1
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Shi J, Bie YQ, Zong A, Fang S, Chen W, Han J, Cao Z, Zhang Y, Taniguchi T, Watanabe K, Fu X, Bulović V, Kaxiras E, Baldini E, Jarillo-Herrero P, Nelson KA. Intrinsic 1[Formula: see text] phase induced in atomically thin 2H-MoTe 2 by a single terahertz pulse. Nat Commun 2023; 14:5905. [PMID: 37737233 PMCID: PMC10516973 DOI: 10.1038/s41467-023-41291-w] [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: 10/15/2022] [Accepted: 08/29/2023] [Indexed: 09/23/2023] Open
Abstract
The polymorphic transition from 2H to 1[Formula: see text]-MoTe2, which was thought to be induced by high-energy photon irradiation among many other means, has been intensely studied for its technological relevance in nanoscale transistors due to the remarkable improvement in electrical performance. However, it remains controversial whether a crystalline 1[Formula: see text] phase is produced because optical signatures of this putative transition are found to be associated with the formation of tellurium clusters instead. Here we demonstrate the creation of an intrinsic 1[Formula: see text] lattice after irradiating a mono- or few-layer 2H-MoTe2 with a single field-enhanced terahertz pulse. Unlike optical pulses, the low terahertz photon energy limits possible structural damages. We further develop a single-shot terahertz-pump-second-harmonic-probe technique and reveal a transition out of the 2H-phase within 10 ns after photoexcitation. Our results not only provide important insights to resolve the long-standing debate over the light-induced polymorphic transition in MoTe2 but also highlight the unique capability of strong-field terahertz pulses in manipulating quantum materials.
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Affiliation(s)
- Jiaojian Shi
- Department of Chemistry, Massachusetts Institute of Technology, Cambridge, MA 02139 USA
| | - Ya-Qing Bie
- Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139 USA
- State Key Lab of Optoelectronic Materials and Technologies, Guangdong Province Key Laboratory of Display Material and Technology, School of Electronics and Information Technology, Sun Yat-sen University, Guangzhou, 510275 People’s Republic of China
| | - Alfred Zong
- Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139 USA
- Department of Chemistry, University of California, Berkeley, CA 94720 USA
| | - Shiang Fang
- Department of Physics, Harvard University, Cambridge, MA 02138 USA
- Department of Physics and Astronomy, Center for Materials Theory, Rutgers University, Piscataway, NJ 08854 USA
- Present Address: Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139 USA
| | - Wei Chen
- Department of Physics, Harvard University, Cambridge, MA 02138 USA
| | - Jinchi Han
- Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA 02139 USA
- School of Integrated Circuits, Peking University, Beijing, 100871 People’s Republic of China
| | - Zhaolong Cao
- State Key Lab of Optoelectronic Materials and Technologies, Guangdong Province Key Laboratory of Display Material and Technology, School of Electronics and Information Technology, Sun Yat-sen University, Guangzhou, 510275 People’s Republic of China
| | - Yong Zhang
- Center for Materials Science & Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139 USA
| | - Takashi Taniguchi
- International Center for Materials Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki, Tsukuba, 305-0044 Japan
| | - Kenji Watanabe
- Research Center for Functional Materials, National Institute for Materials Science, 1-1 Namiki, Tsukuba, 305-0044 Japan
| | - Xuewen Fu
- Ultrafast Electron Microscopy Laboratory, The MOE Key Laboratory of Weak-Light Nonlinear Photonics, School of Physics, Nankai University, Tianjin, 300071 People’s Republic of China
| | - Vladimir Bulović
- Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA 02139 USA
| | | | - Edoardo Baldini
- Department of Physics, Center for Complex Quantum System, The University of Texas at Austin, Austin, TX 78712 USA
| | - Pablo Jarillo-Herrero
- Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139 USA
| | - Keith A. Nelson
- Department of Chemistry, Massachusetts Institute of Technology, Cambridge, MA 02139 USA
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2
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Zong A, Zhang Q, Zhou F, Su Y, Hwangbo K, Shen X, Jiang Q, Liu H, Gage TE, Walko DA, Kozina ME, Luo D, Reid AH, Yang J, Park S, Lapidus SH, Chu JH, Arslan I, Wang X, Xiao D, Xu X, Gedik N, Wen H. Spin-mediated shear oscillators in a van der Waals antiferromagnet. Nature 2023; 620:988-993. [PMID: 37532936 DOI: 10.1038/s41586-023-06279-y] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/01/2022] [Accepted: 06/02/2023] [Indexed: 08/04/2023]
Abstract
Understanding how microscopic spin configuration gives rise to exotic properties at the macroscopic length scale has long been pursued in magnetic materials1-5. One seminal example is the Einstein-de Haas effect in ferromagnets1,6,7, in which angular momentum of spins can be converted into mechanical rotation of an entire object. However, for antiferromagnets without net magnetic moment, how spin ordering couples to macroscopic movement remains elusive. Here we observed a seesaw-like rotation of reciprocal lattice peaks of an antiferromagnetic nanolayer film, whose gigahertz structural resonance exhibits more than an order-of-magnitude amplification after cooling below the Néel temperature. Using a suite of ultrafast diffraction and microscopy techniques, we directly visualize this spin-driven rotation in reciprocal space at the nanoscale. This motion corresponds to interlayer shear in real space, in which individual micro-patches of the film behave as coherent oscillators that are phase-locked and shear along the same in-plane axis. Using time-resolved optical polarimetry, we further show that the enhanced mechanical response strongly correlates with ultrafast demagnetization, which releases elastic energy stored in local strain gradients to drive the oscillators. Our work not only offers the first microscopic view of spin-mediated mechanical motion of an antiferromagnet but it also identifies a new route towards realizing high-frequency resonators8,9 up to the millimetre band, so the capability of controlling magnetic states on the ultrafast timescale10-13 can be readily transferred to engineering the mechanical properties of nanodevices.
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Affiliation(s)
- Alfred Zong
- Department of Chemistry, University of California, Berkeley, Berkeley, CA, USA
- Department of Physics, Massachusetts Institute of Technology, Cambridge, MA, USA
| | - Qi Zhang
- Department of Physics, University of Washington, Seattle, WA, USA
- Advanced Photon Source, Argonne National Laboratory, Lemont, IL, USA
- Department of Physics, Nanjing University, Nanjing, China
| | - Faran Zhou
- Advanced Photon Source, Argonne National Laboratory, Lemont, IL, USA
| | - Yifan Su
- Department of Physics, Massachusetts Institute of Technology, Cambridge, MA, USA
| | - Kyle Hwangbo
- Department of Physics, University of Washington, Seattle, WA, USA
| | - Xiaozhe Shen
- SLAC National Accelerator Laboratory, Menlo Park, CA, USA
| | - Qianni Jiang
- Department of Physics, University of Washington, Seattle, WA, USA
| | - Haihua Liu
- Center for Nanoscale Materials, Argonne National Laboratory, Lemont, IL, USA
| | - Thomas E Gage
- Center for Nanoscale Materials, Argonne National Laboratory, Lemont, IL, USA
| | - Donald A Walko
- Advanced Photon Source, Argonne National Laboratory, Lemont, IL, USA
| | | | - Duan Luo
- SLAC National Accelerator Laboratory, Menlo Park, CA, USA
| | | | - Jie Yang
- SLAC National Accelerator Laboratory, Menlo Park, CA, USA
| | - Suji Park
- SLAC National Accelerator Laboratory, Menlo Park, CA, USA
- Department of Materials Science and Engineering, Stanford University, Stanford, CA, USA
- Center for Functional Nanomaterials, Brookhaven National Laboratory, Upton, NY, USA
| | - Saul H Lapidus
- Advanced Photon Source, Argonne National Laboratory, Lemont, IL, USA
| | - Jiun-Haw Chu
- Department of Physics, University of Washington, Seattle, WA, USA
| | - Ilke Arslan
- Center for Nanoscale Materials, Argonne National Laboratory, Lemont, IL, USA
| | - Xijie Wang
- SLAC National Accelerator Laboratory, Menlo Park, CA, USA
| | - Di Xiao
- Department of Physics, University of Washington, Seattle, WA, USA
- Department of Materials Science and Engineering, University of Washington, Seattle, WA, USA
- Physical Sciences Division, Pacific Northwest National Laboratory, Richland, WA, USA
| | - Xiaodong Xu
- Department of Physics, University of Washington, Seattle, WA, USA.
- Department of Materials Science and Engineering, University of Washington, Seattle, WA, USA.
| | - Nuh Gedik
- Department of Physics, Massachusetts Institute of Technology, Cambridge, MA, USA.
| | - Haidan Wen
- Advanced Photon Source, Argonne National Laboratory, Lemont, IL, USA.
- Materials Science Division, Argonne National Laboratory, Lemont, IL, USA.
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3
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Li WH, Duncan CJR, Andorf MB, Bartnik AC, Bianco E, Cultrera L, Galdi A, Gordon M, Kaemingk M, Pennington CA, Kourkoutis LF, Bazarov IV, Maxson JM. A kiloelectron-volt ultrafast electron micro-diffraction apparatus using low emittance semiconductor photocathodes. STRUCTURAL DYNAMICS (MELVILLE, N.Y.) 2022; 9:024302. [PMID: 35350376 PMCID: PMC8934190 DOI: 10.1063/4.0000138] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/02/2021] [Accepted: 02/16/2022] [Indexed: 06/12/2023]
Abstract
We report the design and performance of a time-resolved electron diffraction apparatus capable of producing intense bunches with simultaneously single digit micrometer probe size, long coherence length, and 200 fs rms time resolution. We measure the 5d (peak) beam brightness at the sample location in micro-diffraction mode to be 7 × 10 13 A / m 2 rad 2 . To generate high brightness electron bunches, the system employs high efficiency, low emittance semiconductor photocathodes driven with a wavelength near the photoemission threshold at a repetition rate up to 250 kHz. We characterize spatial, temporal, and reciprocal space resolution of the apparatus. We perform proof-of-principle measurements of ultrafast heating in single crystal Au samples and compare experimental results with simulations that account for the effects of multiple scattering.
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Affiliation(s)
- W. H. Li
- Cornell Laboratory for Accelerator-Based Sciences and Education, Cornell University, Ithaca, New York 14853, USA
| | - C. J. R. Duncan
- Cornell Laboratory for Accelerator-Based Sciences and Education, Cornell University, Ithaca, New York 14853, USA
| | - M. B. Andorf
- Cornell Laboratory for Accelerator-Based Sciences and Education, Cornell University, Ithaca, New York 14853, USA
| | - A. C. Bartnik
- Cornell Laboratory for Accelerator-Based Sciences and Education, Cornell University, Ithaca, New York 14853, USA
| | - E. Bianco
- Kavli Institute at Cornell for Nanoscale Science, Ithaca, New York 14853, USA
| | - L. Cultrera
- Cornell Laboratory for Accelerator-Based Sciences and Education, Cornell University, Ithaca, New York 14853, USA
| | - A. Galdi
- Cornell Laboratory for Accelerator-Based Sciences and Education, Cornell University, Ithaca, New York 14853, USA
| | - M. Gordon
- University of Chicago, Chicago, Illinois 60637, USA
| | - M. Kaemingk
- Cornell Laboratory for Accelerator-Based Sciences and Education, Cornell University, Ithaca, New York 14853, USA
| | - C. A. Pennington
- Cornell Laboratory for Accelerator-Based Sciences and Education, Cornell University, Ithaca, New York 14853, USA
| | | | - I. V. Bazarov
- Cornell Laboratory for Accelerator-Based Sciences and Education, Cornell University, Ithaca, New York 14853, USA
| | - J. M. Maxson
- Cornell Laboratory for Accelerator-Based Sciences and Education, Cornell University, Ithaca, New York 14853, USA
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4
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Light-induced dimension crossover dictated by excitonic correlations. Nat Commun 2022; 13:963. [PMID: 35181649 PMCID: PMC8857203 DOI: 10.1038/s41467-022-28309-5] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/17/2021] [Accepted: 01/17/2022] [Indexed: 11/08/2022] Open
Abstract
In low-dimensional systems with strong electronic correlations, the application of an ultrashort laser pulse often yields novel phases that are otherwise inaccessible. The central challenge in understanding such phenomena is to determine how dimensionality and many-body correlations together govern the pathway of a non-adiabatic transition. To this end, we examine a layered compound, 1T-TiSe2, whose three-dimensional charge-density-wave (3D CDW) state also features exciton condensation due to strong electron-hole interactions. We find that photoexcitation suppresses the equilibrium 3D CDW while creating a nonequilibrium 2D CDW. Remarkably, the dimension reduction does not occur unless bound electron-hole pairs are broken. This relation suggests that excitonic correlations maintain the out-of-plane CDW coherence, settling a long-standing debate over their role in the CDW transition. Our findings demonstrate how optical manipulation of electronic interaction enables one to control the dimensionality of a broken-symmetry order, paving the way for realizing other emergent states in strongly correlated systems.
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5
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Lv BQ, Zong A, Wu D, Rozhkov AV, Fine BV, Chen SD, Hashimoto M, Lu DH, Li M, Huang YB, Ruff JPC, Walko DA, Chen ZH, Hwang I, Su Y, Shen X, Wang X, Han F, Po HC, Wang Y, Jarillo-Herrero P, Wang X, Zhou H, Sun CJ, Wen H, Shen ZX, Wang NL, Gedik N. Unconventional Hysteretic Transition in a Charge Density Wave. PHYSICAL REVIEW LETTERS 2022; 128:036401. [PMID: 35119886 DOI: 10.1103/physrevlett.128.036401] [Citation(s) in RCA: 8] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/14/2021] [Revised: 08/21/2021] [Accepted: 12/14/2021] [Indexed: 06/14/2023]
Abstract
Hysteresis underlies a large number of phase transitions in solids, giving rise to exotic metastable states that are otherwise inaccessible. Here, we report an unconventional hysteretic transition in a quasi-2D material, EuTe_{4}. By combining transport, photoemission, diffraction, and x-ray absorption measurements, we observe that the hysteresis loop has a temperature width of more than 400 K, setting a record among crystalline solids. The transition has an origin distinct from known mechanisms, lying entirely within the incommensurate charge density wave (CDW) phase of EuTe_{4} with no change in the CDW modulation periodicity. We interpret the hysteresis as an unusual switching of the relative CDW phases in different layers, a phenomenon unique to quasi-2D compounds that is not present in either purely 2D or strongly coupled 3D systems. Our findings challenge the established theories on metastable states in density wave systems, pushing the boundary of understanding hysteretic transitions in a broken-symmetry state.
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Affiliation(s)
- B Q Lv
- Massachusetts Institute of Technology, Department of Physics, Cambridge, Massachusetts 02139, USA
| | - Alfred Zong
- Massachusetts Institute of Technology, Department of Physics, Cambridge, Massachusetts 02139, USA
- University of California at Berkeley, Department of Chemistry, Berkeley, California 94720, USA
| | - D Wu
- International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China
- Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China
| | - A V Rozhkov
- Institute for Theoretical and Applied Electrodynamics, Russian Academy of Sciences, Moscow 125412, Russia
| | - Boris V Fine
- Laboratory for the Physics of Complex Quantum Systems, Moscow Institute of Physics and Technology, Institutsky pereulok 9, Dolgoprudny 141701, Russia
- Institute for Theoretical Physics, University of Leipzig, Brüderstrasse 16, 04103 Leipzig, Germany
| | - Su-Di Chen
- Department of Applied Physics, Stanford University, Stanford, California 94305, USA
- Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory and Stanford University, Menlo Park, California 94025, USA
| | - Makoto Hashimoto
- Stanford Synchrotron Radiation Lightsource, SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA
| | - Dong-Hui Lu
- Stanford Synchrotron Radiation Lightsource, SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA
| | - M Li
- Shanghai Synchrotron Radiation Facility, Shanghai Advanced Research Institute, Chinese Academy of Sciences, Shanghai 201204, China
| | - Y-B Huang
- Shanghai Synchrotron Radiation Facility, Shanghai Advanced Research Institute, Chinese Academy of Sciences, Shanghai 201204, China
| | | | - Donald A Walko
- Advanced Photon Source, Argonne National Laboratory, Lemont, Illinois 60439, USA
| | - Z H Chen
- Shanghai Synchrotron Radiation Facility, Shanghai Advanced Research Institute, Chinese Academy of Sciences, Shanghai 201204, China
| | - Inhui Hwang
- Advanced Photon Source, Argonne National Laboratory, Lemont, Illinois 60439, USA
| | - Yifan Su
- Massachusetts Institute of Technology, Department of Physics, Cambridge, Massachusetts 02139, USA
| | - Xiaozhe Shen
- SLAC National Accelerator Laboratory, Menlo Park, California, USA
| | - Xirui Wang
- Massachusetts Institute of Technology, Department of Physics, Cambridge, Massachusetts 02139, USA
| | - Fei Han
- Massachusetts Institute of Technology, Department of Nuclear Science and Engineering, Cambridge, Massachusetts 02139, USA
| | - Hoi Chun Po
- Massachusetts Institute of Technology, Department of Physics, Cambridge, Massachusetts 02139, USA
- Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China
| | - Yao Wang
- Department of Physics and Astronomy, Clemson University, Clemson, South Carolina 29631, USA
| | - Pablo Jarillo-Herrero
- Massachusetts Institute of Technology, Department of Physics, Cambridge, Massachusetts 02139, USA
| | - Xijie Wang
- SLAC National Accelerator Laboratory, Menlo Park, California, USA
| | - Hua Zhou
- Advanced Photon Source, Argonne National Laboratory, Lemont, Illinois 60439, USA
| | - Cheng-Jun Sun
- Advanced Photon Source, Argonne National Laboratory, Lemont, Illinois 60439, USA
| | - Haidan Wen
- Advanced Photon Source, Argonne National Laboratory, Lemont, Illinois 60439, USA
- Materials Science Division, Argonne National Laboratory, Lemont, Illinois 60439, USA
| | - Zhi-Xun Shen
- Department of Applied Physics, Stanford University, Stanford, California 94305, USA
- Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory and Stanford University, Menlo Park, California 94025, USA
| | - N L Wang
- International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China
- Beijing Academy of Quantum Information Sciences, Beijing 100913, China
| | - Nuh Gedik
- Massachusetts Institute of Technology, Department of Physics, Cambridge, Massachusetts 02139, USA
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6
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Zong A, Dolgirev PE, Kogar A, Su Y, Shen X, Straquadine JAW, Wang X, Luo D, Kozina ME, Reid AH, Li R, Yang J, Weathersby SP, Park S, Sie EJ, Jarillo-Herrero P, Fisher IR, Wang X, Demler E, Gedik N. Role of Equilibrium Fluctuations in Light-Induced Order. PHYSICAL REVIEW LETTERS 2021; 127:227401. [PMID: 34889631 DOI: 10.1103/physrevlett.127.227401] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/02/2021] [Accepted: 10/01/2021] [Indexed: 06/13/2023]
Abstract
Engineering novel states of matter with light is at the forefront of materials research. An intensely studied direction is to realize broken-symmetry phases that are "hidden" under equilibrium conditions but can be unleashed by an ultrashort laser pulse. Despite a plethora of experimental discoveries, the nature of these orders and how they transiently appear remain unclear. To this end, we investigate a nonequilibrium charge density wave (CDW) in rare-earth tritellurides, which is suppressed in equilibrium but emerges after photoexcitation. Using a pump-pump-probe protocol implemented in ultrafast electron diffraction, we demonstrate that the light-induced CDW consists solely of order parameter fluctuations, which bear striking similarities to critical fluctuations in equilibrium despite differences in the length scale. By calculating the dynamics of CDW fluctuations in a nonperturbative model, we further show that the strength of the light-induced order is governed by the amplitude of equilibrium fluctuations. These findings highlight photoinduced fluctuations as an important ingredient for the emergence of transient orders out of equilibrium. Our results further suggest that materials with strong fluctuations in equilibrium are promising platforms to host hidden orders after laser excitation.
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Affiliation(s)
- Alfred Zong
- Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
- Department of Chemistry, University of California at Berkeley, Berkeley, California 94720, USA
| | - Pavel E Dolgirev
- Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
| | - Anshul Kogar
- Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
- Department of Physics and Astronomy, University of California at Los Angeles, Los Angeles, California 90095, USA
| | - Yifan Su
- Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
| | - Xiaozhe Shen
- SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA
| | - Joshua A W Straquadine
- Department of Applied Physics, Stanford University, Stanford, California 94305, USA
- SIMES, SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA
- Geballe Laboratory for Advanced Materials, Stanford University, Stanford, California 94305, USA
| | - Xirui Wang
- Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
| | - Duan Luo
- SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA
| | - Michael E Kozina
- SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA
| | - Alexander H Reid
- SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA
| | - Renkai Li
- SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA
| | - Jie Yang
- SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA
| | | | - Suji Park
- SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA
- Department of Materials Science and Engineering, Stanford University, Stanford, California 94305, USA
| | - Edbert J Sie
- SIMES, SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA
- Geballe Laboratory for Advanced Materials, Stanford University, Stanford, California 94305, USA
| | - Pablo Jarillo-Herrero
- Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
| | - Ian R Fisher
- Department of Applied Physics, Stanford University, Stanford, California 94305, USA
- SIMES, SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA
- Geballe Laboratory for Advanced Materials, Stanford University, Stanford, California 94305, USA
| | - Xijie Wang
- SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA
| | - Eugene Demler
- Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
- Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland
| | - Nuh Gedik
- Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
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