near B_{c}[50]. However, it is noted that Landau quantization should be linked with the quantum mobility, not the classical Drude mobility [19]. In order to further study the observed IQH transition, we analyze the amplitudes of the magnetoresistivity oscillations versus the inverse of B at various temperatures. As shown in Figure 4, there is a good linear fit to Equation 1 which allows us to estimate the quantum mobility to be around 0.12 m^{2}/V/s. Therefore, near μ_{q}B_{c}≈ 0.37 which is considerably smaller than 1. Our results obtained on multilayered graphene are consistent with those obtained in GaAsbased weakly disordered systems [19,21].
Figure 4. as a function of the inverse of the magnetic field 1/B. The solid line corresponds to the best fit to Equation 1.
It has been shown that the elementary neutral excitations in graphene in a high magnetic field are different from those of a standard 2D system [51]. In this case, the particular Landaulevel quantization in graphene yields linear magnetoplasmon modes. Moreover, instability of magnetoplasmons can be observed in layered graphene structures [52]. Therefore, in order to fully understand the observed IQH transition in our multilayer graphene sample, magnetoplasmon modes as well as collective phenomena may need to be considered. The spin effect should not be important in our system [53]. At present, it is unclear whether intra and/or intergraphene layer interactions play an important role in our system. Nevertheless, the fact that the lowfield Hall resistivity is nominally Tindependent suggests that Coulomb interactions do not seem to be dominant in our system.
Conclusion
In conclusion, we have presented magnetoresistivity measurements on a multilayered graphene flake. An approximately temperatureindependent point in ρ_{xx} is ascribed to the direct IQH transition. Near the crossing field B_{c}, ρ_{xx} is close to ρ_{xy}, indicating that at B_{c}, the classical mobility is close to 1/B_{c} such that B_{c} is close to 1. On the other hand, μ_{q}B_{c}≈ 0.37 which is much smaller than 1. Therefore, different mobilities must be considered for the direct IQH transition. Together with existing experimental results obtained on various material systems, our new results obtained in a graphenebased system strongly suggest that the direct IQH transition is a universal effect in 2D.
Abbreviations
2D: Twodimensional; 2DEGs: Twodimensional electron gases; IQH: Insulatorquantum Hall; SdH: Shubnikovde Haas.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
CC and LHL performed the experiments. CC, TO, and AMM fabricated the device. NA, YO, and JPB coordinated the project. TPW and STL provided key interpretation of the data. CC and CTL drafted the paper. All the authors read and agree the final version of the paper.
Acknowledgments
This work was funded by the National Science Council (NSC), Taiwan (grant no: NSC 992911I002126 and NSC 1012811M002096). CC gratefully acknowledges the financial support from Interchange Association, Japan (IAJ) and the NSC, Taiwan for providing a Japan/Taiwan Summer Program student grant.
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