1
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Wang C, Gupta A, Singh SK, Madathil PT, Chung YJ, Pfeiffer LN, Baldwin KW, Winkler R, Shayegan M. Fractional Quantum Hall State at Filling Factor ν=1/4 in Ultra-High-Quality GaAs Two-Dimensional Hole Systems. Phys Rev Lett 2023; 131:266502. [PMID: 38215363 DOI: 10.1103/physrevlett.131.266502] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/24/2023] [Accepted: 12/01/2023] [Indexed: 01/14/2024]
Abstract
Single-component fractional quantum Hall states (FQHSs) at even-denominator filling factors may host non-Abelian quasiparticles that are considered to be building blocks of topological quantum computers. Such states, however, are rarely observed in the lowest-energy Landau level, namely at filling factors ν<1. Here, we report evidence for an even-denominator FQHS at ν=1/4 in ultra-high-quality two-dimensional hole systems confined to modulation-doped GaAs quantum wells. We observe a deep minimum in the longitudinal resistance at ν=1/4, superimposed on a highly insulating background, suggesting a close competition between the ν=1/4 FQHS and the magnetic-field-induced, pinned Wigner solid states. Our experimental observations are consistent with the very recent theoretical calculations that predict that substantial Landau level mixing, caused by the large hole effective mass, can induce composite fermion pairing and lead to a non-Abelian FQHS at ν=1/4. Our results demonstrate that Landau level mixing can provide a very potent means for tuning the interaction between composite fermions and creating new non-Abelian FQHSs.
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Affiliation(s)
- Chengyu Wang
- Department of Electrical and Computer Engineering, Princeton University, Princeton, New Jersey 08544, USA
| | - A Gupta
- Department of Electrical and Computer Engineering, Princeton University, Princeton, New Jersey 08544, USA
| | - S K Singh
- Department of Electrical and Computer Engineering, Princeton University, Princeton, New Jersey 08544, USA
| | - P T Madathil
- Department of Electrical and Computer Engineering, Princeton University, Princeton, New Jersey 08544, USA
| | - Y J Chung
- Department of Electrical and Computer Engineering, Princeton University, Princeton, New Jersey 08544, USA
| | - L N Pfeiffer
- Department of Electrical and Computer Engineering, Princeton University, Princeton, New Jersey 08544, USA
| | - K W Baldwin
- Department of Electrical and Computer Engineering, Princeton University, Princeton, New Jersey 08544, USA
| | - R Winkler
- Department of Physics, Northern Illinois University, DeKalb, Illinois 60115, USA
| | - M Shayegan
- Department of Electrical and Computer Engineering, Princeton University, Princeton, New Jersey 08544, USA
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2
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Wang C, Gupta A, Chung YJ, Pfeiffer LN, West KW, Baldwin KW, Winkler R, Shayegan M. Highly Anisotropic Even-Denominator Fractional Quantum Hall State in an Orbitally Coupled Half-Filled Landau Level. Phys Rev Lett 2023; 131:056302. [PMID: 37595236 DOI: 10.1103/physrevlett.131.056302] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/16/2023] [Accepted: 07/06/2023] [Indexed: 08/20/2023]
Abstract
The even-denominator fractional quantum Hall states (FQHSs) in half-filled Landau levels are generally believed to host non-Abelian quasiparticles and be of potential use in topological quantum computing. Of particular interest is the competition and interplay between the even-denominator FQHSs and other ground states, such as anisotropic phases and composite fermion Fermi seas. Here, we report the observation of an even-denominator fractional quantum Hall state with highly anisotropic in-plane transport coefficients at Landau level filling factor ν=3/2. We observe this state in an ultra-high-quality GaAs two-dimensional hole system when a large in-plane magnetic field is applied. By increasing the in-plane field, we observe a sharp transition from an isotropic composite fermion Fermi sea to an anisotropic even-denominator FQHS. Our data and calculations suggest that a unique feature of two-dimensional holes, namely the coupling between heavy-hole and light-hole states, combines different orbital components in the wave function of one Landau level, and leads to the emergence of a highly anisotropic even-denominator fractional quantum Hall state. Our results demonstrate that the GaAs two-dimensional hole system is a unique platform for the exploration of exotic, many-body ground states.
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Affiliation(s)
- Chengyu Wang
- Department of Electrical and Computer Engineering, Princeton University, Princeton, New Jersey 08544, USA
| | - A Gupta
- Department of Electrical and Computer Engineering, Princeton University, Princeton, New Jersey 08544, USA
| | - Y J Chung
- Department of Electrical and Computer Engineering, Princeton University, Princeton, New Jersey 08544, USA
| | - L N Pfeiffer
- Department of Electrical and Computer Engineering, Princeton University, Princeton, New Jersey 08544, USA
| | - K W West
- Department of Electrical and Computer Engineering, Princeton University, Princeton, New Jersey 08544, USA
| | - K W Baldwin
- Department of Electrical and Computer Engineering, Princeton University, Princeton, New Jersey 08544, USA
| | - R Winkler
- Department of Physics, Northern Illinois University, DeKalb, Illinois 60115, USA
| | - M Shayegan
- Department of Electrical and Computer Engineering, Princeton University, Princeton, New Jersey 08544, USA
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3
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Hossain MS, Ma MK, Chung YJ, Singh SK, Gupta A, West KW, Baldwin KW, Pfeiffer LN, Winkler R, Shayegan M. Valley-Tunable Even-Denominator Fractional Quantum Hall State in the Lowest Landau Level of an Anisotropic System. Phys Rev Lett 2023; 130:126301. [PMID: 37027870 DOI: 10.1103/physrevlett.130.126301] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/06/2022] [Accepted: 02/22/2023] [Indexed: 06/19/2023]
Abstract
Fractional quantum Hall states (FQHSs) at even-denominator Landau level filling factors (ν) are of prime interest as they are predicted to host exotic, topological states of matter. We report here the observation of a FQHS at ν=1/2 in a two-dimensional electron system of exceptionally high quality, confined to a wide AlAs quantum well, where the electrons can occupy multiple conduction-band valleys with an anisotropic effective mass. The anisotropy and multivalley degree of freedom offer an unprecedented tunability of the ν=1/2 FQHS as we can control both the valley occupancy via the application of in-plane strain, and the ratio between the strengths of the short- and long-range Coulomb interaction by tilting the sample in the magnetic field to change the electron charge distribution. Thanks to this tunability, we observe phase transitions from a compressible Fermi liquid to an incompressible FQHS and then to an insulating phase as a function of tilt angle. We find that this evolution and the energy gap of the ν=1/2 FQHS depend strongly on valley occupancy.
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Affiliation(s)
- Md Shafayat Hossain
- Department of Electrical and Computer Engineering, Princeton University, Princeton, New Jersey 08544, USA
| | - Meng K Ma
- Department of Electrical and Computer Engineering, Princeton University, Princeton, New Jersey 08544, USA
| | - Y J Chung
- Department of Electrical and Computer Engineering, Princeton University, Princeton, New Jersey 08544, USA
| | - S K Singh
- Department of Electrical and Computer Engineering, Princeton University, Princeton, New Jersey 08544, USA
| | - A Gupta
- Department of Electrical and Computer Engineering, Princeton University, Princeton, New Jersey 08544, USA
| | - K W West
- Department of Electrical and Computer Engineering, Princeton University, Princeton, New Jersey 08544, USA
| | - K W Baldwin
- Department of Electrical and Computer Engineering, Princeton University, Princeton, New Jersey 08544, USA
| | - L N Pfeiffer
- Department of Electrical and Computer Engineering, Princeton University, Princeton, New Jersey 08544, USA
| | - R Winkler
- Department of Physics, Northern Illinois University, DeKalb, Illinois 60115, USA
| | - M Shayegan
- Department of Electrical and Computer Engineering, Princeton University, Princeton, New Jersey 08544, USA
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4
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Wang C, Gupta A, Singh SK, Chung YJ, Pfeiffer LN, West KW, Baldwin KW, Winkler R, Shayegan M. Even-Denominator Fractional Quantum Hall State at Filling Factor ν=3/4. Phys Rev Lett 2022; 129:156801. [PMID: 36269975 DOI: 10.1103/physrevlett.129.156801] [Citation(s) in RCA: 4] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/15/2022] [Accepted: 08/17/2022] [Indexed: 06/16/2023]
Abstract
Fractional quantum Hall states (FQHSs) exemplify exotic phases of low-disorder two-dimensional (2D) electron systems when electron-electron interaction dominates over the thermal and kinetic energies. Particularly intriguing among the FQHSs are those observed at even-denominator Landau level filling factors, as their quasiparticles are generally believed to obey non-Abelian statistics and be of potential use in topological quantum computing. Such states, however, are very rare and fragile, and are typically observed in the excited Landau level of 2D electron systems with the lowest amount of disorder. Here we report the observation of a new and unexpected even-denominator FQHS at filling factor ν=3/4 in a GaAs 2D hole system with an exceptionally high quality (mobility). Our magnetotransport measurements reveal a strong minimum in the longitudinal resistance at ν=3/4, accompanied by a developing Hall plateau centered at (h/e^{2})/(3/4). This even-denominator FQHS is very unusual as it is observed in the lowest Landau level and in a 2D hole system. While its origin is unclear, it is likely a non-Abelian state, emerging from the residual interaction between composite fermions.
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Affiliation(s)
- Chengyu Wang
- Department of Electrical and Computer Engineering, Princeton University, Princeton, New Jersey 08544, USA
| | - A Gupta
- Department of Electrical and Computer Engineering, Princeton University, Princeton, New Jersey 08544, USA
| | - S K Singh
- Department of Electrical and Computer Engineering, Princeton University, Princeton, New Jersey 08544, USA
| | - Y J Chung
- Department of Electrical and Computer Engineering, Princeton University, Princeton, New Jersey 08544, USA
| | - L N Pfeiffer
- Department of Electrical and Computer Engineering, Princeton University, Princeton, New Jersey 08544, USA
| | - K W West
- Department of Electrical and Computer Engineering, Princeton University, Princeton, New Jersey 08544, USA
| | - K W Baldwin
- Department of Electrical and Computer Engineering, Princeton University, Princeton, New Jersey 08544, USA
| | - R Winkler
- Department of Physics, Northern Illinois University, DeKalb, Illinois 60115, USA
| | - M Shayegan
- Department of Electrical and Computer Engineering, Princeton University, Princeton, New Jersey 08544, USA
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5
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Villegas Rosales KA, Madathil PT, Chung YJ, Pfeiffer LN, West KW, Baldwin KW, Shayegan M. Fractional Quantum Hall Effect Energy Gaps: Role of Electron Layer Thickness. Phys Rev Lett 2021; 127:056801. [PMID: 34397247 DOI: 10.1103/physrevlett.127.056801] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/09/2021] [Accepted: 06/24/2021] [Indexed: 06/13/2023]
Abstract
The fractional quantum Hall effect stands as a quintessential manifestation of an interacting two-dimensional electron system. One of the fractional quantum Hall effect's most fundamental characteristics is the energy gap separating the incompressible ground state from its excitations. Yet, despite nearly four decades of investigations, a quantitative agreement between the theoretically calculated and experimentally measured energy gaps is lacking. Here we report a systematic experimental study that incorporates very high-quality two-dimensional electron systems confined to GaAs quantum wells with fixed density and varying well widths. The results demonstrate a clear decrease of the energy gap as the electron layer is made thicker and the short-range component of the Coulomb interaction is weakened. We also provide a quantitative comparison between the measured energy gaps and the available theoretical calculations that takes into account the role of finite layer thickness and Landau level mixing. All the measured energy gaps fall below the calculations, but as the electron layer thickness increases, the results of experiments and calculations come closer. Accounting for the role of disorder in a phenomenological manner, we find better overall agreement between the measured and calculated energy gaps, although some puzzling discrepancies remain.
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Affiliation(s)
- K A Villegas Rosales
- Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA
| | - P T Madathil
- 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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6
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Barkeshli M, Nayak C, Papić Z, Young A, Zaletel M. Topological Exciton Fermi Surfaces in Two-Component Fractional Quantized Hall Insulators. Phys Rev Lett 2018; 121:026603. [PMID: 30085706 DOI: 10.1103/physrevlett.121.026603] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/09/2016] [Indexed: 06/08/2023]
Abstract
A wide variety of two-dimensional electron systems allow for independent control of the total and relative charge density of two-component fractional quantum Hall (FQH) states. In particular, a recent experiment on bilayer graphene (BLG) observed a continuous transition between a compressible and incompressible phase at total filling ν_{T}=1/2 as charge is transferred between the layers, with the remarkable property that the incompressible phase has a finite interlayer polarizability. We argue that this occurs because the topological order of ν_{T}=1/2 systems supports a novel type of interlayer exciton that carries Fermi statistics. If the fermionic excitons are lower in energy than the conventional bosonic excitons (i.e., electron-hole pairs), they can form an emergent neutral Fermi surface, providing a possible explanation of an incompressible yet polarizable state at ν_{T}=1/2. We perform exact diagonalization studies that demonstrate that fermionic excitons are indeed lower in energy than bosonic excitons. This suggests that a "topological exciton metal" hidden inside a FQH insulator may have been realized experimentally in BLG. We discuss several detection schemes by which the topological exciton metal can be experimentally probed.
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Affiliation(s)
- Maissam Barkeshli
- Department of Physics, Condensed Matter Theory Center, University of Maryland, College Park, Maryland 20742, USA
- Joint Quantum Institute, University of Maryland, College Park, Maryland 20742, USA
- Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106, USA
| | - Chetan Nayak
- Station Q, Microsoft Research, Santa Barbara, California 93106-6105, USA
| | - Zlatko Papić
- School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, United Kingdom
| | - Andrea Young
- Department of Physics, University of California, Santa Barbara, California 93106-6105, USA
| | - Michael Zaletel
- Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106, USA
- Station Q, Microsoft Research, Santa Barbara, California 93106-6105, USA
- Department of Physics, Princeton University, Princeton, New Jersey 08540, USA
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7
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Ghazaryan A, Graß T, Gullans MJ, Ghaemi P, Hafezi M. Light-Induced Fractional Quantum Hall Phases in Graphene. Phys Rev Lett 2017; 119:247403. [PMID: 29286754 DOI: 10.1103/physrevlett.119.247403] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/11/2017] [Indexed: 06/07/2023]
Abstract
We show how to realize two-component fractional quantum Hall phases in monolayer graphene by optically driving the system. A laser is tuned into resonance between two Landau levels, giving rise to an effective tunneling between these two synthetic layers. Remarkably, because of this coupling, the interlayer interaction at nonzero relative angular momentum can become dominant, resembling a hollow-core pseudopotential. In the weak tunneling regime, this interaction favors the formation of singlet states, as we explicitly show by numerical diagonalization, at fillings ν=1/2 and ν=2/3. We discuss possible candidate phases, including the Haldane-Rezayi phase, the interlayer Pfaffian phase, and a Fibonacci phase. This demonstrates that our method may pave the way towards the realization of non-Abelian phases, as well as the control of topological phase transitions, in graphene quantum Hall systems using optical fields and integrated photonic structures.
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Affiliation(s)
- Areg Ghazaryan
- Department of Physics, City College, City University of New York, New York, New York 10031, USA
| | - Tobias Graß
- Joint Quantum Institute, NIST and University of Maryland, College Park, Maryland 20742, USA
- Department of Physics, College Park, Maryland 20742, USA
| | - Michael J Gullans
- Joint Quantum Institute, NIST and University of Maryland, College Park, Maryland 20742, USA
- Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, Maryland 20742, USA
| | - Pouyan Ghaemi
- Department of Physics, City College, City University of New York, New York, New York 10031, USA
- Department of Physics, Graduate Center, City University of New York, New York, New York 10016, USA
| | - Mohammad Hafezi
- Joint Quantum Institute, NIST and University of Maryland, College Park, Maryland 20742, USA
- Department of Physics, College Park, Maryland 20742, USA
- Department of Electrical Engineering and IREAP, University of Maryland, College Park, Maryland 20742, USA
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8
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Mueed MA, Kamburov D, Pfeiffer LN, West KW, Baldwin KW, Shayegan M. Geometric Resonance of Composite Fermions near Bilayer Quantum Hall States. Phys Rev Lett 2016; 117:246801. [PMID: 28009213 DOI: 10.1103/physrevlett.117.246801] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/12/2016] [Indexed: 06/06/2023]
Abstract
Via the application of a parallel magnetic field, we induce a single-layer to bilayer transition in two-dimensional electron systems confined to wide GaAs quantum wells and study the geometric resonance of composite fermions (CFs) with a periodic density modulation in our samples. The measurements reveal that CFs exist close to bilayer quantum Hall states, formed at Landau level filling factors ν=1 and 1/2. Near ν=1, the geometric resonance features are consistent with half the total electron density in the bilayer system, implying that CFs prefer to stay in separate layers and exhibit a two-component behavior. In contrast, close to ν=1/2, CFs appear single-layer-like (single component) as their resonance features correspond to the total density.
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Affiliation(s)
- M A Mueed
- Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA
| | - D Kamburov
- 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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9
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Liu Y, Hasdemir S, Pfeiffer LN, West KW, Baldwin KW, Shayegan M. Observation of an Anisotropic Wigner Crystal. Phys Rev Lett 2016; 117:106802. [PMID: 27636486 DOI: 10.1103/physrevlett.117.106802] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/16/2016] [Indexed: 06/06/2023]
Abstract
We report a new correlated phase of two-dimensional charged carriers in high magnetic fields, manifested by an anisotropic insulating behavior at low temperatures. It appears in a large range of low Landau level fillings 1/3≲ν≲2/3 in hole systems confined to wide GaAs quantum wells when the sample is tilted in magnetic field to an intermediate angle. The parallel field component (B_{∥}) leads to a crossing of the lowest two Landau levels, and an elongated hole wave function in the direction of B_{∥}. Under these conditions, the in-plane resistance exhibits an insulating behavior, with the resistance along B_{∥} about 10 times smaller than the resistance perpendicular to B_{∥}. We interpret this anisotropic insulating phase as a two-component, striped Wigner crystal.
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Affiliation(s)
- Yang Liu
- Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA
| | - S Hasdemir
- 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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10
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Hatke AT, Liu Y, Engel LW, Shayegan M, Pfeiffer LN, West KW, Baldwin KW. Microwave spectroscopy of the low-filling-factor bilayer electron solid in a wide quantum well. Nat Commun 2015; 6:7071. [PMID: 25947282 PMCID: PMC4432649 DOI: 10.1038/ncomms8071] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/23/2014] [Accepted: 03/28/2015] [Indexed: 11/30/2022] Open
Abstract
At the low Landau filling factor termination of the fractional quantum Hall effect series, two-dimensional electron systems exhibit an insulating phase that is understood as a form of pinned Wigner solid. Here we use microwave spectroscopy to probe the transition to the insulator for a wide quantum well sample that can support single-layer or bilayer states depending on its overall carrier density. We find that the insulator exhibits a resonance which is characteristic of a bilayer solid. The resonance also reveals a pair of transitions within the solid, which are not accessible to dc transport measurements. As density is biased deeper into the bilayer solid regime, the resonance grows in specific intensity, and the transitions within the insulator disappear. These behaviours are suggestive of a picture of the insulating phase as an emulsion of liquid and solid components. In 2D electron gases, insulating behaviour at low fractional quantum Hall filling factors is understood by the formation of an electronic Wigner solid. Here, the authors use microwave spectroscopy to evidence an electron liquid–solid mixed phase in bilayer states of GaAs/AlGaAs wide quantum wells.
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Affiliation(s)
- A T Hatke
- National High Magnetic Field Laboratory, Tallahassee, Florida 32310, USA
| | - Y Liu
- Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA
| | - L W Engel
- National High Magnetic Field Laboratory, Tallahassee, Florida 32310, USA
| | - M Shayegan
- 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
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11
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Abstract
The possibility of realizing non-Abelian statistics and utilizing it for topological quantum computation (TQC) has generated widespread interest. However, the non-Abelian statistics that can be realized in most accessible proposals is not powerful enough for universal TQC. In this Letter, we consider a simple bilayer fractional quantum Hall system with the 1/3 Laughlin state in each layer. We show that interlayer tunneling can drive a transition to an exotic non-Abelian state that contains the famous "Fibonacci" anyon, whose non-Abelian statistics is powerful enough for universal TQC. Our analysis rests on startling agreements from a variety of distinct methods, including thin torus limits, effective field theories, and coupled wire constructions. We provide evidence that the transition can be continuous, at which point the charge gap remains open while the neutral gap closes. This raises the question of whether these exotic phases may have already been realized at ν=2/3 in bilayers, as past experiments may not have definitively ruled them out.
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Affiliation(s)
- Abolhassan Vaezi
- Department of Physics, Cornell University, Ithaca, New York 14850, USA
| | - Maissam Barkeshli
- Station Q, Microsoft Research, Santa Barbara, California 93106-6105, USA
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12
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Hatke AT, Liu Y, Magill BA, Moon BH, Engel LW, Shayegan M, Pfeiffer LN, West KW, Baldwin KW. Microwave spectroscopic observation of distinct electron solid phases in wide quantum wells. Nat Commun 2014; 5:4154. [PMID: 24948190 DOI: 10.1038/ncomms5154] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/24/2014] [Accepted: 05/19/2014] [Indexed: 11/08/2022] Open
Abstract
In high magnetic fields, two-dimensional electron systems can form a number of phases in which interelectron repulsion plays the central role, since the kinetic energy is frozen out by Landau quantization. These phases include the well-known liquids of the fractional quantum Hall effect, as well as solid phases with broken spatial symmetry and crystalline order. Solids can occur at the low Landau-filling termination of the fractional quantum Hall effect series but also within integer quantum Hall effects. Here we present microwave spectroscopy studies of wide quantum wells that clearly reveal two distinct solid phases, hidden within what in d.c. transport would be the zero diagonal conductivity of an integer quantum-Hall-effect state. Explanation of these solids is not possible with the simple picture of a Wigner solid of ordinary (quasi) electrons or holes.
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13
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Liu Y, Graninger AL, Hasdemir S, Shayegan M, Pfeiffer LN, West KW, Baldwin KW, Winkler R. Fractional quantum Hall effect at ν=1/2 in hole systems confined to GaAs quantum wells. Phys Rev Lett 2014; 112:046804. [PMID: 24580479 DOI: 10.1103/physrevlett.112.046804] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/29/2013] [Indexed: 06/03/2023]
Abstract
We observe the fractional quantum Hall effect (FQHE) at the even-denominator Landau level filling factor ν=1/2 in two-dimensional hole systems confined to GaAs quantum wells of width 30 to 50 nm and having bilayerlike charge distributions. The ν=1/2 FQHE is stable when the charge distribution is symmetric and only in a range of intermediate densities, qualitatively similar to what is seen in two-dimensional electron systems confined to approximately twice wider GaAs quantum wells. Despite the complexity of the hole Landau level structure, originating from the coexistence and mixing of the heavy- and light-hole states, we find the hole ν=1/2 FQHE to be consistent with a two-component, Halperin-Laughlin (Ψ331) state.
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Affiliation(s)
- Yang Liu
- Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA
| | - A L Graninger
- Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA
| | - S Hasdemir
- Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA
| | - M Shayegan
- 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
| | - R Winkler
- Department of Physics, Northern Illinois University, DeKalb, Illinois 60115, USA and Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA
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14
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Liu Y, Pappas CG, Shayegan M, Pfeiffer LN, West KW, Baldwin KW. Observation of reentrant integer quantum Hall states in the lowest Landau level. Phys Rev Lett 2012; 109:036801. [PMID: 22861882 DOI: 10.1103/physrevlett.109.036801] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/22/2011] [Revised: 02/13/2012] [Indexed: 06/01/2023]
Abstract
Measurements on very low disorder two-dimensional electrons confined to relatively wide GaAs quantum well samples with tunable density reveal a close competition between the electron liquid and solid phases near the Landau level filling factor ν=1. As the density is raised, the fractional quantum Hall liquid at ν=4/5 suddenly disappears at a well-width dependent critical density, and then reappears at higher densities with insulating phases on its flanks. These insulating phases exhibit reentrant ν=1 integer quantum Hall effects and signal the formation of electron Wigner crystal states. Qualitatively similar phenomena are seen near ν=6/5.
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Affiliation(s)
- Yang Liu
- Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA
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Liu Y, Shabani J, Kamburov D, Shayegan M, Pfeiffer LN, West KW, Baldwin KW. Evolution of the 7/2 fractional quantum Hall state in two-subband systems. Phys Rev Lett 2011; 107:266802. [PMID: 22243175 DOI: 10.1103/physrevlett.107.266802] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/19/2011] [Indexed: 05/31/2023]
Abstract
We report the evolution of the fractional quantum Hall state (FQHS) at a total Landau level (LL) filling factor of ν=7/2 in wide GaAs quantum wells in which electrons occupy two electric subbands. The data reveal subtle and distinct evolutions as a function of density, magnetic field tilt angle, or symmetry of the charge distribution. At intermediate tilt angles, for example, we observe a strengthening of the ν=7/2 FQHS. Moreover, in a well with asymmetric change distribution, there is a developing FQHS when the LL filling factor of the symmetric subband ν(S) equals 5/2 while the antisymmetric subband has a filling factor of 1<ν(A)<2.
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Affiliation(s)
- Yang Liu
- Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA
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Liu Y, Kamburov D, Shayegan M, Pfeiffer LN, West KW, Baldwin KW. Anomalous robustness of the ν=5/2 fractional quantum Hall state near a sharp phase boundary. Phys Rev Lett 2011; 107:176805. [PMID: 22107557 DOI: 10.1103/physrevlett.107.176805] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/31/2011] [Indexed: 05/31/2023]
Abstract
We report magnetotransport measurements in wide GaAs quantum wells with a tunable density to probe the stability of the fractional quantum Hall effect at a filling factor of ν=5/2 in the vicinity of the crossing between Landau levels (LLs) belonging to the different (symmetric and antisymmetric) electric subbands. When the Fermi energy (E(F)) lies in the excited-state LL of the symmetric subband, the 5/2 quantum Hall state is surprisingly stable and gets even stronger near this crossing, and then suddenly disappears and turns into a metallic state once E(F) moves to the ground-state LL of the antisymmetric subband. The sharpness of this disappearance suggests a first-order transition.
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Affiliation(s)
- Yang Liu
- Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA
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Shabani J, Liu Y, Shayegan M. Fractional quantum Hall effect at high fillings in a two-subband electron system. Phys Rev Lett 2010; 105:246805. [PMID: 21231548 DOI: 10.1103/physrevlett.105.246805] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/05/2010] [Indexed: 05/30/2023]
Abstract
Magnetotransport measurements in a clean two-dimensional electron system confined to a wide GaAs quantum well reveal that, when the electrons occupy two electric subbands, the sequences of fractional quantum Hall states observed at high fillings (ν>2) are distinctly different from those of a single-subband system. Notably, when the Fermi energy lies in the ground state Landau level of either of the subbands, no quantum Hall states are seen at the even-denominator ν=5/2 and 7/2 fillings; instead, the observed states are at ν=[i+p/(2p±1)], where i=2, 3, 4 and p=1, 2, 3, and include several new states at ν=13/5, 17/5, 18/5, 25/7, and 14/3.
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Affiliation(s)
- J Shabani
- Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA
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Barkeshli M, Wen XG. Anyon condensation and continuous topological phase transitions in non-Abelian fractional quantum Hall states. Phys Rev Lett 2010; 105:216804. [PMID: 21231341 DOI: 10.1103/physrevlett.105.216804] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/30/2010] [Indexed: 05/30/2023]
Abstract
We find a series of possible continuous quantum phase transitions between fractional quantum Hall states at the same filling fraction in two-component quantum Hall systems. These can be driven by tuning the interlayer tunneling and/or interlayer repulsion. One side of the transition is the Halperin (p,p,p-3) Abelian two-component state, while the other side is the non-Abelian Z4 parafermion (Read-Rezayi) state. We predict that the transition is a continuous transition in the 3D Ising class. The critical point is described by a Z2 gauged Ginzburg-Landau theory. These results have implications for experiments on two-component systems at ν=2/3 and single-component systems at ν=8/3.
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Affiliation(s)
- Maissam Barkeshli
- Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
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Abstract
Using exact diagonalization we show that the spin-polarized Coulomb ground state at nu=5/2 is adiabatically connected with the Moore-Read wave function for systems with up to 18 electrons on the surface of a sphere. The ground state is protected by a large gap for all system sizes studied. Furthermore, varying the Haldane pseudopotentials v{1} and v{3}, keeping all others at their value for the Coulomb interaction, energy gap and overlap between ground- and Moore-Read state form hills whose positions and extent in the (v{1},v{3}) plane coincide. We conclude that the physics of the Coulomb ground state at nu=5/2 is captured by the Moore-Read state. Such an adiabatic connection is not found at nu=1/2, unless the width of the interface wave function or Landau level mixing effects are large enough. Yet, a Moore-Read-phase at nu=1/2 appears unlikely in the thermodynamic limit.
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Affiliation(s)
- M Storni
- Condensed Matter Theory, Paul Scherrer Institute, CH-5232 Villigen, Switzerland
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Alicea J, Motrunich OI, Refael G, Fisher MPA. Interlayer coherent composite Fermi liquid phase in quantum Hall bilayers. Phys Rev Lett 2009; 103:256403. [PMID: 20366269 DOI: 10.1103/physrevlett.103.256403] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/26/2009] [Indexed: 05/29/2023]
Abstract
We introduce an interlayer coherent composite Fermi liquid for nu = 1/2 + 1/2 bilayers, in which interlayer Coulomb repulsion drives exciton condensation of composite fermions. As a result, composite fermions propagate coherently between layers--even though electrons do not--and form bonding and antibonding Fermi seas. This phase is compressible with respect to symmetric currents but quantum Hall-like in the counterflow channel. Quantum oscillations of the composite Fermi seas generate a new series of incompressible states at nu = p/[2(p +/- 1)] per layer (p an integer), which is a bilayer analogue of Jain's sequence.
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Affiliation(s)
- Jason Alicea
- Department of Physics, California Institute of Technology, Pasadena, California 91125, USA
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Shabani J, Gokmen T, Chiu YT, Shayegan M. Evidence for developing fractional quantum Hall states at even denominator 1/2 and 1/4 fillings in asymmetric wide quantum wells. Phys Rev Lett 2009; 103:256802. [PMID: 20366273 DOI: 10.1103/physrevlett.103.256802] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/09/2009] [Indexed: 05/29/2023]
Abstract
We report the observation of developing fractional quantum Hall states at Landau level filling factors nu = 1/2 and 1/4 in electron systems confined to wide GaAs quantum wells with significantly asymmetric charge distributions. The very large electric subband separation and the highly asymmetric charge distribution at which we observe these quantum Hall states, together with the fact that they disappear when the charge distribution is made symmetric, suggest that these are one-component states, possibly described by the Moore-Read Pfaffian wave function.
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Affiliation(s)
- J Shabani
- Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA
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Shabani J, Gokmen T, Shayegan M. Correlated states of electrons in wide quantum wells at low fillings: the role of charge distribution symmetry. Phys Rev Lett 2009; 103:046805. [PMID: 19659383 DOI: 10.1103/physrevlett.103.046805] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/24/2009] [Indexed: 05/28/2023]
Abstract
Magnetotransport measurements on electrons confined to a 57-nm-wide, GaAs quantum well reveal that the correlated electron states at low Landau level fillings (nu) display a remarkable dependence on the symmetry of the electron charge distribution. At a density of 1.93 x 10;{11} cm;{-2}, a developing fractional quantum Hall state is observed at the even-denominator filling nu = 1/4 when the distribution is symmetric, but it quickly vanishes when the distribution is made asymmetric. At lower densities, as we make the charge distribution asymmetric, we observe a rapid strengthening of the insulating phases that surround the nu = 1/5 fractional quantum Hall state.
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Affiliation(s)
- J Shabani
- Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA
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Luhman DR, Pan W, Tsui DC, Pfeiffer LN, Baldwin KW, West KW. Observation of a fractional quantum hall state at nu = 1/4 in a wide GaAs quantum well. Phys Rev Lett 2008; 101:266804. [PMID: 19437661 DOI: 10.1103/physrevlett.101.266804] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/27/2023]
Abstract
We report the observation of an even-denominator fractional quantum Hall state at nu = 1/4 in a high quality, wide GaAs quantum well. The sample has a quantum well width of 50 nm and an electron density of n(e) = 2.55 x 10(11) cm(-2). We have performed transport measurements at T - 35 mK in magnetic fields up to 45 T. When the sample is perpendicular to the applied magnetic field, the diagonal resistance displays a kink at nu = 1/4. Upon tilting the sample to an angle of theta = 20.3 degrees a clear fractional quantum Hall state emerges at nu = 1/4 with a plateau in the Hall resistance and a strong minimum in the diagonal resistance.
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Affiliation(s)
- D R Luhman
- Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA
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Wang Z, Chen YP, Engel LW, Tsui DC, Tutuc E, Shayegan M. Pinning modes and interlayer correlation in high-magnetic-field bilayer Wigner solids. Phys Rev Lett 2007; 99:136804. [PMID: 17930621 DOI: 10.1103/physrevlett.99.136804] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/30/2006] [Indexed: 05/25/2023]
Abstract
We report studies of pinning mode resonances in the low total Landau filling (nu) Wigner solid of a series of bilayer hole samples with negligible interlayer tunneling and with varying interlayer separation d. Comparison of states with equal layer densities (p,p) to single layer states (p,0) produced in situ by biasing, indicates that there is interlayer quantum correlation in the solid at small d. Also, the resonance frequency at small d is decreased just near nu = 1/2 and 2/3, indicating the importance in the solid of correlations related to those in the fractional quantum Hall effects.
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Affiliation(s)
- Zhihai Wang
- National High Magnetic Field Laboratory, 1800 East Paul Dirac Drive, Tallahassee, Florida 32310, USA
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Vakili K, Shkolnikov YP, Tutuc E, De Poortere EP, Shayegan M. Realization of an interacting two-valley AlAs bilayer system. Phys Rev Lett 2004; 92:186404. [PMID: 15169516 DOI: 10.1103/physrevlett.92.186404] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/22/2003] [Indexed: 05/24/2023]
Abstract
By using different widths for two AlAs quantum wells comprising a bilayer system, we force the X-point conduction-band electrons in the two layers to occupy valleys with different Fermi contours, electron effective masses, and g factors. Since the occupied valleys are at different X points of the Brillouin zone, the interlayer tunneling is negligibly small despite the close electron layer spacing. We demonstrate the realization of this system via magnetotransport measurements and the observation of a phase-coherent, bilayer nu=1 quantum Hall state flanked by a reentrant insulating phase.
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Affiliation(s)
- K Vakili
- Department of Electrical Engineering, Princeton University, Princeton, NJ 08544, USA
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Tutuc E, Melinte S, De Poortere EP, Pillarisetty R, Shayegan M. Role of density imbalance in an interacting bilayer hole system. Phys Rev Lett 2003; 91:076802. [PMID: 12935040 DOI: 10.1103/physrevlett.91.076802] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/21/2002] [Indexed: 05/24/2023]
Abstract
We study interacting GaAs hole bilayers in the limit of zero interlayer tunneling. When the layers have equal density, we observe a phase-coherent bilayer quantum Hall state (QHS) at a total filling factor nu=1, flanked by a reentrant insulating phase at nearby fillings which suggests the formation of a pinned, bilayer Wigner crystal. As we transfer charge from one layer to another, the phase-coherent QHS becomes stronger, evincing its robustness against charge imbalance, but the insulating phase disappears, suggesting that its stability requires the commensurability of the two layers.
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Affiliation(s)
- E Tutuc
- Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA
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Hamilton AR, Simmons MY, Bolton FM, Patel NK, Millard IS, Nicholls JT, Ritchie DA, Pepper M. Fractional quantum Hall effect in bilayer two-dimensional hole-gas systems. Phys Rev B Condens Matter 1996; 54:R5259-R5262. [PMID: 9986583 DOI: 10.1103/physrevb.54.r5259] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Ihn T, Carmona H, Main PC, Eaves L, Henini M. Magnetic-field-induced resonant tunneling in parallel two-dimensional systems. Phys Rev B Condens Matter 1996; 54:R2315-R2318. [PMID: 9986166 DOI: 10.1103/physrevb.54.r2315] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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McDonald IA, Haldane FD. Topological phase transition in the nu =2/3 quantum Hall effect. Phys Rev B Condens Matter 1996; 53:15845-15855. [PMID: 9983422 DOI: 10.1103/physrevb.53.15845] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Lu JP, Shayegan M. Observation of ballistic transport in the upper subband of a two-dimensional electron system. Phys Rev B Condens Matter 1996; 53:R4217-R4220. [PMID: 9984073 DOI: 10.1103/physrevb.53.r4217] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Nakajima T, Aoki H. Composite-fermion analysis of the double-layer fractional quantum Hall system. Phys Rev B Condens Matter 1995; 52:13780-13783. [PMID: 9980588 DOI: 10.1103/physrevb.52.13780] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Ying X, Parihar SR, Manoharan HC, Shayegan M. Quantitative determination of many-body-induced interlayer charge transfer in bilayer electron systems via Shubnikov-de Haas measurements. Phys Rev B Condens Matter 1995; 52:11611-11614. [PMID: 9980282 DOI: 10.1103/physrevb.52.r11611] [Citation(s) in RCA: 15] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Mancoff FB, Clarke RM, Marcus CM, Zhang SC, Campman K, Gossard AC. Magnetotransport of a two-dimensional electron gas in a spatially random magnetic field. Phys Rev B Condens Matter 1995; 51:13269-13273. [PMID: 9978129 DOI: 10.1103/physrevb.51.13269] [Citation(s) in RCA: 22] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Palacios JJ, Hawrylak P. Correlated few-electron states in vertical double-quantum-dot systems. Phys Rev B Condens Matter 1995; 51:1769-1777. [PMID: 9978898 DOI: 10.1103/physrevb.51.1769] [Citation(s) in RCA: 56] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Lay TS, Suen YW, Manoharan HC, Ying X, Santos MB, Shayegan M. Anomalous temperature dependence of the correlated nu =1 quantum Hall effect in bilayer electron systems. Phys Rev B Condens Matter 1994; 50:17725-17728. [PMID: 9976201 DOI: 10.1103/physrevb.50.17725] [Citation(s) in RCA: 26] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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