1
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Liao L, Chen F, Puebla J, Kishine JI, Kondou K, Luo W, Zhao D, Zhang Y, Ba Y, Otani Y. Nonreciprocal magnetoacoustic waves with out-of-plane phononic angular momenta. SCIENCE ADVANCES 2024; 10:eado2504. [PMID: 38985868 PMCID: PMC11235162 DOI: 10.1126/sciadv.ado2504] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/29/2024] [Accepted: 06/04/2024] [Indexed: 07/12/2024]
Abstract
Surface acoustic wave (SAW) can carry phononic angular momentum, showing great potential as an energy-efficient way to control magnetism. Still, out-of-plane phononic angular momentum in SAW and its interaction with magnetism remain elusive. Here, we studied the SAW-induced magnetoacoustic waves and spin pumping in Ni-based films on LiNbO3 with selected SAW propagation direction. The crystal inversion asymmetry induces circularly polarized phonons with large out-of-plane angular momenta so that up to 60% of the SAW power attenuates nonreciprocally controlled by the out-of-plane magnetization component. The SAW propagation direction dependence of the nonreciprocity verifies the crystal origin of the phononic angular momentum, and a chiral spin pumping demonstrates that the circular polarization can control the spin current generation efficiency. These results provide an additional degree of freedom for the acoustic control of magnetism and open an avenue for applying circularly polarized phonons.
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Affiliation(s)
- Liyang Liao
- Institute for Solid State Physics, University of Tokyo, Kashiwa 277-8581, Japan
| | - Fa Chen
- School of Integrated Circuits, Huazhong University of Science and Technology, Wuhan 430074, China
| | - Jorge Puebla
- Center for Emergent Matter Science, RIKEN, Wako, Saitama 351-0198, Japan
| | - Jun-ichiro Kishine
- The Open University of Japan, Chiba 261-0013, Japan
- Quantum Research Center for Chirality, Institute for Molecular Science, Aichi 444-8585, Japan
| | - Kouta Kondou
- Center for Emergent Matter Science, RIKEN, Wako, Saitama 351-0198, Japan
| | - Wei Luo
- School of Integrated Circuits, Huazhong University of Science and Technology, Wuhan 430074, China
| | - Degang Zhao
- School of Physics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China
| | - Yue Zhang
- School of Integrated Circuits, Huazhong University of Science and Technology, Wuhan 430074, China
| | - You Ba
- Center for Emergent Matter Science, RIKEN, Wako, Saitama 351-0198, Japan
| | - Yoshichika Otani
- Institute for Solid State Physics, University of Tokyo, Kashiwa 277-8581, Japan
- Center for Emergent Matter Science, RIKEN, Wako, Saitama 351-0198, Japan
- Trans-scale Quantum Science Institute, University of Tokyo, Tokyo 113-8654, Japan
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2
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Manley ME, Stonaha PJ, Bruno NM, Karaman I, Arroyave R, Chi S, Abernathy DL, Stone MB, Chumlyakov YI, Lynn JW. Hybrid magnon-phonon localization enhances function near ferroic glassy states. SCIENCE ADVANCES 2024; 10:eadn2840. [PMID: 38875343 PMCID: PMC11177935 DOI: 10.1126/sciadv.adn2840] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/29/2023] [Accepted: 05/09/2024] [Indexed: 06/16/2024]
Abstract
Ferroic materials on the verge of forming ferroic glasses exhibit heightened functionality that is often attributed to competing long- and short-range correlations. However, the physics underlying these enhancements is not well understood. The Ni45Co5Mn36.6In13.4 Heusler alloy is on the edge of forming both spin and strain glasses and exhibits magnetic field-induced shape memory and large magnetocaloric effects, making it a candidate for multicaloric cooling applications. We show using neutron scattering that localized magnon-phonon hybrid modes, which are inherently spread across reciprocal space, act as a bridge between phonons and magnons and result in substantial magnetic field-induced shifts in the phonons, triple the caloric response, and alter phase stability. We attribute these modes to the localization of phonons and magnons by antiphase boundaries coupled to magnetic domains. Because the interplay between short- and long-range correlations is common near ferroic glassy states, our work provides general insights on how glassiness enhances function.
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Affiliation(s)
- Michael E Manley
- Materials Sciences and Technology Division, Oak Ridge National Lab, Oak Ridge, TN 37831, USA
| | - Paul J Stonaha
- Materials Sciences and Technology Division, Oak Ridge National Lab, Oak Ridge, TN 37831, USA
| | - Nickolaus M Bruno
- Department of Materials Science and Engineering, Texas A&M University, College Station, TX 77843, USA
- NASA Glenn Research Center, Cleveland, OH 44135, USA
| | - Ibrahim Karaman
- Department of Materials Science and Engineering, Texas A&M University, College Station, TX 77843, USA
| | - Raymundo Arroyave
- Department of Materials Science and Engineering, Texas A&M University, College Station, TX 77843, USA
| | - Songxue Chi
- Neutron Scattering Division, Oak Ridge National Lab, Oak Ridge, TN 37831, USA
| | - Douglas L Abernathy
- Neutron Scattering Division, Oak Ridge National Lab, Oak Ridge, TN 37831, USA
| | - Matthew B Stone
- Neutron Scattering Division, Oak Ridge National Lab, Oak Ridge, TN 37831, USA
| | - Yuri I Chumlyakov
- Siberian Physical Technical Institute, Tomsk State University, Tomsk, Russia
| | - Jeffrey W Lynn
- NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD, 20899, USA
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3
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Sano R, Ominato Y, Matsuo M. Acoustomagnonic Spin Hall Effect in Honeycomb Antiferromagnets. PHYSICAL REVIEW LETTERS 2024; 132:236302. [PMID: 38905670 DOI: 10.1103/physrevlett.132.236302] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/19/2023] [Accepted: 05/03/2024] [Indexed: 06/23/2024]
Abstract
The recently discovered Van der Waals antiferromagnets have suffered from the lack of a comprehensive method to study their magnetic properties. Here, we propose an ac intrinsic magnon spin Hall current driven by surface acoustic waves as a novel probe for such antiferromagnets. Our results pave the way towards mechanical detection and manipulation of the magnetic order in two-dimensional antiferromagnets. Furthermore, they will overcome the difficulties with weak magnetic responses inherent in the use of antiferromagnets and hence provide a building block for future antiferromagnetic spintronics.
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Affiliation(s)
| | | | - Mamoru Matsuo
- Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China
- CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100190, China
- RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan
- Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan
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4
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Liu S, Wang Q, Long MQ, Wang YP. Theoretical study of the nonlinear magnon-phonon coupling in CoF 2. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2024; 36:355801. [PMID: 38768607 DOI: 10.1088/1361-648x/ad4dfa] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/25/2024] [Accepted: 05/20/2024] [Indexed: 05/22/2024]
Abstract
The coupling and interplay between magnon and phonon are important topics for spintronics and magnonics. In this work we studied the nonlinear magnon-phonon coupling in CoF2. First-principles calculations demonstrate that the antiferromagnetic resonance magnon drives a phonon with B1gcharacter; the oscillating driving force has a frequency twice of that of the magnon. Comparing with similar materials indicates a strong correlation between the strength of nonlinear magnon-phonon coupling and the orbital magnetic moment of the magnetic ion. This work pave the way for theoretical study of nonlinear magnon-phonon coupling.
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Affiliation(s)
- Shuang Liu
- School of Physics, Central South University, Changsha, People's Republic of China
| | - Qian Wang
- School of Physics, Central South University, Changsha, People's Republic of China
| | - Meng-Qiu Long
- School of Physics, Central South University, Changsha, People's Republic of China
| | - Yun-Peng Wang
- School of Physics, Central South University, Changsha, People's Republic of China
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5
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Go G, An D, Lee HW, Kim SK. Magnon Orbital Nernst Effect in Honeycomb Antiferromagnets without Spin-Orbit Coupling. NANO LETTERS 2024; 24:5968-5974. [PMID: 38682941 PMCID: PMC11117403 DOI: 10.1021/acs.nanolett.4c00430] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/26/2024] [Revised: 04/24/2024] [Accepted: 04/25/2024] [Indexed: 05/01/2024]
Abstract
Recently, topological responses of magnons have emerged as a central theme in magnetism and spintronics. However, resulting Hall responses are typically weak and infrequent, since, according to present understanding, they arise from effective spin-orbit couplings, which are weaker compared to the exchange energy. Here, by investigating transport properties of magnon orbital moments, we predict that the magnon orbital Nernst effect is an intrinsic characteristic of the honeycomb antiferromagnet and therefore, it manifests even in the absence of spin-orbit coupling. For the electric detection, we propose an experimental scheme based on the magnetoelectric effect. Our results break the conventional wisdom that the Hall transport of magnons requires spin-orbit coupling by predicting the magnon orbital Nernst effect in a system without it, which leads us to envision that our work initiates the intensive search for various magnon Hall effects in generic magnetic systems with no reliance on spin-orbit coupling.
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Affiliation(s)
- Gyungchoon Go
- Department
of Physics, Korea Advanced Institute of
Science and Technology, Daejeon 34141, Korea
| | - Daehyeon An
- Department
of Physics, Korea Advanced Institute of
Science and Technology, Daejeon 34141, Korea
| | - Hyun-Woo Lee
- Department
of Physics, Pohang University of Science
and Technology, Pohang 37673, Korea
| | - Se Kwon Kim
- Department
of Physics, Korea Advanced Institute of
Science and Technology, Daejeon 34141, Korea
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6
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Fox C, Mao Y, Zhang X, Wang Y, Xiao J. Stacking Order Engineering of Two-Dimensional Materials and Device Applications. Chem Rev 2024; 124:1862-1898. [PMID: 38150266 DOI: 10.1021/acs.chemrev.3c00618] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/28/2023]
Abstract
Stacking orders in 2D van der Waals (vdW) materials dictate the relative sliding (lateral displacement) and twisting (rotation) between atomically thin layers. By altering the stacking order, many new ferroic, strongly correlated and topological orderings emerge with exotic electrical, optical and magnetic properties. Thanks to the weak vdW interlayer bonding, such highly flexible and energy-efficient stacking order engineering has transformed the design of quantum properties in 2D vdW materials, unleashing the potential for miniaturized high-performance device applications in electronics, spintronics, photonics, and surface chemistry. This Review provides a comprehensive overview of stacking order engineering in 2D vdW materials and their device applications, ranging from the typical fabrication and characterization methods to the novel physical properties and the emergent slidetronics and twistronics device prototyping. The main emphasis is on the critical role of stacking orders affecting the interlayer charge transfer, orbital coupling and flat band formation for the design of innovative materials with on-demand quantum properties and surface potentials. By demonstrating a correlation between the stacking configurations and device functionality, we highlight their implications for next-generation electronic, photonic and chemical energy conversion devices. We conclude with our perspective of this exciting field including challenges and opportunities for future stacking order engineering research.
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Affiliation(s)
- Carter Fox
- Department of Materials Science and Engineering, University of Wisconsin─Madison, Madison, Wisconsin 53706, United States
- Department of Physics, University of Wisconsin─Madison, Madison, Wisconsin 53706, United States
| | - Yulu Mao
- Department of Electrical and Computer Engineering, University of Wisconsin─Madison, Madison, Wisconsin 53706, United States
| | - Xiang Zhang
- Faculty of Science, University of Hong Kong, Hong Kong, China
- Faculty of Engineering, University of Hong Kong, Hong Kong, China
| | - Ying Wang
- Department of Materials Science and Engineering, University of Wisconsin─Madison, Madison, Wisconsin 53706, United States
- Department of Physics, University of Wisconsin─Madison, Madison, Wisconsin 53706, United States
- Department of Electrical and Computer Engineering, University of Wisconsin─Madison, Madison, Wisconsin 53706, United States
| | - Jun Xiao
- Department of Materials Science and Engineering, University of Wisconsin─Madison, Madison, Wisconsin 53706, United States
- Department of Physics, University of Wisconsin─Madison, Madison, Wisconsin 53706, United States
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7
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Zou J, Bosco S, Thingstad E, Klinovaja J, Loss D. Dissipative Spin-Wave Diode and Nonreciprocal Magnonic Amplifier. PHYSICAL REVIEW LETTERS 2024; 132:036701. [PMID: 38307041 DOI: 10.1103/physrevlett.132.036701] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/28/2023] [Accepted: 12/01/2023] [Indexed: 02/04/2024]
Abstract
We propose an experimentally feasible dissipative spin-wave diode comprising two magnetic layers coupled via a nonmagnetic spacer. We theoretically demonstrate that the spacer mediates not only coherent interactions but also dissipative coupling. Interestingly, an appropriately engineered dissipation engenders a nonreciprocal device response, facilitating the realization of a spin-wave diode. This diode permits wave propagation in one direction alone, given that the coherent Dzyaloshinskii-Moriya (DM) interaction is balanced with the dissipative coupling. The polarity of the diode is determined by the sign of the DM interaction. Furthermore, we show that when the magnetic layers undergo incoherent pumping, the device operates as a unidirectional spin-wave amplifier. The amplifier gain is augmented by cascading multiple magnetic bilayers. By extending our model to a one-dimensional ring structure, we establish a connection between the physics of spin-wave amplification and non-Hermitian topology. Our proposal opens up a new avenue for harnessing inherent dissipation in spintronic applications.
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Affiliation(s)
- Ji Zou
- Department of Physics, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland
| | - Stefano Bosco
- Department of Physics, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland
| | - Even Thingstad
- Department of Physics, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland
| | - Jelena Klinovaja
- Department of Physics, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland
| | - Daniel Loss
- Department of Physics, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland
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8
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Ueda H, Mankowsky R, Paris E, Sander M, Deng Y, Liu B, Leroy L, Nag A, Skoropata E, Wang C, Ukleev V, Perren GS, Dössegger J, Gurung S, Svetina C, Abreu E, Savoini M, Kimura T, Patthey L, Razzoli E, Lemke HT, Johnson SL, Staub U. Non-equilibrium dynamics of spin-lattice coupling. Nat Commun 2023; 14:7778. [PMID: 38012165 PMCID: PMC10681982 DOI: 10.1038/s41467-023-43581-9] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/03/2023] [Accepted: 11/14/2023] [Indexed: 11/29/2023] Open
Abstract
Quantifying the dynamics of normal modes and how they interact with other excitations is of central importance in condensed matter. Spin-lattice coupling is relevant to several sub-fields of condensed matter physics; examples include spintronics, high-Tc superconductivity, and topological materials. However, experimental approaches that can directly measure it are rare and incomplete. Here we use time-resolved X-ray diffraction to directly access the ultrafast motion of atoms and spins following the coherent excitation of an electromagnon in a multiferroic hexaferrite. One striking outcome is the different phase shifts relative to the driving field of the two different components. This phase shift provides insight into the excitation process of such a coupled mode. This direct observation of combined lattice and magnetization dynamics paves the way to access the mode-selective spin-lattice coupling strength, which remains a missing fundamental parameter for ultrafast control of magnetism and is relevant to a wide variety of materials.
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Affiliation(s)
- Hiroki Ueda
- SwissFEL, Paul Scherrer Institute, 5232, Villigen-PSI, Switzerland.
- Swiss Light Source, Paul Scherrer Institute, 5232, Villigen-PSI, Switzerland.
| | - Roman Mankowsky
- SwissFEL, Paul Scherrer Institute, 5232, Villigen-PSI, Switzerland
| | - Eugenio Paris
- SwissFEL, Paul Scherrer Institute, 5232, Villigen-PSI, Switzerland
| | - Mathias Sander
- SwissFEL, Paul Scherrer Institute, 5232, Villigen-PSI, Switzerland
| | - Yunpei Deng
- SwissFEL, Paul Scherrer Institute, 5232, Villigen-PSI, Switzerland
| | - Biaolong Liu
- SwissFEL, Paul Scherrer Institute, 5232, Villigen-PSI, Switzerland
| | - Ludmila Leroy
- Swiss Light Source, Paul Scherrer Institute, 5232, Villigen-PSI, Switzerland
| | - Abhishek Nag
- SwissFEL, Paul Scherrer Institute, 5232, Villigen-PSI, Switzerland
| | - Elizabeth Skoropata
- Swiss Light Source, Paul Scherrer Institute, 5232, Villigen-PSI, Switzerland
| | - Chennan Wang
- Départment de Physique and Fribourg Center for Nanomaterials, Université de Fribourg, 1700, Fribourg, Switzerland
| | - Victor Ukleev
- Swiss Light Source, Paul Scherrer Institute, 5232, Villigen-PSI, Switzerland
- Helmholtz-Zentrum Berlin für Materialien und Energie, Albert-Einstein-Straße 15, 12489, Berlin, Germany
| | | | - Janine Dössegger
- Institute for Quantum Electronics, Physics Department, ETH Zurich, 8093, Zurich, Switzerland
| | - Sabina Gurung
- Institute for Quantum Electronics, Physics Department, ETH Zurich, 8093, Zurich, Switzerland
| | - Cristian Svetina
- SwissFEL, Paul Scherrer Institute, 5232, Villigen-PSI, Switzerland
- Madrid Institute for Advanced Studies, IMDEA Nanociencia, Ciudad Universitaria de Cantoblanco, Calle Faraday 9, Madrid, 28049, Spain
| | - Elsa Abreu
- Institute for Quantum Electronics, Physics Department, ETH Zurich, 8093, Zurich, Switzerland
| | - Matteo Savoini
- Institute for Quantum Electronics, Physics Department, ETH Zurich, 8093, Zurich, Switzerland
| | - Tsuyoshi Kimura
- Department of Advanced Materials Science, University of Tokyo, Kashiwa, Chiba, 277-8561, Japan
| | - Luc Patthey
- SwissFEL, Paul Scherrer Institute, 5232, Villigen-PSI, Switzerland
| | - Elia Razzoli
- SwissFEL, Paul Scherrer Institute, 5232, Villigen-PSI, Switzerland
| | | | - Steven Lee Johnson
- SwissFEL, Paul Scherrer Institute, 5232, Villigen-PSI, Switzerland
- Institute for Quantum Electronics, Physics Department, ETH Zurich, 8093, Zurich, Switzerland
| | - Urs Staub
- Swiss Light Source, Paul Scherrer Institute, 5232, Villigen-PSI, Switzerland.
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9
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Dos Santos Dias M, Biniskos N, Dos Santos FJ, Schmalzl K, Persson J, Bourdarot F, Marzari N, Blügel S, Brückel T, Lounis S. Topological magnons driven by the Dzyaloshinskii-Moriya interaction in the centrosymmetric ferromagnet Mn 5Ge 3. Nat Commun 2023; 14:7321. [PMID: 37951946 PMCID: PMC10640582 DOI: 10.1038/s41467-023-43042-3] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/09/2023] [Accepted: 10/31/2023] [Indexed: 11/14/2023] Open
Abstract
The phase of the quantum-mechanical wave function can encode a topological structure with wide-ranging physical consequences, such as anomalous transport effects and the existence of edge states robust against perturbations. While this has been exhaustively demonstrated for electrons, properties associated with the elementary quasiparticles in magnetic materials are still underexplored. Here, we show theoretically and via inelastic neutron scattering experiments that the bulk ferromagnet Mn5Ge3 hosts gapped topological Dirac magnons. Although inversion symmetry prohibits a net Dzyaloshinskii-Moriya interaction in the unit cell, it is locally allowed and is responsible for the gap opening in the magnon spectrum. This gap is predicted and experimentally verified to close by rotating the magnetization away from the c-axis with an applied magnetic field. Hence, Mn5Ge3 realizes a gapped Dirac magnon material in three dimensions. Its tunability by chemical doping or by thin film nanostructuring defines an exciting new platform to explore and design topological magnons. More generally, our experimental route to verify and control the topological character of the magnons is applicable to bulk centrosymmetric hexagonal materials, which calls for systematic investigation.
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Affiliation(s)
- M Dos Santos Dias
- Peter Grünberg Institut and Institute for Advanced Simulation, Forschungszentrum Jülich & JARA, D-52425, Jülich, Germany.
- Faculty of Physics, University of Duisburg-Essen and CENIDE, D-47053, Duisburg, Germany.
- Scientific Computing Department, STFC Daresbury Laboratory, Warrington, WA4 4AD, UK.
| | - N Biniskos
- Forschungszentrum Jülich GmbH, Jülich Centre for Neutron Science at MLZ, Lichtenbergstr. 1, D-85748, Garching, Germany.
- Charles University, Faculty of Mathematics and Physics, Department of Condensed Matter Physics, Ke Karlovu 5, 121 16, Praha, Czech Republic.
| | - F J Dos Santos
- Laboratory for Materials Simulations, Paul Scherrer Institut, 5232, Villigen, PSI, Switzerland.
- Theory and Simulation of Materials (THEOS), and National Centre for Computational Design and Discovery of Novel Materials (MARVEL), École Polytechnique Fédérale de Lausanne, 1015, Lausanne, Switzerland.
| | - K Schmalzl
- Forschungszentrum Jülich GmbH, Jülich Centre for Neutron Science at ILL, 71 Avenue des Martyrs, F-38000, Grenoble, France
| | - J Persson
- Forschungszentrum Jülich GmbH, Jülich Centre for Neutron Science (JCNS-2) and Peter Grünberg Institut (PGI-4), JARA-FIT, D-52425, Jülich, Germany
| | - F Bourdarot
- Université Grenoble Alpes, CEA, IRIG, MEM, MDN, F-38000, Grenoble, France
| | - N Marzari
- Laboratory for Materials Simulations, Paul Scherrer Institut, 5232, Villigen, PSI, Switzerland
- Theory and Simulation of Materials (THEOS), and National Centre for Computational Design and Discovery of Novel Materials (MARVEL), École Polytechnique Fédérale de Lausanne, 1015, Lausanne, Switzerland
| | - S Blügel
- Peter Grünberg Institut and Institute for Advanced Simulation, Forschungszentrum Jülich & JARA, D-52425, Jülich, Germany
| | - T Brückel
- Forschungszentrum Jülich GmbH, Jülich Centre for Neutron Science (JCNS-2) and Peter Grünberg Institut (PGI-4), JARA-FIT, D-52425, Jülich, Germany
| | - S Lounis
- Peter Grünberg Institut and Institute for Advanced Simulation, Forschungszentrum Jülich & JARA, D-52425, Jülich, Germany
- Faculty of Physics, University of Duisburg-Essen and CENIDE, D-47053, Duisburg, Germany
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10
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Liao L, Puebla J, Yamamoto K, Kim J, Maekawa S, Hwang Y, Ba Y, Otani Y. Valley-Selective Phonon-Magnon Scattering in Magnetoelastic Superlattices. PHYSICAL REVIEW LETTERS 2023; 131:176701. [PMID: 37955488 DOI: 10.1103/physrevlett.131.176701] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/29/2023] [Accepted: 09/06/2023] [Indexed: 11/14/2023]
Abstract
Phonons and magnons are engineered by periodic potential landscapes in phononic and magnonic crystals, and their combined studies may enable valley phonon transport tunable by the magnetic field. Through nonreciprocal surface acoustic wave transmission, we demonstrate valley-selective phonon-magnon scattering in magnetoelastic superlattices. The lattice symmetry and the out-of-plane magnetization component control the sign of nonreciprocity. The phonons in the valleys play a crucial role in generating nonreciprocal transmission by inducing circularly polarized strains that couple with the magnons. The transmission spectra show a nonreciprocity peak near a transmission gap, matching the phononic band structure. Our results open the way for manipulating valley phonon transport through periodically varying magnon-phonon coupling.
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Affiliation(s)
- Liyang Liao
- Institute for Solid State Physics, University of Tokyo, Kashiwa 277-8581, Japan
| | - Jorge Puebla
- Center for Emergent Matter Science, RIKEN, Wako, Saitama 351-0198, Japan
| | - Kei Yamamoto
- Center for Emergent Matter Science, RIKEN, Wako, Saitama 351-0198, Japan
- Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan
| | - Junyeon Kim
- Center for Emergent Matter Science, RIKEN, Wako, Saitama 351-0198, Japan
| | - Sadamichi Maekawa
- Center for Emergent Matter Science, RIKEN, Wako, Saitama 351-0198, Japan
- Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan
- Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China
| | - Yunyoung Hwang
- Institute for Solid State Physics, University of Tokyo, Kashiwa 277-8581, Japan
- Center for Emergent Matter Science, RIKEN, Wako, Saitama 351-0198, Japan
| | - You Ba
- Center for Emergent Matter Science, RIKEN, Wako, Saitama 351-0198, Japan
| | - Yoshichika Otani
- Institute for Solid State Physics, University of Tokyo, Kashiwa 277-8581, Japan
- Center for Emergent Matter Science, RIKEN, Wako, Saitama 351-0198, Japan
- Trans-scale Quantum Science Institute, University of Tokyo, Tokyo 113-8654, Japan
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11
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Bao S, Gu ZL, Shangguan Y, Huang Z, Liao J, Zhao X, Zhang B, Dong ZY, Wang W, Kajimoto R, Nakamura M, Fennell T, Yu SL, Li JX, Wen J. Direct observation of topological magnon polarons in a multiferroic material. Nat Commun 2023; 14:6093. [PMID: 37773159 PMCID: PMC10541872 DOI: 10.1038/s41467-023-41791-9] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/13/2023] [Accepted: 09/19/2023] [Indexed: 10/01/2023] Open
Abstract
Magnon polarons are novel elementary excitations possessing hybrid magnonic and phononic signatures, and are responsible for many exotic spintronic and magnonic phenomena. Despite long-term sustained experimental efforts in chasing for magnon polarons, direct spectroscopic evidence of their existence is hardly observed. Here, we report the direct observation of magnon polarons using neutron spectroscopy on a multiferroic Fe2Mo3O8 possessing strong magnon-phonon coupling. Specifically, below the magnetic ordering temperature, a gap opens at the nominal intersection of the original magnon and phonon bands, leading to two separated magnon-polaron bands. Each of the bands undergoes mixing, interconverting and reversing between its magnonic and phononic components. We attribute the formation of magnon polarons to the strong magnon-phonon coupling induced by Dzyaloshinskii-Moriya interaction. Intriguingly, we find that the band-inverted magnon polarons are topologically nontrivial. These results uncover exotic elementary excitations arising from the magnon-phonon coupling, and offer a new route to topological states by considering hybridizations between different types of fundamental excitations.
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Affiliation(s)
- Song Bao
- National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing, 210093, China
| | - Zhao-Long Gu
- National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing, 210093, China
| | - Yanyan Shangguan
- National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing, 210093, China
| | - Zhentao Huang
- National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing, 210093, China
| | - Junbo Liao
- National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing, 210093, China
| | - Xiaoxue Zhao
- National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing, 210093, China
| | - Bo Zhang
- National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing, 210093, China
| | - Zhao-Yang Dong
- Department of Applied Physics, Nanjing University of Science and Technology, Nanjing, 210094, China
| | - Wei Wang
- School of Science, Nanjing University of Posts and Telecommunications, Nanjing, 210023, China
| | - Ryoichi Kajimoto
- J-PARC Center, Japan Atomic Energy Agency (JAEA), Tokai, Ibaraki, 319-1195, Japan
| | - Mitsutaka Nakamura
- J-PARC Center, Japan Atomic Energy Agency (JAEA), Tokai, Ibaraki, 319-1195, Japan
| | - Tom Fennell
- Laboratory for Neutron Scattering and Imaging, Paul Scherrer Institute (PSI), CH-5232, Villigen, Switzerland
| | - Shun-Li Yu
- National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing, 210093, China.
- Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing, 210093, China.
| | - Jian-Xin Li
- National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing, 210093, China.
- Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing, 210093, China.
| | - Jinsheng Wen
- National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing, 210093, China.
- Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing, 210093, China.
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12
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Cui J, Boström EV, Ozerov M, Wu F, Jiang Q, Chu JH, Li C, Liu F, Xu X, Rubio A, Zhang Q. Chirality selective magnon-phonon hybridization and magnon-induced chiral phonons in a layered zigzag antiferromagnet. Nat Commun 2023; 14:3396. [PMID: 37296106 DOI: 10.1038/s41467-023-39123-y] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/29/2022] [Accepted: 05/30/2023] [Indexed: 06/12/2023] Open
Abstract
Two-dimensional (2D) magnetic systems possess versatile magnetic order and can host tunable magnons carrying spin angular momenta. Recent advances show angular momentum can also be carried by lattice vibrations in the form of chiral phonons. However, the interplay between magnons and chiral phonons as well as the details of chiral phonon formation in a magnetic system are yet to be explored. Here, we report the observation of magnon-induced chiral phonons and chirality selective magnon-phonon hybridization in a layered zigzag antiferromagnet (AFM) FePSe3. With a combination of magneto-infrared and magneto-Raman spectroscopy, we observe chiral magnon polarons (chiMP), the new hybridized quasiparticles, at zero magnetic field. The hybridization gap reaches 0.25 meV and survives down to the quadrilayer limit. Via first principle calculations, we uncover a coherent coupling between AFM magnons and chiral phonons with parallel angular momenta, which arises from the underlying phonon and space group symmetries. This coupling lifts the chiral phonon degeneracy and gives rise to an unusual Raman circular polarization of the chiMP branches. The observation of coherent chiral spin-lattice excitations at zero magnetic field paves the way for angular momentum-based hybrid phononic and magnonic devices.
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Affiliation(s)
- Jun Cui
- National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, 210093, Nanjing, China
| | - Emil Viñas Boström
- Max Planck Institute for the Structure and Dynamics of Matter, Luruper Chaussee 149, 22761, Hamburg, Germany
| | - Mykhaylo Ozerov
- National High Magnetic Field Laboratory, Florida State University, Tallahassee, FL, 32310, USA.
| | - Fangliang Wu
- National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, 210093, Nanjing, China
| | - Qianni Jiang
- Department of Physics, University of Washington, Seattle, WA, 98195, USA
| | - Jiun-Haw Chu
- Department of Physics, University of Washington, Seattle, WA, 98195, USA
| | - Changcun Li
- School of Optoelectronic Science and Engineering, University of Electronic Science and Technology of China, 611731, Chengdu, China
| | - Fucai Liu
- School of Optoelectronic Science and Engineering, University of Electronic Science and Technology of China, 611731, Chengdu, China
| | - Xiaodong Xu
- Department of Physics, University of Washington, Seattle, WA, 98195, USA
- Department of Materials Science and Engineering, University of Washington, Seattle, WA, 98195, USA
| | - Angel Rubio
- Max Planck Institute for the Structure and Dynamics of Matter, Luruper Chaussee 149, 22761, Hamburg, Germany.
- Center for Computational Quantum Physics, The Flatiron Institute, New York, NY, 10010, USA.
| | - Qi Zhang
- National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, 210093, Nanjing, China.
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13
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Zeng YJ, Ding ZK, Pan H, Feng YX, Chen KQ. Nonequilibrium Green's function method for phonon heat transport in quantum system. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2022; 34:223001. [PMID: 35263716 DOI: 10.1088/1361-648x/ac5c21] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/24/2021] [Accepted: 03/09/2022] [Indexed: 06/14/2023]
Abstract
Phonon heat transport property in quantum devices is of great interesting since it presents significant quantum behaviors. In the past few decades, great efforts have been devoted to establish the theoretical method for phonon heat transport simulation in nanostructures. However, modeling phonon heat transport from wavelike coherent regime to particlelike incoherent regime remains a challenging task. The widely adopted theoretical approach, such as molecular dynamics, semiclassical Boltzmann transport equation, captures quantum mechanical effects within different degrees of approximation. Among them, Non-equilibrium Green's function (NEGF) method has attracted wide attention, as its ability to perform full quantum simulation including many-body interactions. In this review, we summarized recent theoretical advances of phonon NEGF method and the applications on the numerical simulation for phonon heat transport in nanostructures. At last, the challenges of numerical simulation are discussed.
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Affiliation(s)
- Yu-Jia Zeng
- Department of Physics, School of Physics and Electronic Science, Hunan University, Changsha, People's Republic of China
| | - Zhong-Ke Ding
- Department of Physics, School of Physics and Electronic Science, Hunan University, Changsha, People's Republic of China
| | - Hui Pan
- Department of Physics, School of Physics and Electronic Science, Hunan University, Changsha, People's Republic of China
| | - Ye-Xin Feng
- Department of Physics, School of Physics and Electronic Science, Hunan University, Changsha, People's Republic of China
| | - Ke-Qiu Chen
- Department of Physics, School of Physics and Electronic Science, Hunan University, Changsha, People's Republic of China
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14
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Zhang H, Xu C, Carnahan C, Sretenovic M, Suri N, Xiao D, Ke X. Anomalous Thermal Hall Effect in an Insulating van der Waals Magnet. PHYSICAL REVIEW LETTERS 2021; 127:247202. [PMID: 34951793 DOI: 10.1103/physrevlett.127.247202] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/04/2021] [Revised: 10/12/2021] [Accepted: 11/10/2021] [Indexed: 06/14/2023]
Abstract
Two-dimensional (2D) van der Waals (vdW) magnets have been a fertile playground for the discovery and exploration of physical phenomena and new physics. In this Letter, we report the observation of an anomalous thermal Hall effect (THE) with κ_{xy}∼1×10^{-2} W K^{-1} m^{-1} in an insulating van der Waals ferromagnet VI_{3}. The thermal Hall signal persists in the absence of an external magnetic field and flips sign upon the switching of the magnetization. In combination with theoretical calculations, we show that VI_{3} exhibits a dual nature of the THE, i.e., dominated by topological magnons hosted by the ferromagnetic honeycomb lattice at higher temperatures and by phonons induced by the magnon-phonon coupling at lower temperatures. Our results not only position VI_{3} as the first ferromagnetic system to investigate both anomalous magnon and phonon THEs, but also render it as a potential platform for spintronics-magnonics applications.
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Affiliation(s)
- Heda Zhang
- Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824-2320, USA
| | - Chunqiang Xu
- Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824-2320, USA
- School of Physics Southeast University, Nanjing 211189, China
| | - Caitlin Carnahan
- Department of Physics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
| | - Milos Sretenovic
- Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824-2320, USA
| | - Nishchay Suri
- Department of Physics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
| | - Di Xiao
- Department of Physics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
- Department of Materials Science and Engineering, University of Washington, Seattle, Washington 98195, USA
- Department of Physics, University of Washington, Seattle, Washington 98195, USA
| | - Xianglin Ke
- Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824-2320, USA
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15
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Lu YS, Li JL, Wu CT. Topological Phase Transitions of Dirac Magnons in Honeycomb Ferromagnets. PHYSICAL REVIEW LETTERS 2021; 127:217202. [PMID: 34860095 DOI: 10.1103/physrevlett.127.217202] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/01/2021] [Revised: 08/30/2021] [Accepted: 10/13/2021] [Indexed: 06/13/2023]
Abstract
The study of the magnonic thermal Hall effect in magnets with Dzyaloshinskii-Moriya interaction (DMI) has recently drawn attention because of the underlying topology. Topological phase transitions may arise when there exist two or more distinct topological phases, and they are often revealed by a gap-closing phenomenon. In this work, we consider the magnons in honeycomb ferromagnets described by a Heisenberg Hamiltonian containing both an out-of-plane DMI and a Zeeman interaction. We demonstrate that the magnonic system exhibits temperature (or magnetic field) driven topological phase transitions due to magnon-magnon interactions. Specifically, when the temperature increases, the magnonic energy gap at Dirac points closes and reopens at a critical temperature, T_{c}. By showing that the Chern numbers of the magnonic bands are distinct above and below T_{c}, we confirm that the gap-closing phenomenon is indeed a signature for the topological phase transitions. Furthermore, our analysis indicates that the thermal Hall conductivity in the magnonic system exhibits a sign reversal at T_{c}, which can serve as an experimental probe of its topological nature. Our theory predicts that in CrI_{3} such a phenomenon exists and is experimentally accessible.
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Affiliation(s)
- Yu-Shan Lu
- Department of Electrophysics, National Yang Ming Chiao Tung University, Hsinchu, Taiwan
| | - Jian-Lin Li
- Department of Electrophysics, National Yang Ming Chiao Tung University, Hsinchu, Taiwan
| | - Chien-Te Wu
- Department of Electrophysics, National Yang Ming Chiao Tung University, Hsinchu, Taiwan
- Physics Division, National Center for Theoretical Sciences, Taipei, Taiwan
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16
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Liu S, Granados Del Águila A, Bhowmick D, Gan CK, Thu Ha Do T, Prosnikov MA, Sedmidubský D, Sofer Z, Christianen PCM, Sengupta P, Xiong Q. Direct Observation of Magnon-Phonon Strong Coupling in Two-Dimensional Antiferromagnet at High Magnetic Fields. PHYSICAL REVIEW LETTERS 2021; 127:097401. [PMID: 34506201 DOI: 10.1103/physrevlett.127.097401] [Citation(s) in RCA: 23] [Impact Index Per Article: 7.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/02/2021] [Accepted: 07/30/2021] [Indexed: 05/08/2023]
Abstract
We report the direct observation of strong coupling between magnons and phonons in a two-dimensional antiferromagnetic semiconductor FePS_{3}, via magneto-Raman spectroscopy at magnetic fields up to 30 Tesla. A Raman-active magnon at 121 cm^{-1} is identified through Zeeman splitting in an applied magnetic field. At a field-driven resonance with a nearby phonon mode, a hybridized magnon-phonon quasiparticle is formed due to strong coupling between the two modes. We develop a microscopic model of the strong coupling in the two-dimensional magnetic lattice, which enables us to elucidate the nature of the emergent quasiparticle. Our polarized Raman results directly show that the magnons transfer their spin angular momentum to the phonons and generate phonon spin through the strong coupling.
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Affiliation(s)
- Sheng Liu
- Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371
| | - Andrés Granados Del Águila
- Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371
| | - Dhiman Bhowmick
- Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371
| | - Chee Kwan Gan
- Institute of High Performance Computing, 1 Fusionopolis Way, 16-16 Connexis, Singapore 138632
| | - T Thu Ha Do
- Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371
| | - M A Prosnikov
- High Field Magnet Laboratory, HFML-EMFL, Radboud University, 6525 ED Nijmegen, Netherlands
| | - David Sedmidubský
- Department of Inorganic Chemistry, University of Chemistry and Technology Prague, Technická 5, 166 28 Prague 6, Czech Republic
| | - Zdenek Sofer
- Department of Inorganic Chemistry, University of Chemistry and Technology Prague, Technická 5, 166 28 Prague 6, Czech Republic
| | - Peter C M Christianen
- High Field Magnet Laboratory, HFML-EMFL, Radboud University, 6525 ED Nijmegen, Netherlands
| | - Pinaki Sengupta
- Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371
| | - Qihua Xiong
- State Key Laboratory of Low-Dimensional Quantum Physics and Department of Physics, Tsinghua University, Beijing 100084, China
- Beijing Academy of Quantum Information Sciences, Beijing 100193, China
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17
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Zhang S, Go G, Lee KJ, Kim SK. SU(3) Topology of Magnon-Phonon Hybridization in 2D Antiferromagnets. PHYSICAL REVIEW LETTERS 2020; 124:147204. [PMID: 32338956 DOI: 10.1103/physrevlett.124.147204] [Citation(s) in RCA: 12] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/01/2019] [Accepted: 03/27/2020] [Indexed: 06/11/2023]
Abstract
Magnon-phonon hybrid excitations are studied theoretically in a two-dimensional antiferromagnet with an easy axis normal to the plane. We show that two magnon bands and one phonon band are intertwined by the magnetoelastic coupling through a nontrivial SU(3) topology, which can be intuitively perceived by identifying a skyrmion structure in the momentum space. Our results are insensitive to lattice details and generally applicable to two-dimensional antiferromagnets. We show this by developing a continuum theory as the long-wavelength approximation to the tight-binding model. The theoretical results can be probed by measuring the thermal Hall conductance as a function of the temperature and the magnetic field. We envision that the magnetoelastic coupling in antiferromagnets can be a promising venue in search of various topological excitations, which cannot be found in magnetic or elastic models alone.
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Affiliation(s)
- Shu Zhang
- Department of Physics and Astronomy, Johns Hopkins University, Baltimore, Maryland 21218, USA
- Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA
| | - Gyungchoon Go
- Department of Materials Science and Engineering, Korea University, Seoul 02841, Korea
| | - Kyung-Jin Lee
- Department of Materials Science and Engineering, Korea University, Seoul 02841, Korea
- KU-KIST Graduate School of Converging Science and Technology, Korea University, Seoul 02841, Korea
| | - Se Kwon Kim
- Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211, USA
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18
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Park S, Nagaosa N, Yang BJ. Thermal Hall Effect, Spin Nernst Effect, and Spin Density Induced by a Thermal Gradient in Collinear Ferrimagnets from Magnon-Phonon Interaction. NANO LETTERS 2020; 20:2741-2746. [PMID: 32101440 DOI: 10.1021/acs.nanolett.0c00363] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/10/2023]
Abstract
We theoretically study the intrinsic thermal Hall and spin Nernst effect in collinear ferrimagnets on a honeycomb lattice with broken inversion symmetry. The broken inversion symmetry allows in-plane Dzyaloshinskii-Moriya interaction between the nearest neighbors, which does not affect the linear spin wave theory. However, the Dzyaloshinskii-Moriya interaction induces large Berry curvature in the magnetoelastic excitations through the magnon-phonon interaction (MPI) to produce thermal Hall current. Furthermore, the magnetoelastic excitations transport spin, which is inherited from the magnons. Therefore, spin Nernst current accompanies the thermal Hall current. Because the MPI does not conserve the spin, we examine the spatial distribution of spin induced by a thermal gradient in the system having a stripe geometry. We find that spin is accumulated at the edges, reflecting the spin Nernst current. We also find that the total spin of the system-and, therefore, the magnetization-is changed, because of the thermal gradient and MPI.
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Affiliation(s)
- Sungjoon Park
- Department of Physics and Astronomy, Seoul National University, Seoul 08826, Korea
- Center for Correlated Electron Systems, Institute for Basic Science (IBS), Seoul 08826, Korea
- Center for Theoretical Physics (CTP), Seoul National University, Seoul 08826, Korea
| | - Naoto Nagaosa
- Department of Applied Physics, The University of Tokyo, Bunkyo, Tokyo 113-8656, Japan
- RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan
| | - Bohm-Jung Yang
- Department of Physics and Astronomy, Seoul National University, Seoul 08826, Korea
- Center for Correlated Electron Systems, Institute for Basic Science (IBS), Seoul 08826, Korea
- Center for Theoretical Physics (CTP), Seoul National University, Seoul 08826, Korea
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