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Kreisbeck C, Kramer T, Molina RA. Time-dependent wave packet simulations of transport through Aharanov-Bohm rings with an embedded quantum dot. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2017; 29:155301. [PMID: 28195564 DOI: 10.1088/1361-648x/aa605d] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
We have performed time-dependent wave packet simulations of realistic Aharonov-Bohm (AB) devices with a quantum dot embedded in one of the arms of the interferometer. The AB ring can function as a measurement device for the intrinsic transmission phase through the quantum dot, however, care has to be taken in analyzing the influence of scattering processes in the junctions of the interferometer arms. We consider a harmonic quantum dot and show how the Darwin-Fock spectrum emerges as a unique pattern in the interference fringes of the AB oscillations.
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Affiliation(s)
- C Kreisbeck
- Department of Chemistry and Chemical Biology, Harvard University, Cambridge, MA 02138, United States of America
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2
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Takada S, Bäuerle C, Yamamoto M, Watanabe K, Hermelin S, Meunier T, Alex A, Weichselbaum A, von Delft J, Ludwig A, Wieck AD, Tarucha S. Transmission phase in the Kondo regime revealed in a two-path interferometer. PHYSICAL REVIEW LETTERS 2014; 113:126601. [PMID: 25279636 DOI: 10.1103/physrevlett.113.126601] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/21/2013] [Indexed: 06/03/2023]
Abstract
We report on the direct observation of the transmission phase shift through a Kondo correlated quantum dot by employing a new type of two-path interferometer. We observed a clear π/2-phase shift, which persists up to the Kondo temperature TK. Above this temperature, the phase shifts by more than π/2 at each Coulomb peak, approaching the behavior observed for the standard Coulomb blockade regime. These observations are in remarkable agreement with two-level numerical renormalization group calculations. The unique combination of experimental and theoretical results presented here fully elucidates the phase evolution in the Kondo regime.
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Affiliation(s)
- S Takada
- Department of Applied Physics, University of Tokyo, Bunkyo-ku, Tokyo 113-8656, Japan
| | - C Bäuerle
- Université Grenoble Alpes, Institut NEEL, F-38042 Grenoble, France and CNRS, Institut NEEL, F-38042 Grenoble, France
| | - M Yamamoto
- Department of Applied Physics, University of Tokyo, Bunkyo-ku, Tokyo 113-8656, Japan and PRESTO, JST, Kawaguchi-shi, Saitama 331-0012, Japan
| | - K Watanabe
- Department of Applied Physics, University of Tokyo, Bunkyo-ku, Tokyo 113-8656, Japan
| | - S Hermelin
- Université Grenoble Alpes, Institut NEEL, F-38042 Grenoble, France and CNRS, Institut NEEL, F-38042 Grenoble, France
| | - T Meunier
- Université Grenoble Alpes, Institut NEEL, F-38042 Grenoble, France and CNRS, Institut NEEL, F-38042 Grenoble, France
| | - A Alex
- Physics Department, Arnold Sommerfeld Center for Theoretical Physics, and Center for NanoScience, Ludwig-Maximilians-Universität, Theresienstraße 37, 80333 München, Germany
| | - A Weichselbaum
- Physics Department, Arnold Sommerfeld Center for Theoretical Physics, and Center for NanoScience, Ludwig-Maximilians-Universität, Theresienstraße 37, 80333 München, Germany
| | - J von Delft
- Physics Department, Arnold Sommerfeld Center for Theoretical Physics, and Center for NanoScience, Ludwig-Maximilians-Universität, Theresienstraße 37, 80333 München, Germany
| | - A Ludwig
- Lehrstuhl für Angewandte Festkörperphysik, Ruhr-Universität Bochum, Universitätsstraße 150, 44780 Bochum, Germany
| | - A D Wieck
- Lehrstuhl für Angewandte Festkörperphysik, Ruhr-Universität Bochum, Universitätsstraße 150, 44780 Bochum, Germany
| | - S Tarucha
- Department of Applied Physics, University of Tokyo, Bunkyo-ku, Tokyo 113-8656, Japan and RIKEN Center for Emergent Matter Science (CEMS), 2-1 Hirosawa, Wako-shi, Saitama 31-0198, Japan
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3
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Yamauchi Y, Sekiguchi K, Chida K, Arakawa T, Nakamura S, Kobayashi K, Ono T, Fujii T, Sakano R. Evolution of the Kondo effect in a quantum dot probed by shot noise. PHYSICAL REVIEW LETTERS 2011; 106:176601. [PMID: 21635054 DOI: 10.1103/physrevlett.106.176601] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/11/2010] [Indexed: 05/30/2023]
Abstract
We measure the current and shot noise in a quantum dot in the Kondo regime to address the nonequilibrium properties of the Kondo effect. By systematically tuning the temperature and gate voltages to define the level positions in the quantum dot, we observe an enhancement of the shot noise as temperature decreases below the Kondo temperature, which indicates that the two-particle scattering process grows as the Kondo state evolves. Below the Kondo temperature, the Fano factor defined at finite temperature is found to exceed the expected value of unity from the noninteracting model, reaching 1.8±0.2.
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Affiliation(s)
- Yoshiaki Yamauchi
- Institute for Chemical Research, Kyoto University, Uji, Kyoto 611-0011, Japan
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4
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Puller VI, Meir Y. How to measure the transmission phase through a quantum dot in a two-terminal interferometer. PHYSICAL REVIEW LETTERS 2010; 104:256801. [PMID: 20867406 DOI: 10.1103/physrevlett.104.256801] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/17/2010] [Indexed: 05/29/2023]
Abstract
Measurement of the transmission phase through a quantum dot (QD) embedded in an arm of a two-terminal Aharonov-Bohm (AB) interferometer is inhibited by phase symmetry, i.e., the property that the linear response conductance of a two-terminal device is an even function of the magnetic field. It is demonstrated that in a setup consisting of an interferometer with a QD in each of its arms, with one of the QDs capacitively coupled to a nearby quantum point contact (QPC), phase symmetry is broken when a finite voltage bias is applied to the QPC. The transmission phase via the uncoupled QD can then be deduced from the amplitude of the odd component of the AB oscillations.
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Affiliation(s)
- Vadim I Puller
- Department of Physics, Ben-Gurion University of the Negev, Beer Sheva 84105, Israel
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5
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Dias da Silva LGGV, Sandler N, Simon P, Ingersent K, Ulloa SE. Tunable pseudogap Kondo effect and quantum phase transitions in Aharonov-Bohm interferometers. PHYSICAL REVIEW LETTERS 2009; 102:166806. [PMID: 19518741 DOI: 10.1103/physrevlett.102.166806] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/11/2009] [Indexed: 05/27/2023]
Abstract
We study two quantum dots embedded in the arms of an Aharonov-Bohm ring threaded by a magnetic flux. This system can be described by an effective one-impurity Anderson model with an energy- and flux-dependent density of states. For specific values of the flux, this density of states vanishes at the Fermi energy, yielding a controlled realization of the pseudogap Kondo effect. The conductance and transmission phase shifts reflect a nontrivial interplay between wave interference and interactions, providing clear signatures of quantum phase transitions between Kondo and non-Kondo ground states.
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Affiliation(s)
- Luis G G V Dias da Silva
- Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA.
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6
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Hedin ER, Joe YS, Satanin AM. Resonance and phase shift in an open Aharonov-Bohm ring with an embedded quantum dot. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2009; 21:015303. [PMID: 21817218 DOI: 10.1088/0953-8984/21/1/015303] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/31/2023]
Abstract
The transmission and phase properties of electron transport through a quantum dot (QD) with variable coupling to a third-terminal probe are investigated analytically for the case of the QD connected directly to source and drain reservoirs and when the QD is embedded in one arm of an Aharonov-Bohm (AB) ring. Using the tight-binding model, explicit analytical expressions of the transmission through the QD for each case are given. Expressions for the conductance with coupling to the third terminal, which breaks unitarity and phase-locking, are also given. It is shown that in a three-terminal interferometer the zero of the Fano resonance in the transmission moves off the real energy axis for finite values of the coupling parameter. The zero orbits around the pole in the complex energy plane as a function of magnetic flux through the ring, and can be returned to the real energy axis unless the coupling parameter exceeds a critical value. With the QD embedded in one arm of the AB ring, the electron transmission and the transmission phase, and the phase of the AB oscillations, are described in relation to the degree of coupling to the third-terminal probe which opens the interferometer. By tuning the degree of coupling to the probe, it is shown that the phase of the AB oscillations can be made to match the intrinsic phase of the QD, facilitating experimental characterization of the phase response of the QD.
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Affiliation(s)
- Eric R Hedin
- Center for Computational Nanoscience, Department of Physics and Astronomy, Ball State University, Muncie, IN 47306, USA
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7
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Li JL, Li YX. Andreev reflection and Aharonov-Bohm oscillations through a parallel-coupled double quantum dot with spin-flip scattering. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2008; 20:465202. [PMID: 21693842 DOI: 10.1088/0953-8984/20/46/465202] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/30/2023]
Abstract
Using nonequilibrium Green's function techniques, we investigate Andreev reflection and Aharonov-Bohm oscillations through a parallel-coupled double quantum dot connected with a ferromagnetic lead and a superconductor lead. The possibility of controlling Andreev reflection and Aharonov-Bohm oscillations of the system is explored by tuning the interdot coupling, the gate voltage, the magnetic flux, and the intradot spin-flip scattering. When the spin-flip scattering increases, Fano resonant peaks resulting from the asymmetrical levels of the two quantum dots begin to split, and Aharonov-Bohm oscillations are suppressed. Due to the interdot coupling, one strongly and one weakly coupled state of the system can be formed. The magnetic flux can exchange the function of the two states, which leads to a swap effect.
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Affiliation(s)
- Jin-Liang Li
- College of Physics, Hebei Normal University, Shijiazhuang 050016, People's Republic of China. Hebei Advanced Film Laboratory, Shijiazhuang 050016, People's Republic of China
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8
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Zaffalon M, Bid A, Heiblum M, Mahalu D, Umansky V. Transmission phase of a singly occupied quantum dot in the Kondo regime. PHYSICAL REVIEW LETTERS 2008; 100:226601. [PMID: 18643437 DOI: 10.1103/physrevlett.100.226601] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/07/2007] [Indexed: 05/26/2023]
Abstract
We report on the phase measurements on a quantum dot containing a single electron in the Kondo regime. Transport takes place through a single orbital state. Although the conductance is far from the unitary limit, we measure directly, for the first time, a transmission phase as theoretically predicted of pi/2. As the dot's coupling to the leads is decreased, with the dot entering the Coulomb blockade regime, the phase reaches a value of pi. Temperature shows little effect on the phase behavior in the range 30-600 mK, even though both the two-terminal conductance and amplitude of the Aharonov-Bohm oscillations are strongly affected. These results also suggest that previous phase measurements involved transport through more than a single level.
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Affiliation(s)
- M Zaffalon
- Braun Center for Submicron Research, Department of Condensed Matter Physics, Weizmann Institute of Science, 76100 Rehovot, Israel
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9
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Transport Properties of an Aharonov-Bohm Interferometer with an In-line Quantum Dot. E-JOURNAL OF SURFACE SCIENCE AND NANOTECHNOLOGY 2007. [DOI: 10.1380/ejssnt.2007.29] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/20/2022]
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10
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Simon P, Feinberg D. Electronic spin precession and interferometry from spin-orbital entanglement in a double quantum dot. PHYSICAL REVIEW LETTERS 2006; 97:247207. [PMID: 17280318 DOI: 10.1103/physrevlett.97.247207] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/17/2006] [Indexed: 05/13/2023]
Abstract
A double quantum dot inserted in parallel between two metallic leads can entangle the electron spin with the orbital (dot index) degree of freedom. An Aharonov-Bohm orbital phase can be transferred to the spinor wave function, providing a geometrical control of the spin precession around a fixed magnetic field. A fully coherent behavior occurs in a mixed orbital-spin Kondo regime. Evidence for the spin precession can be obtained, either using spin-polarized metallic leads or by placing the double dot in one branch of a metallic loop.
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Affiliation(s)
- P Simon
- Laboratoire de Physique et Modélisation des Milieux Condensés, CNRS and Université Joseph Fourier, BP 166, 38042 Grenoble, France
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11
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Aharony A, Entin-Wohlman O, Oreg Y, van Delft J. Comment on "Theoretical analysis of the transmission phase shift of a quantum dot in the presence of Kondo correlations". PHYSICAL REVIEW LETTERS 2006; 96:189705; author reply 189706. [PMID: 16712412 DOI: 10.1103/physrevlett.96.189705] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/10/2005] [Indexed: 05/09/2023]
Affiliation(s)
- A Aharony
- Department of Physics Ben Gurion University, Beer Sheva 84105, Israel
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12
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Jerez A, Vitushinsky P, Lavagna M. Theoretical analysis of the transmission phase shift of a quantum dot in the presence of Kondo correlations. PHYSICAL REVIEW LETTERS 2005; 95:127203. [PMID: 16197103 DOI: 10.1103/physrevlett.95.127203] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/10/2005] [Indexed: 05/04/2023]
Abstract
We study the effects of Kondo correlations on the transmission phase shift of a quantum dot coupled to two leads in comparison with the experimental determinations made by Aharonov-Bohm (AB) quantum interferometry. We propose here a theoretical interpretation of these results based on scattering theory combined with Bethe ansatz calculations. We show that there is a factor of 2 difference between the phase of the S-matrix responsible for the shift in the AB oscillations and the one controlling the conductance. Quantitative agreement is obtained with experimental results for two different values of the coupling to the leads.
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Affiliation(s)
- A Jerez
- European Synchrotron Radiation Facility, 6, rue Jules Horowitz, 38043 Grenoble Cedex 9, France
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13
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Sato M, Aikawa H, Kobayashi K, Katsumoto S, Iye Y. Observation of the Fano-Kondo antiresonance in a quantum wire with a side-coupled quantum dot. PHYSICAL REVIEW LETTERS 2005; 95:066801. [PMID: 16090971 DOI: 10.1103/physrevlett.95.066801] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/04/2004] [Indexed: 05/03/2023]
Abstract
We have observed the Fano-Kondo antiresonance in a quantum wire with a side-coupled quantum dot. In a weak coupling regime, dips due to the Fano effect appeared. As the coupling strength increased, conductance in the regions between the dips decreased alternately. From the temperature dependence and the response to the magnetic field, we conclude that the conductance reduction is due to the Fano-Kondo antiresonance. At a Kondo valley with the Fano parameter q approximately 0, the phase shift is locked to pi/2 against the gate voltage when the system is close to the unitary limit in agreement with theoretical predictions by Gerland et al. [Phys. Rev. Lett. 84, 3710 (2000)].
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Affiliation(s)
- Masahiro Sato
- Institute for Solid State Physics, University of Tokyo, 5-1-5 Kashiwanoha, Chiba 277-8581, Japan
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14
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Glazman LI, Pustilnik M. Course 7 Low-temperature transport through a quantum dot. NANOPHYSICS: COHERENCE AND TRANSPORT, ÉCOLE D'ÉTÉ DE PHYSIQUE DES HOUCHES SESSION LXXXI 2005. [DOI: 10.1016/s0924-8099(05)80050-2] [Citation(s) in RCA: 15] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/12/2022]
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15
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Fuhrer A, Ihn T, Ensslin K, Wegscheider W, Bichler M. Kondo effect in a many-electron quantum ring. PHYSICAL REVIEW LETTERS 2004; 93:176803. [PMID: 15525104 DOI: 10.1103/physrevlett.93.176803] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/10/2004] [Indexed: 05/24/2023]
Abstract
The Kondo effect is investigated in a many-electron quantum ring as a function of the magnetic field. For fields applied perpendicular to the plane of the ring a modulation of the Kondo effect with the Aharonov-Bohm period is observed. This effect is discussed in terms of the energy spectrum of the ring and the parametrically changing tunnel coupling. In addition, we use gate voltages to modify the ground-state spin of the ring. The observed splitting of the Kondo-related zero-bias anomaly in this configuration is tuned with an in-plane magnetic field.
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Affiliation(s)
- A Fuhrer
- Solid State Physics Laboratory, ETH Zürich, 8093 Zürich, Switzerland
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16
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Sigrist M, Fuhrer A, Ihn T, Ensslin K, Ulloa SE, Wegscheider W, Bichler M. Magnetic-field-dependent transmission phase of a double-dot system in a quantum ring. PHYSICAL REVIEW LETTERS 2004; 93:066802. [PMID: 15323649 DOI: 10.1103/physrevlett.93.066802] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/08/2003] [Indexed: 05/24/2023]
Abstract
The Aharonov-Bohm effect is measured in a four-terminal open ring geometry. Two quantum dots are embedded in the structure, one in each of the two interfering paths. The number of electrons in the two dots can be controlled independently. The transmission phase is measured as electrons are added to or taken away from the individual quantum dots. Although the measured phase shifts are in qualitative agreement with theoretical predictions, the phase evolution exhibits unexpected dependence on the magnetic field. Phase lapses are found only in certain ranges of the magnetic field.
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Affiliation(s)
- M Sigrist
- Solid State Physics, ETH Zürich, 8093, Switzerland
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Aikawa H, Kobayashi K, Sano A, Katsumoto S, Iye Y. Observation of "partial coherence" in an Aharonov-Bohm interferometer with a quantum dot. PHYSICAL REVIEW LETTERS 2004; 92:176802. [PMID: 15169178 DOI: 10.1103/physrevlett.92.176802] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/11/2003] [Indexed: 05/24/2023]
Abstract
We report experiments on the interference through spin states of electrons in a quantum dot (QD) embedded in an Aharonov-Bohm (AB) interferometer. We have picked up a spin-pair state, for which the environmental conditions are ideally similar. The AB amplitude is traced in a range of gate voltage that covers the pair. The behavior of the asymmetry in the amplitude around the two Coulomb peaks agrees with the theoretical prediction that the spin-flip process in a QD is related to the quantum dephasing of electrons. These results constitute evidence of "partial coherence" due to an entanglement of spins in the QD and in the interferometer.
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Affiliation(s)
- Hisashi Aikawa
- Institute for Solid State Physics, University of Tokyo, 5-1-5 Kashiwanoha, Chiba 277-8581, Japan
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18
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Keyser UF, Fühner C, Borck S, Haug RJ, Bichler M, Abstreiter G, Wegscheider W. Kondo effect in a few-electron quantum ring. PHYSICAL REVIEW LETTERS 2003; 90:196601. [PMID: 12785965 DOI: 10.1103/physrevlett.90.196601] [Citation(s) in RCA: 21] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/12/2002] [Indexed: 05/24/2023]
Abstract
A small quantum ring with less than ten electrons was studied by transport spectroscopy. For strong coupling to the leads a Kondo effect is observed and used to characterize the spin structure of the system in a wide range of magnetic fields. At small magnetic fields Aharonov-Bohm oscillations influenced by Coulomb interaction appear. They exhibit phase jumps by pi at the Coulomb-blockade resonances. Inside Coulomb-blockade valleys the Aharonov-Bohm oscillations can also be studied due to the finite conductance caused by the Kondo effect. Astonishingly, the maxima of the oscillations show linear shifts with increasing magnetic field and gate voltage.
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Affiliation(s)
- U F Keyser
- Institut für Festkörperphysik, Universität Hannover, Germany.
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19
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Silvestrov PG, Imry Y. Enhanced sensitivity of the transmission phase of a quantum dot to Kondo correlations. PHYSICAL REVIEW LETTERS 2003; 90:106602. [PMID: 12689020 DOI: 10.1103/physrevlett.90.106602] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/17/2001] [Indexed: 05/24/2023]
Abstract
The strong sensitivity of the transmission phase through a quantum dot embedded into one arm of a two-wave Aharonov-Bohm interferometer to the Kondo effect is explained. The enhancement takes place because of the buildup of the exchange scattering on the dot due to Kondo correlations even much above T(K). The enhanced exchange competes with the potential scattering, which is always weak. Both cases of the Anderson impurity model and a multilevel quantum dot are considered. In the latter case in addition to the description of peculiar phase behavior a mechanism leading to ferromagnetic Kondo coupling in quantum dots is proposed.
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Affiliation(s)
- P G Silvestrov
- Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands
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Entin-Wohlman O, Aharony A, Imry Y, Levinson Y, Schiller A. Broken unitarity and phase measurements in Aharonov-Bohm interferometers. PHYSICAL REVIEW LETTERS 2002; 88:166801. [PMID: 11955245 DOI: 10.1103/physrevlett.88.166801] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/03/2001] [Indexed: 05/23/2023]
Abstract
Aharonov-Bohm mesoscopic solid-state interferometers yield a conductance which contains a term cos(phi+beta), where phi relates to the magnetic flux. Experiments with a quantum dot on one of the interfering paths aim to relate beta to the dot's intrinsic Friedel transmission phase alpha(1). For closed systems, which conserve the electron current (unitarity), the Onsager relation requires that beta = 0 or pi. For open systems, we show that in general beta depends on the details of the broken unitarity. Although it gives information on the resonances of the dot, beta is generally not equal to alpha(1). A direct relation between beta and alpha(1) requires specific ways of opening the system, which are discussed.
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Affiliation(s)
- O Entin-Wohlman
- School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel
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