1
|
Le HA, Lee IH, Kim YH, Eric Yang SR. Phase diagram and crossover phases of topologically ordered graphene zigzag nanoribbons: role of localization effects. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2024; 36:265604. [PMID: 38547530 DOI: 10.1088/1361-648x/ad38f9] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/12/2024] [Accepted: 03/28/2024] [Indexed: 04/06/2024]
Abstract
We computed the phase diagram of zigzag graphene nanoribbons as a function of on-site repulsion, doping, and disorder strength. The topologically ordered phase undergoes topological phase transitions into crossover phases, which are new disordered phases with non-universal topological entanglement entropy that exhibits significant variance. We explored the nature of non-local correlations in both the topologically ordered and crossover phases. In the presence of localization effects, strong on-site repulsion and/or doping weaken non-local correlations between the opposite zigzag edges of the topologically ordered phase. In one of the crossover phases, bothe-/2solitonic fractional charges and spin-charge separation were absent; however, charge-transfer correlations between the zigzag edges were possible. Another crossover phase contains solitonice-/2fractional charges but lacks charge transfer correlations. We also observed properties of non-topological, strongly disordered, and strongly repulsive phases. Each phase on the phase diagram exhibits a different zigzag-edge structure. Additionally, we investigated the tunneling of solitonic fractional charges under an applied voltage between the zigzag edges of undoped topologically ordered zigzag ribbons, and found that it may lead to a zero-bias tunneling anomaly.
Collapse
Affiliation(s)
- Hoang-Anh Le
- Department of Physics, Korea University, Seoul 02841, Republic of Korea
| | - In-Hwan Lee
- Department of Physics, Korea University, Seoul 02841, Republic of Korea
| | - Young Heon Kim
- Department of Physics, Korea University, Seoul 02841, Republic of Korea
| | - S-R Eric Yang
- Department of Physics, Korea University, Seoul 02841, Republic of Korea
| |
Collapse
|
2
|
Kim YH, Lee HJ, Lee HY, Yang SRE. New disordered anyon phase of doped graphene zigzag nanoribbon. Sci Rep 2022; 12:14551. [PMID: 36008453 PMCID: PMC9411593 DOI: 10.1038/s41598-022-18731-6] [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: 06/16/2022] [Accepted: 08/18/2022] [Indexed: 11/28/2022] Open
Abstract
We investigate interacting disordered zigzag nanoribbons at low doping, using the Hubbard model to treat electron interactions within the density matrix renormalization group and Hartree-Fock method. Extra electrons that are inserted into an interacting disordered zigzag nanoribbon divide into anyons. Furthermore, the fractional charges form a new disordered anyon phase with a highly distorted edge spin density wave, containing numerous localized magnetic moments residing on the zigzag edges, thereby displaying spin-charge separation and a strong non-local correlation between the opposite zigzag edges. We make the following new predictions, which can be experimentally tested: (1) In the low doping case and weak disorder regime, the soft gap in the tunneling density of states is replaced by a sharp peak at the midgap energy with two accompanying peaks. The \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$e^-/2$$\end{document}e-/2 fractional charges that reside on the boundary of the zigzag edges are responsible for these peaks. (2) We find that the midgap peak disappears as the doping concentration increases. The presence of \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$e^-/2$$\end{document}e-/2 fractional charges will be strongly supported by the detection of these peaks. Doped zigzag ribbons may also exhibit unusual transport, magnetic, and inter-edge tunneling properties.
Collapse
Affiliation(s)
- Young Heon Kim
- Department of Physics, Korea University, Seoul, 02855, South Korea
| | - Hye Jeong Lee
- Department of Physics, Korea University, Seoul, 02855, South Korea
| | - Hyun-Yong Lee
- Department of Applied Physics, Graduate School, Korea University, Sejong, 30019, South Korea.,Division of Display and Semiconductor Physics, Korea University, Sejong, 30019, South Korea.,Interdisciplinary Program in E.ICT-Culture-Sports Convergence, Korea University, Sejong, 30019, South Korea
| | - S-R Eric Yang
- Department of Physics, Korea University, Seoul, 02855, South Korea.
| |
Collapse
|
3
|
Wu X, Xiao D, Chen CZ, Sun J, Zhang L, Chan MHW, Samarth N, Xie XC, Lin X, Chang CZ. Scaling behavior of the quantum phase transition from a quantum-anomalous-Hall insulator to an axion insulator. Nat Commun 2020; 11:4532. [PMID: 32913228 PMCID: PMC7483742 DOI: 10.1038/s41467-020-18312-z] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/15/2020] [Accepted: 08/18/2020] [Indexed: 11/08/2022] Open
Abstract
The phase transitions from one plateau to the next plateau or to an insulator in quantum Hall and quantum anomalous Hall (QAH) systems have revealed universal scaling behaviors. A magnetic-field-driven quantum phase transition from a QAH insulator to an axion insulator was recently demonstrated in magnetic topological insulator sandwich samples. Here, we show that the temperature dependence of the derivative of the longitudinal resistance on magnetic field at the transition point follows a characteristic power-law that indicates a universal scaling behavior for the QAH to axion insulator phase transition. Similar to the quantum Hall plateau to plateau transition, the QAH to axion insulator transition can also be understood by the Chalker-Coddington network model. We extract a critical exponent κ ~ 0.38 ± 0.02 in agreement with recent high-precision numerical results on the correlation length exponent of the Chalker-Coddington model at ν ~ 2.6, rather than the generally-accepted value of 2.33.
Collapse
Affiliation(s)
- Xinyu Wu
- International Center for Quantum Materials, Peking University, Beijing, 100871, China
| | - Di Xiao
- Department of Physics, The Pennsylvania State University, University Park, PA, 16802, USA
| | - Chui-Zhen Chen
- Institute for Advanced Study and School of Physical Science and Technology, Soochow University, Suzhou, 215006, China
| | - Jian Sun
- International Center for Quantum Materials, Peking University, Beijing, 100871, China
| | - Ling Zhang
- Department of Physics, The Pennsylvania State University, University Park, PA, 16802, USA
| | - Moses H W Chan
- Department of Physics, The Pennsylvania State University, University Park, PA, 16802, USA
| | - Nitin Samarth
- Department of Physics, The Pennsylvania State University, University Park, PA, 16802, USA
| | - X C Xie
- International Center for Quantum Materials, Peking University, Beijing, 100871, China
- Beijing Academy of Quantum Information Sciences, Beijing, 100193, China
- CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing, 100190, China
| | - Xi Lin
- International Center for Quantum Materials, Peking University, Beijing, 100871, China.
- Beijing Academy of Quantum Information Sciences, Beijing, 100193, China.
- CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing, 100190, China.
| | - Cui-Zu Chang
- Department of Physics, The Pennsylvania State University, University Park, PA, 16802, USA.
| |
Collapse
|
4
|
Jeong YH, Eric Yang SR, Cha MC. Soliton fractional charge of disordered graphene nanoribbon. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2019; 31:265601. [PMID: 30921770 DOI: 10.1088/1361-648x/ab146b] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/09/2023]
Abstract
We investigate the properties of the gap-edge states of half-filled interacting disordered zigzag graphene nanoribbons, and find that the midgap states can display a quantized fractional charge of 1/2. These gap-edge states can be represented by topological kinks with their site probability distribution divided between the left and right zigzag edges with different chiralities. In addition, there are numerous spin-split gap-edge states, similar to those in a Mott-Anderson insulator.
Collapse
Affiliation(s)
- Y H Jeong
- Department of Physics, Korea University, Seoul, Republic of Korea
| | | | | |
Collapse
|
5
|
Soliton Fractional Charges in Graphene Nanoribbon and Polyacetylene: Similarities and Differences. NANOMATERIALS 2019; 9:nano9060885. [PMID: 31207969 PMCID: PMC6630912 DOI: 10.3390/nano9060885] [Citation(s) in RCA: 13] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 05/20/2019] [Revised: 06/07/2019] [Accepted: 06/13/2019] [Indexed: 11/17/2022]
Abstract
An introductory overview of current research developments regarding solitons and fractional boundary charges in graphene nanoribbons is presented. Graphene nanoribbons and polyacetylene have chiral symmetry and share numerous similar properties, e.g., the bulk-edge correspondence between the Zak phase and the existence of edge states, along with the presence of chiral boundary states, which are important for charge fractionalization. In polyacetylene, a fermion mass potential in the Dirac equation produces an excitation gap, and a twist in this scalar potential produces a zero-energy chiral soliton. Similarly, in a gapful armchair graphene nanoribbon, a distortion in the chiral gauge field can produce soliton states. In polyacetylene, a soliton is bound to a domain wall connecting two different dimerized phases. In graphene nanoribbons, a domain-wall soliton connects two topological zigzag edges with different chiralities. However, such a soliton does not display spin-charge separation. The existence of a soliton in finite-length polyacetylene can induce formation of fractional charges on the opposite ends. In contrast, for gapful graphene nanoribbons, the antiferromagnetic coupling between the opposite zigzag edges induces integer boundary charges. The presence of disorder in graphene nanoribbons partly mitigates antiferromagnetic coupling effect. Hence, the average edge charge of gap states with energies within a small interval is e/2, with significant charge fluctuations. However, midgap states exhibit a well-defined charge fractionalization between the opposite zigzag edges in the weak-disorder regime. Numerous occupied soliton states in a disorder-free and doped zigzag graphene nanoribbon form a solitonic phase.
Collapse
|
6
|
Krishtopenko SS, Gavrilenko VI, Goiran M. The effect of exchange interaction on quasiparticle Landau levels in narrow-gap quantum well heterostructures. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2012; 24:135601. [PMID: 22406976 DOI: 10.1088/0953-8984/24/13/135601] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/31/2023]
Abstract
Using the 'screened' Hartree-Fock approximation based on the eight-band k·p Hamiltonian, we have extended our previous work (Krishtopenko et al 2011 J. Phys.: Condens. Matter 23 385601) on exchange enhancement of the g-factor in narrow-gap quantum well heterostructures by calculating the exchange renormalization of quasiparticle energies, the density of states at the Fermi level and the quasiparticle g-factor for different Landau levels overlapping. We demonstrate that exchange interaction yields more pronounced Zeeman splitting of the density of states at the Fermi level and leads to the appearance of peak-shaped features in the dependence of the Landau level energies on the magnetic field at integer filling factors. We also find that the quasiparticle g-factor does not reach the maximum value at odd filling factors in the presence of large overlapping of spin-split Landau levels. We advance an argument that the behavior of the quasiparticle g-factor in weak magnetic fields is defined by a random potential of impurities in narrow-gap heterostructures.
Collapse
Affiliation(s)
- S S Krishtopenko
- Institute for Physics of Microstructures RAS, GSP-105, 603950, Nizhny Novgorod, Russia
| | | | | |
Collapse
|
7
|
Becker S, Karrasch C, Mashoff T, Pratzer M, Liebmann M, Meden V, Morgenstern M. Probing electron-electron interaction in quantum Hall systems with scanning tunneling spectroscopy. PHYSICAL REVIEW LETTERS 2011; 106:156805. [PMID: 21568596 DOI: 10.1103/physrevlett.106.156805] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/07/2010] [Indexed: 05/30/2023]
Abstract
Using low-temperature scanning tunneling spectroscopy applied to the Cs-induced two-dimensional electron system (2DES) on p-type InSb(110), we probe electron-electron interaction effects in the quantum Hall regime. The 2DES is decoupled from bulk states and exhibits spreading resistance within the insulating quantum Hall phases. In quantitative agreement with calculations we find an exchange enhancement of the spin splitting. Moreover, we observe that both the spatially averaged as well as the local density of states feature a characteristic Coulomb gap at the Fermi level. These results show that electron-electron interaction can be probed down to a resolution below all relevant length scales.
Collapse
Affiliation(s)
- S Becker
- II. Physikalisches Institut B and JARA-FIT, RWTH Aachen University, 52074 Aachen, Germany
| | | | | | | | | | | | | |
Collapse
|
8
|
Dial OE, Ashoori RC, Pfeiffer LN, West KW. Anomalous structure in the single particle spectrum of the fractional quantum Hall effect. Nature 2010; 464:566-70. [DOI: 10.1038/nature08941] [Citation(s) in RCA: 32] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/15/2009] [Accepted: 02/18/2010] [Indexed: 11/09/2022]
|
9
|
Atkinson WA, Hirschfeld PJ, MacDonald AH. Gap inhomogeneities and the density of states in disordered d-wave superconductors. PHYSICAL REVIEW LETTERS 2000; 85:3922-3925. [PMID: 11041961 DOI: 10.1103/physrevlett.85.3922] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/22/2000] [Indexed: 05/23/2023]
Abstract
We report on a numerical study of disorder effects in 2D d-wave BCS superconductors. We compare exact numerical solutions of the Bogoliubov-de Gennes (BdG) equations for the density of states rho(E) with the standard T-matrix approximation. Local suppression of the order parameter near impurity sites, which occurs in self-consistent solutions of the BdG equations, leads to apparent power-law behavior rho(E) approximately |E|(alpha) with nonuniversal alpha over an energy scale comparable to the single-impurity resonance energy Omega(0). We show that the novel effects arise from static spatial correlations between the order parameter and the impurity distribution.
Collapse
Affiliation(s)
- WA Atkinson
- Department of Physics, University of Florida, P.O. Box 118440, Gainesville, Florida 32611, USA
| | | | | |
Collapse
|
10
|
Turner N, Nicholls JT, Linfield EH, Brown KM, Jones GA, Ritchie DA. Tunneling between parallel two-dimensional electron gases. PHYSICAL REVIEW. B, CONDENSED MATTER 1996; 54:10614-10624. [PMID: 9984858 DOI: 10.1103/physrevb.54.10614] [Citation(s) in RCA: 28] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
|
11
|
Jeon GS, Choi MY, Yang S. Coulomb gaps in one-dimensional spin-polarized electron systems. PHYSICAL REVIEW. B, CONDENSED MATTER 1996; 54:R8341-R8344. [PMID: 9984590 DOI: 10.1103/physrevb.54.r8341] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
|
12
|
Huang D, Manasreh MO. Effects of the screened exchange interaction on the tunneling and Landau gaps in double quantum wells. PHYSICAL REVIEW. B, CONDENSED MATTER 1996; 54:2044-2048. [PMID: 9986056 DOI: 10.1103/physrevb.54.2044] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
|
13
|
Fogler MM, Koulakov AA, Shklovskii BI. Ground state of a two-dimensional electron liquid in a weak magnetic field. PHYSICAL REVIEW. B, CONDENSED MATTER 1996; 54:1853-1871. [PMID: 9986033 DOI: 10.1103/physrevb.54.1853] [Citation(s) in RCA: 68] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
|
14
|
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]
|
15
|
Lee DH, Wang Z. Effects of electron-electron interactions on the integer quantum Hall transitions. PHYSICAL REVIEW LETTERS 1996; 76:4014-4017. [PMID: 10061170 DOI: 10.1103/physrevlett.76.4014] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
|
16
|
Palacios JJ, MacDonald AH. Numerical tests of the chiral Luttinger liquid theory for fractional Hall edges. PHYSICAL REVIEW LETTERS 1996; 76:118-121. [PMID: 10060448 DOI: 10.1103/physrevlett.76.118] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
|
17
|
Aleiner IL, Glazman LI. Two-dimensional electron liquid in a weak magnetic field. PHYSICAL REVIEW. B, CONDENSED MATTER 1995; 52:11296-11312. [PMID: 9980234 DOI: 10.1103/physrevb.52.11296] [Citation(s) in RCA: 31] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
|
18
|
Herbut IF, Tesanovic Z. Anomalous diffusion in a high magnetic field and the quasiparticle density of states. PHYSICAL REVIEW. B, CONDENSED MATTER 1995; 52:5160-5163. [PMID: 9981700 DOI: 10.1103/physrevb.52.5160] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
|
19
|
Pikus FG, Efros AL. Coulomb gap in a two-dimensional electron gas with a close metallic electrode. PHYSICAL REVIEW. B, CONDENSED MATTER 1995; 51:16871-16877. [PMID: 9978697 DOI: 10.1103/physrevb.51.16871] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
|
20
|
Aleiner IL, Baranger HU, Glazman LI. Tunneling into a Two-Dimensional Electron Liquid in a Weak Magnetic Field. PHYSICAL REVIEW LETTERS 1995; 74:3435-3438. [PMID: 10058200 DOI: 10.1103/physrevlett.74.3435] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
|
21
|
Yang S, MacDonald AH, Huckestein B. Interactions, localization, and the integer quantum Hall effect. PHYSICAL REVIEW LETTERS 1995; 74:3229-3232. [PMID: 10058144 DOI: 10.1103/physrevlett.74.3229] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
|
22
|
Ugajin R. Far-infrared absorption in coupled quantum dots. PHYSICAL REVIEW. B, CONDENSED MATTER 1995; 51:11136-11139. [PMID: 9977825 DOI: 10.1103/physrevb.51.11136] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
|
23
|
Gumbs G, Aizin GR. Tunneling density of states and plasmon excitations in double-quantum-well systems. PHYSICAL REVIEW. B, CONDENSED MATTER 1995; 51:7074-7084. [PMID: 9977266 DOI: 10.1103/physrevb.51.7074] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
|
24
|
Brown KM, Turner N, Nicholls JT, Linfield EH, Pepper M, Ritchie DA, Jones GA. Tunneling between two-dimensional electron gases in a strong magnetic field. PHYSICAL REVIEW. B, CONDENSED MATTER 1994; 50:15465-15468. [PMID: 9975912 DOI: 10.1103/physrevb.50.15465] [Citation(s) in RCA: 39] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
|
25
|
Renn SR, Roberts BW. Tunneling between a pair of parallel Hall droplets. PHYSICAL REVIEW. B, CONDENSED MATTER 1994; 50:7626-7634. [PMID: 9974746 DOI: 10.1103/physrevb.50.7626] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
|
26
|
Johansson P, Kinaret JM. Tunneling between two two-dimensional electron systems in a strong magnetic field. PHYSICAL REVIEW. B, CONDENSED MATTER 1994; 50:4671-4686. [PMID: 9976774 DOI: 10.1103/physrevb.50.4671] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
|
27
|
Aleiner IL, Shklovskii BI. Effect of screening of the Coulomb interaction on the conductivity in the quantum Hall regime. PHYSICAL REVIEW. B, CONDENSED MATTER 1994; 49:13721-13727. [PMID: 10010316 DOI: 10.1103/physrevb.49.13721] [Citation(s) in RCA: 30] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
|
28
|
Varma CM, Larkin AI, Abrahams E. Correlated state of double layers of electron fluids. PHYSICAL REVIEW. B, CONDENSED MATTER 1994; 49:13999-14002. [PMID: 10010352 DOI: 10.1103/physrevb.49.13999] [Citation(s) in RCA: 15] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
|
29
|
Johansson P, Kinaret JM. Magnetophonon shakeup in a Wigner crystal: Applications to tunneling spectroscopy in the quantum Hall regime. PHYSICAL REVIEW LETTERS 1993; 71:1435-1438. [PMID: 10055539 DOI: 10.1103/physrevlett.71.1435] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
|