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Zhuo D, Zhou L, Zhao YF, Zhang R, Yan ZJ, Wang AG, Chan MHW, Liu CX, Chen CZ, Chang CZ. Engineering Plateau Phase Transition in Quantum Anomalous Hall Multilayers. NANO LETTERS 2024; 24:6974-6980. [PMID: 38829211 DOI: 10.1021/acs.nanolett.4c01313] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2024]
Abstract
The plateau phase transition in quantum anomalous Hall (QAH) insulators corresponds to a quantum state wherein a single magnetic domain gives way to multiple domains and then reconverges back to a single magnetic domain. The layer structure of the sample provides an external knob for adjusting the Chern number C of the QAH insulators. Here, we employ molecular beam epitaxy to grow magnetic topological insulator multilayers and realize the magnetic field-driven plateau phase transition between two QAH states with odd Chern number change ΔC. We find that critical exponents extracted for the plateau phase transitions with ΔC = 1 and ΔC = 3 in QAH insulators are nearly identical. We construct a four-layer Chalker-Coddington network model to understand the consistent critical exponents for the plateau phase transitions with ΔC = 1 and ΔC = 3. This work will motivate further investigations into the critical behaviors of plateau phase transitions with different ΔC in QAH insulators.
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Affiliation(s)
- Deyi Zhuo
- Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, United States
| | - Lingjie Zhou
- Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, United States
| | - Yi-Fan Zhao
- Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, United States
| | - Ruoxi Zhang
- Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, United States
| | - Zi-Jie Yan
- Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, United States
| | - Annie G Wang
- Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, United States
| | - Moses H W Chan
- Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, United States
| | - Chao-Xing Liu
- Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, United States
| | - Chui-Zhen Chen
- Institute for Advanced Study and School of Physical Science and Technology, Soochow University, Suzhou 215006, China
| | - Cui-Zu Chang
- Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, United States
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Madathil PT, Villegas Rosales KA, Tai CT, Chung YJ, Pfeiffer LN, West KW, Baldwin KW, Shayegan M. Delocalization and Universality of the Fractional Quantum Hall Plateau-to-Plateau Transitions. PHYSICAL REVIEW LETTERS 2023; 130:226503. [PMID: 37327438 DOI: 10.1103/physrevlett.130.226503] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/13/2023] [Accepted: 05/12/2023] [Indexed: 06/18/2023]
Abstract
Disorder and electron-electron interaction play essential roles in the physics of electron systems in condensed matter. In two-dimensional, quantum Hall systems, extensive studies of disorder-induced localization have led to the emergence of a scaling picture with a single extended state, characterized by a power-law divergence of the localization length in the zero-temperature limit. Experimentally, scaling has been investigated via measuring the temperature dependence of plateau-to-plateau transitions between the integer quantum Hall states (IQHSs), yielding a critical exponent κ≃0.42. Here we report scaling measurements in the fractional quantum Hall state (FQHS) regime where interaction plays a dominant role. Our Letter is partly motivated by recent calculations, based on the composite fermion theory, that suggest identical critical exponents in both IQHS and FQHS cases to the extent that the interaction between composite fermions is negligible. The samples used in our experiments are two-dimensional electron systems confined to GaAs quantum wells of exceptionally high quality. We find that κ varies for transitions between different FQHSs observed on the flanks of Landau level filling factor ν=1/2 and has a value close to that reported for the IQHS transitions only for a limited number of transitions between high-order FQHSs with intermediate strength. We discuss possible origins of the nonuniversal κ observed in our experiments.
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Affiliation(s)
- P T Madathil
- Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA
| | - K A Villegas Rosales
- Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA
| | - C T Tai
- Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA
| | - Y J Chung
- Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA
| | - L N Pfeiffer
- Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA
| | - K W West
- Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA
| | - K W Baldwin
- Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA
| | - M Shayegan
- Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA
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Deng P, Eckberg C, Zhang P, Qiu G, Emmanouilidou E, Yin G, Chong SK, Tai L, Ni N, Wang KL. Probing the mesoscopic size limit of quantum anomalous Hall insulators. Nat Commun 2022; 13:4246. [PMID: 35869045 PMCID: PMC9307791 DOI: 10.1038/s41467-022-31105-w] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/18/2022] [Accepted: 06/06/2022] [Indexed: 11/09/2022] Open
Abstract
The inelastic scattering length (Ls) is a length scale of fundamental importance in condensed matters due to the relationship between inelastic scattering and quantum dephasing. In quantum anomalous Hall (QAH) materials, the mesoscopic length scale Ls plays an instrumental role in determining transport properties. Here we examine Ls in three regimes of the QAH system with distinct transport behaviors: the QAH, quantum critical, and insulating regimes. Although the resistance changes by five orders of magnitude when tuning between these distinct electronic phases, scaling analyses indicate a universal Ls among all regimes. Finally, mesoscopic scaled devices with sizes on the order of Ls were fabricated, enabling the direct detection of the value of Ls in QAH samples. Our results unveil the fundamental length scale that governs the transport behavior of QAH materials. In quantum anomalous Hall (QAH) materials, the mesoscopic scattering length (Ls) plays an instrumental role in determining transport properties. Here, the authors examine Ls in three regimes (QAH, quantum critical, and insulating) with distinct transport behaviours, and find a universal Ls across all regimes.
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Zhao PL, Lu HZ, Xie XC. Theory for Magnetic-Field-Driven 3D Metal-Insulator Transitions in the Quantum Limit. PHYSICAL REVIEW LETTERS 2021; 127:046602. [PMID: 34355953 DOI: 10.1103/physrevlett.127.046602] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/11/2021] [Revised: 04/07/2021] [Accepted: 06/17/2021] [Indexed: 06/13/2023]
Abstract
Metal-insulator transitions driven by magnetic fields have been extensively studied in 2D, but a 3D theory is still lacking. Motivated by recent experiments, we develop a scaling theory for the metal-insulator transitions in the strong-magnetic-field quantum limit of a 3D system. By using a renormalization-group calculation to treat electron-electron interactions, electron-phonon interactions, and disorder on the same footing, we obtain the critical exponent that characterizes the scaling relations of the resistivity to temperature and magnetic field. By comparing the critical exponent with those in a recent experiment [F. Tang et al., Nature (London) 569, 537 (2019)NATUAS0028-083610.1038/s41586-019-1180-9], we conclude that the insulating ground state was not only a charge-density wave driven by electron-phonon interactions but also coexisting with strong electron-electron interactions and backscattering disorder. We also propose a current-scaling experiment for further verification. Our theory will be helpful for exploring the emergent territory of 3D metal-insulator transitions under strong magnetic fields.
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Affiliation(s)
- Peng-Lu Zhao
- Shenzhen Institute for Quantum Science and Engineering and Department of Physics, Southern University of Science and Technology (SUSTech), Shenzhen 518055, China
| | - Hai-Zhou Lu
- Shenzhen Institute for Quantum Science and Engineering and Department of Physics, Southern University of Science and Technology (SUSTech), Shenzhen 518055, China
- Shenzhen Key Laboratory of Quantum Science and Engineering, Shenzhen 518055, China
| | - X C Xie
- International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China
- CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100190, China
- Beijing Academy of Quantum Information Sciences, West Building 3, No. 10, Xibeiwang East Road, Haidian District, Beijing 100193, China
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Salehi M, Shapourian H, Rosen IT, Han MG, Moon J, Shibayev P, Jain D, Goldhaber-Gordon D, Oh S. Quantum-Hall to Insulator Transition in Ultra-Low-Carrier-Density Topological Insulator Films and a Hidden Phase of the Zeroth Landau Level. ADVANCED MATERIALS (DEERFIELD BEACH, FLA.) 2019; 31:e1901091. [PMID: 31259439 DOI: 10.1002/adma.201901091] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/16/2019] [Revised: 06/08/2019] [Indexed: 06/09/2023]
Abstract
A key feature of the topological surface state under a magnetic field is the presence of the zeroth Landau level at the zero energy. Nonetheless, it is challenging to probe the zeroth Landau level due to large electron-hole puddles smearing its energy landscape. Here, by developing ultra-low-carrier density topological insulator Sb2 Te3 films, an extreme quantum limit of the topological surface state is reached and a hidden phase at the zeroth Landau level is uncovered. First, an unexpected quantum-Hall-to-insulator-transition near the zeroth Landau level is discovered. Then, through a detailed scaling analysis, it is found that this quantum-Hall-to-insulator-transition belongs to a new universality class, implying that the insulating phase discovered here has a fundamentally different origin from those in nontopological systems.
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Affiliation(s)
- Maryam Salehi
- Department of Materials Science and Engineering, Rutgers, The State University of New Jersey, Piscataway, NJ, 08854, USA
| | - Hassan Shapourian
- James Franck Institute and Kadanoff Center for Theoretical Physics, University of Chicago, IL, 60637, USA
| | - Ilan Thomas Rosen
- Department of Applied Physics, Stanford University, Stanford, CA, 94305, USA
- Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory, Menlo Park, CA, 94025, USA
| | - Myung-Geun Han
- Condensed Matter Physics and Materials Science, Brookhaven National Lab, Upton, NY, 11973, USA
| | - Jisoo Moon
- Department of Physics and Astronomy, Rutgers, The State University of New Jersey, Piscataway, NJ, 08854, USA
| | - Pavel Shibayev
- Department of Physics and Astronomy, Rutgers, The State University of New Jersey, Piscataway, NJ, 08854, USA
| | - Deepti Jain
- Department of Physics and Astronomy, Rutgers, The State University of New Jersey, Piscataway, NJ, 08854, USA
| | - David Goldhaber-Gordon
- Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory, Menlo Park, CA, 94025, USA
- Department of Physics, Stanford University, Stanford, CA, 94305, USA
| | - Seongshik Oh
- Department of Physics and Astronomy, Rutgers, The State University of New Jersey, Piscataway, NJ, 08854, USA
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Dodoo-Amoo NA, Saeed K, Mistry D, Khanna SP, Li L, Linfield EH, Davies AG, Cunningham JE. Non-universality of scaling exponents in quantum Hall transitions. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2014; 26:475801. [PMID: 25351842 DOI: 10.1088/0953-8984/26/47/475801] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
We have investigated experimentally the scaling behaviour of quantum Hall transitions in GaAs/AlGaAs heterostructures of a range of mobility, carrier concentration, and spacer layer width. All three critical scaling exponents γ, κ and p were determined independently for each sample. We measure the localization length exponent to be γ ≈ 2.3, in good agreement with expected predictions from scaling theory, but κ and p are found to possess non-universal values. Results obtained for κ range from κ = 0.16 ± 0.02 to κ = 0.67 ± 0.02, and are found to be Landau level (LL) dependent, whereas p is found to decrease with increasing sample mobility. Our results demonstrate the existence of two transport regimes in the LL conductivity peak; universality is found within the quantum coherent transport regime present in the tails of the conductivity peak, but is absent within the classical transport regime found close to the critical point at the centre of the conductivity peak. We explain these results using a percolation model and show that the critical scaling exponent depends on certain important length scales that correspond to the microscopic description of electron transport in the bulk of a two-dimensional electron system.
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Affiliation(s)
- N A Dodoo-Amoo
- School of Electronic and Electrical Engineering, University of Leeds, Woodhouse Lane, Leeds, LS2 9JT,UK
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Neal AT, Du Y, Liu H, Ye PD. Two-dimensional TaSe2 metallic crystals: spin-orbit scattering length and breakdown current density. ACS NANO 2014; 8:9137-9142. [PMID: 25133691 DOI: 10.1021/nn5027164] [Citation(s) in RCA: 24] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/03/2023]
Abstract
We have determined the spin-orbit scattering length of two-dimensional layered 2H-TaSe2 metallic crystals by detailed characterization of the weak antilocalization phenomena in this strong spin-orbit interaction material. By fitting the observed magneto-conductivity, the spin-orbit scattering length for 2H-TaSe2 is determined to be 17 nm in the few-layer films. This small spin-orbit scattering length is comparable to that of Pt, which is widely used to study the spin Hall effect, and indicates the potential of TaSe2 for use in spin Hall effect devices. A material must also support large charge currents in addition to strong spin-orbit coupling to achieve spin-transfer-torque via the spin Hall effect. Therefore, we have characterized the room temperature breakdown current density of TaSe2 in air, where the best breakdown current density reaches 3.7 × 10(7) A/cm(2). This large breakdown current further indicates the potential of TaSe2 for use in spin-torque devices and two-dimensional device interconnect applications.
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Affiliation(s)
- Adam T Neal
- School of Electrical and Computer Engineering and Birck Nanotechnology Center, Purdue University , West Lafayette, Indiana 47907, United States
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Liu FH, Lo ST, Chuang C, Woo TP, Lee HY, Liu CW, Liu CI, Huang LI, Liu CH, Yang Y, Chang CYS, Li LJ, Mende PC, Feenstra RM, Elmquist RE, Liang CT. Hot carriers in epitaxial graphene sheets with and without hydrogen intercalation: role of substrate coupling. NANOSCALE 2014; 6:10562-10568. [PMID: 25117572 DOI: 10.1039/c4nr02980a] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/03/2023]
Abstract
The development of graphene electronic devices produced by industry relies on efficient control of heat transfer from the graphene sheet to its environment. In nanoscale devices, heat is one of the major obstacles to the operation of such devices at high frequencies. Here we have studied the transport of hot carriers in epitaxial graphene sheets on 6H-SiC (0001) substrates with and without hydrogen intercalation by driving the device into the non-equilibrium regime. Interestingly, we have demonstrated that the energy relaxation time of the device without hydrogen intercalation is two orders of magnitude shorter than that with hydrogen intercalation, suggesting application of epitaxial graphene in high-frequency devices which require outstanding heat exchange with an outside cooling source.
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Affiliation(s)
- Fan-Hung Liu
- Graduate Institute of Applied Physics, National Taiwan University, Taipei 106, Taiwan.
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Liu FH, Hsu CS, Chuang C, Woo TP, Huang LI, Lo ST, Fukuyama Y, Yang Y, Elmquist RE, Liang CT. Dirac fermion heating, current scaling, and direct insulator-quantum Hall transition in multilayer epitaxial graphene. NANOSCALE RESEARCH LETTERS 2013; 8:360. [PMID: 23968131 PMCID: PMC3765374 DOI: 10.1186/1556-276x-8-360] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 07/02/2013] [Accepted: 08/12/2013] [Indexed: 06/02/2023]
Abstract
We have performed magnetotransport measurements on multilayer epitaxial graphene. By increasing the driving current I through our graphene devices while keeping the bath temperature fixed, we are able to study Dirac fermion heating and current scaling in such devices. Using zero-field resistivity as a self thermometer, we are able to determine the effective Dirac fermion temperature (TDF) at various driving currents. At zero field, it is found that TDF ∝ I≈1/2. Such results are consistent with electron heating in conventional two-dimensional systems in the plateau-plateau transition regime. With increasing magnetic field B, we observe an I-independent point in the measured longitudinal resistivity ρxx which is equivalent to the direct insulator-quantum Hall (I-QH) transition characterized by a temperature-independent point in ρxx. Together with recent experimental evidence for direct I-QH transition, our new data suggest that such a transition is a universal effect in graphene, albeit further studies are required to obtain a thorough understanding of such an effect.
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Affiliation(s)
- Fan-Hung Liu
- Graduate Institute of Applied Physics, National Taiwan University, Taipei 106, Taiwan
| | - Chang-Shun Hsu
- Graduate Institute of Applied Physics, National Taiwan University, Taipei 106, Taiwan
| | - Chiashain Chuang
- Department of Physics, National Taiwan University, Taipei 106, Taiwan
| | - Tak-Pong Woo
- Department of Physics, National Taiwan University, Taipei 106, Taiwan
| | - Lung-I Huang
- Department of Physics, National Taiwan University, Taipei 106, Taiwan
| | - Shun-Tsung Lo
- Graduate Institute of Applied Physics, National Taiwan University, Taipei 106, Taiwan
| | - Yasuhiro Fukuyama
- National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba, Ibaraki 305-8568, Japan
| | - Yanfei Yang
- National Institute of Standards and Technology (NIST), Gaithersburg, MD 20899, USA
| | - Randolph E Elmquist
- National Institute of Standards and Technology (NIST), Gaithersburg, MD 20899, USA
| | - Chi-Te Liang
- Graduate Institute of Applied Physics, National Taiwan University, Taipei 106, Taiwan
- Department of Physics, National Taiwan University, Taipei 106, Taiwan
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Amado M, Malyshev AV, Sedrakyan A, Domínguez-Adame F. Numerical study of the localization length critical index in a network model of plateau-plateau transitions in the quantum Hall effect. PHYSICAL REVIEW LETTERS 2011; 107:066402. [PMID: 21902346 DOI: 10.1103/physrevlett.107.066402] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/12/2011] [Indexed: 05/31/2023]
Abstract
We calculate numerically the localization length critical index within the Chalker-Coddington model of the plateau-plateau transitions in the quantum Hall effect. We report a finite-size scaling analysis using both the traditional power-law corrections to the scaling function and the inverse logarithmic ones, which provided a more stable fit resulting in the localization length critical index ν = 2.616 ± 0.014. We observe an increase of the critical exponent ν with the system size, which is possibly the origin of discrepancies with early results obtained for smaller systems.
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Affiliation(s)
- M Amado
- GISC, Departamento de Física de Materiales, Universidad Complutense, Madrid, Spain
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Li W, Vicente CL, Xia JS, Pan W, Tsui DC, Pfeiffer LN, West KW. Scaling in plateau-to-plateau transition: a direct connection of quantum hall systems with the Anderson localization model. PHYSICAL REVIEW LETTERS 2009; 102:216801. [PMID: 19519123 DOI: 10.1103/physrevlett.102.216801] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/19/2005] [Indexed: 05/27/2023]
Abstract
The quantum Hall-plateau transition was studied at temperatures down to 1 mK in a random alloy disordered high mobility two-dimensional electron gas. A perfect power-law scaling with kappa=0.42 was observed from 1.2 K down to 12 mK. This perfect scaling terminates sharply at a saturation temperature of Ts approximately 10 mK. The saturation is identified as a finite-size effect when the quantum phase coherence length (Lphi proportional, T(-p/2)) reaches the sample size (W) of millimeter scale. From a size dependent study, Ts proportional, W(-1) was observed and p=2 was obtained. The exponent of the localization length, determined directly from the measured kappa and p, is nu=2.38, and the dynamic critical exponent z=1.
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Affiliation(s)
- Wanli Li
- Princeton University, Princeton, New Jersey 08544, USA.
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Visser AD, Ponomarenko LA, Galistu G, Lang DTND, Pruisken AMM, Zeitler U, Maude D. Quantum critical behaviour of the plateau-insulator transition in the quantum Hall regime. ACTA ACUST UNITED AC 2006. [DOI: 10.1088/1742-6596/51/1/088] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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13
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Li W, Csáthy GA, Tsui DC, Pfeiffer LN, West KW. Scaling and universality of integer quantum Hall plateau-to-plateau transitions. PHYSICAL REVIEW LETTERS 2005; 94:206807. [PMID: 16090272 DOI: 10.1103/physrevlett.94.206807] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/30/2004] [Indexed: 05/03/2023]
Abstract
We have investigated the integer quantum Hall plateau-to-plateau transition in two-dimensional electrons confined to AlxGa(1-x)As-Al0.33Ga0.67As heterostructures over a broad range of Al concentration x. For x between 0.65% and 1.6%, where the dominant contribution to disorder is from the short-range alloy potential fluctuations, we observe a perfect power-law scaling in the temperature range from 30 mK to 1 K with a critical exponent kappa = 0.42 +/- 0.01.
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Affiliation(s)
- Wanli Li
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
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14
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Polyakov DG. Spin-flip scattering in the quantum Hall regime. PHYSICAL REVIEW. B, CONDENSED MATTER 1996; 53:15777-15788. [PMID: 9983414 DOI: 10.1103/physrevb.53.15777] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Chow E, Wei HP. Experiments on inelastic scattering in the integer quantum Hall effect. PHYSICAL REVIEW. B, CONDENSED MATTER 1995; 52:13749-13752. [PMID: 9980580 DOI: 10.1103/physrevb.52.13749] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Brandes T. Inelastic scattering, multifractality, and scaling in the integer quantum Hall effect. PHYSICAL REVIEW. B, CONDENSED MATTER 1995; 52:8391-8399. [PMID: 9979843 DOI: 10.1103/physrevb.52.8391] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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17
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Hanna CB, Arovas DP, Mullen K, Girvin SM. Effect of spin degeneracy on scaling in the quantum Hall regime. PHYSICAL REVIEW. B, CONDENSED MATTER 1995; 52:5221-5232. [PMID: 9981707 DOI: 10.1103/physrevb.52.5221] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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