# Experimental evidence for direct insulator-quantum Hall transition in multi-layer graphene

^{1,}

^{2}Li-Hung Lin,

^{3}Nobuyuki Aoki,

^{}

^{2}Takahiro Ouchi,

^{2}Akram M Mahjoub,

^{2}Tak-Pong Woo,

^{1}Jonathan P Bird,

^{2,}

^{4}Yuichi Ochiai,

^{2}Shun-Tsung Lo,

^{5}and Chi-Te Liang

^{}

^{1,}

^{5}

^{1}Department of Physics, National Taiwan University, Taipei 106, Taiwan

^{2}Graduate School of Advanced Integration Science, Chiba University, Chiba 263-8522, Japan

^{3}Department of Electrophysics, National Chiayi University, Chiayi 600, Taiwan

^{4}Department of Electrical Engineering, University at Buffalo, The State University of New York, Buffalo, NY 14206-1500, USA

^{5}Graduate Institute of Applied Physics, National Taiwan University, Taipei 106, Taiwan

^{}Corresponding author.

## Abstract

We have performed magnetotransport measurements on a multi-layer graphene flake. At the crossing magnetic field *B*_{c}, an approximately temperature-independent point in the measured longitudinal resistivity *ρ*_{xx}, which is ascribed to the direct insulator-quantum Hall (I-QH) 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 I-QH 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 I-QH transition observed in multi-layered graphene. Combined with existing experimental results obtained in various material systems, our data obtained on graphene suggest that the direct I-QH transition is a universal effect in 2D.

**Keywords:**Insulator-quantum Hall transition, Graphene flake, Multi-layer graphene

## Background

Graphene, which is an ideal two-dimensional system [1], has attracted a great deal of worldwide interest. Interesting effects such as Berry's phase [2,3] and fractional quantum Hall effect [4-6] 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 insulator-quantum Hall (I-QH) transition [10-13] is a fascinating physical phenomenon in the field of two-dimensional (2D) physics. In particular, a direct transition from an insulator to a high Landau-level filling factor *ν* > 2 QH state which is normally dubbed as the direct I-QH transition continues to attract interest [14]. The direct I-QH transition has been observed in various systems such as SiGe hole gas [14], GaAs multiple quantum well devices [15], GaAs two-dimensional electron gases (2DEGs) containing InAs quantum dots [16-18], a delta-doped GaAs quantum well with additional modulation doping [19,20], GaN-based 2DEGs grown on sapphire [21] and on Si [22], InAs-based 2DEGs [23], and even some conventional GaAs-based 2DEGs [24], suggesting that it is a universal effect. Although some quantum phase transitions, such as plateau-plateau transitions [25] and metal-to-insulator transitions [26-29], have been observed in single-layer graphene and insulating behavior has been observed in disordered graphene such as hydrogenated graphene [30-33], graphene exposed to ozone [34], reduced graphene oxide [35], and fluorinated graphene [36,37], the direct I-QH transition has not been observed in a graphene-based 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 double-layer graphene heterostructure [38], but not in single-layer pristine graphene. Moreover, the disorder of single graphene is normally lower than those of multi-layer graphene devices. Since one needs sufficient disorder in order to see the I-QH transition [11], multi-layer graphene seems to be a suitable choice for studying such a transition in a pristine graphene-based system. Besides, the top and bottom layers may isolate the environmental impurities [39-42], making multi-layer graphene a stable and suitable system for observing the I-QH transition.

In this paper, we report magnetotransport measurements on a multi-layer graphene flake. We observe an approximately temperature-independent point in the measured longitudinal resistivity *ρ*_{xx} which can be ascribed to experimental evidence for the direct I-QH transition. At the crossing field *B*_{c} in which *ρ*_{xx} is approximately *T*-independent, *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 I-QH transition in graphene-based devices.

## Methods

A multi-layer graphene flake, mechanically exfoliated from natural graphite, was deposited onto a 300-nm-thick SiO_{2}/Si substrate. Optical microscopy was used to locate the graphene flakes, and the thickness of multi-layer 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 inter-layer distance [1,43]. Ti/Au contacts were deposited on the multi-layer graphene flake by electron-beam lithography and lift-off process. The multi-layer graphene flake was made into a Hall bar pattern with a length-to-width 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 post-exfoliation annealing treatment [45,46]. The magnetoresistivity of the graphene device was measured using standard AC lock-in 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 4*e*^{2}/*h*, they allow us to determine the Landau-level 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].

**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 4

*e*

^{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 *T*-independent 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 Landau-level filling factor is about 125 which is much bigger than 1. Therefore, we have observed evidence for a direct insulator-quantum Hall transition in our multi-layer graphene. The crossing points for *B* > 5.43 T can be ascribed to approximately *T*-independent points near half filling factors in the conventional Shubnikov-de Haas (SdH) model [17].

**Longitudinal and Hall resistivity**

*ρ*_{xx}

**(**

*B***) and**

*ρ*_{xy}

**(**

*B***) at various temperatures**

*T***.**An approximately

*T*-independent 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 $D\left(B,T\right)=\frac{4{\mathit{\pi}}^{3}{k}_{\mathrm{B}}{m}^{*}T}{\mathit{heB}}/sinh\frac{4{\mathit{\pi}}^{3}{k}_{\mathrm{B}}{m}^{*}T}{\mathit{heB}}$, *ρ*_{0}, *k*_{B}, *h*, and *e* are a constant, the Boltzmann constant, Plank's constant, and electron charge, respectively. When $\frac{\mathit{4}{\mathit{\pi}}^{3}{k}_{\mathrm{B}}{m}^{*}T}{\mathit{heB}}\mathit{>}\mathit{1}$, 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 0.06*m*_{0} to 0.07*m*_{0} where *m*_{0} is the rest mass of an electron. Interestingly, the measured effective mass is quite close to that in GaAs (0.067*m*_{0}).

**Amplitudes of the observed oscillations Δ**

*ρ*_{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 I-QH 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 I-QH 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 multi-layered graphene are consistent with those obtained in GaAs-based weakly disordered systems [19,21].

**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 Landau-level 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 I-QH transition in our multi-layer 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 inter-graphene layer interactions play an important role in our system. Nevertheless, the fact that the low-field Hall resistivity is nominally *T*-independent suggests that Coulomb interactions do not seem to be dominant in our system.

## Conclusion

In conclusion, we have presented magnetoresistivity measurements on a multi-layered graphene flake. An approximately temperature-independent point in *ρ*_{xx} is ascribed to the direct I-QH 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 I-QH transition. Together with existing experimental results obtained on various material systems, our new results obtained in a graphene-based system strongly suggest that the direct I-QH transition is a universal effect in 2D.

## Abbreviations

2D: Two-dimensional; 2DEGs: Two-dimensional electron gases; I-QH: Insulator-quantum Hall; SdH: Shubnikov-de 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 99-2911-I-002-126 and NSC 101-2811-M-002-096). CC gratefully acknowledges the financial support from Interchange Association, Japan (IAJ) and the NSC, Taiwan for providing a Japan/Taiwan Summer Program student grant.

## References

- Novoselov KS, Geim AK, Morozov SV, Jiang D, Zhang Y, Dubonos SV, Grigorieva IV, Firsov AA. Electric field effect in atomically thin carbon films. Science. 2004;8:666. doi: 10.1126/science.1102896. [PubMed] [Cross Ref]
- Zhang Y, Tan Y-W, Stormer HL, Kim P. Experimental observation of the quantum Hall effect and Berry's phase in graphene. Nature. 2005;8:201. doi: 10.1038/nature04235. [PubMed] [Cross Ref]
- Novoselov KS, Geim AK, Morozov SV, Jiang D, Katsnelson MI, Grigorieva IV, Dubonos SV, Firsov AA. Two-dimensional gas of massless Dirac fermions in graphene. Nature. 2005;8:197. doi: 10.1038/nature04233. [PubMed] [Cross Ref]
- Bolotin KI, Ghahari F, Shulman MD, Stormer HL, Kim P. Observation of the fractional quantum Hall effect in graphene. Nature. 2009;8:196. doi: 10.1038/nature08582. [PubMed] [Cross Ref]
- Du X, Skachko I, Duerr F, Luican A, Andrei EY. Fractional quantum Hall effect and insulating phase of Dirac electrons in graphene. Nature. 2009;8:192. doi: 10.1038/nature08522. [PubMed] [Cross Ref]
- Feldman BE, Krauss B, Smet JH, Yacoby A. Unconventional sequence of fractional quantum Hall states in suspended graphene. Science. 2012;8:1196. doi: 10.1126/science.1224784. [PubMed] [Cross Ref]
- Lee C, Wei X, Kysar JW, Hone J. Measurement of the elastic properties and intrinsic strength of monolayer graphene. Science. 2008;8:385. doi: 10.1126/science.1157996. [PubMed] [Cross Ref]
- Nair PR, Blake P, Grigorenko AN, Novoselov KS, Booth TJ, Stauber T, Peres NMR, Geim AK. Fine structure constant defines visual transparency of graphene. Science. 2008;8:1308. doi: 10.1126/science.1156965. [PubMed] [Cross Ref]
- Balandin AA, Ghosh S, Bao W, Calizo I, Teweldebrhan D, Miao F, Lau CN. Superior thermal conductivity of single-layer graphene. Nano Lett. 2008;8:902. doi: 10.1021/nl0731872. [PubMed] [Cross Ref]
- Kivelson S, Lee DH, Zhang SC. Global phase diagram in the quantum Hall effect. Phys Rev B. 1992;8:2223. doi: 10.1103/PhysRevB.46.2223. [PubMed] [Cross Ref]
- Jiang HW, Johnson CE, Wang KL, Hannahs ST. Observation of magnetic-field-induced delocalization: transition from Anderson insulator to quantum Hall conductor. Phys Rev Lett. 1993;8:1439. doi: 10.1103/PhysRevLett.71.1439. [PubMed] [Cross Ref]
- Wang T, Clark KP, Spencer GF, Mack AM, Kirk WP. Magnetic-field-induced metal-insulator transition in two dimensions. Phys Rev Lett. 1994;8:709. doi: 10.1103/PhysRevLett.72.709. [PubMed] [Cross Ref]
- Hughes RJF, Nicholls JT, Frost JEF, Linfield EH, Pepper M, Ford CJB, Ritchie DA, Jones GAC, Kogan E, Kaveh M. Magnetic-field-induced insulator-quantum Hall-insulator transition in a disordered two-dimensional electron gas. J Phys Condens Matter. 1994;8:4763. doi: 10.1088/0953-8984/6/25/014. [Cross Ref]
- Song S-H, Shahar D, Tsui DC, Xie YH, Monroe D. New universality at the magnetic field driven insulator to integer quantum Hall effect transitions. Phys Rev Lett. 1997;8:2200. doi: 10.1103/PhysRevLett.78.2200. [Cross Ref]
- Lee CH, Chang YH, Suen YW, Lin HH. Magnetic-field-induced delocalization in center-doped GaAs/Al
_{x}Ga_{1-}_{x }As multiple quantum wells. Phys Rev B. 1998;8:10629. doi: 10.1103/PhysRevB.58.10629. [Cross Ref] - Huang T-Y, Juang JR, Huang CF, Kim G-H, Huang C-P, Liang C-T, Chang YH, Chen YF, Lee Y, Ritchie DA. On the low-field insulator-quantum Hall conductor transitions. Physica E. 2004;8:240. doi: 10.1016/j.physe.2003.11.258. [Cross Ref]
- Huang T-Y, Liang C-T, Kim G-H, Huang CF, Huang C-P, Lin J-Y, Goan H-S, Ritchie DA. From insulator to quantum Hall liquid at low magnetic fields. Phys Rev B. 2008;8:113305.
- Liang C-T, Lin L-H, Chen KY, Lo S-T, Wang Y-T, Lou D-S, Kim G-H, Chang Y-H, Ochiai Y, Aoki N, Chen J-C, Lin Y, Huang C-F, Lin S-D, Ritchie DA. On the direct insulator-quantum Hall transition in two-dimensional electron systems in the vicinity of nanoscaled scatterers. Nanoscale Res Lett. 2011;8:131. doi: 10.1186/1556-276X-6-131. [PMC free article] [PubMed] [Cross Ref]
- Chen KY, Chang YH, Liang C-T, Aoki N, Ochiai Y, Huang CF, Lin L-H, Cheng KA, Cheng HH, Lin HH, Wu J-Y, Lin S-D. Probing Landau quantization with the presence of insulator–quantum Hall transition in a GaAs two-dimensional electron system. J Phys Condens Matter. 2008;8:295223. doi: 10.1088/0953-8984/20/29/295223. [Cross Ref]
- Lo S-T, Chen KY, Lin TL, Lin L-H, Luo D-S, Ochiai Y, Aoki N, Wang Y-T, Peng ZF, Lin Y, Chen JC, Lin S-D, Huang CF, Liang C-T. Probing the onset of strong localization and electron–electron interactions with the presence of a direct insulator–quantum Hall transition. Solid State Commun. 2010;8:1902. doi: 10.1016/j.ssc.2010.07.040. [Cross Ref]
- Lin J-Y, Chen J-H, Kim G-H, Park H, Youn DH, Jeon CM, Baik JM, Lee J-L, Liang C-T, Chen YF. Magnetotransport measurements on an AlGaN/GaN two-dimensional electron system. J Korea Phys Soc. 2006;8:1130.
- Kannan ES, Kim GH, Lin JY, Chen JH, Chen KY, Zhang ZY, Liang CT, Lin LH, Youn DH, Kang KY, Chen NC. Experimental evidence for weak insulator-quantum Hall transitions in GaN/AlGaN two-dimensional electron systems. J Korean Phys Soc. 2007;8:1643. doi: 10.3938/jkps.50.1643. [Cross Ref]
- Gao KH, Yu G, Zhou YM, Wei LM, Lin T, Shang LY, Sun L, Yang R, Zhou WZ, Dai N, Chu JH, Austing DG, Gu Y, Zhang YG. Insulator-quantum Hall conductor transition in high electron density gated InGaAs/InAlAs quantum wells. J Appl Phys. 2010;8:063701. doi: 10.1063/1.3486081. [Cross Ref]
- Lo S-T, Wang Y-T, Bohra G, Comfort E, Lin T-Y, Kang M-G, Strasser G, Bird JP, Huang CF, Lin L-H, Chen JC, Liang C-T. Insulator, semiclassical oscillations and quantum Hall liquids at low magnetic fields. J Phys Condens Matter. 2012;8:405601. doi: 10.1088/0953-8984/24/40/405601. [PubMed] [Cross Ref]
- Giesbers AJM, Zeitler U, Ponomarenko LA, Yang R, Novoselov KS. Scaling of the quantum Hall plateau-plateau transition in graphene. Phys Rev B. 2009;8:241411.
- Amado M, Diez E, Rossela F, Bellani V, López-Romero D, Maude DK. Magneto-transport of graphene and quantum phase transitions in the quantum Hall regime. J Phys Condens Matter. 2012;8:305302. doi: 10.1088/0953-8984/24/30/305302. [PubMed] [Cross Ref]
- Amado M, Diez E, López-Romero D, Rossella F, Caridad JM, Dionigi F, Bellani V, Maude DK. Plateau–insulator transition in graphene. New J Phys. 2010;8:053004. doi: 10.1088/1367-2630/12/5/053004. [Cross Ref]
- Zhu W, Yuan HY, Shi QW, Hou JG, Wang XR. Topological transition of graphene from a quantum Hall metal to a quantum Hall insulator at
*ν*= 0. New J Phys. 2011;8:113008. doi: 10.1088/1367-2630/13/11/113008. [Cross Ref] - Checkelsky JG, Li L, Ong NP. Zero-energy state in graphene in a high magnetic field. Phys Rev Lett. 2008;8:206801. [PubMed]
- Elias DC, Nair RR, Mohiuddin TMG, Morozov SV, Blake P, Halsall MP, Ferrari AC, Boukhvalov DW, Katsnelson MI, Geim AK, Novoselov KS. Control of graphene's properties by reversible hydrogenation: evidence for graphane. Science. 2009;8:610. doi: 10.1126/science.1167130. [PubMed] [Cross Ref]
- Chuang C, Puddy RK, Lin H-D, Lo S-T, Chen T-M, Smith CG, Linag C-T. Experimental evidence for Efros-Shklovskii variable range hopping in hydrogenated graphene. Solid State Commun. 2012;8:905. doi: 10.1016/j.ssc.2012.02.002. [Cross Ref]
- Chuang C, Puddy RK, Connolly MR, Lo S-T, Lin H-D, Chen T-M, Smith CG, Liang C-T. Evidence for formation of multi-quantum dots in hydrogenated graphene. Nano Res. 2012;8:459. Lett. [PMC free article] [PubMed]
- Lo S-T, Chuang C, Puddy RK, Chen T-M, Smith CG, Liang C-T. Non-ohmic behavior of carrier transport in highly disordered graphene. Nanotechnology. 2013;8:165201. doi: 10.1088/0957-4484/24/16/165201. [PubMed] [Cross Ref]
- Moser J, Tao H, Roche S, Alzina F, Torres CMS, Bachtold A. Magnetotransport in disordered graphene exposed to ozone: from weak to strong localization. Phys Rev B. 2010;8:205445.
- Wang S-W, Lin HE, Lin H-D, Chen KY, Tu K-H, Chen CW, Chen J-Y, Liu C-H, Liang C-T, Chen YF. Transport behavior and negative magnetoresistance in chemically reduced graphene oxide nanofilms. Nanotechnology. 2011;8:335701. doi: 10.1088/0957-4484/22/33/335701. [PubMed] [Cross Ref]
- Hong X, Cheng S-H, Herding C, Zhu J. Colossal negative magnetoresistance in dilute fluorinated graphene. Phys Rev B. 2011;8:085410.
- Withers F, Russo S, Dubois M, Craciun MF. Tuning the electronic transport properties of graphene through functionalisation with fluorine. Nanoscale Res Lett. 2011;8:526. doi: 10.1186/1556-276X-6-526. [PMC free article] [PubMed] [Cross Ref]
- Ponomarenko LA, Geim AK, Zhukov AA, Jalil R, Morozov SV, Novoselov KS, Grigorieva IV, Hill EH, Cheianov VV, Falko VI, Watanabe K, Taniguchi T, Gorbachev RV. Tunable metal–insulator transition in double-layer graphene heterostructures. Nat Phys. 2011;8:958. doi: 10.1038/nphys2114. [Cross Ref]
- Hass J, de Heer WA, Conrad EH. The growth and morphology of epitaxial multilayer graphene. J Phys Condens Matter. 2008;8:323202. doi: 10.1088/0953-8984/20/32/323202. [Cross Ref]
- Sui Y, Appenzeller J. Screening and interlayer coupling in multilayer graphene field-effect transistors. Nano Lett. 2009;8:2973. doi: 10.1021/nl901396g. [PubMed] [Cross Ref]
- Kim K, Park HJ, Woo B-C, Kim KJ, Kim GT, Yun WS. Electric property evolution of structurally defected multilayer graphene. Nano Lett. 2008;8:3092. doi: 10.1021/nl8010337. [PubMed] [Cross Ref]
- Hass J, Varchon F, Millán-Otoya JE, Sprinkle M, Sharma N, de Heer WA, Berger C, First PN, Magaud L, Conrad EH. Why multilayer graphene on 4
*H*-SiC(0001) behaves like a single sheet of graphene. Phy Rev Lett. 2008;8:125504. [PubMed] - Dresselhaus MS, Dresselhaus G. Intercalation compounds of graphite. Adv Phys. 2002;8:1. doi: 10.1080/00018730110113644. [Cross Ref]
- Ponomarenko LA, Schedin F, Katsnelson MI, Yang R, Hill EW, Novoselov KS, Geim AK. Chaotic Dirac billiard in graphene quantum dots. Science. 2008;8:356. doi: 10.1126/science.1154663. [PubMed] [Cross Ref]
- Bohra G, Somphonsane R, Aoki N, Ochiai Y, Ferry DK, Bird JP. Robust mesoscopic fluctuations in disordered graphene. Appl Phys Lett. 2012;8:093110. doi: 10.1063/1.4748167. [Cross Ref]
- Bohra G, Somphonsane R, Aoki N, Ochiai Y, Akis R, Ferry DK, Bird JP. Nonergodicity and microscopic symmetry breaking of the conductance fluctuations in disordered mesoscopic graphene. Phys Rev B. 2012;8:161405(R).
- Sharapov SG, Gusynin VP, Beck H. Magnetic oscillations in planar systems with the Dirac-like spectrum of quasiparticle excitations. Phys Rev B. 2004;8:075104.
- Berger C, Song Z, Li X, Wu X, Brown N, Naud C, Mayou D, Li T, Hass J, Marchenkov AN, Conrad EH, First PN, de Heer WA. Electronic confinement and coherence in patterned epitaxial graphene. Science. 2006;8:1191. doi: 10.1126/science.1125925. [PubMed] [Cross Ref]
- Coleridge PT, Stoner R, Fletcher R. Low-field transport coefficients in GaAs/Ga
_{1-x}Al_{x}As heterostructures. Phys Rev B. 1989;8:1120. doi: 10.1103/PhysRevB.39.1120. [PubMed] [Cross Ref] - Huckestein B. Quantum Hall effect at low magnetic fields. Phys Rev Lett. 2000;8:3141. doi: 10.1103/PhysRevLett.84.3141. [PubMed] [Cross Ref]
- Roldán R, Fuchs J-N, Goerbig MO. Collective modes of doped graphene and a standard two-dimensional electron gas in a strong magnetic field: linear magnetoplasmons versus magnetoexcitons. Phys Rev B. 2009;8:085408.
- Berman OL, Gumbs G, Lozovik YE. Magnetoplasmons in layered graphene structures. Phys Rev B. 2008;8:085401.
- Cho KS, Liang C-T, Chen YF, Tang YQ, Shen B. Spin-dependent photocurrent induced by Rashba-type spin splitting in Al
_{0.25}Ga_{0.75}N/GaN heterostructures. Phys Rev B. 2007;8:085327.

**Springer**

## Formats:

- Article |
- PubReader |
- ePub (beta) |
- PDF (417K) |
- Citation

- Dirac fermion heating, current scaling, and direct insulator-quantum Hall transition in multilayer epitaxial graphene.[Nanoscale Res Lett. 2013]
*Liu FH, Hsu CS, Chuang C, Woo TP, Huang LI, Lo ST, Fukuyama Y, Yang Y, Elmquist RE, Liang CT.**Nanoscale Res Lett. 2013 Aug 22; 8(1):360. Epub 2013 Aug 22.* - Tunable insulator-quantum Hall transition in a weakly interacting two-dimensional electron system.[Nanoscale Res Lett. 2013]
*Lo ST, Wang YT, Lin SD, Strasser G, Bird JP, Chen YF, Liang CT.**Nanoscale Res Lett. 2013 Jul 3; 8(1):307. Epub 2013 Jul 3.* - Metal to insulator transition on the N=0 Landau level in graphene.[Phys Rev Lett. 2010]
*Zhang L, Zhang Y, Khodas M, Valla T, Zaliznyak IA.**Phys Rev Lett. 2010 Jul 23; 105(4):046804. Epub 2010 Jul 22.* - A delta-doped quantum well system with additional modulation doping.[Nanoscale Res Lett. 2011]
*Luo DS, Lin LH, Su YC, Wang YT, Peng ZF, Lo ST, Chen KY, Chang YH, Wu JY, Lin Y, et al.**Nanoscale Res Lett. 2011 Feb 14; 6(1):139. Epub 2011 Feb 14.* - Insulator, semiclassical oscillations and quantum Hall liquids at low magnetic fields.[J Phys Condens Matter. 2012]
*Lo ST, Wang YT, Bohra G, Comfort E, Lin TY, Kang MG, Strasser G, Bird JP, Huang CF, Lin LH, et al.**J Phys Condens Matter. 2012 Oct 10; 24(40):405601. Epub 2012 Sep 12.*

- Dirac fermion heating, current scaling, and direct insulator-quantum Hall transition in multilayer epitaxial graphene[Nanoscale Research Letters. ]
*Liu FH, Hsu CS, Chuang C, Woo TP, Huang LI, Lo ST, Fukuyama Y, Yang Y, Elmquist RE, Liang CT.**Nanoscale Research Letters. 8(1)360*

- PubMedPubMedPubMed citations for these articles

- Experimental evidence for direct insulator-quantum Hall transition in multi-laye...Experimental evidence for direct insulator-quantum Hall transition in multi-layer grapheneNanoscale Research Letters. 2013; 8(1)214

Your browsing activity is empty.

Activity recording is turned off.

See more...