1
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Yazdani A, von Oppen F, Halperin BI, Yacoby A. Hunting for Majoranas. Science 2023; 380:eade0850. [PMID: 37347870 DOI: 10.1126/science.ade0850] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/25/2022] [Accepted: 05/22/2023] [Indexed: 06/24/2023]
Abstract
Over the past decade, there have been considerable efforts to observe non-abelian quasiparticles in novel quantum materials and devices. These efforts are motivated by the goals of demonstrating quantum statistics of quasiparticles beyond those of fermions and bosons and of establishing the underlying science for the creation of topologically protected quantum bits. In this Review, we focus on efforts to create topological superconducting phases that host Majorana zero modes. We consider the lessons learned from existing experimental efforts, which are motivating both improvements to present platforms and the exploration of new approaches. Although the experimental detection of non-abelian quasiparticles remains challenging, the knowledge gained thus far and the opportunities ahead offer high potential for discovery and advances in this exciting area of quantum physics.
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Affiliation(s)
- Ali Yazdani
- Joseph Henry Laboratories and Department of Physics, Princeton University, Princeton, NJ 08540, USA
| | - Felix von Oppen
- Dahlem Center for Complex Quantum Systems and Fachbereich Physik, Freie Universität Berlin, 14195 Berlin, Germany
| | | | - Amir Yacoby
- Department of Physics, Harvard University, Cambridge, MA 02138, USA
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2
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Chiu CK, Wang Z. Yu-Shiba-Rusinov States in a Superconductor with Topological Z_{2} Bands. PHYSICAL REVIEW LETTERS 2022; 128:237001. [PMID: 35749202 DOI: 10.1103/physrevlett.128.237001] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/23/2021] [Revised: 04/19/2022] [Accepted: 04/26/2022] [Indexed: 06/15/2023]
Abstract
A Yu-Shiba-Rusinov (YSR) state is a localized in-gap state induced by a magnetic impurity in a superconductor. Recent experiments used an STM tip to manipulate the exchange coupling between an Fe adatom and the FeTe_{0.55}Se_{0.45} superconductor possessing a Z_{2} nontrivial band structure with topological surface states. As the tip moves close to the single Fe adatom, the energy of the in-gap state modulates and exhibits a zero-energy crossing followed by an unusual return to zero energy, which cannot be understood by coupling the magnetic impurity to the superconducting topological surface Dirac cone. Here, we numerically and analytically study the YSR states in superconductors with nontrivial Z_{2} bands and show the emergence of the two zero-energy crossings as a function of the exchange coupling between the magnetic impurity and the bulk states. We analyze the role of the topological surface states and compare in-gap states to systems with trivial Z_{2} bands. The spin polarization of the YSR states is further studied for future experimental measurement.
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Affiliation(s)
- Ching-Kai Chiu
- RIKEN Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS), Wako, Saitama 351-0198, Japan
- Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China
| | - Ziqiang Wang
- Department of Physics, Boston College, Chestnut Hill, Massachusetts 02467, USA
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3
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Marra P, Nigro A. Majorana/Andreev crossover and the fate of the topological phase transition in inhomogeneous nanowires. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2022; 34:124001. [PMID: 34929683 DOI: 10.1088/1361-648x/ac44d2] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/31/2021] [Accepted: 12/20/2021] [Indexed: 06/14/2023]
Abstract
Majorana bound states (MBS) and Andreev bound states (ABS) in realistic Majorana nanowires setups have similar experimental signatures which make them hard to distinguishing one from the other. Here, we characterize the continuous Majorana/Andreev crossover interpolating between fully-separated, partially-separated, and fully-overlapping Majorana modes, in terms of global and local topological invariants, fermion parity, quasiparticle densities, Majorana pseudospin and spin polarizations, density overlaps and transition probabilities between opposite Majorana components. We found that in inhomogeneous wires, the transition between fully-overlapping trivial ABS and nontrivial MBS does not necessarily mandate the closing of the bulk gap of quasiparticle excitations, but a simple parity crossing of partially-separated Majorana modes (ps-MM) from trivial to nontrivial regimes. We demonstrate that fully-separated and fully-overlapping Majorana modes correspond to the two limiting cases at the opposite sides of a continuous crossover: the only distinction between the two can be obtained by estimating the degree of separations of the Majorana components. This result does not contradict the bulk-edge correspondence: indeed, the field inhomogeneities driving the Majorana/Andreev crossover have a length scale comparable with the nanowire length, and therefore correspond to a nonlocal perturbation which breaks the topological protection of the MBS.
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Affiliation(s)
- Pasquale Marra
- Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8914, Japan
- Department of Physics, and Research and Education Center for Natural Sciences, Keio University, 4-1-1 Hiyoshi, Yokohama, Kanagawa 223-8521, Japan
| | - Angela Nigro
- Dipartimento di Fisica 'E. R. Caianiello', Università degli Studi di Salerno, 84084 Fisciano (Salerno), Italy
- Consiglio Nazionale delle Ricerche CNR-SPIN, UOS Salerno, 84084 Fisciano (Salerno), Italy
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4
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Sedlmayr N, Bena C. Instability of Majorana states in Shiba chains due to leakage into a topological substrate. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2021; 34:104004. [PMID: 34879363 DOI: 10.1088/1361-648x/ac413f] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/03/2021] [Accepted: 12/08/2021] [Indexed: 06/13/2023]
Abstract
We revisit the problem of Majorana states in chains of scalar impurities deposited on a superconductor with a mixed s-wave and p-wave pairing. We also study the formation of Majorana states for magnetic impurity chains. We find that the magnetic impurity chains exhibit well-localized Majorana states when the substrate is trivial, but these states hybridize and get dissolved in the bulk when the substrate is topological. Most surprisingly, and contrary to previous predictions, the scalar impurity chain does not support fully localized Majorana states except for very small and finely tuned parameter regimes, mostly for a non-topological substrate close to the topological transition. Our results indicate that a purely p-wave or a dominant p-wave substrate are not good candidates to support either magnetic or scalar impurity topological Shiba chains.
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Affiliation(s)
- Nicholas Sedlmayr
- Institute of Physics, Maria Curie-Skłodowska University, Plac Marii Skłodowskiej-Curie 1, PL-20031 Lublin, Poland
| | - Cristina Bena
- Université Paris Saclay, CNRS, CEA, Institut de Physique Théorique, 91191, Gif-sur-Yvette, France
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5
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Kamlapure A, Cornils L, Žitko R, Valentyuk M, Mozara R, Pradhan S, Fransson J, Lichtenstein AI, Wiebe J, Wiesendanger R. Correlation of Yu-Shiba-Rusinov States and Kondo Resonances in Artificial Spin Arrays on an s-Wave Superconductor. NANO LETTERS 2021; 21:6748-6755. [PMID: 34351781 PMCID: PMC8392378 DOI: 10.1021/acs.nanolett.1c00387] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 01/28/2021] [Revised: 07/16/2021] [Indexed: 06/13/2023]
Abstract
Mutually interacting magnetic atoms coupled to a superconductor have gained enormous interest due to their potential for the realization of topological superconductivity. Individual magnetic impurities produce states within the superconducting energy gap known as Yu-Shiba-Rusinov (YSR) states. Here, using the tip of a scanning tunneling microscope, we artificially craft spin arrays consisting of an Fe adatom interacting with an assembly of interstitial Fe atoms (IFA) on a superconducting oxygen-reconstructed Ta(100) surface and show that the magnetic interaction between the adatom and the IFA assembly can be tuned by adjusting the number of IFAs in the assembly. The YSR state experiences a characteristic crossover in its energetic position and particle-hole spectral weight asymmetry when the Kondo resonance shows spectral depletion around the Fermi energy. By the help of slave-boson mean-field theory (SBMFT) and numerical renormalization group (NRG) calculations we associate the crossover with the transition from decoupled Kondo singlets to an antiferromagnetic ground state of the Fe adatom spin and the IFA assembly effective spin.
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Affiliation(s)
- Anand Kamlapure
- Department
of Physics, University of Hamburg, Jungiusstrasse 11, D-20355 Hamburg, Germany
| | - Lasse Cornils
- Department
of Physics, University of Hamburg, Jungiusstrasse 11, D-20355 Hamburg, Germany
| | - Rok Žitko
- Jožef
Stefan Institute, Jamova 39, SI-1000 Ljubljana, Slovenia
- Faculty
of Mathematics and Physics, University of
Ljubljana, Jadranska
19, SI-1000 Ljubljana, Slovenia
| | - Maria Valentyuk
- Department
of Physics, University of Hamburg, Jungiusstrasse 11, D-20355 Hamburg, Germany
- Department
of Theoretical Physics and Applied Mathematics, Ural Federal University, 19 Mira Street, Yekaterinburg 620002, Russia
| | - Roberto Mozara
- Department
of Physics, University of Hamburg, Jungiusstrasse 11, D-20355 Hamburg, Germany
| | - Saurabh Pradhan
- Department
of Physics and Astronomy, Uppsala University, P.O. Box 516, Uppsala SE-751 21, Sweden
| | - Jonas Fransson
- Department
of Physics and Astronomy, Uppsala University, P.O. Box 516, Uppsala SE-751 21, Sweden
| | - Alexander I. Lichtenstein
- Department
of Physics, University of Hamburg, Jungiusstrasse 11, D-20355 Hamburg, Germany
- Department
of Theoretical Physics and Applied Mathematics, Ural Federal University, 19 Mira Street, Yekaterinburg 620002, Russia
| | - Jens Wiebe
- Department
of Physics, University of Hamburg, Jungiusstrasse 11, D-20355 Hamburg, Germany
| | - Roland Wiesendanger
- Department
of Physics, University of Hamburg, Jungiusstrasse 11, D-20355 Hamburg, Germany
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6
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Lászlóffy A, Palotás K, Rózsa L, Szunyogh L. Electronic and Magnetic Properties of Building Blocks of Mn and Fe Atomic Chains on Nb(110). NANOMATERIALS 2021; 11:nano11081933. [PMID: 34443761 PMCID: PMC8401957 DOI: 10.3390/nano11081933] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 06/17/2021] [Revised: 07/15/2021] [Accepted: 07/16/2021] [Indexed: 11/16/2022]
Abstract
We present results for the electronic and magnetic structure of Mn and Fe clusters on Nb(110) surface, focusing on building blocks of atomic chains as possible realizations of topological superconductivity. The magnetic ground states of the atomic dimers and most of the monatomic chains are determined by the nearest-neighbor isotropic interaction. To gain physical insight, the dependence on the crystallographic direction as well as on the atomic coordination number is analyzed via an orbital decomposition of this isotropic interaction based on the spin-cluster expansion and the difference in the local density of states between ferromagnetic and antiferromagnetic configurations. A spin-spiral ground state is obtained for Fe chains along the [11¯0] direction as a consequence of the frustration of the isotropic interactions. Here, a flat spin-spiral dispersion relation is identified, which can stabilize spin spirals with various wave vectors together with the magnetic anisotropy. This may lead to the observation of spin spirals of different wave vectors and chiralities in longer chains instead of a unique ground state.
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Affiliation(s)
- András Lászlóffy
- Wigner Research Centre for Physics, Institute for Solid State Physics and Optics, H-1525 Budapest, Hungary;
- Department of Theoretical Physics, Budapest University of Technology and Economics, H-1111 Budapest, Hungary;
| | - Krisztián Palotás
- Wigner Research Centre for Physics, Institute for Solid State Physics and Optics, H-1525 Budapest, Hungary;
- Department of Theoretical Physics, Budapest University of Technology and Economics, H-1111 Budapest, Hungary;
- MTA-SZTE Reaction Kinetics and Surface Chemistry Research Group, University of Szeged, H-6720 Szeged, Hungary
- Correspondence:
| | - Levente Rózsa
- Department of Physics, University of Konstanz, D-78457 Konstanz, Germany;
| | - László Szunyogh
- Department of Theoretical Physics, Budapest University of Technology and Economics, H-1111 Budapest, Hungary;
- MTA-BME Condensed Matter Research Group, Budapest University of Technology and Economics, H-1111 Budapest, Hungary
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7
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Interaction-induced topological phase transition and Majorana edge states in low-dimensional orbital-selective Mott insulators. Nat Commun 2021; 12:2955. [PMID: 34011947 PMCID: PMC8134496 DOI: 10.1038/s41467-021-23261-2] [Citation(s) in RCA: 10] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/27/2020] [Accepted: 04/21/2021] [Indexed: 11/08/2022] Open
Abstract
Topological phases of matter are among the most intriguing research directions in Condensed Matter Physics. It is known that superconductivity induced on a topological insulator's surface can lead to exotic Majorana modes, the main ingredient of many proposed quantum computation schemes. In this context, the iron-based high critical temperature superconductors are a promising platform to host such an exotic phenomenon in real condensed-matter compounds. The Coulomb interaction is commonly believed to be vital for the magnetic and superconducting properties of these systems. This work bridges these two perspectives and shows that the Coulomb interaction can also drive a canonical superconductor with orbital degrees of freedom into the topological state. Namely, we show that above a critical value of the Hubbard interaction the system simultaneously develops spiral spin order, a highly unusual triplet amplitude in superconductivity, and, remarkably, Majorana fermions at the edges of the system.
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8
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Ding H, Hu Y, Randeria MT, Hoffman S, Deb O, Klinovaja J, Loss D, Yazdani A. Tuning interactions between spins in a superconductor. Proc Natl Acad Sci U S A 2021; 118:e2024837118. [PMID: 33782131 PMCID: PMC8040815 DOI: 10.1073/pnas.2024837118] [Citation(s) in RCA: 7] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022] Open
Abstract
Novel many-body and topological electronic phases can be created in assemblies of interacting spins coupled to a superconductor, such as one-dimensional topological superconductors with Majorana zero modes (MZMs) at their ends. Understanding and controlling interactions between spins and the emergent band structure of the in-gap Yu-Shiba-Rusinov (YSR) states they induce in a superconductor are fundamental for engineering such phases. Here, by precisely positioning magnetic adatoms with a scanning tunneling microscope (STM), we demonstrate both the tunability of exchange interaction between spins and precise control of the hybridization of YSR states they induce on the surface of a bismuth (Bi) thin film that is made superconducting with the proximity effect. In this platform, depending on the separation of spins, the interplay among Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction, spin-orbit coupling, and surface magnetic anisotropy stabilizes different types of spin alignments. Using high-resolution STM spectroscopy at millikelvin temperatures, we probe these spin alignments through monitoring the spin-induced YSR states and their energy splitting. Such measurements also reveal a quantum phase transition between the ground states with different electron number parity for a pair of spins in a superconductor tuned by their separation. Experiments on larger assemblies show that spin-spin interactions can be mediated in a superconductor over long distances. Our results show that controlling hybridization of the YSR states in this platform provides the possibility of engineering the band structure of such states for creating topological phases.
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Affiliation(s)
- Hao Ding
- Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544
- Department of Physics, Princeton University, Princeton, NJ 08544
| | - Yuwen Hu
- Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544
- Department of Physics, Princeton University, Princeton, NJ 08544
| | - Mallika T Randeria
- Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544
- Department of Physics, Princeton University, Princeton, NJ 08544
| | - Silas Hoffman
- Department of Physics, University of Basel, CH-4056 Basel, Switzerland
- Department of Physics, University of Florida, Gainesville, FL 32611
| | - Oindrila Deb
- Department of Physics, University of Basel, CH-4056 Basel, Switzerland
| | - Jelena Klinovaja
- Department of Physics, University of Basel, CH-4056 Basel, Switzerland
| | - Daniel Loss
- Department of Physics, University of Basel, CH-4056 Basel, Switzerland
| | - Ali Yazdani
- Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544;
- Department of Physics, Princeton University, Princeton, NJ 08544
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9
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Schneider L, Brinker S, Steinbrecher M, Hermenau J, Posske T, Dos Santos Dias M, Lounis S, Wiesendanger R, Wiebe J. Controlling in-gap end states by linking nonmagnetic atoms and artificially-constructed spin chains on superconductors. Nat Commun 2020; 11:4707. [PMID: 32948776 PMCID: PMC7501864 DOI: 10.1038/s41467-020-18540-3] [Citation(s) in RCA: 11] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/10/2020] [Accepted: 08/25/2020] [Indexed: 11/09/2022] Open
Abstract
Chains of magnetic atoms with either strong spin-orbit coupling or spiral magnetic order which are proximity-coupled to superconducting substrates can host topologically non-trivial Majorana bound states. The experimental signature of these states consists of spectral weight at the Fermi energy which is spatially localized near the ends of the chain. However, topologically trivial Yu-Shiba-Rusinov in-gap states localized near the ends of the chain can lead to similar spectra. Here, we explore a protocol to disentangle these contributions by artificially augmenting a candidate Majorana spin chain with orbitally-compatible nonmagnetic atoms. Combining scanning tunneling spectroscopy with ab-initio and tight-binding calculations, we realize a sharp spatial transition between the proximity-coupled spiral magnetic order and the non-magnetic superconducting wire termination, with persistent zero-energy spectral weight localized at either end of the magnetic spiral. Our findings open a new path towards the control of the spatial position of in-gap end states, trivial or Majorana, via different chain terminations, and the realization of designer Majorana chain networks for demonstrating topological quantum computation.
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Affiliation(s)
- Lucas Schneider
- Department of Physics, Universität Hamburg, D-20355, Hamburg, Germany
| | - Sascha Brinker
- Peter Grünberg Institut and Institute for Advanced Simulation, Forschungszentrum Jülich & JARA, D-52425, Jülich, Germany.,Department of Physics, RWTH Aachen University, 52056, Aachen, Germany
| | - Manuel Steinbrecher
- Department of Physics, Universität Hamburg, D-20355, Hamburg, Germany.,Institute for Molecules and Materials (IMM), Radboud University, Nijmegen, The Netherlands
| | - Jan Hermenau
- Department of Physics, Universität Hamburg, D-20355, Hamburg, Germany
| | - Thore Posske
- I. Institute for Theoretical Physics, Universität Hamburg, D-20355, Hamburg, Germany
| | - Manuel Dos Santos Dias
- Peter Grünberg Institut and Institute for Advanced Simulation, Forschungszentrum Jülich & JARA, D-52425, Jülich, Germany
| | - Samir Lounis
- Peter Grünberg Institut and Institute for Advanced Simulation, Forschungszentrum Jülich & JARA, D-52425, Jülich, Germany
| | | | - Jens Wiebe
- Department of Physics, Universität Hamburg, D-20355, Hamburg, Germany.
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10
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Kim H, Rózsa L, Schreyer D, Simon E, Wiesendanger R. Long-range focusing of magnetic bound states in superconducting lanthanum. Nat Commun 2020; 11:4573. [PMID: 32917904 PMCID: PMC7486372 DOI: 10.1038/s41467-020-18406-8] [Citation(s) in RCA: 14] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/27/2020] [Accepted: 08/21/2020] [Indexed: 11/08/2022] Open
Abstract
Quantum mechanical systems with long-range interactions between quasiparticles provide a promising platform for coherent quantum information technology. Superconductors are a natural choice for solid-state based quantum devices, while magnetic impurities inside superconductors give rise to quasiparticle excitations of broken Cooper pairs that provide characteristic information about the host superconductor. Here, we reveal that magnetic impurities embedded below a superconducting La(0001) surface interact via quasiparticles extending to very large distances, up to several tens of nanometers. Using low-temperature scanning probe techniques, we observe the corresponding anisotropic and giant oscillations in the LDOS. Theoretical calculations indicate that the quasi-two-dimensional surface states with their strongly anisotropic Fermi surface play a crucial role for the focusing and long-range extension of the magnetic bound states. The quasiparticle focusing mechanism should facilitate the design of versatile magnetic structures with tunable and directed magnetic interactions over large distances, thereby paving the way toward the design of low-dimensional magnet-superconductor hybrid systems exhibiting topologically non-trivial quantum states as possible elements of quantum computation schemes based on Majorana quasiparticles.
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Affiliation(s)
- Howon Kim
- Department of Physics, University of Hamburg, D-20355, Hamburg, Germany.
| | - Levente Rózsa
- Department of Physics, University of Hamburg, D-20355, Hamburg, Germany
- Department of Physics, University of Konstanz, D-78457, Konstanz, Germany
| | - Dominik Schreyer
- Department of Physics, University of Hamburg, D-20355, Hamburg, Germany
| | - Eszter Simon
- Department of Theoretical Physics, Budapest University of Technology and Economics, Budafoki út 8, H-1111, Budapest, Hungary
- Department Chemie, Physikalische Chemie, Ludwig-Maximilians-Universität München, Butenandtstr. 5-13, D-81377, München, Germany
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11
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Perrin V, Santos FLN, Ménard GC, Brun C, Cren T, Civelli M, Simon P. Unveiling Odd-Frequency Pairing around a Magnetic Impurity in a Superconductor. PHYSICAL REVIEW LETTERS 2020; 125:117003. [PMID: 32975960 DOI: 10.1103/physrevlett.125.117003] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/07/2020] [Accepted: 08/17/2020] [Indexed: 06/11/2023]
Abstract
We study the unconventional superconducting correlations caused by a single isolated magnetic impurity in a conventional s-wave superconductor. Because of the local breaking of time-reversal symmetry, the impurity induces unconventional superconductivity, which is even in both space and spin variables but odd under time inversion. We derive an exact proportionality relation between the even-frequency component of the local electron density of states and the imaginary part of the odd-frequency local pairing function. By applying this relation to scanning tunneling microscopy spectra taken on top of magnetic impurities immersed in a Pb/Si(111) monolayer, we show experimental evidence of the occurrence of the odd-frequency pairing in these systems and explicitly extract its superconducting function from the data.
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Affiliation(s)
- Vivien Perrin
- Université Paris-Saclay, CNRS, Laboratoire de Physique des Solides, 91405 Orsay, France
| | - Flávio L N Santos
- Université Paris-Saclay, CNRS, Laboratoire de Physique des Solides, 91405 Orsay, France
- Departamento de Fisica, Universidade Federal de Minas Gerais, Caixa Postal 702, Belo Horizonte, MG 30123-970, Brazil
| | - Gerbold C Ménard
- Institut des NanoSciences de Paris, Sorbonne Université and CNRS-UMR, 7588, 75005 Paris, France
| | - Christophe Brun
- Institut des NanoSciences de Paris, Sorbonne Université and CNRS-UMR, 7588, 75005 Paris, France
| | - Tristan Cren
- Institut des NanoSciences de Paris, Sorbonne Université and CNRS-UMR, 7588, 75005 Paris, France
| | - Marcello Civelli
- Université Paris-Saclay, CNRS, Laboratoire de Physique des Solides, 91405 Orsay, France
| | - Pascal Simon
- Université Paris-Saclay, CNRS, Laboratoire de Physique des Solides, 91405 Orsay, France
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12
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Maiti M, Smotlacha J. Theory of universal differential conductance of magnetic Weyl type-II junctions. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2020; 32:405301. [PMID: 32639952 DOI: 10.1088/1361-648x/ab926e] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/03/2020] [Accepted: 05/12/2020] [Indexed: 06/11/2023]
Abstract
We study the transport properties of junctions of normal and superconducting Weyl semimetal with tilted dispersion, in the presence of magnetization induced by magnetic strips. The sub gap tunnelling conductance shows robust signatures in the presence of different orientation and strength of magnetization of the magnetic strips. We obtain the analytical results for the normal-magnetic-superconducting junction in the thin barrier limit and demonstrate that these results have no analogues to their conventional counterparts and junctions with Dirac electrons in two-dimensions. We discuss possible experimental setups to test our theoretical predictions.
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Affiliation(s)
- M Maiti
- Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Moscow Region, Russia
| | - J Smotlacha
- Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Moscow Region, Russia
- Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University, Brehova 7, 110 00 Prague, Czech Republic
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13
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Wadhwa P, Kumar TJD, Shukla A, Kumar R. Signatures of non-trivial band topology in LaAs/LaBi heterostructure. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2020; 32:395703. [PMID: 32470966 DOI: 10.1088/1361-648x/ab97e2] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/20/2020] [Accepted: 05/29/2020] [Indexed: 06/11/2023]
Abstract
In this article, we investigate non-trivial topological features in a heterostructure of extreme magnetoresistance (XMR) materials LaAs and LaBi using density functional theory. The proposed heterostructure is found to be dynamically stable and shows bulk band inversion with non-trivial Z2topological invariant and a Dirac cone at the surface. In addition, its electron and hole carrier densities ratio is also calculated to investigate the possibility to possess XMR effect. Electrons and holes in the heterostructure are found to be nearly compensated, thereby facilitating it to be a suitable candidate for XMR studies.
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Affiliation(s)
- Payal Wadhwa
- TGraMS Laboratory, Department of Physics, Indian Institute of Technology Ropar, Rupnagar-140001, Punjab, India
| | - T J Dhilip Kumar
- Department of Chemistry, Indian Institute of Technology Ropar, Rupnagar, Punjab-140001, India
| | - Alok Shukla
- Department of Physics, Indian Institute of Technology Bombay, Powai-400076, Mumbai, India
| | - Rakesh Kumar
- TGraMS Laboratory, Department of Physics, Indian Institute of Technology Ropar, Rupnagar-140001, Punjab, India
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14
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Yang X, Yuan Y, Peng Y, Minamitani E, Peng L, Xian JJ, Zhang WH, Fu YS. Observation of short-range Yu-Shiba-Rusinov states with threefold symmetry in layered superconductor 2H-NbSe 2. NANOSCALE 2020; 12:8174-8179. [PMID: 32242592 DOI: 10.1039/d0nr01383h] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/11/2023]
Abstract
Yu-Shiba-Rusinov (YSR) states arise when magnetic impurities interact with superconductivity. The intricacy of coupling and the nature of the superconductivity determine the behavior of the YSR state, whose detailed correlations are not yet fully understood. Here, we study the YSR state of a single Fe adatom on the surface of 2H-NbSe2 with combined low temperature scanning tunneling microscopy/spectroscopy, density functional theory calculations and tight-binding modeling. It is found that the Fe adatom occupies the hollow site of the Se surface layer. A prominent YSR state close to the Fermi level is observed. The YSR state exhibits a threefold symmetry along the diagonal direction of the Se lattice. The spatial decay of the YSR state follows a behavior in three-dimensional superconductivity. This behavior contrasts with a previous study of imbedded Fe impurities, whose YSR state shows a six-fold symmetry and a two-dimensional long-range decay. According to our theoretical modeling, the coupling configurations affect the adatom-substrate hopping and the interlayer coupling of the substrate. Both factors are crucial for the consequent behavior of the YSR state.
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Affiliation(s)
- Xing Yang
- School of Physics and Wuhan National High Magnetic Field Center, Huazhong University of Science and Technology, Wuhan 430074, China.
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15
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Large spatial extension of the zero-energy Yu-Shiba-Rusinov state in a magnetic field. Nat Commun 2020; 11:1834. [PMID: 32286260 PMCID: PMC7156378 DOI: 10.1038/s41467-020-15322-9] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/19/2019] [Accepted: 03/03/2020] [Indexed: 11/30/2022] Open
Abstract
Various promising qubit concepts have been put forward recently based on engineered superconductor subgap states like Andreev bound states, Majorana zero modes or the Yu-Shiba-Rusinov (Shiba) states. The coupling of these subgap states via a superconductor strongly depends on their spatial extension and is an essential next step for future quantum technologies. Here we investigate the spatial extension of a Shiba state in a semiconductor quantum dot coupled to a superconductor. With detailed transport measurements and numerical renormalization group calculations we find a remarkable more than 50 nm extension of the zero energy Shiba state, much larger than the one observed in very recent scanning tunneling microscopy measurements. Moreover, we demonstrate that its spatial extension increases substantially in a magnetic field. Local magnetic moments coupled to superconductors can form subgap Yu-Shiba-Rusinov states. Here the authors show that Shiba states made with an InAs nanowire quantum dot have large spatial extent, which is beneficial for making Shiba chains that are predicted to host Majorana zero modes.
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16
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Legg HF, Braunecker B. Spin liquid mediated RKKY interaction. Sci Rep 2019; 9:17697. [PMID: 31776372 PMCID: PMC6881398 DOI: 10.1038/s41598-019-53842-7] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/23/2019] [Accepted: 10/30/2019] [Indexed: 12/02/2022] Open
Abstract
We propose an RKKY-type interaction that is mediated by a spin liquid. If a spin liquid exists such an interaction could leave a fingerprint by ordering underlying localised moments such as nuclear spins. This interaction has a unique phenomenology that is distinct from the RKKY interaction found in fermionic systems; most notably the lack of a Fermi surface and absence of the requirement for itinerant electrons, since most spin liquids are insulators. We demonstrate that the interaction is predominately shaped by the lattice symmetries of the underlying spin liquid. As a working example we investigate the possible ordering of nuclear spins that interact through an underlying lattice of the two-dimensional spin-1/2 kagome antiferromagnet (KHAF), although the treatment remains general and can be extended to other spin liquids and dimensions. We find that several different nuclear spin orderings minimise the RKKY-type energy induced by the KHAF but are unstable due to a zero-energy flat magnon band in linear spin-wave theory. Despite this we show that a small magnetic field is able to gap out this magnon spectrum resulting in an intricate nuclear magnetism.
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Affiliation(s)
- Henry F Legg
- Institut für Theoretische Physik, Universität zu Köln, D-50937, Cologne, Germany.
| | - Bernd Braunecker
- SUPA, School of Physics and Astronomy, University of St Andrews, North Haugh, St Andrews, Fife KY16 9SS, United Kingdom
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17
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Kotetes P, Mercaldo MT, Cuoco M. Synthetic Weyl Points and Chiral Anomaly in Majorana Devices with Nonstandard Andreev-Bound-State Spectra. PHYSICAL REVIEW LETTERS 2019; 123:126802. [PMID: 31633965 DOI: 10.1103/physrevlett.123.126802] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/12/2019] [Revised: 08/10/2019] [Indexed: 06/10/2023]
Abstract
We demonstrate how to design various nonstandard types of Andreev-bound-state (ABS) dispersions, via a composite construction relying on Majorana bound states (MBSs). Here, the MBSs appear at the interface of a Josephson junction consisting of two topological superconductors (TSCs). Each TSC harbors multiple MBSs per edge by virtue of a chiral or unitary symmetry. We find that, while the ABS dispersions are 2π periodic, they still contain multiple crossings which are protected by the conservation of fermion parity. A single junction with four interface MBSs and all MBS couplings fully controllable, or networks of such coupled junctions with partial coupling tunability, open the door for topological band structures with Weyl points or nodes in synthetic dimensions, which in turn allow for fermion-parity (FP) pumping with a cycle set by the ABS-dispersion details. In fact, in the case of nodes, the FP pumping is a manifestation of chiral anomaly in 2D synthetic spacetime. The possible experimental demonstration of ABS engineering in these devices further promises to unveil new paths for the detection of MBSs and higher-dimensional chiral anomaly.
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Affiliation(s)
- Panagiotis Kotetes
- CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
- Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark
| | - Maria Teresa Mercaldo
- Dipartimento di Fisica "E. R. Caianiello," Università di Salerno, IT-84084 Fisciano (SA), Italy
| | - Mario Cuoco
- Dipartimento di Fisica "E. R. Caianiello," Università di Salerno, IT-84084 Fisciano (SA), Italy
- CNR-SPIN, IT-84084 Fisciano (SA), Italy
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18
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Volpez Y, Loss D, Klinovaja J. Second-Order Topological Superconductivity in π-Junction Rashba Layers. PHYSICAL REVIEW LETTERS 2019; 122:126402. [PMID: 30978045 DOI: 10.1103/physrevlett.122.126402] [Citation(s) in RCA: 24] [Impact Index Per Article: 4.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/06/2018] [Indexed: 06/09/2023]
Abstract
We consider a Josephson junction bilayer consisting of two tunnel-coupled two-dimensional electron gas layers with Rashba spin-orbit interaction, proximitized by a top and bottom s-wave superconductor with phase difference ϕ close to π. We show that, in the presence of a finite weak in-plane Zeeman field, the bilayer can be driven into a second order topological superconducting phase, hosting two Majorana corner states (MCSs). If ϕ=π, in a rectangular geometry, these zero-energy bound states are located at two opposite corners determined by the direction of the Zeeman field. If the phase difference ϕ deviates from π by a critical value, one of the two MCSs gets relocated to an adjacent corner. As the phase difference ϕ increases further, the system becomes trivially gapped. The obtained MCSs are robust against static and magnetic disorder. We propose two setups that could realize such a model: one is based on controlling ϕ by magnetic flux, the other involves an additional layer of randomly oriented magnetic impurities responsible for the phase shift of π in the proximity-induced superconducting pairing.
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Affiliation(s)
- Yanick Volpez
- Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland
| | - Daniel Loss
- Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland
| | - Jelena Klinovaja
- Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland
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Posske T, Thorwart M. Winding Up Quantum Spin Helices: How Avoided Level Crossings Exile Classical Topological Protection. PHYSICAL REVIEW LETTERS 2019; 122:097204. [PMID: 30932535 DOI: 10.1103/physrevlett.122.097204] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/18/2018] [Indexed: 06/09/2023]
Abstract
A magnetic helix can be wound into a classical Heisenberg chain by fixing one end while rotating the other one. We show that in quantum Heisenberg chains of finite length, the magnetization slips back to the trivial state beyond a finite turning angle. Avoided level crossings thus undermine classical topological protection. Yet, for special values of the axial Heisenberg anisotropy, stable spin helices form again, which are nonlocally entangled. Away from these sweet spots, spin helices can be stabilized dynamically or by dissipation. For half-integer spin chains of odd length, a spin slippage state and its Kramers partner define a qubit with a nontrivial Berry connection.
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Affiliation(s)
- Thore Posske
- I. Institut für Theoretische Physik, Universität Hamburg, Jungiusstraße 9, 20355 Hamburg, Germany
| | - Michael Thorwart
- I. Institut für Theoretische Physik, Universität Hamburg, Jungiusstraße 9, 20355 Hamburg, Germany
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20
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Ren C, Wu Y, Sun M, Wang S, Tian H. Measuring the nonlocality of different types of Majorana bound states in a topological superconducting wire. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2019; 31:045501. [PMID: 30524102 DOI: 10.1088/1361-648x/aaf149] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/09/2023]
Abstract
According to the degree of topological protection, Majorana bound states (MBSs) can be divided into three types: ideal zero-energy MBSs (IZMs), finite-energy MBSs (FEMs) and zero-energy MBSs at parity crossing points (PZMs). Herein, we investigate the nonlocality of these three types of MBSs by comparing the conductance spectra of a normal lead-topological superconducting wire-normal lead (NSN) junction and an NS junction. We find that for the FEM-related tunnelling process, the decrease in the nonlocal processes is trivially accompanied by an increase in the local processes, whereas for the IZM-related tunnelling process, the left and right tunnelling processes are completely independent. Remarkably, PZMs induce a nonlocal electron-blocking effect in which incoming electrons from the left lead cannot participate in local Andreev reflection unless the right lead is present, even though no nonlocal tunnelling processes occur in the right lead of an NSN junction. We show that this PZM-mediated nonlocal electron-blocking effect is due to the nonlocal coupling of the left lead to the more distant PZM and that the phase difference between the two end PZMs is [Formula: see text]. Our findings provide an experimentally accessible method for characterizing MBSs by probing their different nonlocal signatures.
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Affiliation(s)
- Chongdan Ren
- Department of Physics, Zunyi Normal University, Kweichow 563002, People's Republic of China
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21
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Hsu CH, Stano P, Klinovaja J, Loss D. Majorana Kramers Pairs in Higher-Order Topological Insulators. PHYSICAL REVIEW LETTERS 2018; 121:196801. [PMID: 30468588 DOI: 10.1103/physrevlett.121.196801] [Citation(s) in RCA: 40] [Impact Index Per Article: 6.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/30/2018] [Revised: 08/24/2018] [Indexed: 06/09/2023]
Abstract
We propose a tune-free scheme to realize Kramers pairs of Majorana bound states in recently discovered higher-order topological insulators (HOTIs). We show that, by bringing two hinges of a HOTI into the proximity of an s-wave superconductor, the competition between local and crossed Andreev pairing leads to the formation of Majorana Kramers pairs, when the latter pairing dominates over the former. We demonstrate that such a topological superconductivity is stabilized by moderate electron-electron interactions. The proposed setup avoids the application of a magnetic field or local voltage gates, and requires weaker interactions compared with nonhelical nanowires.
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Affiliation(s)
- Chen-Hsuan Hsu
- RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan
| | - Peter Stano
- RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan
- Department of Applied Physics, School of Engineering, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
- Institute of Physics, Slovak Academy of Sciences, 845 11 Bratislava, Slovakia
| | - Jelena Klinovaja
- RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan
- Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland
| | - Daniel Loss
- RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan
- Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland
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22
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Yan Z, Song F, Wang Z. Majorana Corner Modes in a High-Temperature Platform. PHYSICAL REVIEW LETTERS 2018; 121:096803. [PMID: 30230860 DOI: 10.1103/physrevlett.121.096803] [Citation(s) in RCA: 52] [Impact Index Per Article: 8.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/25/2018] [Revised: 06/20/2018] [Indexed: 06/08/2023]
Abstract
We introduce two-dimensional topological insulators in proximity to high-temperature cuprate or iron-based superconductors as high-temperature platforms of Majorana Kramers pairs of zero modes. The proximity-induced pairing at the helical edge state of the topological insulator serves as a Dirac mass, whose sign changes at the sample corner because of the pairing symmetry of high-T_{c} superconductors. This sign changing naturally creates at each corner a pair of Majorana zero modes protected by time-reversal symmetry. Conceptually, this is a topologically trivial superconductor-based approach for Majorana zero modes. We provide quantitative criteria and suggest candidate materials for this proposal.
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Affiliation(s)
- Zhongbo Yan
- Institute for Advanced Study, Tsinghua University, Beijing 100084, China
| | - Fei Song
- Institute for Advanced Study, Tsinghua University, Beijing 100084, China
| | - Zhong Wang
- Institute for Advanced Study, Tsinghua University, Beijing 100084, China
- Collaborative Innovation Center of Quantum Matter, Beijing 100871, China
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23
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Ibañez-Azpiroz J, Dos Santos Dias M, Blügel S, Lounis S. Spin-fluctuation and spin-relaxation effects of single adatoms from first principles. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2018; 30:343002. [PMID: 30020083 DOI: 10.1088/1361-648x/aad43d] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
Single adatoms offer an exceptional playground for studying magnetism and its associated dynamics at the atomic scale. Here we review recent results on single adatoms deposited on metallic substrates, based on time-dependent density functional theory. First we analyze quantum zero-point spin-fluctuations (ZPSF) as calculated from the fluctuation-dissipation theorem, and show how they affect the magnetic stability by modifying the magnetic anisotropy energy. We also assess the impact of ZPSF in the limit of small hybridization to the substrate characteristic of semi-insulating substrates, connecting to recent experimental investigations where magnetic stability of a single adatom was achieved for the first time. Secondly, we inspect further the dynamics of single adatoms by considering the longitudinal and transverse spin-relaxation processes, whose time-scales are analyzed and related to the underlying electronic structure of both the adatom and the substrate. Thirdly, we analyze spin-fluctuation modes of paramagnetic adatoms, i.e. adatoms where the Stoner criterion for magnetism is almost fulfilled. Interestingly, such modes can develop well-defined peaks in the meV range, their main characteristics being determined by two fundamental electronic properties, namely the Stoner parameter and the density of states at the Fermi level. Furthermore, simulated inelastic scanning tunneling spectroscopy curves reveal that these spin-fluctuation modes can be triggered by tunneling electrons, opening up potential applications also for paramagnetic adatoms. Lastly, an overview of the outstanding issues and future directions is given.
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Affiliation(s)
- Julen Ibañez-Azpiroz
- Centro de Física de Materiales, Universidad del País Vasco, 20018 San Sebastián, Spain
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24
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Abstract
Topological states of matter support quantised nondissipative responses and exotic quantum particles that cannot be accessed in common materials. The exceptional properties and application potential of topological materials have triggered a large-scale search for new realisations. Breaking away from the popular trend focusing almost exclusively on crystalline symmetries, we introduce the Shiba glass as a platform for amorphous topological quantum matter. This system consists of an ensemble of randomly distributed magnetic atoms on a superconducting surface. We show that subgap Yu–Shiba–Rusinov states on the magnetic moments form a topological superconducting phase at critical density despite a complete absence of spatial order. Experimental signatures of the amorphous topological state can be obtained by scanning tunnelling microscopy measurements probing the topological edge mode. Our discovery demonstrates the physical feasibility of amorphous topological quantum matter, presenting a concrete route to fabricating new topological systems from nontopological materials with random dopants. Apart from the intensive efforts to explore topological properties in crystalline materials, the study of such properties in amorphous materials has been rare. Here, Pöyhönen et al. predict a topological superconducting phase in an ensemble of randomly distributed magnetic atoms on a superconducting surface.
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25
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Kim H, Palacio-Morales A, Posske T, Rózsa L, Palotás K, Szunyogh L, Thorwart M, Wiesendanger R. Toward tailoring Majorana bound states in artificially constructed magnetic atom chains on elemental superconductors. SCIENCE ADVANCES 2018; 4:eaar5251. [PMID: 29756034 PMCID: PMC5947976 DOI: 10.1126/sciadv.aar5251] [Citation(s) in RCA: 61] [Impact Index Per Article: 10.2] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/17/2017] [Accepted: 03/23/2018] [Indexed: 05/22/2023]
Abstract
Realizing Majorana bound states (MBS) in condensed matter systems is a key challenge on the way toward topological quantum computing. As a promising platform, one-dimensional magnetic chains on conventional superconductors were theoretically predicted to host MBS at the chain ends. We demonstrate a novel approach to design of model-type atomic-scale systems for studying MBS using single-atom manipulation techniques. Our artificially constructed atomic Fe chains on a Re surface exhibit spin spiral states and a remarkable enhancement of the local density of states at zero energy being strongly localized at the chain ends. Moreover, the zero-energy modes at the chain ends are shown to emerge and become stabilized with increasing chain length. Tight-binding model calculations based on parameters obtained from ab initio calculations corroborate that the system resides in the topological phase. Our work opens new pathways to design MBS in atomic-scale hybrid structures as a basis for fault-tolerant topological quantum computing.
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Affiliation(s)
- Howon Kim
- Department of Physics, University of Hamburg, D-20355 Hamburg, Germany
- Corresponding author. (H.K.); (R.W.)
| | | | - Thore Posske
- Department of Physics, University of Hamburg, D-20355 Hamburg, Germany
| | - Levente Rózsa
- Department of Physics, University of Hamburg, D-20355 Hamburg, Germany
| | - Krisztián Palotás
- Institute of Physics, Slovak Academy of Sciences, 84511 Bratislava, Slovakia
- MTA-SZTE Reaction Kinetics and Surface Chemistry Research Group, University of Szeged, 6720 Szeged, Hungary
| | - László Szunyogh
- Department of Theoretical Physics and MTA-BME Condensed Matter Research Group, Budapest University of Technology and Economics, 1111 Budapest, Hungary
| | - Michael Thorwart
- Department of Physics, University of Hamburg, D-20355 Hamburg, Germany
| | - Roland Wiesendanger
- Department of Physics, University of Hamburg, D-20355 Hamburg, Germany
- Corresponding author. (H.K.); (R.W.)
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26
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Reeg C, Loss D, Klinovaja J. Proximity effect in a two-dimensional electron gas coupled to a thin superconducting layer. BEILSTEIN JOURNAL OF NANOTECHNOLOGY 2018; 9:1263-1271. [PMID: 29765804 PMCID: PMC5942388 DOI: 10.3762/bjnano.9.118] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 01/27/2018] [Accepted: 03/23/2018] [Indexed: 06/08/2023]
Abstract
There have recently been several experiments studying induced superconductivity in semiconducting two-dimensional electron gases that are strongly coupled to thin superconducting layers, as well as probing possible topological phases supporting Majorana bound states in such setups. We show that a large band shift is induced in the semiconductor by the superconductor in this geometry, thus making it challenging to realize a topological phase. Additionally, we show that while increasing the thickness of the superconducting layer reduces the magnitude of the band shift, it also leads to a more significant renormalization of the semiconducting material parameters and does not reduce the challenge of tuning into a topological phase.
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Affiliation(s)
- Christopher Reeg
- Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland
| | - Daniel Loss
- Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland
| | - Jelena Klinovaja
- Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland
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27
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Xiao J, An J. Boundary states in the chiral symmetric systems with a spatial symmetry. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2018; 30:055002. [PMID: 29256433 DOI: 10.1088/1361-648x/aaa2d5] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
We study topological systems with both a chiral and a spatial symmetry which result in an additional spatial chiral symmetry. We distinguish the topologically nontrivial states according to the chiral symmetries protecting them and study several models in 1D and 3D systems. The perturbations breaking the spatial symmetry can break only one of the two chiral symmetries while the perturbations preserving the spatial symmetry always break or preserve both of them. In 3D systems, besides the 3D symmetries, the topologically nontrivial boundary modes may also be protected by the hidden lower dimensional symmetries. We then figure out the corresponding topological invariants and connect them with the 3D invariants.
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Affiliation(s)
- Jinpeng Xiao
- National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, People's Republic of China
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28
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Hatter N, Heinrich BW, Rolf D, Franke KJ. Scaling of Yu-Shiba-Rusinov energies in the weak-coupling Kondo regime. Nat Commun 2017; 8:2016. [PMID: 29222411 PMCID: PMC5722882 DOI: 10.1038/s41467-017-02277-7] [Citation(s) in RCA: 38] [Impact Index Per Article: 5.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/28/2017] [Accepted: 11/16/2017] [Indexed: 11/26/2022] Open
Abstract
The competition of the free-spin state of a paramagnetic impurity on a superconductor with its screened counterpart is characterized by the energy scale of Kondo screening compared to the superconducting pairing energy Δ. When the experimental temperature suppresses Kondo screening, but preserves superconductivity, i.e., when Δ/kB > T > TK (kB is Boltzmann’s constant and TK the Kondo temperature), this description fails. Here, we explore this temperature range in a set of manganese phthalocyanine molecules decorated with ammonia on Pb(111). We show that these molecules suffice the required energy conditions by exhibiting weak-coupling Kondo resonances. We correlate the Yu-Shiba-Rusinov bound states energy inside the superconducting gap with the intensity of the Kondo resonance. The observed correlation follows the expectations for a classical spin on a superconductor. This finding is important in view of many theoretical predictions using a classical spin model, in particular for the description of Majorana bound states in magnetic nanostructures on superconducting substrates. The description of a paramagnetic impurity on a superconductor remains elusive in the weak-coupling Kondo regime. Here, Hatter et al. correlate the energy of the Yu-Shiba-Rusinov bound states with the intensity of the Kondo resonances in such a regime, revealing a behavior well described by classical spin models.
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Affiliation(s)
- Nino Hatter
- Fachbereich Physik, Freie Universität Berlin, Arnimallee 14, 14195, Berlin, Germany
| | - Benjamin W Heinrich
- Fachbereich Physik, Freie Universität Berlin, Arnimallee 14, 14195, Berlin, Germany.
| | - Daniela Rolf
- Fachbereich Physik, Freie Universität Berlin, Arnimallee 14, 14195, Berlin, Germany
| | - Katharina J Franke
- Fachbereich Physik, Freie Universität Berlin, Arnimallee 14, 14195, Berlin, Germany
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29
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Shao B, Schüler M, Schönhoff G, Frauenheim T, Czycholl G, Wehling TO. Optically and Electrically Controllable Adatom Spin-orbital Dynamics in Transition Metal Dichalcogenides. NANO LETTERS 2017; 17:6721-6726. [PMID: 28978200 DOI: 10.1021/acs.nanolett.7b02785] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
We analyze the interplay of spin-valley coupling, orbital physics, and magnetic anisotropy taking place at single magnetic atoms adsorbed on semiconducting transition metal dichalcogenides, MX2 (M = Mo, W; X = S, Se). Orbital selection rules turn out to govern the kinetic exchange coupling between the adatom and charge carriers in the MX2 and lead to highly orbitally dependent spin-flip scattering rates, as we illustrate for the example of transition metal adatoms with d9 configuration. Our ab initio calculations suggest that d9 configurations are realizable by single Co, Rh, or Ir adatoms on MoS2, which additionally exhibit a sizable magnetic anisotropy. We find that the interaction of the adatom with carriers in the MX2 allows to tune its behavior from a quantum regime with full Kondo screening to a regime of "Ising spintronics" where its spin-orbital moment acts as classical bit, which can be erased and written electronically and optically.
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Affiliation(s)
- Bin Shao
- Bremen Center for Computational Materials Science and ‡Institut für Theoretische Physik, Universität Bremen , 28359 Bremen, Germany
| | - Malte Schüler
- Bremen Center for Computational Materials Science and ‡Institut für Theoretische Physik, Universität Bremen , 28359 Bremen, Germany
| | - Gunnar Schönhoff
- Bremen Center for Computational Materials Science and ‡Institut für Theoretische Physik, Universität Bremen , 28359 Bremen, Germany
| | - Thomas Frauenheim
- Bremen Center for Computational Materials Science and ‡Institut für Theoretische Physik, Universität Bremen , 28359 Bremen, Germany
| | - Gerd Czycholl
- Bremen Center for Computational Materials Science and ‡Institut für Theoretische Physik, Universität Bremen , 28359 Bremen, Germany
| | - Tim O Wehling
- Bremen Center for Computational Materials Science and ‡Institut für Theoretische Physik, Universität Bremen , 28359 Bremen, Germany
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30
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Ibañez-Azpiroz J, Dias MDS, Schweflinghaus B, Blügel S, Lounis S. Tuning Paramagnetic Spin Excitations of Single Adatoms. PHYSICAL REVIEW LETTERS 2017; 119:017203. [PMID: 28731747 DOI: 10.1103/physrevlett.119.017203] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/13/2017] [Indexed: 06/07/2023]
Abstract
We predict the existence of paramagnetic spin excitations (PSE) in nonmagnetic single adatoms. Our calculations demonstrate that PSE develop a well-defined structure in the meV region when the adatom's Stoner criterion for magnetism is close to the critical point. We further reveal a subtle tunability and enhancement of PSE by external magnetic fields. Finally, we show how PSE can be detected as moving steps in the dI/dV signal of inelastic scanning tunneling spectroscopy, opening a potential route for experimentally accessing electronic properties of nonmagnetic adatoms, such as the Stoner parameter.
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Affiliation(s)
- Julen Ibañez-Azpiroz
- Peter Grünberg Institute and Institute for Advanced Simulation, Forschungszentrum Jülich & JARA, D-52425 Jülich, Germany
| | - Manuel Dos Santos Dias
- Peter Grünberg Institute and Institute for Advanced Simulation, Forschungszentrum Jülich & JARA, D-52425 Jülich, Germany
| | - Benedikt Schweflinghaus
- Peter Grünberg Institute and Institute for Advanced Simulation, Forschungszentrum Jülich & JARA, D-52425 Jülich, Germany
| | - Stefan Blügel
- Peter Grünberg Institute and Institute for Advanced Simulation, Forschungszentrum Jülich & JARA, D-52425 Jülich, Germany
| | - Samir Lounis
- Peter Grünberg Institute and Institute for Advanced Simulation, Forschungszentrum Jülich & JARA, D-52425 Jülich, Germany
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31
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Yan Z, Bi R, Wang Z. Majorana Zero Modes Protected by a Hopf Invariant in Topologically Trivial Superconductors. PHYSICAL REVIEW LETTERS 2017; 118:147003. [PMID: 28430502 DOI: 10.1103/physrevlett.118.147003] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/10/2017] [Indexed: 06/07/2023]
Abstract
Majorana zero modes are usually attributed to topological superconductors. We study a class of two-dimensional topologically trivial superconductors without chiral edge modes, which nevertheless host robust Majorana zero modes in topological defects. The construction of this minimal single-band model is facilitated by the Hopf map and the Hopf invariant. This work will stimulate investigations of Majorana zero modes in superconductors in the topologically trivial regime.
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Affiliation(s)
- Zhongbo Yan
- Institute for Advanced Study, Tsinghua University, Beijing 100084, China
| | - Ren Bi
- Institute for Advanced Study, Tsinghua University, Beijing 100084, China
| | - Zhong Wang
- Institute for Advanced Study, Tsinghua University, Beijing 100084, China
- Collaborative Innovation Center of Quantum Matter, Beijing 100871, China
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32
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Deng MT, Vaitiekėnas S, Hansen EB, Danon J, Leijnse M, Flensberg K, Nygård J, Krogstrup P, Marcus CM. Majorana bound state in a coupled quantum-dot hybrid-nanowire system. Science 2016; 354:1557-1562. [DOI: 10.1126/science.aaf3961] [Citation(s) in RCA: 687] [Impact Index Per Article: 85.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/03/2016] [Accepted: 11/16/2016] [Indexed: 11/02/2022]
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33
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Čadež T, Sacramento PD. Zero energy modes in a superconductor with ferromagnetic adatom chains and quantum phase transitions. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2016; 28:495703. [PMID: 27754979 DOI: 10.1088/0953-8984/28/49/495703] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
We study Majorana zero energy modes (MZEM) that occur in an s-wave superconducting surface, at the ends of a ferromagnetic (FM) chain of adatoms, in the presence of Rashba spin-orbit interaction (SOI) considering both non self-consistent and self-consistent superconducting order. We find that in the self-consistent solution, the average superconducting gap function over the adatom sites has a discontinuous drop with increasing exchange interaction at the same critical value where the topological phase transition occurs. We also study the MZEM for both treatments of superconducting order and find that the decay length is a linear function of the exchange coupling strength, chemical potential and superconducting order. For wider FM chains the MZEM occur at smaller exchange couplings and the slope of the decay length as a function of exchange coupling grows with chain width. Thus we suggest experimental detection of different delocalization of MZEM in chains of varying widths. We discuss similarities and differences between the MZEM for the two treatments of the superconducting order.
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Affiliation(s)
- Tilen Čadež
- Beijing Computational Science Research Center, Zhongguancun Software Park II, No. 10 West Dongbeiwang Road, Haidian District, Beijing, 100094, People's Republic of China. CeFEMA, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal. Jožef Stefan Institute, 1000 Ljubljana, Slovenia
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34
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Kaladzhyan V, Bena C, Simon P. Asymptotic behavior of impurity-induced bound states in low-dimensional topological superconductors. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2016; 28:485701. [PMID: 27705962 DOI: 10.1088/0953-8984/28/48/485701] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
We study theoretically the asymptotic behavior of the Shiba bound states associated with magnetic impurities embedded in both 2D and 1D anomalous superconductors. We calculate analytically the spatial dependence of the local density of states together with the spin polarization associated with the Shiba bound states. We show that the latter quantity exhibits drastic differences between s-wave and different types of p-wave superconductors. Such properties, which could be measured using spin-polarized STM, offer therefore a way to discriminate between singlet and triplet pairing in low-dimensional superconductors, as well as a way to estimate the amplitude of the triplet pairing in these systems.
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Affiliation(s)
- V Kaladzhyan
- Institut de Physique Théorique, CEA/Saclay, Orme des Merisiers, 91190 Gif-sur-Yvette Cedex, France. Laboratoire de Physique des Solides, CNRS, University of Paris-Sud, Université Paris-Saclay, 91405 Orsay Cedex, France
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35
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Chen YQ, Wu YM, Liu XJ. Symmetry reduction and boundary modes for Fe chains on an s-wave superconductor. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2016; 28:475701. [PMID: 27633482 DOI: 10.1088/0953-8984/28/47/475701] [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 investigate the superconducting phases and boundary modes for a quasi-1D system formed by up to three Fe chains on an s-wave superconductor, motivated by a recent experiment. While the Rashba type spin-orbit coupling together with a magnetic ordering is necessary to drive the system to be of nontrivial topology, we show that the onsite [Formula: see text] spin-orbit term, inter-chain diagonal hopping couplings, and magnetic disorders in the Fe chains are crucial in determining the symmetry classes of superconducting phases, which can be topologically trivial or nontrivial in different parameter regimes. In general multiple low-energy Andreev bound states, as well as a single Majorana zero mode if the phase is topological, are obtained in the ends of Fe chains. The nontrivial symmetry reduction mechanism is uncovered to provide an understanding of the present results, and may explain the zero-bias peak observed in the experiment. The present study can be applied to generic multiple-chain system.
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Affiliation(s)
- Yu-Qin Chen
- International Center for Quantum Materials and School of Physics, Peking University, Beijing 100871, People's Republic of China. Collaborative Innovation Center of Quantum Matter, Beijing 100871, People's Republic of China
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36
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Chen HJ, Fang XW, Chen CZ, Li Y, Tang XD. Robust signatures detection of Majorana fermions in superconducting iron chains. Sci Rep 2016; 6:36600. [PMID: 27857149 PMCID: PMC5114679 DOI: 10.1038/srep36600] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/06/2016] [Accepted: 10/17/2016] [Indexed: 11/21/2022] Open
Abstract
We theoretically propose an optical means to detect Majorana fermions in superconducting iron (Fe) chains with a hybrid quantum dot-nanomechanical resonator system driven by two-tone fields, which is very different from the current tunneling spectroscopy experiments with electrical means. Based on the scheme, the phenomenon of Majorana modes induced transparency is demonstrated and a straightforward method to determine the quantum dot-Majorana fermions coupling strength is also presented. We further investigate the role of the nanomechanical resonator, and the resonator behaving as a phonon cavity enhances the exciton resonance spectrum, which is robust for detecting of Majorana fermions. The coherent optical spectrum affords a potential supplement to detecte Majorana fermions and supports Majorana fermions-based topological quantum computation in superconducting iron chains.
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Affiliation(s)
- Hua-Jun Chen
- School of Science, Anhui University of Science and Technology, Huainan Anhui, 232001, China
| | - Xian-Wen Fang
- School of Science, Anhui University of Science and Technology, Huainan Anhui, 232001, China
| | - Chang-Zhao Chen
- School of Science, Anhui University of Science and Technology, Huainan Anhui, 232001, China
| | - Yang Li
- School of Science, Anhui University of Science and Technology, Huainan Anhui, 232001, China
| | - Xu-Dong Tang
- School of Science, Anhui University of Science and Technology, Huainan Anhui, 232001, China
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37
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Chen W, Sigrist M, Schnyder AP. Scaling theory of [Formula: see text] topological invariants. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2016; 28:365501. [PMID: 27400801 DOI: 10.1088/0953-8984/28/36/365501] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
For inversion-symmetric topological insulators and superconductors characterized by [Formula: see text] topological invariants, two scaling schemes are proposed to judge topological phase transitions driven by an energy parameter. The scaling schemes renormalize either the phase gradient or the second derivative of the Pfaffian of the time-reversal operator, through which the renormalization group flow of the driving energy parameter can be obtained. The Pfaffian near the time-reversal invariant momentum is revealed to display a universal critical behavior for a great variety of models examined.
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Affiliation(s)
- Wei Chen
- Theoretische Physik, ETH-Zürich, CH-8093 Zürich, Switzerland
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38
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Two-dimensional chiral topological superconductivity in Shiba lattices. Nat Commun 2016; 7:12297. [PMID: 27465127 PMCID: PMC4974475 DOI: 10.1038/ncomms12297] [Citation(s) in RCA: 31] [Impact Index Per Article: 3.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/20/2015] [Accepted: 06/22/2016] [Indexed: 11/28/2022] Open
Abstract
The chiral p-wave superconductor is the archetypal example of a state of matter that supports non-Abelian anyons, a highly desired type of exotic quasiparticle. With this, it is foundational for the distant goal of building a topological quantum computer. While some candidate materials for bulk chiral superconductors exist, they are subject of an ongoing debate about their actual paring state. Here we propose an alternative route to chiral superconductivity, consisting of the surface of an ordinary superconductor decorated with a two-dimensional lattice of magnetic impurities. We furthermore identify a promising experimental platform to realize this proposal. Chiral p-wave superconductors may form the basis for topological quantum computers however one has yet to be identified. Here, the authors propose that such a state may be created by a two-dimensional array of magnetic adatoms on the surface of a superconductor with strong spin-orbit coupling.
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39
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Li J, Pan W, Bernevig BA, Lutchyn RM. Detection of Majorana Kramers Pairs Using a Quantum Point Contact. PHYSICAL REVIEW LETTERS 2016; 117:046804. [PMID: 27494493 DOI: 10.1103/physrevlett.117.046804] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/17/2015] [Indexed: 06/06/2023]
Abstract
We propose a setup that integrates a quantum point contact (QPC) and a Josephson junction on a quantum spin Hall sample, experimentally realizable in InAs/GaSb quantum wells. The confinement due to both the QPC and the superconductor results in a Kramers pair of Majorana zero-energy bound states when the superconducting phases in the two arms differ by an odd multiple of π across the Josephson junction. We investigate the detection of these Majorana pairs with the integrated QPC, and find a robust switching from normal to Andreev scattering across the edges due to the presence of Majorana Kramers pairs. Such a switching of the current represents a qualitative signature where multiterminal differential conductances oscillate with alternating signs when the external magnetic field is tuned. We show that this qualitative signature is also present in current cross-correlations. Thus, the change of the backscattering current nature affects both conductance and shot noise, the measurement of which offers a significant advantage over quantitative signatures such as conductance quantization in realistic measurements.
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Affiliation(s)
- Jian Li
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
| | - Wei Pan
- Sandia National Laboratories, Albuquerque, New Mexico 87185, USA
| | - B Andrei Bernevig
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
| | - Roman M Lutchyn
- Station Q, Microsoft Research, Santa Barbara, California 93106-6105, USA
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40
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Hals KMD, Schecter M, Rudner MS. Composite Topological Excitations in Ferromagnet-Superconductor Heterostructures. PHYSICAL REVIEW LETTERS 2016; 117:017001. [PMID: 27419584 DOI: 10.1103/physrevlett.117.017001] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/24/2016] [Indexed: 06/06/2023]
Abstract
We investigate the formation of a new type of composite topological excitation-the Skyrmion-vortex pair (SVP)-in hybrid systems consisting of coupled ferromagnetic and superconducting layers. Spin-orbit interaction in the superconductor mediates a magnetoelectric coupling between the vortex and the Skyrmion, with a sign (attractive or repulsive) that depends on the topological indices of the constituents. We determine the conditions under which a bound SVP is formed and characterize the range and depth of the effective binding potential through analytical estimates and numerical simulations. Furthermore, we develop a semiclassical description of the coupled Skyrmion-vortex dynamics and discuss how SVPs can be controlled by applied spin currents.
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Affiliation(s)
- Kjetil M D Hals
- Niels Bohr International Academy, Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark
- The Center for Quantum Devices, Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark
| | - Michael Schecter
- The Center for Quantum Devices, Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark
| | - Mark S Rudner
- Niels Bohr International Academy, Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark
- The Center for Quantum Devices, Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark
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41
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Klinovaja J, Stano P, Loss D. Topological Floquet Phases in Driven Coupled Rashba Nanowires. PHYSICAL REVIEW LETTERS 2016; 116:176401. [PMID: 27176529 DOI: 10.1103/physrevlett.116.176401] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/13/2015] [Indexed: 06/05/2023]
Abstract
We consider periodically driven arrays of weakly coupled wires with conduction and valence bands of Rashba type and study the resulting Floquet states. This nonequilibrium system can be tuned into nontrivial phases such as topological insulators, Weyl semimetals, and dispersionless zero-energy edge mode regimes. In the presence of strong electron-electron interactions, we generalize these regimes to the fractional case, where elementary excitations have fractional charges e/m with m being an odd integer.
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Affiliation(s)
- Jelena Klinovaja
- Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland
| | - Peter Stano
- RIKEN Center for Emergent Matter Science, Wako, Saitama 351-0198, Japan
- Institute of Physics, Slovak Academy of Sciences, 845 11 Bratislava, Slovakia
| | - Daniel Loss
- Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland
- RIKEN Center for Emergent Matter Science, Wako, Saitama 351-0198, Japan
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42
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Chen W. Scaling theory of topological phase transitions. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2016; 28:055601. [PMID: 26790004 DOI: 10.1088/0953-8984/28/5/055601] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
Topologically ordered systems are characterized by topological invariants that are often calculated from the momentum space integration of a certain function that represents the curvature of the many-body state. The curvature function may be Berry curvature, Berry connection, or other quantities depending on the system. Akin to stretching a messy string to reveal the number of knots it contains, a scaling procedure is proposed for the curvature function in inversion symmetric systems, from which the topological phase transition can be identified from the flow of the driving energy parameters that control the topology (hopping, chemical potential, etc) under scaling. At an infinitesimal operation, one obtains the renormalization group (RG) equations for the driving energy parameters. A length scale defined from the curvature function near the gap-closing momentum is suggested to characterize the scale invariance at critical points and fixed points, and displays a universal critical behavior in a variety of systems examined.
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Affiliation(s)
- Wei Chen
- Institut für Theoretische Physik, ETH Zürich, 8093 Zürich, Switzerland
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43
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Manipulating Majorana zero modes on atomic rings with an external magnetic field. Nat Commun 2016; 7:10395. [PMID: 26791080 PMCID: PMC4735899 DOI: 10.1038/ncomms10395] [Citation(s) in RCA: 49] [Impact Index Per Article: 6.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/29/2014] [Accepted: 12/07/2015] [Indexed: 11/09/2022] Open
Abstract
Non-Abelian quasiparticles have been predicted to exist in a variety of condensed matter systems. Their defining property is that an adiabatic braid between two of them results in a non-trivial change of the quantum state of the system. The simplest non-Abelian quasiparticles--the Majorana bound states--can occur in one-dimensional electronic nano-structures proximity-coupled to a bulk superconductor. Here we propose a set-up, based on chains of magnetic adatoms on the surface of a thin-film superconductor, in which the control over an externally applied magnetic field suffices to create and manipulate Majorana bound states. We consider specifically rings of adatoms and show that they allow for the creation, annihilation, adiabatic motion and braiding of pairs of Majorana bound states by varying the magnitude and orientation of the external magnetic field.
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44
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Peng Y, Pientka F, Vinkler-Aviv Y, Glazman LI, von Oppen F. Robust Majorana Conductance Peaks for a Superconducting Lead. PHYSICAL REVIEW LETTERS 2015; 115:266804. [PMID: 26765015 DOI: 10.1103/physrevlett.115.266804] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/29/2015] [Indexed: 06/05/2023]
Abstract
Experimental evidence for Majorana bound states largely relies on measurements of the tunneling conductance. While the conductance into a Majorana state is in principle quantized to 2e^{2}/h, observation of this quantization has been elusive, presumably due to temperature broadening in the normal-metal lead. Here, we propose to use a superconducting lead instead, whose gap strongly suppresses thermal excitations. For a wide range of tunneling strengths and temperatures, a Majorana state is then signaled by symmetric conductance peaks at eV=±Δ of a universal height G=(4-π)2e(2)/h. For a superconducting scanning tunneling microscope tip, Majorana states appear as spatial conductance plateaus while the conductance varies with the local wave function for trivial Andreev bound states. We discuss effects of nonresonant (bulk) Andreev reflections and quasiparticle poisoning.
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Affiliation(s)
- Yang Peng
- Dahlem Center for Complex Quantum Systems and Fachbereich Physik, Freie Universität Berlin, 14195 Berlin, Germany
| | - Falko Pientka
- Dahlem Center for Complex Quantum Systems and Fachbereich Physik, Freie Universität Berlin, 14195 Berlin, Germany
| | - Yuval Vinkler-Aviv
- Dahlem Center for Complex Quantum Systems and Fachbereich Physik, Freie Universität Berlin, 14195 Berlin, Germany
| | - Leonid I Glazman
- Department of Physics, Yale University, New Haven, Connecticut 06520, USA
| | - Felix von Oppen
- Dahlem Center for Complex Quantum Systems and Fachbereich Physik, Freie Universität Berlin, 14195 Berlin, Germany
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45
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Gao Y, Zhou T, Huang H, Huang R. Majorana zero modes in the hopping-modulated one-dimensional p-wave superconducting model. Sci Rep 2015; 5:17049. [PMID: 26586205 PMCID: PMC4653629 DOI: 10.1038/srep17049] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/19/2015] [Accepted: 10/23/2015] [Indexed: 11/23/2022] Open
Abstract
We investigate the one-dimensional p-wave superconducting model with
periodically modulated hopping and show that under time-reversal symmetry, the
number of the Majorana zero modes (MZMs) strongly depends on the modulation period.
If the modulation period is odd, there can be at most one MZM. However if the period
is even, the number of the MZMs can be zero, one and two. In addition, the MZMs will
disappear as the chemical potential varies. We derive the condition for the
existence of the MZMs and show that the topological properties in this model are
dramatically different from the one with periodically modulated potential.
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Affiliation(s)
- Yi Gao
- Department of Physics and Institute of Theoretical Physics, Nanjing Normal University, Nanjing, 210023, China
| | - Tao Zhou
- College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China
| | - Huaixiang Huang
- Department of Physics, Shanghai University, Shanghai, 200444, China
| | - Ran Huang
- Department of Physics and Institute of Theoretical Physics, Nanjing Normal University, Nanjing, 210023, China
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46
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Sedlmayr N, Guigou M, Simon P, Bena C. Majoranas with and without a 'character': hybridization, braiding and chiral Majorana number. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2015; 27:455601. [PMID: 26490497 DOI: 10.1088/0953-8984/27/45/455601] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
In this paper we demonstrate under what conditions a pseudo-spin degree of freedom or character can be ascribed to Majorana bound states (MBS). These exotic states can be created at the boundaries of non-interacting systems, corresponding to D, DIII and BDI in the usual classification scheme, and we focus on one dimension. We have found that such a character is directly related to the class of the topological superconductor and its description by a Z, rather than a Zs, invariant which corresponds to the BDI class. We have also found that the DIII case with mirror symmetry, which supports multiple MBS, is in fact equivalent to the BDI class with an additional time-reversal symmetry. In all cases where a character can be given to the Majorana states we show how to construct the appropriate local operator explicitly with various examples. We also examine the consequences of the Majorana character by considering possible hybridization of MBS brought into proximity and find that two MBS with the same character do not hybridize. Finally, we show that having this character or not has no consequence on the braiding properties of MBS.
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Affiliation(s)
- N Sedlmayr
- Institut de Physique Théorique, CEA/Saclay, Orme des Merisiers, 91190 Gif-sur-Yvette Cedex, France
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47
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Ruby M, Pientka F, Peng Y, von Oppen F, Heinrich BW, Franke KJ. End States and Subgap Structure in Proximity-Coupled Chains of Magnetic Adatoms. PHYSICAL REVIEW LETTERS 2015; 115:197204. [PMID: 26588411 DOI: 10.1103/physrevlett.115.197204] [Citation(s) in RCA: 33] [Impact Index Per Article: 3.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/21/2015] [Indexed: 05/27/2023]
Abstract
A recent experiment [Nadj-Perge et al., Science 346, 602 (2014)] provides evidence for Majorana zero modes in iron (Fe) chains on the superconducting Pb(110) surface. Here, we study this system by scanning tunneling microscopy using superconducting tips. This high-resolution technique resolves a rich subgap structure, including zero-energy excitations in some chains. We compare the symmetry properties of the data under voltage reversal against theoretical expectations and provide evidence that the putative Majorana signature overlaps with a previously unresolved low-energy resonance. Interpreting the data within a Majorana framework suggests that the topological gap is smaller than previously extracted from experiment. Aided by model calculations, we also analyze higher-energy features of the subgap spectrum and their relation to high-bias peaks which we associate with the Fe d bands.
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Affiliation(s)
- Michael Ruby
- Fachbereich Physik, Freie Universität Berlin, 14195 Berlin, Germany
| | - Falko Pientka
- Dahlem Center for Complex Quantum Systems and Fachbereich Physik, Freie Universität Berlin, 14195 Berlin, Germany
| | - Yang Peng
- Dahlem Center for Complex Quantum Systems and Fachbereich Physik, Freie Universität Berlin, 14195 Berlin, Germany
| | - Felix von Oppen
- Dahlem Center for Complex Quantum Systems and Fachbereich Physik, Freie Universität Berlin, 14195 Berlin, Germany
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48
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Pershoguba SS, Björnson K, Black-Schaffer AM, Balatsky AV. Currents induced by magnetic impurities in superconductors with spin-orbit coupling. PHYSICAL REVIEW LETTERS 2015; 115:116602. [PMID: 26406845 DOI: 10.1103/physrevlett.115.116602] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/21/2015] [Indexed: 06/05/2023]
Abstract
We show that superconducting currents are generated around magnetic impurities and ferromagnetic islands proximity coupled to superconductors with finite spin-orbit coupling. Using the Ginzburg-Landau theory, T-matrix calculation, as well as self-consistent numerical simulation on a lattice, we find a strong dependence of the current on the direction and magnitude of the magnetic moment. We establish that in the case of point magnetic impurities, the current is carried by the induced Yu-Shiba-Rusinov (YSR) subgap states. In the vicinity of the phase transition, where the YSR states cross at zero energy, the current increases dramatically. Furthermore, we show that the currents are orthogonal to the local spin polarization and, thus, can be probed by measuring the spin-polarized local density of states.
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Affiliation(s)
- Sergey S Pershoguba
- Nordita, Center for Quantum Materials, KTH Royal Institute of Technology, and Stockholm University, Roslagstullsbacken 23, S-106 91 Stockholm, Sweden
| | - Kristofer Björnson
- Department of Physics and Astronomy, Uppsala University, Box 516, S-751 20 Uppsala, Sweden
| | | | - Alexander V Balatsky
- Nordita, Center for Quantum Materials, KTH Royal Institute of Technology, and Stockholm University, Roslagstullsbacken 23, S-106 91 Stockholm, Sweden
- Institute for Materials Science, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
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49
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Röntynen J, Ojanen T. Topological Superconductivity and High Chern Numbers in 2D Ferromagnetic Shiba Lattices. PHYSICAL REVIEW LETTERS 2015; 114:236803. [PMID: 26196820 DOI: 10.1103/physrevlett.114.236803] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/18/2014] [Indexed: 05/27/2023]
Abstract
Inspired by the recent experimental observation of topological superconductivity in ferromagnetic chains, we consider a dilute 2D lattice of magnetic atoms deposited on top of a superconducting surface with a Rashba spin-orbit coupling. We show that the studied system supports a generalization of px+ipy superconductivity and that its topological phase diagram contains Chern numbers higher than ξ/a(≫1), where ξ is the superconducting coherence length and a is the distance between the magnetic atoms. The signatures of nontrivial topology can be observed by STM spectroscopy in finite-size islands.
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Affiliation(s)
- Joel Röntynen
- O. V. Lounasmaa Laboratory (LTL), Aalto University, P. O. Box 15100, FI-00076 AALTO, Finland
| | - Teemu Ojanen
- O. V. Lounasmaa Laboratory (LTL), Aalto University, P. O. Box 15100, FI-00076 AALTO, Finland
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50
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Kim Y, Zhang J, Rossi E, Lutchyn RM. Impurity-Induced Bound States in Superconductors with Spin-Orbit Coupling. PHYSICAL REVIEW LETTERS 2015; 114:236804. [PMID: 26196821 DOI: 10.1103/physrevlett.114.236804] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/22/2014] [Indexed: 05/27/2023]
Abstract
We study the effect of strong spin-orbit coupling (SOC) on bound states induced by impurities in superconductors. The presence of SOC breaks the SU(2)-spin symmetry and causes the superconducting order parameter to have generically both singlet (s-wave) and triplet (p-wave) components. We find that in the presence of SOC the spectrum of Yu-Shiba-Rusinov (YSR) states is qualitatively different in s-wave and p-wave superconductors, a fact that can be used to identify the superconducting pairing symmetry of the host system. We also predict that, in the presence of SOC, the spectrum of the impurity-induced bound states depends on the orientation of the magnetic moment S of the impurity and, in particular, that by changing the orientation of S, the fermion-parity of the lowest energy bound state can be tuned. We then study the case of a dimer of magnetic impurities and show that, in this case, the YSR spectrum for a p-wave superconductor is qualitatively very different from the one for an s-wave superconductor even in the limit of vanishing SOC.
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Affiliation(s)
- Younghyun Kim
- Department of Physics, University of California, Santa Barbara, California 93106, USA
| | - Junhua Zhang
- Department of Physics, College of William and Mary, Williamsburg, Virginia 23187, USA
| | - E Rossi
- Department of Physics, College of William and Mary, Williamsburg, Virginia 23187, USA
| | - Roman M Lutchyn
- Station Q, Microsoft Research, Santa Barbara, California 93106-6105, USA
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