1
|
Mandal I, Freire H. Transport properties in non-Fermi liquid phases of nodal-point semimetals. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2024; 36:443002. [PMID: 39038487 DOI: 10.1088/1361-648x/ad665e] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/13/2024] [Accepted: 07/22/2024] [Indexed: 07/24/2024]
Abstract
In this review, we survey the current progress in computing transport properties in semimetals which harbour non-Fermi liquid (NFL) phases. We first discuss the widely-used Kubo formalism, which can be applied to the effective theory describing the stable NFL phase obtained via a renormalization group procedure and, hence, is applicable for temperatures close to zero (e.g. optical conductivity). For finite-temperature regimes, which apply to the computations of the generalized DC conductivity tensors, we elucidate the memory matrix approach. This approach is based on an effective hydrodynamic description of the system, and is especially suited for tackling transport calculations in strongly-interacting quantum field theories, because it does not rely on the existence of long-lived quasiparticles. As a concrete example, we apply these two approaches to find the response of the so-calledLuttinger-Abrikosov-Benelavskii phaseof isotropic three-dimensional Luttinger semimetals, which arises under the effects of long-ranged (unscreened) Coulomb interactions, with the chemical potential fine-tuned to cut exactly the nodal point. In particular, we focus on the electric conductivity tensors, thermal and thermoelectric response, Raman response, free energy, entropy density, and shear viscosity.
Collapse
Affiliation(s)
- Ipsita Mandal
- Department of Physics, Shiv Nadar Institution of Eminence (SNIoE), Gautam Buddha Nagar, Uttar Pradesh 201314, India
- Freiburg Institute for Advanced Studies (FRIAS), University of Freiburg, D-79104 Freiburg, Germany
| | - Hermann Freire
- Instituto de Física, Universidade Federal de Goiás, 74.001-970, Goiânia, GO, Brazil
| |
Collapse
|
2
|
Sellati N, Fiore J, Castellani C, Benfatto L. Optical Absorption in Tilted Geometries as an Indirect Measurement of Longitudinal Plasma Waves in Layered Cuprates. NANOMATERIALS (BASEL, SWITZERLAND) 2024; 14:1021. [PMID: 38921897 PMCID: PMC11206324 DOI: 10.3390/nano14121021] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/12/2024] [Revised: 05/31/2024] [Accepted: 06/09/2024] [Indexed: 06/27/2024]
Abstract
Electromagnetic waves propagating in a layered superconductor with arbitrary momentum, with respect to the main crystallographic directions, exhibit an unavoidable mixing between longitudinal and transverse degrees of freedom. Here we show that this basic physical mechanism explains the emergence of a well-defined absorption peak in the in-plane optical conductivity when light propagates at small tilting angles relative to the stacking direction in layered cuprates. More specifically, we show that this peak, often interpreted as a spurious leakage of the c-axis Josephson plasmon, is instead a signature of the true longitudinal plasma mode occurring at larger momenta. By combining a classical approach based on Maxwell's equations with a full quantum derivation of the plasma modes based on modeling the superconducting phase degrees of freedom, we provide an analytical expression for the absorption peak as a function of the tilting angle and light polarization. We suggest that an all-optical measurement in tilted geometry can be used as an alternative way to access plasma-wave dispersion, usually measured by means of large-momenta scattering techniques like resonant inelastic X-ray scattering (RIXS) or electron energy loss spectroscopy (EELS).
Collapse
Affiliation(s)
| | | | | | - Lara Benfatto
- Department of Physics and ISC-CNR, “Sapienza” University of Rome, P.le Aldo Moro 5, 00185 Rome, Italy; (N.S.); (J.F.); (C.C.)
| |
Collapse
|
3
|
Setty C, Baggioli M, Zaccone A. Anharmonic theory of superconductivity and its applications to emerging quantum materials. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2024; 36:173002. [PMID: 38252997 DOI: 10.1088/1361-648x/ad2159] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/20/2023] [Accepted: 01/22/2024] [Indexed: 01/24/2024]
Abstract
The role of anharmonicity on superconductivity has often been disregarded in the past. Recently, it has been recognized that anharmonic decoherence could play a fundamental role in determining the superconducting properties (electron-phonon coupling, critical temperature, etc) of a large class of materials, including systems close to structural soft-mode instabilities, amorphous solids and metals under extreme high-pressure conditions. Here, we review recent theoretical progress on the role of anharmonic effects, and in particular certain universal properties of anharmonic damping, on superconductivity. Our focus regards the combination of microscopic-agnostic effective theories for bosonic mediators with the well-established BCS theory and Migdal-Eliashberg theory for superconductivity. We discuss in detail the theoretical frameworks, their possible implementation within first-principles methods, and the experimental probes for anharmonic decoherence. Finally, we present several concrete applications to emerging quantum materials, including hydrides, ferroelectrics and systems with charge density wave instabilities.
Collapse
Affiliation(s)
- Chandan Setty
- Department of Physics and Astronomy, Rice Center for Quantum Materials, Rice University, Houston, TX 77005, United States of America
| | - Matteo Baggioli
- Wilczek Quantum Center, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, People's Republic of China
- Shanghai Research Center for Quantum Sciences, Shanghai 201315, People's Republic of China
| | - Alessio Zaccone
- Department of Physics 'A. Pontremoli', University of Milan, via Celoria 16, 20133 Milan, Italy
- Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue, CB30HE Cambridge, United Kingdom
| |
Collapse
|
4
|
Moradifar P, Liu Y, Shi J, Siukola Thurston ML, Utzat H, van Driel TB, Lindenberg AM, Dionne JA. Accelerating Quantum Materials Development with Advances in Transmission Electron Microscopy. Chem Rev 2023. [PMID: 37979189 DOI: 10.1021/acs.chemrev.2c00917] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/20/2023]
Abstract
Quantum materials are driving a technology revolution in sensing, communication, and computing, while simultaneously testing many core theories of the past century. Materials such as topological insulators, complex oxides, superconductors, quantum dots, color center-hosting semiconductors, and other types of strongly correlated materials can exhibit exotic properties such as edge conductivity, multiferroicity, magnetoresistance, superconductivity, single photon emission, and optical-spin locking. These emergent properties arise and depend strongly on the material's detailed atomic-scale structure, including atomic defects, dopants, and lattice stacking. In this review, we describe how progress in the field of electron microscopy (EM), including in situ and in operando EM, can accelerate advances in quantum materials and quantum excitations. We begin by describing fundamental EM principles and operation modes. We then discuss various EM methods such as (i) EM spectroscopies, including electron energy loss spectroscopy (EELS), cathodoluminescence (CL), and electron energy gain spectroscopy (EEGS); (ii) four-dimensional scanning transmission electron microscopy (4D-STEM); (iii) dynamic and ultrafast EM (UEM); (iv) complementary ultrafast spectroscopies (UED, XFEL); and (v) atomic electron tomography (AET). We describe how these methods could inform structure-function relations in quantum materials down to the picometer scale and femtosecond time resolution, and how they enable precision positioning of atomic defects and high-resolution manipulation of quantum materials. For each method, we also describe existing limitations to solve open quantum mechanical questions, and how they might be addressed to accelerate progress. Among numerous notable results, our review highlights how EM is enabling identification of the 3D structure of quantum defects; measuring reversible and metastable dynamics of quantum excitations; mapping exciton states and single photon emission; measuring nanoscale thermal transport and coupled excitation dynamics; and measuring the internal electric field and charge density distribution of quantum heterointerfaces- all at the quantum materials' intrinsic atomic and near atomic-length scale. We conclude by describing open challenges for the future, including achieving stable sample holders for ultralow temperature (below 10K) atomic-scale spatial resolution, stable spectrometers that enable meV energy resolution, and high-resolution, dynamic mapping of magnetic and spin fields. With atomic manipulation and ultrafast characterization enabled by EM, quantum materials will be poised to integrate into many of the sustainable and energy-efficient technologies needed for the 21st century.
Collapse
Affiliation(s)
- Parivash Moradifar
- Department of Materials Science and Engineering, Stanford University, Stanford, California 94305, United States
| | - Yin Liu
- Department of Materials Science and Engineering, Stanford University, Stanford, California 94305, United States
- Department of Materials Science and Engineering, North Carolina State University, Raleigh, North Carolina 27695, United States
| | - Jiaojian Shi
- Department of Materials Science and Engineering, Stanford University, Stanford, California 94305, United States
- SLAC National Accelerator Laboratory, 2575 Sand Hill Road MS69, Menlo Park, California 94025, United States
| | | | - Hendrik Utzat
- Department of Materials Science and Engineering, Stanford University, Stanford, California 94305, United States
- Department of Chemistry, University of California Berkeley, Berkeley, California 94720, United States
| | - Tim B van Driel
- Linac Coherent Light Source, SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, California 94025, United States
| | - Aaron M Lindenberg
- Department of Materials Science and Engineering, Stanford University, Stanford, California 94305, United States
- SLAC National Accelerator Laboratory, 2575 Sand Hill Road MS69, Menlo Park, California 94025, United States
| | - Jennifer A Dionne
- Department of Materials Science and Engineering, Stanford University, Stanford, California 94305, United States
- Department of Radiology, Stanford University, Stanford, California 94305, United States
| |
Collapse
|
5
|
Arpaia R, Martinelli L, Sala MM, Caprara S, Nag A, Brookes NB, Camisa P, Li Q, Gao Q, Zhou X, Garcia-Fernandez M, Zhou KJ, Schierle E, Bauch T, Peng YY, Di Castro C, Grilli M, Lombardi F, Braicovich L, Ghiringhelli G. Signature of quantum criticality in cuprates by charge density fluctuations. Nat Commun 2023; 14:7198. [PMID: 37938250 PMCID: PMC10632404 DOI: 10.1038/s41467-023-42961-5] [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: 08/19/2022] [Accepted: 10/25/2023] [Indexed: 11/09/2023] Open
Abstract
The universality of the strange metal phase in many quantum materials is often attributed to the presence of a quantum critical point (QCP), a zero-temperature phase transition ruled by quantum fluctuations. In cuprates, where superconductivity hinders direct QCP observation, indirect evidence comes from the identification of fluctuations compatible with the strange metal phase. Here we show that the recently discovered charge density fluctuations (CDF) possess the right properties to be associated to a quantum phase transition. Using resonant x-ray scattering, we studied the CDF in two families of cuprate superconductors across a wide doping range (up to p = 0.22). At p* ≈ 0.19, the putative QCP, the CDF intensity peaks, and the characteristic energy Δ is minimum, marking a wedge-shaped region in the phase diagram indicative of a quantum critical behavior, albeit with anomalies. These findings strengthen the role of charge order in explaining strange metal phenomenology and provide insights into high-temperature superconductivity.
Collapse
Affiliation(s)
- Riccardo Arpaia
- Quantum Device Physics Laboratory, Department of Microtechnology and Nanoscience, Chalmers University of Technology, SE-41296, Göteborg, Sweden.
| | - Leonardo Martinelli
- Dipartimento di Fisica, Politecnico di Milano, Piazza Leonardo da Vinci 32, I-20133, Milano, Italy
| | - Marco Moretti Sala
- Dipartimento di Fisica, Politecnico di Milano, Piazza Leonardo da Vinci 32, I-20133, Milano, Italy
| | - Sergio Caprara
- Dipartimento di Fisica, Università di Roma "La Sapienza", P.le Aldo Moro 5, I-00185, Roma, Italy
- CNR-ISC, via dei Taurini 19, I-00185, Roma, Italy
| | - Abhishek Nag
- Diamond Light Source, Harwell Campus, Didcot, OX11 0DE, United Kingdom
| | - Nicholas B Brookes
- ESRF, The European Synchrotron, 71 Avenue des Martyrs, F-38000, Grenoble, France
| | - Pietro Camisa
- Dipartimento di Fisica, Politecnico di Milano, Piazza Leonardo da Vinci 32, I-20133, Milano, Italy
| | - Qizhi Li
- International Center for Quantum Materials, School of Physics, Peking University, CN-100871, Beijing, China
| | - Qiang Gao
- Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, CN-100190, Beijing, China
| | - Xingjiang Zhou
- Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, CN-100190, Beijing, China
| | | | - Ke-Jin Zhou
- Diamond Light Source, Harwell Campus, Didcot, OX11 0DE, United Kingdom
| | - Enrico Schierle
- Helmholtz-Zentrum Berlin für Materialien und Energie, Albert-Einstein-Straße 15, D-12489, Berlin, Germany
| | - Thilo Bauch
- Quantum Device Physics Laboratory, Department of Microtechnology and Nanoscience, Chalmers University of Technology, SE-41296, Göteborg, Sweden
| | - Ying Ying Peng
- International Center for Quantum Materials, School of Physics, Peking University, CN-100871, Beijing, China
| | - Carlo Di Castro
- Dipartimento di Fisica, Università di Roma "La Sapienza", P.le Aldo Moro 5, I-00185, Roma, Italy
| | - Marco Grilli
- Dipartimento di Fisica, Università di Roma "La Sapienza", P.le Aldo Moro 5, I-00185, Roma, Italy
- CNR-ISC, via dei Taurini 19, I-00185, Roma, Italy
| | - Floriana Lombardi
- Quantum Device Physics Laboratory, Department of Microtechnology and Nanoscience, Chalmers University of Technology, SE-41296, Göteborg, Sweden
| | - Lucio Braicovich
- Dipartimento di Fisica, Politecnico di Milano, Piazza Leonardo da Vinci 32, I-20133, Milano, Italy
- ESRF, The European Synchrotron, 71 Avenue des Martyrs, F-38000, Grenoble, France
| | - Giacomo Ghiringhelli
- Dipartimento di Fisica, Politecnico di Milano, Piazza Leonardo da Vinci 32, I-20133, Milano, Italy.
- CNR-SPIN, Dipartimento di Fisica, Politecnico di Milano, Piazza Leonardo da Vinci 32, I-20133, Milano, Italy.
| |
Collapse
|
6
|
Silkin VM, Drechsler SL, Efremov DV. Unusual Low-Energy Collective Charge Excitations in High- Tc Cuprate Superconductors. J Phys Chem Lett 2023; 14:8060-8068. [PMID: 37655950 PMCID: PMC10510710 DOI: 10.1021/acs.jpclett.3c01871] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/08/2023] [Accepted: 08/14/2023] [Indexed: 09/02/2023]
Abstract
Despite decades of intensive experimental and theoretical efforts, the physics of cuprate high-temperature superconductors in general, and, in particular, their normal state, is still under debate. Here, we report our investigation of low-energy charge excitations in the normal state. We find that the peculiarities of the electronic band structure at low energies have a profound impact on the nature of the intraband collective modes. It gives rise to a new kind of mode with huge intensity and non-Lorentzian spectral function in addition to well-known collective excitations like conventional plasmons and spin fluctuation. We predict two such modes with maximal spectral weight in the nodal and antinodal directions. Additionally, we found a long-living quasi-one-dimensional plasmon becoming an intense soft mode over an extended momentum range along the antinodal direction. These modes might explain some of the resonant inelastic X-ray scattering spectroscopy data.
Collapse
Affiliation(s)
- Vyacheslav M. Silkin
- Donostia
International Physics Center (DIPC), 20018 San Sebastián/Donostia, Basque Country, Spain
- Departamento
de Polímeros y Materiales Avanzados: Física,
Química y Tecnología, Facultad de Ciencias
Químicas, Universidad del País
Vasco UPV/EHU, 20080 San Sebastián/Donostia, Basque Country, Spain
- IKERBASQUE,
Basque Foundation for Science, 48013 Bilbao, Basque Country, Spain
| | - Stefan-Ludwig Drechsler
- Leibniz
Institute for Solid State and Materials Research IFW Dresden, Helmholtzstrasse 20, 01069 Dresden, Germany
| | - Dmitry V. Efremov
- Leibniz
Institute for Solid State and Materials Research IFW Dresden, Helmholtzstrasse 20, 01069 Dresden, Germany
| |
Collapse
|
7
|
Husain AA, Huang EW, Mitrano M, Rak MS, Rubeck SI, Guo X, Yang H, Sow C, Maeno Y, Uchoa B, Chiang TC, Batson PE, Phillips PW, Abbamonte P. Pines' demon observed as a 3D acoustic plasmon in Sr 2RuO 4. Nature 2023; 621:66-70. [PMID: 37558882 PMCID: PMC10482684 DOI: 10.1038/s41586-023-06318-8] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/04/2022] [Accepted: 06/13/2023] [Indexed: 08/11/2023]
Abstract
The characteristic excitation of a metal is its plasmon, which is a quantized collective oscillation of its electron density. In 1956, David Pines predicted that a distinct type of plasmon, dubbed a 'demon', could exist in three-dimensional (3D) metals containing more than one species of charge carrier1. Consisting of out-of-phase movement of electrons in different bands, demons are acoustic, electrically neutral and do not couple to light, so have never been detected in an equilibrium, 3D metal. Nevertheless, demons are believed to be critical for diverse phenomena including phase transitions in mixed-valence semimetals2, optical properties of metal nanoparticles3, soundarons in Weyl semimetals4 and high-temperature superconductivity in, for example, metal hydrides3,5-7. Here, we present evidence for a demon in Sr2RuO4 from momentum-resolved electron energy-loss spectroscopy. Formed of electrons in the β and γ bands, the demon is gapless with critical momentum qc = 0.08 reciprocal lattice units and room-temperature velocity v = (1.065 ± 0.12) × 105 m s-1 that undergoes a 31% renormalization upon cooling to 30 K because of coupling to the particle-hole continuum. The momentum dependence of the intensity of the demon confirms its neutral character. Our study confirms a 67-year old prediction and indicates that demons may be a pervasive feature of multiband metals.
Collapse
Affiliation(s)
- Ali A Husain
- Department of Physics and Materials Research Laboratory, University of Illinois, Urbana, IL, USA.
| | - Edwin W Huang
- Department of Physics and Institute for Condensed Matter Theory, University of Illinois, Urbana, IL, USA
| | - Matteo Mitrano
- Department of Physics, Harvard University, Cambridge, MA, USA
| | - Melinda S Rak
- Department of Physics and Materials Research Laboratory, University of Illinois, Urbana, IL, USA
| | - Samantha I Rubeck
- Department of Physics and Materials Research Laboratory, University of Illinois, Urbana, IL, USA
| | - Xuefei Guo
- Department of Physics and Materials Research Laboratory, University of Illinois, Urbana, IL, USA
| | - Hongbin Yang
- Department of Chemistry and Chemical Biology, Rutgers University, Piscataway, NJ, USA
| | - Chanchal Sow
- Department of Physics, Kyoto University, Kyoto, Japan
- Department of Physics, Indian Institute of Technology, Kanpur, India
| | - Yoshiteru Maeno
- Department of Physics, Kyoto University, Kyoto, Japan
- Toyota Riken - Kyoto Univ. Research Center (TRiKUC), KUIAS, Kyoto University, Kyoto, Japan
| | - Bruno Uchoa
- Department of Physics and Astronomy, University of Oklahoma, Norman, OK, USA
| | - Tai C Chiang
- Department of Physics and Materials Research Laboratory, University of Illinois, Urbana, IL, USA
| | - Philip E Batson
- Department of Physics, Rutgers University, Piscataway, NJ, USA
| | - Philip W Phillips
- Department of Physics and Institute for Condensed Matter Theory, University of Illinois, Urbana, IL, USA
| | - Peter Abbamonte
- Department of Physics and Materials Research Laboratory, University of Illinois, Urbana, IL, USA.
| |
Collapse
|
8
|
Scott K, Kisiel E, Boyle TJ, Basak R, Jargot G, Das S, Agrestini S, Garcia-Fernandez M, Choi J, Pelliciari J, Li J, Chuang YD, Zhong R, Schneeloch JA, Gu G, Légaré F, Kemper AF, Zhou KJ, Bisogni V, Blanco-Canosa S, Frano A, Boschini F, da Silva Neto EH. Low-energy quasi-circular electron correlations with charge order wavelength in Bi 2Sr 2CaCu 2O 8+δ. SCIENCE ADVANCES 2023; 9:eadg3710. [PMID: 37467326 DOI: 10.1126/sciadv.adg3710] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/24/2022] [Accepted: 06/16/2023] [Indexed: 07/21/2023]
Abstract
Most resonant inelastic x-ray scattering (RIXS) studies of dynamic charge order correlations in the cuprates have focused on the high-symmetry directions of the copper oxide plane. However, scattering along other in-plane directions should not be ignored as it may help understand, for example, the origin of charge order correlations or the isotropic scattering resulting in strange metal behavior. Our RIXS experiments reveal dynamic charge correlations over the qx-qy scattering plane in underdoped Bi2Sr2CaCu2O8+δ. Tracking the softening of the RIXS-measured bond-stretching phonon, we show that these dynamic correlations exist at energies below approximately 70 meV and are centered around a quasi-circular manifold in the qx-qy scattering plane with radius equal to the magnitude of the charge order wave vector, qCO. This phonon-tracking procedure also allows us to rule out fluctuations of short-range directional charge order (i.e., centered around [qx = ±qCO, qy = 0] and [qx = 0, qy = ±qCO]) as the origin of the observed correlations.
Collapse
Affiliation(s)
- Kirsty Scott
- Department of Physics, Yale University, New Haven, CT 06520, USA
- Energy Sciences Institute, Yale University, West Haven, CT 06516, USA
| | - Elliot Kisiel
- Department of Physics, University of California San Diego, La Jolla, CA 92093, USA
| | - Timothy J Boyle
- Department of Physics, Yale University, New Haven, CT 06520, USA
- Energy Sciences Institute, Yale University, West Haven, CT 06516, USA
- Department of Physics and Astronomy, University of California, Davis, CA 95616, USA
| | - Rourav Basak
- Department of Physics, University of California San Diego, La Jolla, CA 92093, USA
| | - Gaëtan Jargot
- Centre Énergie Matériaux Télécommunications, Institut National de la Recherche Scientifique, Varennes, Québec J3X 1S2, Canada
| | - Sarmistha Das
- Department of Physics, University of California San Diego, La Jolla, CA 92093, USA
| | | | | | - Jaewon Choi
- Diamond Light Source, Harwell Campus, Didcot OX11 0DE, UK
| | - Jonathan Pelliciari
- National Synchrotron Light Source II, Brookhaven National Laboratory, Upton, NY 11973, USA
| | - Jiemin Li
- National Synchrotron Light Source II, Brookhaven National Laboratory, Upton, NY 11973, USA
| | - Yi-De Chuang
- Advanced Light Source, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
| | - Ruidan Zhong
- Condensed Matter Physics and Materials Science, Brookhaven National Laboratory, Upton, NY 11973, USA
| | - John A Schneeloch
- Condensed Matter Physics and Materials Science, Brookhaven National Laboratory, Upton, NY 11973, USA
| | - Genda Gu
- Condensed Matter Physics and Materials Science, Brookhaven National Laboratory, Upton, NY 11973, USA
| | - François Légaré
- Centre Énergie Matériaux Télécommunications, Institut National de la Recherche Scientifique, Varennes, Québec J3X 1S2, Canada
| | - Alexander F Kemper
- Department of Physics, North Carolina State University, Raleigh, NC 27695, USA
| | - Ke-Jin Zhou
- Diamond Light Source, Harwell Campus, Didcot OX11 0DE, UK
| | - Valentina Bisogni
- National Synchrotron Light Source II, Brookhaven National Laboratory, Upton, NY 11973, USA
| | - Santiago Blanco-Canosa
- Donostia International Physics Center (DIPC), 20018 Donostia-San Sebastian, Basque Country, Spain
- IKERBASQUE, Basque Foundation for Science, 48013 Bilbao, Spain
| | - Alex Frano
- Department of Physics, University of California San Diego, La Jolla, CA 92093, USA
- Canadian Institute for Advanced Research, Toronto, ON M5G 1M1, Canada
| | - Fabio Boschini
- Centre Énergie Matériaux Télécommunications, Institut National de la Recherche Scientifique, Varennes, Québec J3X 1S2, Canada
- Quantum Matter Institute, University of British Columbia, Vancouver, BC V6T 1Z4, Canada
| | - Eduardo H da Silva Neto
- Department of Physics, Yale University, New Haven, CT 06520, USA
- Energy Sciences Institute, Yale University, West Haven, CT 06516, USA
- Department of Physics and Astronomy, University of California, Davis, CA 95616, USA
- Department of Applied Physics, Yale University, New Haven, CT 06520, USA
| |
Collapse
|
9
|
Thornton SJ, Liarte DB, Abbamonte P, Sethna JP, Chowdhury D. Jamming and unusual charge density fluctuations of strange metals. Nat Commun 2023; 14:3919. [PMID: 37400449 DOI: 10.1038/s41467-023-39499-x] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/15/2022] [Accepted: 06/08/2023] [Indexed: 07/05/2023] Open
Abstract
The strange metallic regime across a number of high-temperature superconducting materials presents numerous challenges to the classic theory of Fermi liquid metals. Recent measurements of the dynamical charge response of strange metals, including optimally doped cuprates, have revealed a broad, featureless continuum of excitations, extending over much of the Brillouin zone. The collective density oscillations of this strange metal decay into the continuum in a manner that is at odds with the expectations of Fermi liquid theory. Inspired by these observations, we investigate the phenomenology of bosonic collective modes and the particle-hole excitations in a class of strange metals by making an analogy to the phonons of classical lattices falling apart across an unconventional jamming-like transition associated with the onset of rigidity. By making comparisons to the experimentally measured dynamical response functions, we reproduce many of the qualitative features using the above framework. We conjecture that the dynamics of electronic charge density over an intermediate range of energy scales in a class of strongly correlated metals can be at the brink of a jamming-like transition.
Collapse
Affiliation(s)
| | - Danilo B Liarte
- Department of Physics, Cornell University, Ithaca, NY, 14853, USA
- ICTP South American Institute for Fundamental Research, São Paulo, SP, Brazil
- Institute of Theoretical Physics, São Paulo State University, São Paulo, SP, Brazil
| | - Peter Abbamonte
- Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA
| | - James P Sethna
- Department of Physics, Cornell University, Ithaca, NY, 14853, USA
| | | |
Collapse
|
10
|
Kobayashi H, Sakaguchi Y, Kitagawa H, Oura M, Ikeda S, Kuga K, Suzuki S, Nakatsuji S, Masuda R, Kobayashi Y, Seto M, Yoda Y, Tamasaku K, Komijani Y, Chandra P, Coleman P. Observation of a critical charge mode in a strange metal. Science 2023. [PMID: 36862771 DOI: 10.1126/science.abc4787] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 03/04/2023]
Abstract
Understanding the strange metallic behavior that develops at the brink of localization in quantum materials requires probing the underlying electronic charge dynamics. Using synchrotron radiation-based Mössbauer spectroscopy, we studied the charge fluctuations of the strange metal phase of β-YbAlB4 as a function of temperature and pressure. We found that the usual single absorption peak in the Fermi-liquid regime splits into two peaks upon entering the critical regime. We interpret this spectrum as a single nuclear transition, modulated by nearby electronic valence fluctuations whose long time scales are further enhanced by the formation of charged polarons. These critical charge fluctuations may prove to be a distinct signature of strange metals.
Collapse
Affiliation(s)
- Hisao Kobayashi
- Graduate School of Material Science, University of Hyogo, 3-2-1 Koto, Hyogo 678-1297, Japan.,RIKEN SPring-8 Center, Hyogo 679-5148, Japan
| | - Yui Sakaguchi
- Graduate School of Material Science, University of Hyogo, 3-2-1 Koto, Hyogo 678-1297, Japan
| | - Hayato Kitagawa
- Graduate School of Material Science, University of Hyogo, 3-2-1 Koto, Hyogo 678-1297, Japan.,RIKEN SPring-8 Center, Hyogo 679-5148, Japan
| | - Momoko Oura
- Graduate School of Material Science, University of Hyogo, 3-2-1 Koto, Hyogo 678-1297, Japan.,RIKEN SPring-8 Center, Hyogo 679-5148, Japan
| | - Shugo Ikeda
- Graduate School of Material Science, University of Hyogo, 3-2-1 Koto, Hyogo 678-1297, Japan.,RIKEN SPring-8 Center, Hyogo 679-5148, Japan
| | - Kentaro Kuga
- Institute for Solid State Physics, University of Tokyo, Kashiwa 277-8581, Japan
| | - Shintaro Suzuki
- Institute for Solid State Physics, University of Tokyo, Kashiwa 277-8581, Japan
| | - Satoru Nakatsuji
- Institute for Solid State Physics, University of Tokyo, Kashiwa 277-8581, Japan.,Department of Physics, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan.,Trans-scale Quantum Science Institute, University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan.,Institute for Quantum Matter and Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218, USA
| | - Ryo Masuda
- RIKEN SPring-8 Center, Hyogo 679-5148, Japan.,Institute for Integrated Radiation and Nuclear Science, Kyoto University, Osaka 590-0494, Japan.,Graduate School of Science and Technology, Hirosaki University, Aomori 036-8561 Japan
| | - Yasuhiro Kobayashi
- RIKEN SPring-8 Center, Hyogo 679-5148, Japan.,Institute for Integrated Radiation and Nuclear Science, Kyoto University, Osaka 590-0494, Japan
| | - Makoto Seto
- RIKEN SPring-8 Center, Hyogo 679-5148, Japan.,Institute for Integrated Radiation and Nuclear Science, Kyoto University, Osaka 590-0494, Japan
| | - Yoshitaka Yoda
- Japan Synchrotron Radiation Research Institute, Hyogo 679-5198, Japan
| | | | - Yashar Komijani
- Department of Physics, University of Cincinnati, Cincinnati, OH 45221-0011, USA.,Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08854, USA
| | - Premala Chandra
- Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08854, USA
| | - Piers Coleman
- Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08854, USA.,Hubbard Theory Consortium, Department of Physics, Royal Holloway, University of London, Egham, Surrey TW20 0EX, UK
| |
Collapse
|
11
|
Liarte DB, Thornton SJ, Schwen E, Cohen I, Chowdhury D, Sethna JP. Universal scaling for disordered viscoelastic matter near the onset of rigidity. Phys Rev E 2022; 106:L052601. [PMID: 36559468 DOI: 10.1103/physreve.106.l052601] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/01/2021] [Accepted: 10/06/2022] [Indexed: 06/17/2023]
Abstract
The onset of rigidity in interacting liquids, as they undergo a transition to a disordered solid, is associated with a rearrangement of the low-frequency vibrational spectrum. In this Letter, we derive scaling forms for the singular dynamical response of disordered viscoelastic networks near both jamming and rigidity percolation. Using effective-medium theory, we extract critical exponents, invariant scaling combinations, and analytical formulas for universal scaling functions near these transitions. Our scaling forms describe the behavior in space and time near the various onsets of rigidity, for rigid and floppy phases and the crossover region, including diverging length scales and timescales at the transitions.
Collapse
Affiliation(s)
- Danilo B Liarte
- ICTP South American Institute for Fundamental Research, São Paulo, SP 01140-070, Brazil
- Institute of Theoretical Physics, São Paulo State University, São Paulo, SP 01140-070, Brazil
- Department of Physics, Cornell University, Ithaca, New York 14853, USA
| | | | - Eric Schwen
- Department of Physics, Cornell University, Ithaca, New York 14853, USA
| | - Itai Cohen
- Department of Physics, Cornell University, Ithaca, New York 14853, USA
| | | | - James P Sethna
- Department of Physics, Cornell University, Ithaca, New York 14853, USA
| |
Collapse
|
12
|
Abstract
In traditional metals, the temperature (
T
) dependence of electrical resistivity vanishes at low or high
T
, albeit for different reasons. Here, we review a class of materials, known as “strange” metals, that can violate both of these principles. In strange metals, the change in slope of the resistivity as the mean free path drops below the lattice constant, or as
T
→ 0, can be imperceptible, suggesting continuity between the charge carriers at low and high
T
. We focus on transport and spectroscopic data on candidate strange metals in an effort to isolate and identify a unifying physical principle. Special attention is paid to quantum criticality, Planckian dissipation, Mottness, and whether a new gauge principle is needed to account for the nonlocal transport seen in these materials.
Collapse
Affiliation(s)
- Philip W. Phillips
- Department of Physics and Institute for Condensed Matter Theory, University of Illinois, Urbana, IL 61801, USA
| | - Nigel E. Hussey
- H. H. Wills Physics Laboratory, University of Bristol, Bristol BS8 1TL, UK
- High Field Magnet Laboratory (HFML-EMFL) and Institute for Molecules and Materials, Radboud University, 6525 ED Nijmegen, Netherlands
| | - Peter Abbamonte
- Department of Physics, University of Illinois, Urbana, IL 61801, USA
| |
Collapse
|
13
|
Evolution of Charge-Lattice Dynamics across the Kuramoto Synchronization Phase Diagram of Quantum Tunneling Polarons in Cuprate Superconductors. CONDENSED MATTER 2021. [DOI: 10.3390/condmat6040052] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/17/2022]
Abstract
Because of its sensitivity to the instantaneous structure factor, S(Q,t = 0), Extended X-ray Absorption Fine Structure (EXAFS) is a powerful tool for probing the dynamic structure of condensed matter systems in which the charge and lattice dynamics are coupled. When applied to hole-doped cuprate superconductors, EXAFS has revealed the presence of internal quantum tunneling polarons (IQTPs). An IQTP arises in EXAFS as a two-site distribution for certain Cu–O pairs, which is also duplicated in inelastic scattering but not observed in standard diffraction measurements. The Cu–Sr pair distribution has been found to be highly anharmonic and strongly correlated to both the IQTPs and to superconductivity, as, for example, in YSr2Cu2.75Mo0.25O7.54(Tc=84 K). In order to describe such nontrivial, anharmonic charge-lattice dynamics, we have proposed a model Hamiltonian for a prototype six-atom cluster, in which two Cu-apical-O IQTPs are charge-transfer bridged through Cu atoms by an O atom in the CuO2 plane and are anharmonically coupled via a Sr atom. By applying an exact diagonalization procedure to this cluster, we have verified that our model indeed produces an intricate interplay between charge and lattice dynamics. Then, by using the Kuramoto model for the synchronization of coupled quantum oscillators, we have found a first-order phase transition for the IQTPs into a synchronized, phase-locked phase. Most importantly, we have shown that this transition results specifically from the anharmonicity. Finally, we have provided a phase diagram showing the onset of the phase-locking of IQTPs as a function of the charge-lattice and anharmonic couplings in our model. We have found that the charge, initially confined to the apical oxygens, is partially pumped into the CuO2 plane in the synchronized phase, which suggests a possible connection between the synchronized dynamic structure and high-temperature superconductivity (HTSC) in doped cuprates.
Collapse
|
14
|
Thermomechanical Multifunctionality in 3D-Printed Polystyrene-Boron Nitride Nanotubes (BNNT) Composites. JOURNAL OF COMPOSITES SCIENCE 2021. [DOI: 10.3390/jcs5020061] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
Abstract
In this work, polystyrene (PS) and boron nitrides nanotubes (BNNT) composites were fabricated, prepared, and characterized using modified direct mixing and sonication processes. The polymer composites were extruded into filaments (BNNTs at 10 wt. %) for 3D printing, utilizing the fused deposition modeling (FDM) technique to fabricate parts for mechanical and thermal applications. Using a direct mixing process, we found that the thermal conductivity and the mechanical strength of the PS-BNNT composite were respectively four times and two times higher compared to the sonication method. The thermal stability and glass transition temperatures were positively affected. A 2D microstructural mechanical entanglement model captured the exact geometry of the nanotubes using the MultiMechanics software, and the performance of the additive manufactured (AM) PS-BNNT composites part for thermomechanical application was simulated in COMSOL. The modified direct mixing process for PS-BNNT, which affects morphology, proved to be effective in achieving better interfacial bonding, indicating that BNNTs are promising fillers for improving thermal and mechanical properties, and are applicable for thermal management and electronic packaging.
Collapse
|
15
|
Dynamic electron correlations with charge order wavelength along all directions in the copper oxide plane. Nat Commun 2021; 12:597. [PMID: 33500415 PMCID: PMC7838423 DOI: 10.1038/s41467-020-20824-7] [Citation(s) in RCA: 13] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/21/2020] [Accepted: 12/21/2020] [Indexed: 11/08/2022] Open
Abstract
In strongly correlated systems the strength of Coulomb interactions between electrons, relative to their kinetic energy, plays a central role in determining their emergent quantum mechanical phases. We perform resonant x-ray scattering on Bi2Sr2CaCu2O8+δ, a prototypical cuprate superconductor, to probe electronic correlations within the CuO2 plane. We discover a dynamic quasi-circular pattern in the x-y scattering plane with a radius that matches the wave vector magnitude of the well-known static charge order. Along with doping- and temperature-dependent measurements, our experiments reveal a picture of charge order competing with superconductivity where short-range domains along x and y can dynamically rotate into any other in-plane direction. This quasi-circular spectrum, a hallmark of Brazovskii-type fluctuations, has immediate consequences to our understanding of rotational and translational symmetry breaking in the cuprates. We discuss how the combination of short- and long-range Coulomb interactions results in an effective non-monotonic potential that may determine the quasi-circular pattern. Knowledge of effective Coulomb interactions is central to understand emergent quantum phases in strongly correlated systems. Here, Boschini et al. report a dynamic quasi-circular spectrum of charge density wave fluctuations in the CuO2 plane of Bi2Sr2CaCu2O8+δ, shedding a light on understanding how Coulomb interactions can lead to rotational and translational symmetry breaking in the cuprates.
Collapse
|
16
|
Nag A, Zhu M, Bejas M, Li J, Robarts HC, Yamase H, Petsch AN, Song D, Eisaki H, Walters AC, García-Fernández M, Greco A, Hayden SM, Zhou KJ. Detection of Acoustic Plasmons in Hole-Doped Lanthanum and Bismuth Cuprate Superconductors Using Resonant Inelastic X-Ray Scattering. PHYSICAL REVIEW LETTERS 2020; 125:257002. [PMID: 33416344 DOI: 10.1103/physrevlett.125.257002] [Citation(s) in RCA: 11] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/01/2020] [Revised: 10/18/2020] [Accepted: 11/13/2020] [Indexed: 06/12/2023]
Abstract
High T_{c} superconductors show a rich variety of phases associated with their charge degrees of freedom. Valence charges can give rise to charge ordering or acoustic plasmons in these layered cuprate superconductors. While charge ordering has been observed for both hole- and electron-doped cuprates, acoustic plasmons have only been found in electron-doped materials. Here, we use resonant inelastic x-ray scattering to observe the presence of acoustic plasmons in two families of hole-doped cuprate superconductors (La_{1.84}Sr_{0.16}CuO_{4} and Bi_{2}Sr_{1.6}La_{0.4}CuO_{6+δ}), crucially completing the picture. Interestingly, in contrast to the quasistatic charge ordering which manifests at both Cu and O sites, the observed acoustic plasmons are predominantly associated with the O sites, revealing a unique dichotomy in the behavior of valence charges in hole-doped cuprates.
Collapse
Affiliation(s)
- Abhishek Nag
- Diamond Light Source, Harwell Campus, Didcot OX11 0DE, United Kingdom
| | - M Zhu
- H. H. Wills Physics Laboratory, University of Bristol, Bristol BS8 1TL, United Kingdom
| | - Matías Bejas
- Facultad de Ciencias Exactas, Ingeniería y Agrimensura and Instituto de Física de Rosario (UNR-CONICET), Avenida Pellegrini 250, 2000 Rosario, Argentina
| | - J Li
- Diamond Light Source, Harwell Campus, Didcot OX11 0DE, United Kingdom
- Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
| | - H C Robarts
- Diamond Light Source, Harwell Campus, Didcot OX11 0DE, United Kingdom
- H. H. Wills Physics Laboratory, University of Bristol, Bristol BS8 1TL, United Kingdom
| | - Hiroyuki Yamase
- International Center of Materials Nanoarchitectonics, National Institute for Materials Science, Tsukuba 305-0047, Japan
- Department of Condensed Matter Physics, Graduate School of Science, Hokkaido University, Sapporo 060-0810, Japan
| | - A N Petsch
- H. H. Wills Physics Laboratory, University of Bristol, Bristol BS8 1TL, United Kingdom
| | - D Song
- National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba, Ibaraki 305-8560, Japan
| | - H Eisaki
- National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba, Ibaraki 305-8560, Japan
| | - A C Walters
- Diamond Light Source, Harwell Campus, Didcot OX11 0DE, United Kingdom
| | | | - Andrés Greco
- Facultad de Ciencias Exactas, Ingeniería y Agrimensura and Instituto de Física de Rosario (UNR-CONICET), Avenida Pellegrini 250, 2000 Rosario, Argentina
| | - S M Hayden
- H. H. Wills Physics Laboratory, University of Bristol, Bristol BS8 1TL, United Kingdom
| | - Ke-Jin Zhou
- Diamond Light Source, Harwell Campus, Didcot OX11 0DE, United Kingdom
| |
Collapse
|
17
|
Hepting M, Chaix L, Huang EW, Fumagalli R, Peng YY, Moritz B, Kummer K, Brookes NB, Lee WC, Hashimoto M, Sarkar T, He JF, Rotundu CR, Lee YS, Greene RL, Braicovich L, Ghiringhelli G, Shen ZX, Devereaux TP, Lee WS. Three-dimensional collective charge excitations in electron-doped copper oxide superconductors. Nature 2018; 563:374-378. [PMID: 30429543 DOI: 10.1038/s41586-018-0648-3] [Citation(s) in RCA: 23] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/02/2018] [Accepted: 08/22/2018] [Indexed: 11/09/2022]
Abstract
High-temperature copper oxide superconductors consist of stacked CuO2 planes, with electronic band structures and magnetic excitations that are primarily two-dimensional1,2, but with superconducting coherence that is three-dimensional. This dichotomy highlights the importance of out-of-plane charge dynamics, which has been found to be incoherent in the normal state3,4 within the limited range of momenta accessible by optics. Here we use resonant inelastic X-ray scattering to explore the charge dynamics across all three dimensions of the Brillouin zone. Polarization analysis of recently discovered collective excitations (modes) in electron-doped copper oxides5-7 reveals their charge origin, that is, without mixing with magnetic components5-7. The excitations disperse along both the in-plane and out-of-plane directions, revealing its three-dimensional nature. The periodicity of the out-of-plane dispersion corresponds to the distance between neighbouring CuO2 planes rather than to the crystallographic c-axis lattice constant, suggesting that the interplane Coulomb interaction is responsible for the coherent out-of-plane charge dynamics. The observed properties are hallmarks of the long-sought 'acoustic plasmon', which is a branch of distinct charge collective modes predicted for layered systems8-12 and argued to play a substantial part in mediating high-temperature superconductivity10-12.
Collapse
Affiliation(s)
- M Hepting
- Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory and Stanford University, Menlo Park, CA, USA
| | - L Chaix
- Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory and Stanford University, Menlo Park, CA, USA.,Université Grenoble Alpes, CNRS, Institut Néel, Grenoble, France
| | - E W Huang
- Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory and Stanford University, Menlo Park, CA, USA.,Department of Physics, Stanford University, Stanford, CA, USA
| | - R Fumagalli
- Dipartimento di Fisica, Politecnico di Milano, Milan, Italy
| | - Y Y Peng
- Dipartimento di Fisica, Politecnico di Milano, Milan, Italy.,Department of Physics and Seitz Materials Research Lab, University of Illinois, Urbana, IL, USA
| | - B Moritz
- Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory and Stanford University, Menlo Park, CA, USA
| | - K Kummer
- European Synchrotron Radiation Facility (ESRF), Grenoble, France
| | - N B Brookes
- European Synchrotron Radiation Facility (ESRF), Grenoble, France
| | - W C Lee
- Department of Physics, Binghamton University, Binghamton, NY, USA
| | - M Hashimoto
- Stanford Synchrotron Radiation Lightsource, SLAC National Accelerator Laboratory, Menlo Park, CA, USA
| | - T Sarkar
- Department of Physics, Center for Nanophysics and Advanced Materials, University of Maryland, College Park, MD, USA
| | - J-F He
- Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory and Stanford University, Menlo Park, CA, USA.,Department of Physics, University of Science and Technology of China, Hefei, China
| | - C R Rotundu
- Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory and Stanford University, Menlo Park, CA, USA
| | - Y S Lee
- Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory and Stanford University, Menlo Park, CA, USA
| | - R L Greene
- Department of Physics, Center for Nanophysics and Advanced Materials, University of Maryland, College Park, MD, USA
| | - L Braicovich
- Dipartimento di Fisica, Politecnico di Milano, Milan, Italy.,European Synchrotron Radiation Facility (ESRF), Grenoble, France
| | - G Ghiringhelli
- Dipartimento di Fisica, Politecnico di Milano, Milan, Italy.,CNR-SPIN, Politecnico di Milano, Milan, Italy
| | - Z X Shen
- Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory and Stanford University, Menlo Park, CA, USA.
| | - T P Devereaux
- Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory and Stanford University, Menlo Park, CA, USA.
| | - W S Lee
- Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory and Stanford University, Menlo Park, CA, USA.
| |
Collapse
|