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Wang S, Zhan Z, Fan X, Li Y, Pantaleón PA, Ye C, He Z, Wei L, Li L, Guinea F, Yuan S, Zeng C. Dispersion-Selective Band Engineering in an Artificial Kagome Superlattice. PHYSICAL REVIEW LETTERS 2024; 133:066302. [PMID: 39178462 DOI: 10.1103/physrevlett.133.066302] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/14/2024] [Revised: 04/29/2024] [Accepted: 06/17/2024] [Indexed: 08/25/2024]
Abstract
The relentless pursuit of band structure engineering continues to be a fundamental aspect in solid-state research. Here, we meticulously construct an artificial kagome potential to generate and control multiple Dirac bands of graphene. This unique high-order potential harbors natural multiperiodic components, enabling the reconstruction of band structures through different potential contributions. As a result, the band components, each characterized by distinct dispersions, shift in energy at different velocities in response to the variation of artificial potential. Thereby, we observe a significant spectral weight redistribution of the multiple Dirac peaks. Furthermore, the magnetic field can effectively weaken the superlattice effect and reactivate the intrinsic Dirac band. Overall, we achieve actively dispersion-selective band engineering, a functionality that would substantially increase the freedom in band design.
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Affiliation(s)
- Shuai Wang
- CAS Key Laboratory of Strongly-Coupled Quantum Matter Physics, and Department of Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
- International Center for Quantum Design of Functional Materials (ICQD), Hefei National Research Center for Physical Sciences at the Microscale, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Zhen Zhan
- Key Laboratory of Artificial Micro- and Nano-structures of the Ministry of Education and School of Physics and Technology, Wuhan University, Wuhan, Hubei 430072, China
- Imdea Nanoscience, Madrid 28015, Spain
| | - Xiaodong Fan
- CAS Key Laboratory of Strongly-Coupled Quantum Matter Physics, and Department of Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
- International Center for Quantum Design of Functional Materials (ICQD), Hefei National Research Center for Physical Sciences at the Microscale, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Yonggang Li
- Key Laboratory of Artificial Micro- and Nano-structures of the Ministry of Education and School of Physics and Technology, Wuhan University, Wuhan, Hubei 430072, China
| | | | - Chaochao Ye
- CAS Key Laboratory of Strongly-Coupled Quantum Matter Physics, and Department of Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
- International Center for Quantum Design of Functional Materials (ICQD), Hefei National Research Center for Physical Sciences at the Microscale, University of Science and Technology of China, Hefei, Anhui 230026, China
| | | | | | - Lin Li
- CAS Key Laboratory of Strongly-Coupled Quantum Matter Physics, and Department of Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
- International Center for Quantum Design of Functional Materials (ICQD), Hefei National Research Center for Physical Sciences at the Microscale, University of Science and Technology of China, Hefei, Anhui 230026, China
- Hefei National Laboratory, University of Science and Technology of China, Hefei, Anhui 230088, China
| | | | - Shengjun Yuan
- Key Laboratory of Artificial Micro- and Nano-structures of the Ministry of Education and School of Physics and Technology, Wuhan University, Wuhan, Hubei 430072, China
- Wuhan Institute of Quantum Technology, Wuhan, Hubei 430206, China
| | - Changgan Zeng
- CAS Key Laboratory of Strongly-Coupled Quantum Matter Physics, and Department of Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
- International Center for Quantum Design of Functional Materials (ICQD), Hefei National Research Center for Physical Sciences at the Microscale, University of Science and Technology of China, Hefei, Anhui 230026, China
- Hefei National Laboratory, University of Science and Technology of China, Hefei, Anhui 230088, China
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2
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An image interaction approach to quantum-phase engineering of two-dimensional materials. Nat Commun 2022; 13:5175. [PMID: 36056011 PMCID: PMC9440131 DOI: 10.1038/s41467-022-32508-5] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/26/2022] [Accepted: 08/02/2022] [Indexed: 11/09/2022] Open
Abstract
Tuning electrical, optical, and thermal material properties is central for engineering and understanding solid-state systems. In this scenario, atomically thin materials are appealing because of their sensitivity to electric and magnetic gating, as well as to interlayer hybridization. Here, we introduce a radically different approach to material engineering relying on the image interaction experienced by electrons in a two-dimensional material when placed in proximity of an electrically neutral structure. We theoretically show that electrons in a semiconductor atomic layer acquire a quantum phase resulting from the image potential induced by the presence of a neighboring periodic array of conducting ribbons, which in turn modifies the optical, electrical, and thermal properties of the monolayer, giving rise to additional interband optical absorption, plasmon hybridization, and metal-insulator transitions. Beyond its fundamental interest, material engineering based on the image interaction represents a disruptive approach to tailor the properties of atomic layers for application in nanodevices.
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3
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Freeney SE, Slot MR, Gardenier TS, Swart I, Vanmaekelbergh D. Electronic Quantum Materials Simulated with Artificial Model Lattices. ACS NANOSCIENCE AU 2022; 2:198-224. [PMID: 35726276 PMCID: PMC9204828 DOI: 10.1021/acsnanoscienceau.1c00054] [Citation(s) in RCA: 4] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 11/24/2021] [Revised: 01/24/2022] [Accepted: 01/28/2022] [Indexed: 11/29/2022]
Abstract
![]()
The
band structure and electronic properties of a material are
defined by the sort of elements, the atomic registry in the crystal,
the dimensions, the presence of spin–orbit coupling, and the
electronic interactions. In natural crystals, the interplay of these
factors is difficult to unravel, since it is usually not possible
to vary one of these factors in an independent way, keeping the others
constant. In other words, a complete understanding of complex electronic
materials remains challenging to date. The geometry of two- and one-dimensional
crystals can be mimicked in artificial lattices. Moreover, geometries
that do not exist in nature can be created for the sake of further
insight. Such engineered artificial lattices can be better controlled
and fine-tuned than natural crystals. This makes it easier to vary
the lattice geometry, dimensions, spin–orbit coupling, and
interactions independently from each other. Thus, engineering and
characterization of artificial lattices can provide unique insights.
In this Review, we focus on artificial lattices that are built atom-by-atom
on atomically flat metals, using atomic manipulation in a scanning
tunneling microscope. Cryogenic scanning tunneling microscopy allows
for consecutive creation, microscopic characterization, and band-structure
analysis by tunneling spectroscopy, amounting in the analogue quantum
simulation of a given lattice type. We first review the physical elements
of this method. We then discuss the creation and characterization
of artificial atoms and molecules. For the lattices, we review works
on honeycomb and Lieb lattices and lattices that result in crystalline
topological insulators, such as the Kekulé and “breathing”
kagome lattice. Geometric but nonperiodic structures such as electronic
quasi-crystals and fractals are discussed as well. Finally, we consider
the option to transfer the knowledge gained back to real materials,
engineered by geometric patterning of semiconductor quantum wells.
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Affiliation(s)
- Saoirsé E. Freeney
- Condensed Matter and Interfaces, Debye Institute of Nanomaterial Science, University of Utrecht, Princetonplein 5, 3584 CC Utrecht, The Netherlands
| | - Marlou R. Slot
- Condensed Matter and Interfaces, Debye Institute of Nanomaterial Science, University of Utrecht, Princetonplein 5, 3584 CC Utrecht, The Netherlands
| | - Thomas S. Gardenier
- Condensed Matter and Interfaces, Debye Institute of Nanomaterial Science, University of Utrecht, Princetonplein 5, 3584 CC Utrecht, The Netherlands
| | - Ingmar Swart
- Condensed Matter and Interfaces, Debye Institute of Nanomaterial Science, University of Utrecht, Princetonplein 5, 3584 CC Utrecht, The Netherlands
| | - Daniel Vanmaekelbergh
- Condensed Matter and Interfaces, Debye Institute of Nanomaterial Science, University of Utrecht, Princetonplein 5, 3584 CC Utrecht, The Netherlands
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4
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Levy AL, Zimmerman NM. Broadband Microwave Electrical Transport Spectroscopy for Two-Dimensional Material Systems. JOURNAL OF APPLIED PHYSICS 2022; 131:10.1063/5.0087285. [PMID: 38524783 PMCID: PMC10960354 DOI: 10.1063/5.0087285] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 03/26/2024]
Abstract
In recent years, interesting materials have emerged which are only available as μm-scale flakes, and whose novel physics might be better understood through broadband microwave spectroscopy; examples include twisted bilayer graphene [Y. Cao S. Fang, K. Watanabe, T. Taniguchi, E. Kaxiras and P. Jarillo-Herrero Nature 556, 43 (2018).], 2D materials in which many-body phases are observed [S. Chen R. Ribeiro-Palau, K. Yang, T. Taniguchi, J. Hone, M. O. Goerbig and C. R. Dean Physical Review Letters 122¸ 026802 (2019)], and artificial lattices for analog quantum simulations [J. Salfi J. A. Mol, R. Rahman, G. Klimeck, M. Y. Simmons, L. C. L. Hollenberg and S. Rogge Nature Communications 7, 1 (2016)]. Most previous techniques are unfortunately not sensitive for flakes below mm lateral sizes. We propose a simple technique which does not require sophisticated sample preparation nor Ohmic contact and show through theory and simulations that one will be able to qualitatively measure spectral features of interest, and quantitatively measure the frequency-dependent complex conductivity.
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Affiliation(s)
- Antonio L. Levy
- Physical Measurement Laboratory, National Institute of Standards and Technology (NIST), Gaithersburg, Maryland, 20899-8171, USA
| | - Neil M. Zimmerman
- Physical Measurement Laboratory, National Institute of Standards and Technology (NIST), Gaithersburg, Maryland, 20899-8171, USA
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5
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Quantum fluids of light in all-optical scatterer lattices. Nat Commun 2021; 12:5571. [PMID: 34552069 PMCID: PMC8458361 DOI: 10.1038/s41467-021-25845-4] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/17/2020] [Accepted: 09/02/2021] [Indexed: 11/08/2022] Open
Abstract
One of the recently established paradigms in condensed matter physics is examining a system's behaviour in artificial potentials, giving insight into phenomena of quantum fluids in hard-to-reach settings. A prominent example is the matter-wave scatterer lattice, where high energy matter waves undergo transmission and reflection through narrow width barriers leading to stringent phase matching conditions with lattice band formation. In contrast to evanescently coupled lattice sites, the realisation of a scatterer lattice for macroscopic matter-wave fluids has remained elusive. Here, we implement a system of exciton-polariton condensates in a non-Hermitian Lieb lattice of scatterer potentials. By fine tuning the lattice parameters, we reveal a nonequilibrium phase transition between distinct regimes of polariton condensation: a scatterer lattice of gain guided polaritons condensing on the lattice potential maxima, and trapped polaritons condensing in the potential minima. Our results pave the way towards unexplored physics of non-Hermitian fluids in non-stationary mixtures of confined and freely expanding waves.
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6
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Chen P, Zhang N, Peng K, Zhang L, Yan J, Jiang Z, Zhong Z. Artificial Graphene on Si Substrates: Fabrication and Transport Characteristics. ACS NANO 2021; 15:13703-13711. [PMID: 34286957 DOI: 10.1021/acsnano.1c04995] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/13/2023]
Abstract
Artificial graphene (AG) based on a honeycomb lattice of semiconductor quantum dots (QDs) has been of great interest for exploration and applications of massless Dirac Fermions in semiconductors thanks to the tunable interplay between the carrier interactions and the honeycomb topology. Here, an innovative strategy to realize AG on Si substrates is developed by fabricating a honeycomb lattice of Au nanodisks on a Si/GeSi quantum well. The lateral potential modulation induced by the nanoscale Au/Si Schottky junction results in the formation of quantum dots arranged in a honeycomb lattice to form AG. Nonlinear current-voltage curves of the AG reveal conductance phase transitions with switch on/off voltages, a large electric hysteresis loop, and a strong sharp current peak accompanied by a group of differential-conductance peaks and negative differential conductance around the switch-on voltage, which can be modulated by temperature and light. These features are interpreted by a model based on the Coulomb blockade effect, the collective resonant tunneling, and the coupling of holes in the AG. Our results not only demonstrate an approach to the formation but also will greatly stimulate the characterizations and the applications of innovative semiconductor-based AG.
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Affiliation(s)
- Peizong Chen
- State Key Laboratory of Surface Physics and Department of Physics, Fudan University, Shanghai 200438, People's Republic of China
| | - Ningning Zhang
- State Key Laboratory of Surface Physics and Department of Physics, Fudan University, Shanghai 200438, People's Republic of China
| | - Kun Peng
- State Key Laboratory of Surface Physics and Department of Physics, Fudan University, Shanghai 200438, People's Republic of China
| | - Lijian Zhang
- State Key Laboratory of Surface Physics and Department of Physics, Fudan University, Shanghai 200438, People's Republic of China
| | - Jia Yan
- State Key Laboratory of Surface Physics and Department of Physics, Fudan University, Shanghai 200438, People's Republic of China
| | - Zuimin Jiang
- State Key Laboratory of Surface Physics and Department of Physics, Fudan University, Shanghai 200438, People's Republic of China
| | - Zhenyang Zhong
- State Key Laboratory of Surface Physics and Department of Physics, Fudan University, Shanghai 200438, People's Republic of China
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7
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Miao Z, Zheng CY, Schatz GC, Lee B, Mirkin CA. Low‐Density 2D Superlattices Assembled via Directional DNA Bonding. Angew Chem Int Ed Engl 2021. [DOI: 10.1002/ange.202105796] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
Affiliation(s)
- Ziyi Miao
- Department of Materials Science and Engineering Northwestern University 2220 Campus Drive Evanston IL 60208 USA
- International Institute for Nanotechnology Northwestern University USA
| | - Cindy Y. Zheng
- Department of Chemistry Northwestern University 2145 Sheridan Road Evanston IL 60208 USA
- International Institute for Nanotechnology Northwestern University USA
| | - George C. Schatz
- Department of Chemistry Northwestern University 2145 Sheridan Road Evanston IL 60208 USA
- International Institute for Nanotechnology Northwestern University USA
| | - Byeongdu Lee
- X-ray Science Division Argonne National Laboratory 9700 South Cass Avenue Argonne IL 60439 USA
| | - Chad A. Mirkin
- Department of Materials Science and Engineering Northwestern University 2220 Campus Drive Evanston IL 60208 USA
- Department of Chemistry Northwestern University 2145 Sheridan Road Evanston IL 60208 USA
- International Institute for Nanotechnology Northwestern University USA
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8
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Miao Z, Zheng CY, Schatz GC, Lee B, Mirkin CA. Low-Density 2D Superlattices Assembled via Directional DNA Bonding. Angew Chem Int Ed Engl 2021; 60:19035-19040. [PMID: 34310029 DOI: 10.1002/anie.202105796] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/28/2021] [Revised: 06/08/2021] [Indexed: 11/11/2022]
Abstract
It is critical to assemble nanoparticles (NPs) into superlattices with controlled symmetries and spacings on substrates for metamaterials applications, where such structural parameters dictate their properties. Here, we use DNA to assemble anisotropic NPs of three shapes-cubes, octahedra, and rhombic dodecahedra-on substrates and investigate their thermally induced reorganization into two-dimensional (2D) crystalline films. We report two new low-density 2D structures, including a honeycomb lattice based on octahedral NPs. The low-density lattices favored here are not usually seen when particles are crystallized via other bottom-up assembly techniques. Furthermore, we show that, consistent with the complementary contact model, a primary driving force for crystallization is the formation of directional, face-to-face DNA bonds between neighboring NPs and between NPs and the substrate. Our results can be used to deliberately prepare crystalline NP films with novel morphologies.
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Affiliation(s)
- Ziyi Miao
- Department of Materials Science and Engineering, Northwestern University, 2220 Campus Drive, Evanston, IL, 60208, USA.,International Institute for Nanotechnology, Northwestern University, USA
| | - Cindy Y Zheng
- Department of Chemistry, Northwestern University, 2145 Sheridan Road, Evanston, IL, 60208, USA.,International Institute for Nanotechnology, Northwestern University, USA
| | - George C Schatz
- Department of Chemistry, Northwestern University, 2145 Sheridan Road, Evanston, IL, 60208, USA.,International Institute for Nanotechnology, Northwestern University, USA
| | - Byeongdu Lee
- X-ray Science Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, IL, 60439, USA
| | - Chad A Mirkin
- Department of Materials Science and Engineering, Northwestern University, 2220 Campus Drive, Evanston, IL, 60208, USA.,Department of Chemistry, Northwestern University, 2145 Sheridan Road, Evanston, IL, 60208, USA.,International Institute for Nanotechnology, Northwestern University, USA
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9
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Wang Z, Cheng S, Liu X, Jiang H. Topological kink states in graphene. NANOTECHNOLOGY 2021; 32:402001. [PMID: 34161935 DOI: 10.1088/1361-6528/ac0dd8] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/18/2021] [Accepted: 06/23/2021] [Indexed: 06/13/2023]
Abstract
Due to the unique band structure, graphene exhibits a number of exotic electronic properties that have not been observed in other materials. Among them, it has been demonstrated that there exist the one-dimensional valley-polarized topological kink states localized in the vicinity of the domain wall of graphene systems, where a bulk energy gap opens due to the inversion symmetry breaking. Notably, the valley-momentum locking nature makes the topological kink states attractive to the property manipulation in valleytronics. This paper systematically reviews both the theoretical research and experimental progress on topological kink states in monolayer graphene, bilayer graphene and graphene-like classical wave systems. Besides, various applications of topological kink states, including the valley filter, current partition, current manipulation, Majorana zero modes and etc, are also introduced.
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Affiliation(s)
- Zibo Wang
- College of Physics and Electronic Engineering, Sichuan Normal University, Chengdu 610068, People's Republic of China
- Center for Computational Sciences, Sichuan Normal University, Chengdu 610068, People's Republic of China
| | - Shuguang Cheng
- Department of Physics, Northwest University, Xi'an 710069, People's Republic of China
| | - Xiao Liu
- School of Physical Science and Technology, Soochow University, Suzhou 215006, People's Republic of China
| | - Hua Jiang
- School of Physical Science and Technology, Soochow University, Suzhou 215006, People's Republic of China
- Institute for Advanced Study of Soochow University, Suzhou 215006, People's Republic of China
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10
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Capiod P, van der Sluijs M, de Boer J, Delerue C, Swart I, Vanmaekelbergh D. Electronic properties of atomically coherent square PbSe nanocrystal superlattice resolved by Scanning Tunneling Spectroscopy. NANOTECHNOLOGY 2021; 32:325706. [PMID: 33930872 DOI: 10.1088/1361-6528/abfd57] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/16/2021] [Accepted: 04/30/2021] [Indexed: 06/12/2023]
Abstract
Rock-salt lead selenide nanocrystals can be used as building blocks for large scale square superlattices via two-dimensional assembly of nanocrystals at a liquid-air interface followed by oriented attachment. Here we report Scanning Tunneling Spectroscopy measurements of the local density of states of an atomically coherent superlattice with square geometry made from PbSe nanocrystals. Controlled annealing of the sample permits the imaging of a clean structure and to reproducibly probe the band gap and the valence hole and conduction electron states. The measured band gap and peak positions are compared to the results of optical spectroscopy and atomistic tight-binding calculations of the square superlattice band structure. In spite of the crystalline connections between nanocrystals that induce significant electronic couplings, the electronic structure of the superlattices remains very strongly influenced by the effects of disorder and variability.
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Affiliation(s)
- Pierre Capiod
- Debye Institute for Nanomaterials Science, Utrecht University, PO Box 80 000, 3508 TA Utrecht, The Netherlands
| | - Maaike van der Sluijs
- Debye Institute for Nanomaterials Science, Utrecht University, PO Box 80 000, 3508 TA Utrecht, The Netherlands
| | - Jeroen de Boer
- Debye Institute for Nanomaterials Science, Utrecht University, PO Box 80 000, 3508 TA Utrecht, The Netherlands
| | - Christophe Delerue
- Univ. Lille, CNRS, Centrale Lille, Univ. Polytechnique Hauts-de-France, Junia, UMR 8520-IEMN, F-59000 Lille, France
| | - Ingmar Swart
- Debye Institute for Nanomaterials Science, Utrecht University, PO Box 80 000, 3508 TA Utrecht, The Netherlands
| | - Daniel Vanmaekelbergh
- Debye Institute for Nanomaterials Science, Utrecht University, PO Box 80 000, 3508 TA Utrecht, The Netherlands
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11
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Zhang Q, Wu TC, Kuang G, Xie A, Lin N. Investigation of edge states in artificial graphene nano-flakes. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2021; 33:225003. [PMID: 33607633 DOI: 10.1088/1361-648x/abe819] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/13/2020] [Accepted: 02/19/2021] [Indexed: 06/12/2023]
Abstract
Graphene nano-flakes (GNFs) are predicted to host spin-polarized metallic edge states, which are envisioned for exploration of spintronics at the nanometer scale. To date, experimental realization of GNFs is only in its infancy because of the limitation of precise cutting or synthesizing methods at the nanometer scale. Here, we use low temperature scanning tunneling microscope to manipulate coronene molecules on a Cu(111) surface to build artificial triangular and hexagonal GNFs with either zigzag or armchair type of edges. We observe that an electronic state at the Dirac point emerges only in the GNFs with zigzag edges and localizes at the outmost lattice sites. The experimental results agree well with the tight-binding calculations. Our work renders an experimental confirmation of the predicated edge states of the GNFs.
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Affiliation(s)
- Qiushi Zhang
- Department of Physics, The Hong Kong University of Science and Technology, Hong Kong, People's Republic of China
- Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN, United States of America
| | - Tsz Chun Wu
- Department of Physics, The Hong Kong University of Science and Technology, Hong Kong, People's Republic of China
| | - Guowen Kuang
- Department of Physics, The Hong Kong University of Science and Technology, Hong Kong, People's Republic of China
| | - A'yu Xie
- Department of Physics, The Hong Kong University of Science and Technology, Hong Kong, People's Republic of China
| | - Nian Lin
- Department of Physics, The Hong Kong University of Science and Technology, Hong Kong, People's Republic of China
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12
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Du L, Liu Z, Wind SJ, Pellegrini V, West KW, Fallahi S, Pfeiffer LN, Manfra MJ, Pinczuk A. Observation of Flat Bands in Gated Semiconductor Artificial Graphene. PHYSICAL REVIEW LETTERS 2021; 126:106402. [PMID: 33784167 DOI: 10.1103/physrevlett.126.106402] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/17/2020] [Accepted: 02/15/2021] [Indexed: 06/12/2023]
Abstract
Flat bands near M points in the Brillouin zone are key features of honeycomb symmetry in artificial graphene (AG) where electrons may condense into novel correlated phases. Here we report the observation of van Hove singularity doublet of AG in GaAs quantum well transistors, which presents the evidence of flat bands in semiconductor AG. Two emerging peaks in photoluminescence spectra tuned by backgate voltages probe the singularity doublet of AG flat bands and demonstrate their accessibility to the Fermi level. As the Fermi level crosses the doublet, the spectra display dramatic stability against electron density, indicating interplays between electron-electron interactions and honeycomb symmetry. Our results provide a new flexible platform to explore intriguing flat band physics.
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Affiliation(s)
- Lingjie Du
- School of Physics, and National Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China
- Department of Applied Physics and Applied Mathematics, Columbia University, New York, New York 10027, USA
| | - Ziyu Liu
- Department of Physics, Columbia University, New York, New York 10027, USA
| | - Shalom J Wind
- Department of Applied Physics and Applied Mathematics, Columbia University, New York, New York 10027, USA
| | - Vittorio Pellegrini
- Istituto Italiano di Tecnologia, Graphene Labs, Via Morego 30, I-16163 Genova, Italy
| | - Ken W West
- Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA
| | - Saeed Fallahi
- Department of Physics and Astronomy, and School of Materials Engineering, and School of Electrical and Computer Engineering, Purdue University, West Lafayette, Indiana 47907, USA
| | - Loren N Pfeiffer
- Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA
| | - Michael J Manfra
- Department of Physics and Astronomy, and School of Materials Engineering, and School of Electrical and Computer Engineering, Purdue University, West Lafayette, Indiana 47907, USA
| | - Aron Pinczuk
- Department of Applied Physics and Applied Mathematics, Columbia University, New York, New York 10027, USA
- Department of Physics, Columbia University, New York, New York 10027, USA
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13
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Wu HR, Wu BL, Cheng SG, Jiang H. The realization of quantum anomalous Hall effect in two dimensional electron gas. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2021; 33:105701. [PMID: 33232942 DOI: 10.1088/1361-648x/abcd7e] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/11/2023]
Abstract
The quantum anomalous Hall effect (QAHE), carrying dissipationless chiral edge states, occurs without any magnetic field. Two main strategies were proposed to host QAHE: the magnetic topological insulator thin films and graphene systems. Only the former one was realized in experiment at low temperature. In this paper, by dealing with the two-dimensional electron gas with an anti-dot lattice, a realistic platform is proposed to host the QAHE with both Chern number [Formula: see text] and [Formula: see text]. Based on the calculation of the Berry curvature integral and spacial wave function, the topological nature of the QAH edge states is well demonstrated. In the QAH region, the conductance shows quantized plateaus and their values are robust against Anderson disorder. In addition, we have also studied the effects of the size and shape of the anti-dot lattice on QAHE and they provide extra manners to adjust the system parameters. Taking the advantages of the well developed micro-manufacture technique in semiconductors, the proposal is experimentally accessible in micro-scale.
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Affiliation(s)
- Hua-Rui Wu
- School of Physical Science and Technology, Soochow University, Suzhou 215006, People's Republic of China
| | - Bing-Lan Wu
- School of Physical Science and Technology, Soochow University, Suzhou 215006, People's Republic of China
| | - Shu-Guang Cheng
- Department of Physics, Northwest University, Xi'an 710069, People's Republic of China
| | - Hua Jiang
- School of Physical Science and Technology, Soochow University, Suzhou 215006, People's Republic of China
- Institute for Advanced Study, Soochow University, Suzhou, 215006, People's Republic of China
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14
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Franchina Vergel NA, Post LC, Sciacca D, Berthe M, Vaurette F, Lambert Y, Yarekha D, Troadec D, Coinon C, Fleury G, Patriarche G, Xu T, Desplanque L, Wallart X, Vanmaekelbergh D, Delerue C, Grandidier B. Engineering a Robust Flat Band in III-V Semiconductor Heterostructures. NANO LETTERS 2021; 21:680-685. [PMID: 33337891 DOI: 10.1021/acs.nanolett.0c04268] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/12/2023]
Abstract
Electron states in semiconductor materials can be modified by quantum confinement. Adding to semiconductor heterostructures the concept of lateral geometry offers the possibility to further tailor the electronic band structure with the creation of unique flat bands. Using block copolymer lithography, we describe the design, fabrication, and characterization of multiorbital bands in a honeycomb In0.53Ga0.47As/InP heterostructure quantum well with a lattice constant of 21 nm. Thanks to an optimized surface quality, scanning tunnelling spectroscopy reveals the existence of a strong resonance localized between the lattice sites, signature of a p-orbital flat band. Together with theoretical computations, the impact of the nanopatterning imperfections on the band structure is examined. We show that the flat band is protected against the lateral and vertical disorder, making this industry-standard system particularly attractive for the study of exotic phases of matter.
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Affiliation(s)
- Nathali A Franchina Vergel
- Univ. Lille, CNRS, Centrale Lille, Univ. Polytechnique Hauts-de-France, Junia-ISEN, UMR 8520 - IEMN, 59000 Lille, France
| | - L Christiaan Post
- Debye Institute for Nanomaterials Science, Utrecht University, 3508 TA Utrecht, The Netherlands
| | - Davide Sciacca
- Univ. Lille, CNRS, Centrale Lille, Univ. Polytechnique Hauts-de-France, Junia-ISEN, UMR 8520 - IEMN, 59000 Lille, France
| | - Maxime Berthe
- Univ. Lille, CNRS, Centrale Lille, Univ. Polytechnique Hauts-de-France, Junia-ISEN, UMR 8520 - IEMN, 59000 Lille, France
| | - François Vaurette
- Univ. Lille, CNRS, Centrale Lille, Univ. Polytechnique Hauts-de-France, Junia-ISEN, UMR 8520 - IEMN, 59000 Lille, France
| | - Yannick Lambert
- Univ. Lille, CNRS, Centrale Lille, Univ. Polytechnique Hauts-de-France, Junia-ISEN, UMR 8520 - IEMN, 59000 Lille, France
| | - Dmitri Yarekha
- Univ. Lille, CNRS, Centrale Lille, Univ. Polytechnique Hauts-de-France, Junia-ISEN, UMR 8520 - IEMN, 59000 Lille, France
| | - David Troadec
- Univ. Lille, CNRS, Centrale Lille, Univ. Polytechnique Hauts-de-France, Junia-ISEN, UMR 8520 - IEMN, 59000 Lille, France
| | - Christophe Coinon
- Univ. Lille, CNRS, Centrale Lille, Univ. Polytechnique Hauts-de-France, Junia-ISEN, UMR 8520 - IEMN, 59000 Lille, France
| | - Guillaume Fleury
- Univ. Bordeaux, CNRS, Bordeaux INP, LCPO, UMR 5629, F-33600 Pessac, France
| | - Gilles Patriarche
- CNRS, Centre de Nanosciences et de Nanotechnologies (C2N), University Paris-Saclay, 91120 Palaiseau, France
| | - Tao Xu
- Sino-European School of Technology, Shanghai University, 200444 Shanghai, China
| | - Ludovic Desplanque
- Univ. Lille, CNRS, Centrale Lille, Univ. Polytechnique Hauts-de-France, Junia-ISEN, UMR 8520 - IEMN, 59000 Lille, France
| | - Xavier Wallart
- Univ. Lille, CNRS, Centrale Lille, Univ. Polytechnique Hauts-de-France, Junia-ISEN, UMR 8520 - IEMN, 59000 Lille, France
| | - Daniel Vanmaekelbergh
- Debye Institute for Nanomaterials Science, Utrecht University, 3508 TA Utrecht, The Netherlands
| | - Christophe Delerue
- Univ. Lille, CNRS, Centrale Lille, Univ. Polytechnique Hauts-de-France, Junia-ISEN, UMR 8520 - IEMN, 59000 Lille, France
| | - Bruno Grandidier
- Univ. Lille, CNRS, Centrale Lille, Univ. Polytechnique Hauts-de-France, Junia-ISEN, UMR 8520 - IEMN, 59000 Lille, France
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15
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Wang J, Santos LH. Classification of Topological Phase Transitions and van Hove Singularity Steering Mechanism in Graphene Superlattices. PHYSICAL REVIEW LETTERS 2020; 125:236805. [PMID: 33337183 DOI: 10.1103/physrevlett.125.236805] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/02/2020] [Accepted: 10/21/2020] [Indexed: 06/12/2023]
Abstract
We study quantum phase transitions in graphene superlattices in external magnetic fields, where a framework is presented to classify multiflavor Dirac fermion critical points describing hopping-tuned topological phase transitions of integer and fractional Hofstadter-Chern insulators. We argue and provide numerical support for the existence of transitions that can be explained by a nontrivial interplay of Chern bands and van Hove singularities near charge neutrality. This work provides a route to critical phenomena beyond conventional quantum Hall plateau transitions.
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Affiliation(s)
- Jian Wang
- Department of Physics, Emory University, Atlanta, Georgia 30322, USA
| | - Luiz H Santos
- Department of Physics, Emory University, Atlanta, Georgia 30322, USA
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16
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Gardenier T, van den Broeke JJ, Moes JR, Swart I, Delerue C, Slot MR, Smith CM, Vanmaekelbergh D. p Orbital Flat Band and Dirac Cone in the Electronic Honeycomb Lattice. ACS NANO 2020; 14:13638-13644. [PMID: 32991147 PMCID: PMC7596780 DOI: 10.1021/acsnano.0c05747] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 07/10/2020] [Accepted: 09/29/2020] [Indexed: 06/11/2023]
Abstract
Theory anticipates that the in-plane px, py orbitals in a honeycomb lattice lead to potentially useful quantum electronic phases. So far, p orbital bands were only realized for cold atoms in optical lattices and for light and exciton-polaritons in photonic crystals. For electrons, in-plane p orbital physics is difficult to access since natural electronic honeycomb lattices, such as graphene and silicene, show strong s-p hybridization. Here, we report on electronic honeycomb lattices prepared on a Cu(111) surface in a scanning tunneling microscope that, by design, show (nearly) pure orbital bands, including the p orbital flat band and Dirac cone.
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Affiliation(s)
- Thomas
S. Gardenier
- Debye
Institute for Nanomaterials Science, Utrecht University, P.O. Box 80.000, 3508 TA Utrecht, The Netherlands
| | - Jette J. van den Broeke
- Institute
for Theoretical Physics, Utrecht University, P.O. Box 80.089, 3508 TB Utrecht, The Netherlands
| | - Jesper R. Moes
- Debye
Institute for Nanomaterials Science, Utrecht University, P.O. Box 80.000, 3508 TA Utrecht, The Netherlands
| | - Ingmar Swart
- Debye
Institute for Nanomaterials Science, Utrecht University, P.O. Box 80.000, 3508 TA Utrecht, The Netherlands
| | - Christophe Delerue
- Université
de Lille, CNRS, Centrale Lille, Yncréa-ISEN,
Université Polytechnique Hauts-de-France, UMR 8520−IEMN, F-59000 Lille, France
| | - Marlou R. Slot
- Debye
Institute for Nanomaterials Science, Utrecht University, P.O. Box 80.000, 3508 TA Utrecht, The Netherlands
| | - C. Morais Smith
- Institute
for Theoretical Physics, Utrecht University, P.O. Box 80.089, 3508 TB Utrecht, The Netherlands
| | - Daniel Vanmaekelbergh
- Debye
Institute for Nanomaterials Science, Utrecht University, P.O. Box 80.000, 3508 TA Utrecht, The Netherlands
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17
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Yang Y, Jia Z, Wu Y, Xiao RC, Hang ZH, Jiang H, Xie XC. Gapped topological kink states and topological corner states in honeycomb lattice. Sci Bull (Beijing) 2020; 65:531-537. [PMID: 36659184 DOI: 10.1016/j.scib.2020.01.024] [Citation(s) in RCA: 12] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/09/2019] [Revised: 12/30/2019] [Accepted: 01/15/2020] [Indexed: 01/21/2023]
Abstract
Based on the tight-binding calculations on honeycomb lattice and photonic experimental visualization on artificial graphene (AG), we report the domain-wall-induced gapped topological kink states and topological corner states. In honeycomb lattice, domain walls (DWs) with gapless topological kink states could be induced either by sublattice symmetry breaking or by lattice deformation. We find that the coexistence of these two mechanisms will induce DWs with gapped topological kink states. Significantly, the intersection of these two types of DWs gives rise to topological corner state localized at the crossing point. Through the manipulation of the DWs, we show AG with honeycomb lattice structure not only a versatile platform supporting multiple topological corner modes in a controlled manner, but also possessing promising applications such as fabricating topological quantum dots composed of gapped topological kink states and topological corner states.
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Affiliation(s)
- Yuting Yang
- School of Physical Science and Technology, Soochow University, Suzhou 215006, China
| | - Ziyuan Jia
- School of Physical Science and Technology, Soochow University, Suzhou 215006, China
| | - Yijia Wu
- International Center for Quantum Materials, Peking University, Beijing 100871, China
| | - Rui-Chun Xiao
- School of Physical Science and Technology, Soochow University, Suzhou 215006, China
| | - Zhi Hong Hang
- School of Physical Science and Technology, Soochow University, Suzhou 215006, China; Institute for Advanced Study, Soochow University, Suzhou 215006, China.
| | - Hua Jiang
- School of Physical Science and Technology, Soochow University, Suzhou 215006, China; Institute for Advanced Study, Soochow University, Suzhou 215006, China.
| | - X C Xie
- International Center for Quantum Materials, Peking University, Beijing 100871, China; Beijing Academy of Quantum Information Sciences, Beijing 100193, China; CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100190, China
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18
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Post LC, Xu T, Franchina Vergel NA, Tadjine A, Lambert Y, Vaurette F, Yarekha D, Desplanque L, Stiévenard D, Wallart X, Grandidier B, Delerue C, Vanmaekelbergh D. Triangular nanoperforation and band engineering of InGaAs quantum wells: a lithographic route toward Dirac cones in III-V semiconductors. NANOTECHNOLOGY 2019; 30:155301. [PMID: 30630145 DOI: 10.1088/1361-6528/aafd3f] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/16/2023]
Abstract
The design of two-dimensional periodic structures at the nanoscale has renewed attention for band structure engineering. Here, we investigate the nanoperforation of InGaAs quantum wells epitaxially grown on InP substrates using high-resolution e-beam lithography and highly plasma based dry etching. We report on the fabrication of a honeycomb structure with an effective lattice constant down to 23 nm by realising triangular antidot lattice with an ultimate periodicity of 40 nm in a 10 nm thick InGaAs quantum well on a p-type InP. The quality of the honeycomb structures is discussed in detail, and calculations show the possibility to measure Dirac physics in these type of samples. Based on the statistical analysis of the fluctuations in pore size and periodicity, calculations of the band structure are performed to assess the robustness of the Dirac cones with respect to distortions of the honeycomb lattice.
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Affiliation(s)
- L C Post
- Condensed Matter and Interfaces, Debye Institute for Nanomaterials Science, Utrecht University, 3584 CC Utrecht, The Netherlands
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19
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Klages R, Gallegos SSG, Solanpää J, Sarvilahti M, Räsänen E. Normal and Anomalous Diffusion in Soft Lorentz Gases. PHYSICAL REVIEW LETTERS 2019; 122:064102. [PMID: 30822076 DOI: 10.1103/physrevlett.122.064102] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/15/2018] [Revised: 11/02/2018] [Indexed: 06/09/2023]
Abstract
Motivated by electronic transport in graphenelike structures, we study the diffusion of a classical point particle in Fermi potentials situated on a triangular lattice. We call this system a soft Lorentz gas, as the hard disks in the conventional periodic Lorentz gas are replaced by soft repulsive scatterers. A thorough computational analysis yields both normal and anomalous (super)diffusion with an extreme sensitivity on model parameters. This is due to an intricate interplay between trapped and ballistic periodic orbits, whose existence is characterized by tonguelike structures in parameter space. These results hold even for small softness, showing that diffusion in the paradigmatic hard Lorentz gas is not robust for realistic potentials, where we find an entirely different type of diffusion.
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Affiliation(s)
- Rainer Klages
- Queen Mary University of London, School of Mathematical Sciences, Mile End Road, London E1 4NS, United Kingdom
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
- Institute for Theoretical Physics, University of Cologne, Zülpicher Straße 77, 50937 Cologne, Germany
| | - Sol Selene Gil Gallegos
- Queen Mary University of London, School of Mathematical Sciences, Mile End Road, London E1 4NS, United Kingdom
| | - Janne Solanpää
- Computational Physics Laboratory, Tampere University, P.O. Box 692, FI-33014 Tampere, Finland
| | - Mika Sarvilahti
- Computational Physics Laboratory, Tampere University, P.O. Box 692, FI-33014 Tampere, Finland
| | - Esa Räsänen
- Computational Physics Laboratory, Tampere University, P.O. Box 692, FI-33014 Tampere, Finland
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20
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Liu QL, Zhao ZY, Yi JH, Zhang ZY. A High-Throughput Study of the Electronic Structure and Physical Properties of Short-Period (GaAs) m(AlAs) n (m, n ≤ 10) Superlattices Based on Density Functional Theory Calculations. NANOMATERIALS 2018; 8:nano8090709. [PMID: 30201917 PMCID: PMC6164511 DOI: 10.3390/nano8090709] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 08/13/2018] [Revised: 09/05/2018] [Accepted: 09/06/2018] [Indexed: 11/28/2022]
Abstract
As important functional materials, the electronic structure and physical properties of (GaAs)m(AlAs)n superlattices (SLs) have been extensively studied. However, due to limitations of computational methods and computational resources, it is sometimes difficult to thoroughly understand how and why the modification of their structural parameters affects their electronic structure and physical properties. In this article, a high-throughput study based on density functional theory calculations has been carried out to obtain detailed information and to further provide the underlying intrinsic mechanisms. The band gap variations of (GaAs)m(AlAs)n superlattices have been systematically investigated and summarized. They are very consistent with the available reported experimental measurements. Furthermore, the direct-to-indirect-gap transition of (GaAs)m(AlAs)n superlattices has been predicted and explained. For certain thicknesses of the GaAs well (m), the band gap value of (GaAs)m(AlAs)n SLs exponentially increases (increasing n), while for certain thicknesses of the AlAs barrier (n), the band gap value of (GaAs)m(AlAs)n SLs exponentially decreases (increasing m). In both cases, the band gap values converge to certain values. Furthermore, owing to the energy eigenvalues at different k-points showing different variation trends, (GaAs)m(AlAs)n SLs transform from a Γ-Γ direct band gap to Γ-M indirect band gap when the AlAs barrier is thick enough. The intrinsic reason for these variations is that the contributions and positions of the electronic states of the GaAs well and the AlAs barrier change under altered thickness conditions. Moreover, we have found that the binding energy can be used as a detector to estimate the band gap value in the design of (GaAs)m(AlAs)n devices. Our findings are useful for the design of novel (GaAs)m(AlAs)n superlattices-based optoelectronic devices.
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Affiliation(s)
- Qing-Lu Liu
- Faculty of Materials Science and Engineering, Kunming University of Science and Technology, Kunming 650093, China.
| | - Zong-Yan Zhao
- Faculty of Materials Science and Engineering, Kunming University of Science and Technology, Kunming 650093, China.
| | - Jian-Hong Yi
- Faculty of Materials Science and Engineering, Kunming University of Science and Technology, Kunming 650093, China.
| | - Zi-Yang Zhang
- Key Laboratory of Nanodevices and Applications, Suzhou Institute of Nano-Tech and Nano-Bionics, Chinese Academy of Sciences, Suzhou 215123, China.
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21
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Fu HH, Wu R. New topological states in HgTe quantum wells from defect patterning. NANOSCALE 2018; 10:15462-15467. [PMID: 30105337 DOI: 10.1039/c8nr04878a] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
To explore new methods for the realization of the quantum spin Hall (QSH) effect in two-dimensional (2D) materials, we have constructed a honeycomb geometry (HG) by etching rows of hexagonal holes in HgTe quantum wells (QWs). Theoretical calculations show that multiple Dirac cones can be produced by HG, regardless of whether the band inversion occurs or not. Furthermore, the topological states originating from a narrow HG region in a wide ribbon show strong localization at the physical edges of the ribbon, making them easy to manipulate and exploit. When the band inversion condition for QW states is satisfied, the topological states generated by two different mechanisms may coexist. Our studies pave the way to produce and control multiple QSH states in 2D materials as desired for the design of innovative spintronic materials.
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Affiliation(s)
- Hua-Hua Fu
- Department of Physics and Astronomy, University of California, Irvine, California 92697-4575, USA.
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22
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Emerging many-body effects in semiconductor artificial graphene with low disorder. Nat Commun 2018; 9:3299. [PMID: 30120251 PMCID: PMC6098128 DOI: 10.1038/s41467-018-05775-4] [Citation(s) in RCA: 18] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/13/2018] [Accepted: 07/18/2018] [Indexed: 11/08/2022] Open
Abstract
The interplay between electron-electron interactions and the honeycomb topology is expected to produce exotic quantum phenomena and find applications in advanced devices. Semiconductor-based artificial graphene (AG) is an ideal system for these studies that combines high-mobility electron gases with AG topology. However, to date, low-disorder conditions that reveal the interplay of electron-electron interaction with AG symmetry have not been achieved. Here, we report the creation of low-disorder AG that preserves the near-perfection of the pristine electron layer by fabricating small period triangular antidot lattices on high-quality quantum wells. Resonant inelastic light scattering spectra show collective spin-exciton modes at the M-point's nearly flatband saddle-point singularity in the density of states. The observed Coulomb exchange interaction energies are comparable to the gap of Dirac bands at the M-point, demonstrating interplay between quasiparticle interactions and the AG potential. The saddle-point exciton energies are in the terahertz range, making low-disorder AG suitable for contemporary optoelectronic applications.
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