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Scott JW, Nguyen MD, Park D, Halperin WP. Magnetic Susceptibility of Andreev Bound States in Superfluid ^{3}He-B. Phys Rev Lett 2023; 131:046001. [PMID: 37566829 DOI: 10.1103/physrevlett.131.046001] [Citation(s) in RCA: 2] [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: 02/03/2023] [Revised: 05/23/2023] [Accepted: 06/12/2023] [Indexed: 08/13/2023]
Abstract
Nuclear magnetic resonance measurements of the magnetic susceptibility of superfluid ^{3}He imbibed in anisotropic aerogel reveal anomalous behavior at low temperatures. Although the frequency shift clearly identifies a low-temperature phase as the B phase, the magnetic susceptibility does not display the expected decrease associated with the formation of the opposite-spin Cooper pairs. This susceptibility anomaly appears to be the predicted high-field behavior corresponding to the Ising-like magnetic character of surface Andreev bound states within the planar aerogel structures.
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Affiliation(s)
- J W Scott
- Northwestern University, Evanston, Illinois 60208, USA
| | - M D Nguyen
- Northwestern University, Evanston, Illinois 60208, USA
| | - D Park
- Northwestern University, Evanston, Illinois 60208, USA
| | - W P Halperin
- Northwestern University, Evanston, Illinois 60208, USA
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Heikkinen PJ, Casey A, Levitin LV, Rojas X, Vorontsov A, Sharma P, Zhelev N, Parpia JM, Saunders J. Fragility of surface states in topological superfluid 3He. Nat Commun 2021; 12:1574. [PMID: 33692358 DOI: 10.1038/s41467-021-21831-y] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/20/2020] [Accepted: 01/28/2021] [Indexed: 11/23/2022] Open
Abstract
Superfluid 3He, with unconventional spin-triplet p-wave pairing, provides a model system for topological superconductors, which have attracted significant interest through potential applications in topologically protected quantum computing. In topological insulators and quantum Hall systems, the surface/edge states, arising from bulk-surface correspondence and the momentum space topology of the band structure, are robust. Here we demonstrate that in topological superfluids and superconductors the surface Andreev bound states, which depend on the momentum space topology of the emergent order parameter, are fragile with respect to the details of surface scattering. We confine superfluid 3He within a cavity of height D comparable to the Cooper pair diameter ξ0. We precisely determine the superfluid transition temperature Tc and the suppression of the superfluid energy gap, for different scattering conditions tuned in situ, and compare to the predictions of quasiclassical theory. We discover that surface magnetic scattering leads to unexpectedly large suppression of Tc, corresponding to an increased density of low energy bound states. Surface states in topological quantum matter are protected by their band structure. Here, on confined superfluid 3He as a model for topological superconductors, the authors report experimental evidence on the fragility of surface Andreev bound states with respect to the details of surface scattering.
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Bunkov YM, Gazizulin RR. Direct observation of the specific heat of Majorana quasiparticles in superfluid 3He-B. Sci Rep 2020; 10:20120. [PMID: 33208835 DOI: 10.1038/s41598-020-77128-5] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/30/2020] [Accepted: 11/06/2020] [Indexed: 11/08/2022] Open
Abstract
The existence of Majorana quasiparticles was predicted in the edge state in topological insulators, especially in the p-wave superfluid medium [Formula: see text]He-B. Due to its purity and coherent quantum state, [Formula: see text]He-B is an ideal platform for searching for Majorana fermions in condensed matter systems. In the limit of extremely low temperatures, the density of Bogolyubov quasiparticles and the heat capacity of [Formula: see text]He-B decrease exponentially. In this article, we present the first observation of the deviation of its heat capacity from exponential dependence in the limit of record low cooling. We found an additional heat capacity that more than doubled the heat capacity of bulk [Formula: see text]He-B and changes as T[Formula: see text]. The additional heat capacity is in good agreement with the predicted heat capacity of 2D gas of Majorana. This observation is a direct proof of the existence of Majorana quasiparticles in [Formula: see text]He-B.
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Vorontsov AB. Andreev bound states in superconducting films and confined superfluid 3He. Philos Trans A Math Phys Eng Sci 2018; 376:rsta.2015.0144. [PMID: 29941623 PMCID: PMC6030148 DOI: 10.1098/rsta.2015.0144] [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] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 04/27/2018] [Indexed: 06/08/2023]
Abstract
This paper reviews confinement-driven phase transitions in superconductors and Bardeen-Cooper-Schrieffer superfluids, and the appearance in thin films of new phases that break the time-reversal or translational symmetry. The origins of the new phases are closely tied to the Andreev scattering processes involving particle-hole conversions that create surface quasiparticle states with energies inside the superconducting gap. Restructuring of the low-energy spectrum in the surface region of several coherence lengths ξ0 results in large spatial variations of the superconducting order parameter. In confined geometry, such as slabs, films, pores or nano-dots, with one or more physical dimensions D∼10ξ0, the Andreev bound states can dominate properties of a superconductor, leading to modified experimental signatures. They can significantly change the energy landscape, and drive transitions into new superconducting phases. The new phases are expected in a variety of materials, from singlet d-wave superconductors to multi-component triplet superfluid 3He, but properties of the new phases will depend on the symmetry of the parent state. I will highlight the connection between the Andreev surface states and confinement-stabilized phases with additional broken symmetries, describe recent progress and open questions in the theoretical and experimental investigation of superfluids in confined geometry.This article is part of the theme issue 'Andreev bound states'.
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Affiliation(s)
- Anton B Vorontsov
- Department of Physics, Montana State University, Bozeman, MT 59717, USA
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Mizushima T, Machida K. Multifaceted properties of Andreev bound states: interplay of symmetry and topology. Philos Trans A Math Phys Eng Sci 2018; 376:20150355. [PMID: 29941630 PMCID: PMC6030149 DOI: 10.1098/rsta.2015.0355] [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] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 05/11/2018] [Indexed: 06/08/2023]
Abstract
Andreev bound states (ABSs) ubiquitously emerge as a consequence of non-trivial topological structures of the order parameter of superfluids and superconductors and significantly contribute to thermodynamics and low-energy quantum transport phenomena. We here share the current status of our knowledge on their multifaceted properties such as Majorana fermions and odd-frequency pairing. A unified concept behind ABSs originates from a soliton state in the one-dimensional Dirac equation with mass domain wall and interplay of ABSs with symmetry and topology enrich their physical characteristics. We make an overview of ABSs with a special focus on superfluid 3He. The quantum liquid confined to restricted geometries serves as a rich repository of noteworthy quantum phenomena, such as the mass acquisition of Majorana fermions driven by spontaneous symmetry breaking, topological quantum criticality, Weyl superfluidity and the anomalous magnetic response. The marriage of the superfluid 3He and nano-fabrication techniques will take one to a new horizon of topological quantum phenomena associated with ABSs.This article is part of the theme issue 'Andreev bound states'.
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Affiliation(s)
- T Mizushima
- Department of Materials Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan
| | - K Machida
- Department of Physics, Ritsumeikan University, Kusatsu 525-8577, Japan
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Tsutsumi Y. Scattering Theory on Surface Majorana Fermions by an Impurity in ^{3}He-B. Phys Rev Lett 2017; 118:145301. [PMID: 28430515 DOI: 10.1103/physrevlett.118.145301] [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: 09/02/2016] [Indexed: 06/07/2023]
Abstract
We have formulated the scattering theory on Majorana fermions emerging in the surface bound state of the superfluid ^{3}He B phase (^{3}He-B) by an impurity. By applying the theory to the electron bubble, which is regarded as the impurity, trapped below a free surface of ^{3}He-B, the observed mobility of the electron bubble [J. Phys. Soc. Jpn. 82, 124607 (2013)JUPSAU0031-901510.7566/JPSJ.82.124607] is quantitatively reproduced. The mobility is suppressed in low temperatures from the expected value in the bulk ^{3}He-B by the contribution from the surface Majorana fermions. By contrast, the mobility does not depend on the trapped depth of the electron bubble in spite of the spatial variation of the wave function of the surface Majorana fermions. Our formulated theory demonstrates the depth-independent mobility by considering intermediate states in the scattering process. Therefore, we conclude that the experiment has succeeded in observing Majorana fermions in the surface bound state.
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Affiliation(s)
- Yasumasa Tsutsumi
- Department of Basic Science, University of Tokyo, Meguro, Tokyo 153-8902, Japan
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Zheng P, Jiang WG, Barquist CS, Lee Y, Chan HB. Critical Velocity in the Presence of Surface Bound States in Superfluid ^{3}He-B. Phys Rev Lett 2017; 118:065301. [PMID: 28234542 DOI: 10.1103/physrevlett.118.065301] [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: 10/06/2016] [Indexed: 06/06/2023]
Abstract
A microelectromechanical oscillator with a gap of 1.25 μm was immersed in superfluid ^{3}He-B and cooled below 250 μK at various pressures. Mechanical resonances of its shear motion were measured at various levels of driving force. The oscillator enters into a nonlinear regime above a certain threshold velocity. The damping increases rapidly in the nonlinear region and eventually prevents the velocity of the oscillator from increasing beyond the critical velocity which is much lower than the Landau critical velocity. We propose that this peculiar nonlinear behavior stems from the escape of quasiparticles from the surface bound states into the bulk fluid.
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Affiliation(s)
- P Zheng
- Department of Physics, University of Florida, Gainesville, Florida 32611-8440, USA
| | - W G Jiang
- Department of Physics, University of Florida, Gainesville, Florida 32611-8440, USA
| | - C S Barquist
- Department of Physics, University of Florida, Gainesville, Florida 32611-8440, USA
| | - Y Lee
- Department of Physics, University of Florida, Gainesville, Florida 32611-8440, USA
| | - H B Chan
- Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
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Zheng P, Jiang WG, Barquist CS, Lee Y, Chan HB. Anomalous Damping of a Microelectromechanical Oscillator in Superfluid ^{3}He-B. Phys Rev Lett 2016; 117:195301. [PMID: 27858447 DOI: 10.1103/physrevlett.117.195301] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/13/2016] [Indexed: 06/06/2023]
Abstract
The mechanical resonance properties of a microelectromechanical oscillator with a gap of 1.25 μm was studied in superfluid ^{3}He-B at various pressures. The oscillator was driven in the linear damping regime where the damping coefficient is independent of the oscillator velocity. The quality factor of the oscillator remains low (Q≈80) down to 0.1T_{c}, 4 orders of magnitude less than the intrinsic quality factor measured in vacuum at 4 K. In addition to the Boltzmann temperature dependent contribution to the damping, a damping proportional to temperature was found to dominate at low temperatures. We propose a multiple scattering mechanism of the surface Andreev bound states to be a possible cause for the anomalous damping.
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Affiliation(s)
- P Zheng
- Department of Physics, University of Florida, Gainesville, Florida 32611-8440, USA
| | - W G Jiang
- Department of Physics, University of Florida, Gainesville, Florida 32611-8440, USA
| | - C S Barquist
- Department of Physics, University of Florida, Gainesville, Florida 32611-8440, USA
| | - Y Lee
- Department of Physics, University of Florida, Gainesville, Florida 32611-8440, USA
| | - H B Chan
- Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
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Abstract
Owing to the richness of symmetry and well-established knowledge of bulk superfluidity, the superfluid (3)He has offered a prototypical system to study intertwining of topology and symmetry. This article reviews recent progress in understanding the topological superfluidity of (3)He in a multifaceted manner, including symmetry considerations, the Jackiw-Rebbi's index theorem, and the quasiclassical theory. Special focus is placed on the symmetry protected topological superfuidity of the (3)He-B confined in a slab geometry. The (3)He-B under a magnetic field is separated to two different sub-phases: the symmetry protected topological phase and non-topological phase. The former phase is characterized by the existence of symmetry protected Majorana fermions. The topological phase transition between them is triggered by the spontaneous breaking of a hidden discrete symmetry. The critical field is quantitatively determined from the microscopic calculation that takes account of magnetic dipole interaction of the (3)He nucleus. It is also demonstrated that odd-frequency even-parity Cooper pair amplitudes are emergent in low-lying quasiparticles. The key ingredients, symmetry protected Majorana fermions and odd-frequency pairing, bring an important consequence that the coupling of the surface states to an applied field is prohibited by the hidden discrete symmetry, while the topological phase transition with the spontaneous symmetry breaking is accompanied by anomalous enhancement and anisotropic quantum criticality of surface spin susceptibility. We also illustrate common topological features between topological crystalline superconductors and symmetry protected topological superfluids, taking UPt3 and Rashba superconductors as examples.
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Murakawa S, Yamaguchi A, Arai M, Wasai M, Aoki Y, Ishimoto H, Nomura R, Okuda Y, Nagato Y, Higashitani S, Nagai K. Spin-dependent acoustic response in the nonunitary A1 and A2 phases of superfluid 3He under high magnetic fields. Phys Rev Lett 2015; 114:105304. [PMID: 25815943 DOI: 10.1103/physrevlett.114.105304] [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: 10/14/2014] [Indexed: 06/04/2023]
Abstract
The transverse acoustic impedance of superfluid ^{3}He was measured in the A1 and A2 phases up to 13 T to investigate the surface states in nonunitary superfluids. The temperature dependence of the impedance was much larger in the A1 phase than in the A2 phase. This nonsymmetric behavior indicates that momentum exchange with walls for spin-down surface states is quite different from that for spin-up surface states. The spin-dependent response might be a reflection of an essential feature of the nonunitary states where gap amplitudes depend on spin states. Weak-coupling theories ignore any spin-dependent processes and do not account for the nonsymmetric behavior.
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Affiliation(s)
- S Murakawa
- Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan
| | - A Yamaguchi
- Institute for Solid State Physics, University of Tokyo, Chiba 277-8581, Japan
| | - M Arai
- Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan
| | - M Wasai
- Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan
| | - Y Aoki
- Institute for Solid State Physics, University of Tokyo, Chiba 277-8581, Japan
| | - H Ishimoto
- Institute for Solid State Physics, University of Tokyo, Chiba 277-8581, Japan
| | - R Nomura
- Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan
| | - Y Okuda
- Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan
| | - Y Nagato
- Graduate School of Integrated Arts and Sciences, Hiroshima University, Hiroshima 739-8521, Japan
| | - S Higashitani
- Graduate School of Integrated Arts and Sciences, Hiroshima University, Hiroshima 739-8521, Japan
| | - K Nagai
- Graduate School of Integrated Arts and Sciences, Hiroshima University, Hiroshima 739-8521, Japan
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Levitin LV, Bennett RG, Surovtsev EV, Parpia JM, Cowan B, Casey AJ, Saunders J. Surface-induced order parameter distortion in superfluid ³He-B measured by nonlinear NMR. Phys Rev Lett 2013; 111:235304. [PMID: 24476290 DOI: 10.1103/physrevlett.111.235304] [Citation(s) in RCA: 3] [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: 07/30/2013] [Indexed: 06/03/2023]
Abstract
The B phase of superfluid 3He is a three-dimensional time-reversal invariant topological superfluid, predicted to support gapless Majorana surface states. We confine superfluid 3He into a thin nanofluidic slab geometry. In the presence of a weak symmetry-breaking magnetic field, we have observed two possible states of the confined 3He-B phase manifold, through the small tipping angle NMR response. Large tipping angle nonlinear NMR has allowed the identification of the order parameter of these states and enabled a measurement of the surface-induced gap distortion. The results for two different quasiparticle surface scattering boundary conditions are compared with the predictions of weak-coupling quasiclassical theory. We identify a textural domain wall between the two B phase states, the edge of which at the cavity surface is predicted to host gapless states, protected in the magnetic field.
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Affiliation(s)
- Lev V Levitin
- Department of Physics, Royal Holloway University of London, Egham, TW20 0EX Surrey, United Kingdom
| | - Robert G Bennett
- Department of Physics, Royal Holloway University of London, Egham, TW20 0EX Surrey, United Kingdom and Department of Physics, Cornell University, Ithaca, New York 14853, USA
| | - Evgeny V Surovtsev
- Kapitza Institute for Physical Problems, ul. Kosygina 2, Moscow 119334, Russia
| | - Jeevak M Parpia
- Department of Physics, Cornell University, Ithaca, New York 14853, USA
| | - Brian Cowan
- Department of Physics, Royal Holloway University of London, Egham, TW20 0EX Surrey, United Kingdom
| | - Andrew J Casey
- Department of Physics, Royal Holloway University of London, Egham, TW20 0EX Surrey, United Kingdom
| | - John Saunders
- Department of Physics, Royal Holloway University of London, Egham, TW20 0EX Surrey, United Kingdom
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Abstract
Superfluid (3)He is an intensively investigated and well characterized p-wave superfluid. In the bulk Balian-Werthamer state, which is commonly called the (3)He B phase, the superfluid gap is opened isotropically but near a flat boundary such as a wall of a container it can harbor interesting quasi-particle states inside the gap. These states are called surface Andreev bound states, and have not been experimentally explored in detail. Transverse acoustic impedance measurement has revealed their existence and provided spectroscopic details of the dispersion of the bound states. Recent theoretical arguments claim that the surface Andreev bound states of the superfluid (3)He B phase can be recognized as the edge states of the topological superfluid and be regarded as a Majorana fermion, a fancy particle which has not been confirmed in elementary particle physics. In this review, we present up-to-date knowledge on the surface Andreev bound states of the (3)He B phase revealed by acoustic spectroscopy and the possible realization of a Majorana fermion, along with related studies on this topic.
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Affiliation(s)
- Y Okuda
- Department of Physics, Tokyo Institute of Technology, 2-12-1, O-okayama, Meguro-ku, Tokyo 152-8551, Japan.
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Murakawa S, Wasai M, Akiyama K, Wada Y, Tamura Y, Nomura R, Okuda Y. Strong suppression of the Kosterlitz-Thouless transition in a 4He film under high pressure. Phys Rev Lett 2012; 108:025302. [PMID: 22324695 DOI: 10.1103/physrevlett.108.025302] [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: 09/07/2011] [Indexed: 05/31/2023]
Abstract
We have found that the surface specularity for 3He quasiparticle scattering is closely related to the superfluidity and the Kosterlitz-Thouless (KT) transition of 4He film adsorbed on the surface. The specularity is determined by measurements of the transverse acoustic impedance of bulk liquid 3He. The unique point of our system is that we can control the correlation among 4He atoms in the film by changing the pressure of the bulk 3He. The observed KT transition temperature is significantly suppressed by increasing the pressure, which suggests a strong correlation effect on KT transition.
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Affiliation(s)
- S Murakawa
- Department of Physics, Tokyo Institute of Technology, Tokyo, Japan
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Chung SB, Zhang SC. Detecting the Majorana fermion surface state of 3He-B through spin relaxation. Phys Rev Lett 2009; 103:235301. [PMID: 20366156 DOI: 10.1103/physrevlett.103.235301] [Citation(s) in RCA: 5] [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: 08/14/2009] [Revised: 10/20/2009] [Indexed: 05/29/2023]
Abstract
The Majorana fermion, which can be useful for topological quantum computation, has eluded detection. (3)He-B, recently shown to be a time-reversal invariant topological superfluid, has a gapless Majorana fermion surface state. We show here that an electron spin relaxation experiment can detect this surface state--its Majorana nature through the Zeeman field direction dependence of the relaxation time 1/T(1) proportional, proportional to sin(2)theta, where theta is the angle between the field and the surface normal. We propose an experimental setup where an electron inside a nanobubble is injected below the (3)He liquid surface.
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Affiliation(s)
- Suk Bum Chung
- Department of Physics, McCullough Building, Stanford University, Stanford, California 94305, USA
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