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Qiao J, Ushakov IV, Safronov IS, Oshorov AD, Wang Z, Andrukhova OV, Rychkova OV. Physical Mechanism of Nanocrystalline Composite Deformation Responsible for Fracture Plastic Nature at Cryogenic Temperatures. Nanomaterials (Basel) 2024; 14:723. [PMID: 38668217 PMCID: PMC11053807 DOI: 10.3390/nano14080723] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/18/2024] [Revised: 04/15/2024] [Accepted: 04/17/2024] [Indexed: 04/29/2024]
Abstract
In this work, we consider the physical basis of deformation and fracture in layered composite nanocrystalline/amorphous material-low-melting crystalline alloy in a wide temperature range. Deformation and fracture at the crack tip on the boundary of such materials as nanocrystalline alloy of the trademark 5BDSR, amorphous alloy of the trademark 82K3XSR and low-melting crystalline alloy were experimentally investigated. The crack was initiated by uniaxial stretching in a temperature range of 77-293 K. A theoretical description of the processes of deformation and fracture at the crack tip is proposed, with the assumption that these processes lead to local heating and ensure the plastic character of crack growth at liquid nitrogen temperatures. The obtained results improve the theoretical understanding of the physics of fracture at the boundary of nanocrystalline and crystalline alloys in a wide temperature range. The possibility of preserving the plastic nature of fracture in a thin boundary layer of crystalline-nanocrystalline material at cryogenic temperatures has been experimentally shown.
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Affiliation(s)
- Jianyong Qiao
- School of Energy and Mining Engineering, China University of Mining and Technology (Beijing), Beijing 100083, China;
| | - Ivan Vladimirovich Ushakov
- Physics Department, National University of Science and Technology “MISIS”, Moscow 119049, Russia; (I.S.S.)
| | - Ivan Sergeevich Safronov
- Physics Department, National University of Science and Technology “MISIS”, Moscow 119049, Russia; (I.S.S.)
| | - Ayur Dasheevich Oshorov
- Physics Department, National University of Science and Technology “MISIS”, Moscow 119049, Russia; (I.S.S.)
| | - Zhiqiang Wang
- China-Russia Dynamics Research Center, China University of Mining and Technology (Beijing), Beijing 100083, China;
| | - Olga Vitalievna Andrukhova
- Physics Department, National University of Science and Technology “MISIS”, Moscow 119049, Russia; (I.S.S.)
| | - Olga Vladimirovna Rychkova
- Physics Department, National University of Science and Technology “MISIS”, Moscow 119049, Russia; (I.S.S.)
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2
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Zhou L, Yang F, Zhang S, Zhang T. Chemical Rules for Stacked Kagome and Honeycomb Topological Semimetals. Adv Mater 2024; 36:e2309803. [PMID: 38281121 DOI: 10.1002/adma.202309803] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/21/2023] [Revised: 01/15/2024] [Indexed: 01/29/2024]
Abstract
The chemical rules for predicting and understanding topological states in stacked kagome and honeycomb lattices are studied in both analytical and numerical ways. Starting with a minimal five-band tight-binding model, all the topological states are sorted into five groups, which are determined by the interlayer and intralayer hopping parameters. Combined with the model, an algorithm is designed to obtain a series of experimentally synthesized topological semimetals with kagome and honeycomb layers, i.e., IAMX family (IA = Alkali metal element, M = Rare earth metal element, X = Carbon group element), in the inorganic crystal structure database. A follow-up high-throughput calculation shows that IAMX family materials are all nodal-line semimetals and they will be Weyl semimetals after taking spin-orbit coupling into consideration. To have further insights into the topology of the IAMX family, a detailed chemical rule analysis is carried out on the high-throughput calculations, including the lattice constants of the structure, intralayer and interlayer couplings, bond strengths, electronegativity, and so on, which are consistent with the tight-binding model. This study provides a way to discover and modulate topological properties in stacked kagome and honeycomb crystals and offers candidates for studying topology-related properties like topological superconductors and axion insulators.
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Affiliation(s)
- Liqin Zhou
- Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing, 100190, China
| | - Fazhi Yang
- Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing, 100190, China
| | - Shuai Zhang
- Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, 100190, China
| | - Tiantian Zhang
- Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, 100190, China
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Fukami M, Marcks JC, Candido DR, Weiss LR, Soloway B, Sullivan SE, Delegan N, Heremans FJ, Flatté ME, Awschalom DD. Magnon-mediated qubit coupling determined via dissipation measurements. Proc Natl Acad Sci U S A 2024; 121:e2313754120. [PMID: 38165926 PMCID: PMC10786302 DOI: 10.1073/pnas.2313754120] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/09/2023] [Accepted: 11/14/2023] [Indexed: 01/04/2024] Open
Abstract
Controlled interaction between localized and delocalized solid-state spin systems offers a compelling platform for on-chip quantum information processing with quantum spintronics. Hybrid quantum systems (HQSs) of localized nitrogen-vacancy (NV) centers in diamond and delocalized magnon modes in ferrimagnets-systems with naturally commensurate energies-have recently attracted significant attention, especially for interconnecting isolated spin qubits at length-scales far beyond those set by the dipolar coupling. However, despite extensive theoretical efforts, there is a lack of experimental characterization of the magnon-mediated interaction between NV centers, which is necessary to develop such hybrid quantum architectures. Here, we experimentally determine the magnon-mediated NV-NV coupling from the magnon-induced self-energy of NV centers. Our results are quantitatively consistent with a model in which the NV center is coupled to magnons by dipolar interactions. This work provides a versatile tool to characterize HQSs in the absence of strong coupling, informing future efforts to engineer entangled solid-state systems.
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Affiliation(s)
- Masaya Fukami
- Pritzker School of Molecular Engineering, University of Chicago, Chicago, IL60637
| | - Jonathan C. Marcks
- Pritzker School of Molecular Engineering, University of Chicago, Chicago, IL60637
- Center for Molecular Engineering and Materials Science Division, Argonne National Laboratory, Lemont, IL60439
| | - Denis R. Candido
- Department of Physics and Astronomy, University of Iowa, Iowa City, IA52242
| | - Leah R. Weiss
- Pritzker School of Molecular Engineering, University of Chicago, Chicago, IL60637
- Advanced Institute for Materials Research, Tohoku University, Sendai980-8577, Japan
| | - Benjamin Soloway
- Pritzker School of Molecular Engineering, University of Chicago, Chicago, IL60637
| | - Sean E. Sullivan
- Center for Molecular Engineering and Materials Science Division, Argonne National Laboratory, Lemont, IL60439
| | - Nazar Delegan
- Center for Molecular Engineering and Materials Science Division, Argonne National Laboratory, Lemont, IL60439
| | - F. Joseph Heremans
- Pritzker School of Molecular Engineering, University of Chicago, Chicago, IL60637
- Center for Molecular Engineering and Materials Science Division, Argonne National Laboratory, Lemont, IL60439
| | - Michael E. Flatté
- Department of Physics and Astronomy, University of Iowa, Iowa City, IA52242
- Department of Applied Physics, Eindhoven University of Technology, Eindhoven5600 MB, Netherlands
| | - David D. Awschalom
- Pritzker School of Molecular Engineering, University of Chicago, Chicago, IL60637
- Center for Molecular Engineering and Materials Science Division, Argonne National Laboratory, Lemont, IL60439
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4
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Trachenko K. Viscosity and diffusion in life processes and tuning of fundamental constants. Rep Prog Phys 2023; 86:112601. [PMID: 37811635 DOI: 10.1088/1361-6633/acfd3e] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/28/2023] [Accepted: 09/26/2023] [Indexed: 10/10/2023]
Abstract
Viewed as one of the grandest questions in modern science, understanding fundamental physical constants has been discussed in high-energy particle physics, astronomy and cosmology. Here, I review how condensed matter and liquid physics gives new insights into fundamental constants and their tuning. This is based on two observations: first, cellular life and the existence of observers depend on viscosity and diffusion. Second, the lower bound on viscosity and upper bound on diffusion are set by fundamental constants, and I briefly review this result and related recent developments in liquid physics. I will subsequently show that bounds on viscosity, diffusion and the newly introduced fundamental velocity gradient in a biochemical machine can all be varied while keeping the fine-structure constant and the proton-to-electron mass ratio intact. This implies that it is possible to produce heavy elements in stars but have a viscous planet where all liquids have very high viscosity (for example that of tar or higher) and where life may not exist. Knowing the range of bio-friendly viscosity and diffusion, we will be able to calculate the range of fundamental constants which favour cellular life and observers and compare this tuning with that discussed in high-energy physics previously. This invites an inter-disciplinary research between condensed matter physics and life sciences, and I formulate several questions that life science can address. I finish with a conjecture of multiple tuning and an evolutionary mechanism.
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Affiliation(s)
- K Trachenko
- School of Physical and Chemical Sciences, Queen Mary University of London, Mile End Road, London E1 4NS, United Kingdom
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5
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Liu P, Zhang Y, Li K, Li Y, Pu Y. Recent advances in 2D van der Waals magnets: Detection, modulation, and applications. iScience 2023; 26:107584. [PMID: 37664598 PMCID: PMC10470320 DOI: 10.1016/j.isci.2023.107584] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 09/05/2023] Open
Abstract
The emergence of two-dimensional (2D) van der Waals magnets provides an exciting platform for exploring magnetism in the monolayer limit. Exotic quantum phenomena and significant potential for spintronic applications are demonstrated in 2D magnetic crystals and heterostructures, which offer unprecedented possibilities in advanced formation technology with low power and high efficiency. In this review, we summarize recent advances in 2D van der Waals magnetic crystals. We focus mainly on van der Waals materials of truly 2D nature with intrinsic magnetism. The detection methods of 2D magnetic materials are first introduced in detail. Subsequently, the effective strategies to modulate the magnetic behavior of 2D magnets (e.g., Curie temperature, magnetic anisotropy, magnetic exchange interaction) are presented. Then, we list the applications of 2D magnets in the spintronic devices. We also highlight current challenges and broad space for the development of 2D magnets in further studies.
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Affiliation(s)
- Ping Liu
- School of Science & New Energy Technology Engineering Laboratory of Jiangsu Province, Nanjing University of Posts and Telecommunications, Nanjing 210023, China
| | - Ying Zhang
- Department of Materials Science & Engineering, CAS Key Lab of Materials for Energy Conversion, University of Science and Technology of China, Hefei 230026, China
| | - Kehan Li
- School of Science & New Energy Technology Engineering Laboratory of Jiangsu Province, Nanjing University of Posts and Telecommunications, Nanjing 210023, China
| | - Yongde Li
- School of Science & New Energy Technology Engineering Laboratory of Jiangsu Province, Nanjing University of Posts and Telecommunications, Nanjing 210023, China
| | - Yong Pu
- School of Science & New Energy Technology Engineering Laboratory of Jiangsu Province, Nanjing University of Posts and Telecommunications, Nanjing 210023, China
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6
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Liu Y, Cui T, Li D. Emerging d- d orbital coupling between non- d-block main-group elements Mg and I at high pressure. iScience 2023; 26:106113. [PMID: 36879798 PMCID: PMC9984552 DOI: 10.1016/j.isci.2023.106113] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/27/2022] [Revised: 11/30/2022] [Accepted: 01/28/2023] [Indexed: 02/05/2023] Open
Abstract
d-d orbital coupling, which increases anisotropic and directional bonding, commonly occurs between d-block transition metals. Here, we report an unexpected d-d orbital coupling in the non-d-block main-group element compound Mg2I based on first-principles calculations. The unfilled d orbitals of Mg and I atoms under ambient conditions become part of the valence orbitals and couple with each other under high pressures, resulting in the formation of highly symmetric I-Mg-I covalent bonding in Mg2I, which forces the valence electrons of Mg atoms into the lattice voids to form interstitial quasi-atoms (ISQs). In turn, the ISQs highly interact with the crystal lattice, contributing to lattice stability. This study greatly enriches the fundamental understanding of chemical bonding between non-d-block main-group elements at high pressures.
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Affiliation(s)
- Yan Liu
- State Key Laboratory of Superhard Materials, College of Physics, Jilin University, Changchun 130012, P.R. China
| | - Tian Cui
- State Key Laboratory of Superhard Materials, College of Physics, Jilin University, Changchun 130012, P.R. China.,School of Physical Science and Technology, Ningbo University, Ningbo 315211, P.R. China
| | - Da Li
- State Key Laboratory of Superhard Materials, College of Physics, Jilin University, Changchun 130012, P.R. China
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Wu J, González-Cataldo F, Soubiran F, Militzer B. The phase diagrams of beryllium and magnesium oxide at megabar pressures. J Phys Condens Matter 2022; 34:144003. [PMID: 35026747 DOI: 10.1088/1361-648x/ac4b2a] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/15/2021] [Accepted: 01/13/2022] [Indexed: 06/14/2023]
Abstract
We performab initiosimulations of beryllium (Be) and magnesium oxide (MgO) at megabar pressures and compare their structural and thermodynamic properties. We make a detailed comparison of our two recently derived phase diagrams of Be (Wuet al2021Phys. Rev.B104014103) and MgO (Soubiran and Militzer 2020Phys. Rev. Lett.125175701) using the thermodynamic integration technique, as they exhibit striking similarities regarding their shape. We explore whether the Lindemann criterion can explain the melting temperatures of these materials through the calculation of the Debye temperature at high pressure. From our free energy calculations, we find that the melting line of both materials is well represented by the Simon-Glazel fitTm(P) =T0(1 +P/a)1/c, whereT0= 1564 K,a= 15.8037 GPa andc= 2.4154 for Be, whileT0= 3010 K,a= 10.5797 GPa andc= 2.8683 for the MgO in the B1. For the B2 phase, we use the valuesa= 26.1163 GPa andc= 2.2426. Both materials exhibit negative Clapeyron slopes on the boundaries between the two solid phases that are strongly affected by anharmonic effects, which also influence the location of the solid-solid-liquid triple point. We find that the quasi-harmonic approximation underestimates the stability range of the low-pressure phases, namely hcp for Be and B1 for MgO. We also compute the phonon dispersion relations at low and high pressure for each of the phases of these materials, and also explore how the phonon density of states is modified by temperature. Finally, we derive secondary shock Hugoniot curves in addition to the principal Hugoniot curve for both materials, and study their offsets in pressure between solid and liquid branches.
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Affiliation(s)
- Jizhou Wu
- Department of Earth and Planetary Science, University of California, Berkeley, CA 94720, United States of America
| | - Felipe González-Cataldo
- Department of Earth and Planetary Science, University of California, Berkeley, CA 94720, United States of America
| | | | - Burkhard Militzer
- Department of Earth and Planetary Science, University of California, Berkeley, CA 94720, United States of America
- Department of Astronomy, University of California, Berkeley, CA 94720, United States of America
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8
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Zhu L, Yuan H, Wu K, Wang X, Liu G, Sun J, Liao X, Chen X. Curvature-controlled delamination patterns of thin films on spherical substrates. iScience 2021; 24:102616. [PMID: 34151230 PMCID: PMC8188561 DOI: 10.1016/j.isci.2021.102616] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/08/2021] [Revised: 05/10/2021] [Accepted: 05/19/2021] [Indexed: 11/17/2022] Open
Abstract
Periodic delamination patterns in multilayer structures have exhibited extensive applications in microelectronics and optics devices. However, delamination behaviors of a closed thin shell on spherical substrates are still elusive. Herein, a unique instability mechanism of buckle delamination in a closed thin film weakly bonded to spherical substrates is studied by experiments, simulations, and theoretical analyses. The system of an Al film depositing on polystyrene spheres subjected to thermal mismatch strain is used for demonstration. Unlike traditional phenomena of wrinkling and wrinkle-induced delamination under increasing misfit strain, the weak adhesion between the core and shell results in a periodic pattern of delaminated hexagonal dimples that emerges directly from the smooth sphere configuration, before which no wrinkling occurs. Both substrate curvature and interfacial adhesion are revealed to control the dimple size and delamination width. These findings open a new venue for manifesting new controllable features for surface microfabrication.
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Affiliation(s)
- Liangliang Zhu
- School of Chemical Engineering, Northwest University, Xi'an 710069, China
| | - Haozhi Yuan
- State Key Laboratory for Mechanical Behavior of Materials, Xi'an Jiaotong University, Xi'an 710049, China
| | - Kai Wu
- State Key Laboratory for Mechanical Behavior of Materials, Xi'an Jiaotong University, Xi'an 710049, China
| | - Xueru Wang
- State Key Laboratory for Mechanical Behavior of Materials, Xi'an Jiaotong University, Xi'an 710049, China
| | - Gang Liu
- State Key Laboratory for Mechanical Behavior of Materials, Xi'an Jiaotong University, Xi'an 710049, China
| | - Jun Sun
- State Key Laboratory for Mechanical Behavior of Materials, Xi'an Jiaotong University, Xi'an 710049, China
| | - Xiangbiao Liao
- Department of Earth and Environmental Engineering, Columbia University, New York, NY 10027, USA
| | - Xi Chen
- Department of Earth and Environmental Engineering, Columbia University, New York, NY 10027, USA
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9
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Doi H, Takahashi KZ, Aoyagi T. Searching for local order parameters to classify water structures at triple points. J Comput Chem 2021; 42:1720-1727. [PMID: 34169566 DOI: 10.1002/jcc.26707] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/26/2021] [Revised: 06/11/2021] [Accepted: 06/15/2021] [Indexed: 11/10/2022]
Abstract
The diversity of ice polymorphs is of interest in condensed-matter physics, engineering, astronomy, and biosphere and climate studies. In particular, their triple points are critical to elucidate the formation of each phase and transitions among phases. However, an approach to distinguish their molecular structures is lacking. When precise molecular geometries are given, order parameters are often computed to quantify the degree of structural ordering and to classify the structures. Many order parameters have been developed for specific or multiple purposes, but their capabilities have not been exhaustively investigated for distinguishing ice polymorphs. Here, 493 order parameters and their combinations are considered for two triple points involving the ice polymorphs ice III-V-liquid and ice V-VI-liquid. Supervised machine learning helps automatic and systematic searching of the parameters. For each triple point, the best set of two order parameters was found that distinguishes three structures with high accuracy. A set of three order parameters is also suggested for better accuracy.
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Affiliation(s)
- Hideo Doi
- Research Center for Computational Design of Advanced Functional Materials, National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba, Ibaraki, Japan
| | - Kazuaki Z Takahashi
- Research Center for Computational Design of Advanced Functional Materials, National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba, Ibaraki, Japan
| | - Takeshi Aoyagi
- Research Center for Computational Design of Advanced Functional Materials, National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba, Ibaraki, Japan
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Zhou TX, Carmiggelt JJ, Gächter LM, Esterlis I, Sels D, Stöhr RJ, Du C, Fernandez D, Rodriguez-Nieva JF, Büttner F, Demler E, Yacoby A. A magnon scattering platform. Proc Natl Acad Sci U S A 2021; 118:e2019473118. [PMID: 34131074 DOI: 10.1073/pnas.2019473118] [Citation(s) in RCA: 7] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022] Open
Abstract
This work describes a general scattering platform that uses magnons to explore the underlying properties of target materials. In this work we show how both phase and amplitude of magnons can be imaged using a nitrogen vacancy center magnetometer and how the scattered pattern of waves can be used to infer geometric and magnetic properties of a target material. To demonstrate this new experimental methodology we use a permalloy disk as our target and show that even with such a simple target unexpected behavior is observed. In addition, we provide a theoretical framework to reconstruct the properties of the target. Scattering experiments have revolutionized our understanding of nature. Examples include the discovery of the nucleus [R. G. Newton, Scattering Theory of Waves and Particles (1982)], crystallography [U. Pietsch, V. Holý, T. Baumback, High-Resolution X-Ray Scattering (2004)], and the discovery of the double-helix structure of DNA [J. D. Watson, F. H. C. Crick, Nature 171, 737–738]. Scattering techniques differ by the type of particles used, the interaction these particles have with target materials, and the range of wavelengths used. Here, we demonstrate a two-dimensional table-top scattering platform for exploring magnetic properties of materials on mesoscopic length scales. Long-lived, coherent magnonic excitations are generated in a thin film of yttrium iron garnet and scattered off a magnetic target deposited on its surface. The scattered waves are then recorded using a scanning nitrogen vacancy center magnetometer that allows subwavelength imaging and operation under conditions ranging from cryogenic to ambient environment. While most scattering platforms measure only the intensity of the scattered waves, our imaging method allows for spatial determination of both amplitude and phase of the scattered waves, thereby allowing for a systematic reconstruction of the target scattering potential. Our experimental results are consistent with theoretical predictions for such a geometry and reveal several unusual features of the magnetic response of the target, including suppression near the target edges and a gradient in the direction perpendicular to the direction of surface wave propagation. Our results establish magnon scattering experiments as a platform for studying correlated many-body systems.
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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: 18] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [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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12
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Jäck B, Xie Y, Andrei Bernevig B, Yazdani A. Observation of backscattering induced by magnetism in a topological edge state. Proc Natl Acad Sci U S A 2020; 117:16214-8. [PMID: 32601184 DOI: 10.1073/pnas.2005071117] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022] Open
Abstract
The boundary modes of topological insulators are protected by the symmetries of the nontrivial bulk electronic states. Unless these symmetries are broken, they can give rise to novel phenomena, such as the quantum spin Hall effect in one-dimensional (1D) topological edge states, where quasiparticle backscattering is suppressed by time-reversal symmetry (TRS). Here, we investigate the properties of the 1D topological edge state of bismuth in the absence of TRS, where backscattering is predicted to occur. Using spectroscopic imaging and spin-polarized measurements with a scanning tunneling microscope, we compared quasiparticle interference (QPI) occurring in the edge state of a pristine bismuth bilayer with that occurring in the edge state of a bilayer, which is terminated by ferromagnetic iron clusters that break TRS. Our experiments on the decorated bilayer edge reveal an additional QPI branch, which can be associated with spin-flip scattering across the Brioullin zone center between time-reversal band partners. The observed QPI characteristics exactly match with theoretical expectations for a topological edge state, having one Kramer's pair of bands. Together, our results provide further evidence for the nontrivial nature of bismuth and in particular, demonstrate backscattering inside a helical topological edge state induced by broken TRS through local magnetism.
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13
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Ma J, Wang H, Nie S, Yi C, Xu Y, Li H, Jandke J, Wulfhekel W, Huang Y, West D, Richard P, Chikina A, Strocov VN, Mesot J, Weng H, Zhang S, Shi Y, Qian T, Shi M, Ding H. Emergence of Nontrivial Low-Energy Dirac Fermions in Antiferromagnetic EuCd 2 As 2. Adv Mater 2020; 32:e1907565. [PMID: 32091144 DOI: 10.1002/adma.201907565] [Citation(s) in RCA: 12] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/18/2019] [Revised: 01/20/2020] [Indexed: 06/10/2023]
Abstract
Parity-time symmetry plays an essential role for the formation of Dirac states in Dirac semimetals. So far, all of the experimentally identified topologically nontrivial Dirac semimetals (DSMs) possess both parity and time reversal symmetry. The realization of magnetic topological DSMs remains a major issue in topological material research. Here, combining angle-resolved photoemission spectroscopy with density functional theory calculations, it is ascertained that band inversion induces a topologically nontrivial ground state in EuCd2 As2 . As a result, ideal magnetic Dirac fermions with simplest double cone structure near the Fermi level emerge in the antiferromagnetic (AFM) phase. The magnetic order breaks time reversal symmetry, but preserves inversion symmetry. The double degeneracy of the Dirac bands is protected by a combination of inversion, time-reversal, and an additional translation operation. Moreover, the calculations show that a deviation of the magnetic moments from the c-axis leads to the breaking of C3 rotation symmetry, and thus, a small bandgap opens at the Dirac point in the bulk. In this case, the system hosts a novel state containing three different types of topological insulator: axion insulator, AFM topological crystalline insulator (TCI), and higher order topological insulator. The results provide an enlarged platform for the quest of topological Dirac fermions in a magnetic system.
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Affiliation(s)
- Junzhang Ma
- Paul Scherrer Institute, Swiss Light Source, CH-5232, Villigen, PSI, Switzerland
- Institute of Condensed Matter Physics, École Polytechnique Fédérale de Lausanne, CH-10 15, Lausanne, Switzerland
| | - Han Wang
- Department of Physics, Applied Physics, and Astronomy, Rensselaer Polytechnic Institute, Troy, NY, 12180, USA
| | - Simin Nie
- Department of Materials Science and Engineering, Stanford University, Stanford, CA, 94305, USA
| | - Changjiang Yi
- Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing, 100190, China
- School of Physics, University of Chinese Academy of Sciences, Beijing, 100190, China
| | - Yuanfeng Xu
- Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing, 100190, China
- School of Physics, University of Chinese Academy of Sciences, Beijing, 100190, China
| | - Hang Li
- Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing, 100190, China
- School of Physics, University of Chinese Academy of Sciences, Beijing, 100190, China
| | - Jasmin Jandke
- Paul Scherrer Institute, Swiss Light Source, CH-5232, Villigen, PSI, Switzerland
| | - Wulf Wulfhekel
- Physikalisches Institut, Karlsruhe Institute of Technology, 76131, Karlsruhe, Germany
| | - Yaobo Huang
- Shanghai Synchrotron Radiation Facility, Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai, 201204, China
| | - Damien West
- Department of Physics, Applied Physics, and Astronomy, Rensselaer Polytechnic Institute, Troy, NY, 12180, USA
| | - Pierre Richard
- Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing, 100190, China
- School of Physics, University of Chinese Academy of Sciences, Beijing, 100190, China
- Institut quantique, Université de Sherbrooke, 2500 boulevard de l'Université, Sherbrooke, Québec, J1K 2R1, Canada
| | - Alla Chikina
- Paul Scherrer Institute, Swiss Light Source, CH-5232, Villigen, PSI, Switzerland
| | - Vladimir N Strocov
- Paul Scherrer Institute, Swiss Light Source, CH-5232, Villigen, PSI, Switzerland
| | - Joël Mesot
- Paul Scherrer Institute, Swiss Light Source, CH-5232, Villigen, PSI, Switzerland
- Institute of Condensed Matter Physics, École Polytechnique Fédérale de Lausanne, CH-10 15, Lausanne, Switzerland
| | - Hongming Weng
- Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing, 100190, China
- Songshan Lake Materials Laboratory Dongguan, Guangdong, 523808, China
| | - Shengbai Zhang
- Department of Physics, Applied Physics, and Astronomy, Rensselaer Polytechnic Institute, Troy, NY, 12180, USA
| | - Youguo Shi
- Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing, 100190, China
- Songshan Lake Materials Laboratory Dongguan, Guangdong, 523808, China
| | - Tian Qian
- Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing, 100190, China
- Songshan Lake Materials Laboratory Dongguan, Guangdong, 523808, China
| | - Ming Shi
- Paul Scherrer Institute, Swiss Light Source, CH-5232, Villigen, PSI, Switzerland
| | - Hong Ding
- Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing, 100190, China
- School of Physics, University of Chinese Academy of Sciences, Beijing, 100190, China
- Songshan Lake Materials Laboratory Dongguan, Guangdong, 523808, China
- CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing, 100190, China
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Abstract
Can the properties of the thermodynamic limit of a many-body quantum system be extrapolated by analyzing a sequence of finite-size cases? We present models for which such an approach gives completely misleading results: translationally invariant, local Hamiltonians on a square lattice with open boundary conditions and constant spectral gap, which have a classical product ground state for all system sizes smaller than a particular threshold size, but a ground state with topological degeneracy for all system sizes larger than this threshold. Starting from a minimal case with spins of dimension 6 and threshold lattice size [Formula: see text], we show that the latter grows faster than any computable function with increasing local spin dimension. The resulting effect may be viewed as a unique type of quantum phase transition that is driven by the size of the system rather than by an external field or coupling strength. We prove that the construction is thermally robust, showing that these effects are in principle accessible to experimental observation.
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15
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Maharaj AV, Rosenberg EW, Hristov AT, Berg E, Fernandes RM, Fisher IR, Kivelson SA. Transverse fields to tune an Ising-nematic quantum phase transition. Proc Natl Acad Sci U S A 2017; 114:13430-4. [PMID: 29208710 DOI: 10.1073/pnas.1712533114] [Citation(s) in RCA: 15] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022] Open
Abstract
The paradigmatic example of a continuous quantum phase transition is the transverse field Ising ferromagnet. In contrast to classical critical systems, whose properties depend only on symmetry and the dimension of space, the nature of a quantum phase transition also depends on the dynamics. In the transverse field Ising model, the order parameter is not conserved, and increasing the transverse field enhances quantum fluctuations until they become strong enough to restore the symmetry of the ground state. Ising pseudospins can represent the order parameter of any system with a twofold degenerate broken-symmetry phase, including electronic nematic order associated with spontaneous point-group symmetry breaking. Here, we show for the representative example of orbital-nematic ordering of a non-Kramers doublet that an orthogonal strain or a perpendicular magnetic field plays the role of the transverse field, thereby providing a practical route for tuning appropriate materials to a quantum critical point. While the transverse fields are conjugate to seemingly unrelated order parameters, their nontrivial commutation relations with the nematic order parameter, which can be represented by a Berry-phase term in an effective field theory, intrinsically intertwine the different order parameters.
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16
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Ellis J, Pike R, Zayats A. Unifying physics and technology in light of Maxwell's equations. Philos Trans A Math Phys Eng Sci 2016; 374:rsta.2015.0264. [PMID: 27458262 PMCID: PMC4971388 DOI: 10.1098/rsta.2015.0264] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 06/13/2016] [Indexed: 06/06/2023]
Affiliation(s)
| | - Roy Pike
- King's College London, London, UK
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17
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Bok JM, Bae JJ, Choi HY, Varma CM, Zhang W, He J, Zhang Y, Yu L, Zhou XJ. Quantitative determination of pairing interactions for high-temperature superconductivity in cuprates. Sci Adv 2016; 2:e1501329. [PMID: 26973872 PMCID: PMC4783123 DOI: 10.1126/sciadv.1501329] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 09/24/2015] [Accepted: 01/05/2016] [Indexed: 06/05/2023]
Abstract
A profound problem in modern condensed matter physics is discovering and understanding the nature of fluctuations and their coupling to fermions in cuprates, which lead to high-temperature superconductivity and the invariably associated strange metal state. We report the quantitative determination of normal and pairing self-energies, made possible by laser-based angle-resolved photoemission measurements of unprecedented accuracy and stability. Through a precise inversion procedure, both the effective interactions in the attractive d-wave symmetry and the repulsive part in the full symmetry are determined. The latter is nearly angle-independent. Near T c, both interactions are nearly independent of frequency and have almost the same magnitude over the complete energy range of up to about 0.4 eV, except for a low-energy feature at around 50 meV that is present only in the repulsive part, which has less than 10% of the total spectral weight. Well below T c, they both change similarly, with superconductivity-induced features at low energies. Besides finding the pairing self-energy and the attractive interactions for the first time, these results expose the central paradox of the problem of high T c: how the same frequency-independent fluctuations can dominantly scatter at angles ±π/2 in the attractive channel to give d-wave pairing and lead to angle-independent repulsive scattering. The experimental results are compared with available theoretical calculations based on antiferromagnetic fluctuations, the Hubbard model, and quantum-critical fluctuations of the loop-current order.
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Affiliation(s)
- Jin Mo Bok
- Department of Physics and Institute for Basic Science Research, SungKyunKwan University, Suwon 440-746, Korea
- National Laboratory for Superconductivity, Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
| | - Jong Ju Bae
- Department of Physics and Institute for Basic Science Research, SungKyunKwan University, Suwon 440-746, Korea
| | - Han-Yong Choi
- Department of Physics and Institute for Basic Science Research, SungKyunKwan University, Suwon 440-746, Korea
- Asia Pacific Center for Theoretical Physics, Pohang 790-784, Korea
| | - Chandra M. Varma
- Department of Physics and Astronomy, University of California, Riverside, CA 92521, USA
| | - Wentao Zhang
- National Laboratory for Superconductivity, Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
- Department of Physics and Astronomy, Shanghai JiaoTong University, Shanghai 200240, China
| | - Junfeng He
- National Laboratory for Superconductivity, Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
| | - Yuxiao Zhang
- National Laboratory for Superconductivity, Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
| | - Li Yu
- National Laboratory for Superconductivity, Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
| | - X. J. Zhou
- National Laboratory for Superconductivity, Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
- Collaborative Innovation Center of Quantum Matter, Beijing 100871, China
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18
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Wortmann B, van Vörden D, Graf P, Robles R, Abufager P, Lorente N, Bobisch CA, Möller R. Reversible 2D Phase Transition Driven By an Electric Field: Visualization and Control on the Atomic Scale. Nano Lett 2016; 16:528-533. [PMID: 26645498 DOI: 10.1021/acs.nanolett.5b04174] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
We report on a reversible structural phase transition of a two-dimensional system that can be locally induced by an external electric field. Two different structural configurations may coexist within a CO monolayer on Cu(111). The balance between the two phases can be shifted by an external electric field, causing the domain boundaries to move, increasing the area of the favored phase controllable both in location and size. If the field is further enhanced new domains nucleate. The arrangement of the CO molecules on the Cu surface is observed in real time and real space with atomic resolution while the electric field driving the phase transition is easily varied over a broad range. Together with the well-known molecular manipulation of CO adlayers, our findings open exciting prospects for combining spontaneous long-range order with man-made CO structures such as "molecule cascades" or "molecular graphene". Our new manipulation mode permits us to bridge the gap between fundamental concepts and the fabrication of arbitrary atomic patterns in large scale, by providing unprecedented insight into the physics of structural phase transitions on the atomic scale.
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Affiliation(s)
- B Wortmann
- Faculty of Physics and Center for Nanointegration Duisburg-Essen (CENIDE), University of Duisburg-Essen , Lotharstraße1-21, 47048 Duisburg, Germany
| | - D van Vörden
- Faculty of Physics and Center for Nanointegration Duisburg-Essen (CENIDE), University of Duisburg-Essen , Lotharstraße1-21, 47048 Duisburg, Germany
| | - P Graf
- Faculty of Physics and Center for Nanointegration Duisburg-Essen (CENIDE), University of Duisburg-Essen , Lotharstraße1-21, 47048 Duisburg, Germany
| | - R Robles
- ICN2 Catalan Institute of Nanoscience and Nanotechnology, CSIC and The Barcelona Institute of Science and Technology, Campus UAB, Bellaterra, 08193 Barcelona, Spain
| | - P Abufager
- ICN2 Catalan Institute of Nanoscience and Nanotechnology, CSIC and The Barcelona Institute of Science and Technology, Campus UAB, Bellaterra, 08193 Barcelona, Spain
- Instituto de Física de Rosario, Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), and Universidad Nacional de Rosario, Avenidas Pellegrini 250, 2000 Rosario, Santa Fe, Argentina
| | - N Lorente
- ICN2 Catalan Institute of Nanoscience and Nanotechnology, CSIC and The Barcelona Institute of Science and Technology, Campus UAB, Bellaterra, 08193 Barcelona, Spain
- Centro de Física de Materiales, CFM/MPC (CSIC-UPV/EHU) , Paseo Manuel de Lardizabal 5, 20018 Donostia-San Sebastián, Spain
- Donostia International Physics Center (DIPC) , Paseo Manuel de Lardizabal 4, 20018 Donostia-San Sebastián, Spain
| | - C A Bobisch
- Faculty of Physics and Center for Nanointegration Duisburg-Essen (CENIDE), University of Duisburg-Essen , Lotharstraße1-21, 47048 Duisburg, Germany
| | - R Möller
- Faculty of Physics and Center for Nanointegration Duisburg-Essen (CENIDE), University of Duisburg-Essen , Lotharstraße1-21, 47048 Duisburg, Germany
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19
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Koumoulis D, Morris GD, He L, Kou X, King D, Wang D, Hossain MD, Wang KL, Fiete GA, Kanatzidis MG, Bouchard LS. Nanoscale β-nuclear magnetic resonance depth imaging of topological insulators. Proc Natl Acad Sci U S A 2015; 112:E3645-50. [PMID: 26124141 PMCID: PMC4507211 DOI: 10.1073/pnas.1502330112] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022] Open
Abstract
Considerable evidence suggests that variations in the properties of topological insulators (TIs) at the nanoscale and at interfaces can strongly affect the physics of topological materials. Therefore, a detailed understanding of surface states and interface coupling is crucial to the search for and applications of new topological phases of matter. Currently, no methods can provide depth profiling near surfaces or at interfaces of topologically inequivalent materials. Such a method could advance the study of interactions. Herein, we present a noninvasive depth-profiling technique based on β-detected NMR (β-NMR) spectroscopy of radioactive (8)Li(+) ions that can provide "one-dimensional imaging" in films of fixed thickness and generates nanoscale views of the electronic wavefunctions and magnetic order at topological surfaces and interfaces. By mapping the (8)Li nuclear resonance near the surface and 10-nm deep into the bulk of pure and Cr-doped bismuth antimony telluride films, we provide signatures related to the TI properties and their topological nontrivial characteristics that affect the electron-nuclear hyperfine field, the metallic shift, and magnetic order. These nanoscale variations in β-NMR parameters reflect the unconventional properties of the topological materials under study, and understanding the role of heterogeneities is expected to lead to the discovery of novel phenomena involving quantum materials.
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Affiliation(s)
- Dimitrios Koumoulis
- Department of Chemistry and Biochemistry, University of California, Los Angeles, CA 90095
| | - Gerald D Morris
- TRI University Meson Facility (TRIUMF), Vancouver, BC, Canada V6T 2A3
| | - Liang He
- Department of Electrical Engineering, University of California, Los Angeles, CA 90095
| | - Xufeng Kou
- Department of Electrical Engineering, University of California, Los Angeles, CA 90095
| | - Danny King
- Department of Chemistry and Biochemistry, University of California, Los Angeles, CA 90095
| | - Dong Wang
- Department of Physics & Astronomy, University of British Columbia, Vancouver, BC,Canada V6T 1Z1
| | - Masrur D Hossain
- TRI University Meson Facility (TRIUMF), Vancouver, BC, Canada V6T 2A3; Department of Physics & Astronomy, University of British Columbia, Vancouver, BC,Canada V6T 1Z1
| | - Kang L Wang
- Department of Electrical Engineering, University of California, Los Angeles, CA 90095
| | - Gregory A Fiete
- Department of Physics, University of Texas at Austin, Austin, TX 78712
| | | | - Louis-S Bouchard
- Department of Chemistry and Biochemistry, University of California, Los Angeles, CA 90095; California NanoSystems Institute, University of California, Los Angeles, CA 90095
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20
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Haravifard S, Banerjee A, van Wezel J, Silevitch DM, dos Santos AM, Lang JC, Kermarrec E, Srajer G, Gaulin BD, Molaison JJ, Dabkowska HA, Rosenbaum TF. Emergence of long-range order in sheets of magnetic dimers. Proc Natl Acad Sci U S A 2014; 111:14372-7. [PMID: 25246541 DOI: 10.1073/pnas.1413318111] [Citation(s) in RCA: 21] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022] Open
Abstract
Quantum spins placed on the corners of a square lattice can dimerize and form singlets, which then can be transformed into a magnetic state as the interactions between dimers increase beyond threshold. This is a strictly 2D transition in theory, but real-world materials often need the third dimension to stabilize long-range order. We use high pressures to convert sheets of Cu(2+) spin 1/2 dimers from local singlets to global antiferromagnet in the model system SrCu2(BO3)2. Single-crystal neutron diffraction measurements at pressures above 5 GPa provide a direct signature of the antiferromagnetic ordered state, whereas high-resolution neutron powder and X-ray diffraction at commensurate pressures reveal a tilting of the Cu spins out of the plane with a critical exponent characteristic of 3D transitions. The addition of anisotropic, interplane, spin-orbit terms in the venerable Shastry-Sutherland Hamiltonian accounts for the influence of the third dimension.
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