1
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Klein D, Michaeli K. One-Shot GW Transport Calculations: A Charge-Conserving Solution. J Phys Chem Lett 2023; 14:897-904. [PMID: 36662651 PMCID: PMC11163472 DOI: 10.1021/acs.jpclett.2c03362] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/06/2022] [Accepted: 01/10/2023] [Indexed: 06/17/2023]
Abstract
Transport measurements are a common method of characterizing small systems in chemistry and physics. When interactions are negligible, the current through submicrometer structures can be obtained using the Landauer formula. Meir and Wingreen derived an exact expression for the current in the presence of interactions. This powerful tool requires knowledge of the exact Green's function. Alternatively, self-consistent approximations for the Green's function are frequently sufficient for calculating the current while crucially satisfying all conservation laws. We provide here yet another alternative, circumventing the high computational cost of these methods. We present expressions for the electric and thermal currents in which the lowest-order self-energy is summed to all orders (one-shot GW approximation). We account for both self-energy and vertex corrections such that current is conserved. Our formulas for the currents capture important features due to interactions and, hence, provide a powerful tool for cases in which the exact solution cannot be found.
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Affiliation(s)
- Dan Klein
- Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel
| | - Karen Michaeli
- Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel
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2
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Rath H, Halder N, Narayanasamy R, Usharani D. Chemical transformation of Doubly N-Confused Porphodimethene to variants of (anti)Aromatic Doubly N-Confused Porphyrinoids and σ aromatic Doubly N-Confused Isophlorinoid. Org Chem Front 2022. [DOI: 10.1039/d2qo00160h] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
Chemical conversion of non-aromatic trans-doubly N-confused porphodimethene to hitherto unknown variants of doubly N-confused porphyrinoids/isophlorinoid have been unravelled by precise interplay between the types of oxidants, the types of meso-aryl...
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3
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Czap G, Han Z, Wagner PJ, Ho W. Detection and Characterization of Anharmonic Overtone Vibrations of Single Molecules on a Metal Surface. PHYSICAL REVIEW LETTERS 2019; 122:106801. [PMID: 30932655 DOI: 10.1103/physrevlett.122.106801] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/22/2018] [Indexed: 06/09/2023]
Abstract
Inelastic electron tunneling spectroscopy (IETS) with the scanning tunneling microscope (STM) is a powerful technique used to characterize the vibration and spin states at the single-molecule level. While IETS lacks hard selection rules, historically it has been assumed that vibrational overtones are rarely seen or even absent. Here we provide definitive experimental evidence that the hindered rotation overtone excitation of carbon monoxide molecules adsorbed on Ag(110) can be detected with STM-IETS via isotope substitution. We also demonstrate that the anharmonicity of the overtone excitation can be characterized and compared between adsorption sites and find evidence of anisotropy in the vibrational anharmonicity for CO adsorbed on the [11[over ¯]0] step edge.
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Affiliation(s)
- Gregory Czap
- Department of Physics and Astronomy, University of California, Irvine, California 92697-4575, USA
| | - Zhumin Han
- Department of Physics and Astronomy, University of California, Irvine, California 92697-4575, USA
| | - Peter J Wagner
- Department of Physics and Astronomy, University of California, Irvine, California 92697-4575, USA
| | - W Ho
- Department of Physics and Astronomy, University of California, Irvine, California 92697-4575, USA
- Department of Chemistry, University of California, Irvine, California 92697-2025, USA
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4
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Tsuji Y, Yoshizawa K. Effects of electron-phonon coupling on quantum interference in polyenes. J Chem Phys 2018; 149:134115. [DOI: 10.1063/1.5048955] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Yuta Tsuji
- Institute for Materials Chemistry and Engineering and IRCCS, Kyushu University, Nishi-ku, Fukuoka 819-0395, Japan
| | - Kazunari Yoshizawa
- Institute for Materials Chemistry and Engineering and IRCCS, Kyushu University, Nishi-ku, Fukuoka 819-0395, Japan
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5
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Cuny J, Tarrat N, Spiegelman F, Huguenot A, Rapacioli M. Density-functional tight-binding approach for metal clusters, nanoparticles, surfaces and bulk: application to silver and gold. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2018; 30:303001. [PMID: 29916820 DOI: 10.1088/1361-648x/aacd6c] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
Density-functional based tight-binding (DFTB) is an efficient quantum mechanical method that can describe a variety of systems, going from organic and inorganic compounds to metallic and hybrid materials. The present topical review addresses the ability and performance of DFTB to investigate energetic, structural, spectroscopic and dynamical properties of gold and silver materials. After a brief overview of the theoretical basis of DFTB, its parametrization and its transferability, we report its past and recent applications to gold and silver systems, including small clusters, nanoparticles, bulk and surfaces, bare and interacting with various organic and inorganic compounds. The range of applications covered by those studies goes from plasmonics and molecular electronics, to energy conversion and surface chemistry. Finally, perspectives of DFTB in the field of gold and silver surfaces and NPs are outlined.
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Affiliation(s)
- Jérôme Cuny
- Laboratoire de Chimie et Physique Quantiques (LCPQ), Université de Toulouse III [UPS] and CNRS, 118 Route de Narbonne, F-31062 Toulouse, France
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6
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Gurunarayanan SP, Verellen N, Zharinov VS, James Shirley F, Moshchalkov VV, Heyns M, Van de Vondel J, Radu IP, Van Dorpe P. Electrically Driven Unidirectional Optical Nanoantennas. NANO LETTERS 2017; 17:7433-7439. [PMID: 29068692 DOI: 10.1021/acs.nanolett.7b03312] [Citation(s) in RCA: 21] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/26/2023]
Abstract
Directional antennas revolutionized modern day telecommunication by enabling precise beaming of radio and microwave signals with minimal loss of energy. Similarly, directional optical nanoantennas are expected to pave the way toward on-chip wireless communication and information processing. Currently, on-chip integration of such antennas is hampered by their multielement design or the requirement of complicated excitation schemes. Here, we experimentally demonstrate electrical driving of in-plane tunneling nanoantennas to achieve broadband unidirectional emission of light. Far-field interference, as a result of the spectral overlap between the dipolar emission of the tunnel junction and the fundamental quadrupole-like resonance of the nanoantenna, gives rise to a directional radiation pattern. By tuning this overlap using the applied voltage, we record directivities as high as 5 dB. In addition to electrical tunability, we also demonstrate passive tunability of the directivity using the antenna geometry. These fully configurable electrically driven nanoantennas provide a simple way to direct optical energy on-chip using an extremely small device footprint.
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Affiliation(s)
- Surya Prakash Gurunarayanan
- Department of Materials Engineering, KU Leuven , B-3001 Leuven, Belgium
- IMEC , Kapeldreef 75, B-3001 Leuven, Belgium
| | - Niels Verellen
- IMEC , Kapeldreef 75, B-3001 Leuven, Belgium
- INPAC-Institute for Nanoscale Physics and Chemistry, Department of Physics and Astronomy, KU Leuven , Celestijnenlaan 200D, B-3001 Leuven, Belgium
| | - Vyacheslav S Zharinov
- INPAC-Institute for Nanoscale Physics and Chemistry, Department of Physics and Astronomy, KU Leuven , Celestijnenlaan 200D, B-3001 Leuven, Belgium
| | - Finub James Shirley
- IMEC , Kapeldreef 75, B-3001 Leuven, Belgium
- INPAC-Institute for Nanoscale Physics and Chemistry, Department of Physics and Astronomy, KU Leuven , Celestijnenlaan 200D, B-3001 Leuven, Belgium
| | - Victor V Moshchalkov
- INPAC-Institute for Nanoscale Physics and Chemistry, Department of Physics and Astronomy, KU Leuven , Celestijnenlaan 200D, B-3001 Leuven, Belgium
| | - Marc Heyns
- Department of Materials Engineering, KU Leuven , B-3001 Leuven, Belgium
- IMEC , Kapeldreef 75, B-3001 Leuven, Belgium
| | - Joris Van de Vondel
- INPAC-Institute for Nanoscale Physics and Chemistry, Department of Physics and Astronomy, KU Leuven , Celestijnenlaan 200D, B-3001 Leuven, Belgium
| | | | - Pol Van Dorpe
- IMEC , Kapeldreef 75, B-3001 Leuven, Belgium
- INPAC-Institute for Nanoscale Physics and Chemistry, Department of Physics and Astronomy, KU Leuven , Celestijnenlaan 200D, B-3001 Leuven, Belgium
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7
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Reddy BK, Rawson J, Gadekar SC, Kögerler P, Anand VG. A naphthalene-fused dimer of an anti-aromatic expanded isophlorin. Chem Commun (Camb) 2017; 53:8211-8214. [PMID: 28681895 DOI: 10.1039/c7cc04050d] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
We report the first synthesis of a covalent expanded isophlorin dimer from two 24-π doubly S-confused sapphyrin-like pentathiaisophlorins. It exhibits marginal peripheral aromaticity rather than strong global diatropicity or paratropicity and weak intermacrocycle electronic communication. Quantum chemical methods discern that cross-conjugation is responsible for these unusual electronic features.
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Affiliation(s)
- Baddigam Kiran Reddy
- Department of Chemistry, Indian Institute of Science Education and Research (IISER), Pune, 411008, Maharashtra, India.
| | - Jeff Rawson
- PGI-6, Forschungszentrum Jülich, 52428 Jülich, Germany and Institute of Inorganic Chemistry, RWTH Aachen University, 52074 Aachen, Germany.
| | - Santosh C Gadekar
- Department of Chemistry, Indian Institute of Science Education and Research (IISER), Pune, 411008, Maharashtra, India.
| | - Paul Kögerler
- PGI-6, Forschungszentrum Jülich, 52428 Jülich, Germany and Institute of Inorganic Chemistry, RWTH Aachen University, 52074 Aachen, Germany.
| | - Venkataramanarao G Anand
- Department of Chemistry, Indian Institute of Science Education and Research (IISER), Pune, 411008, Maharashtra, India.
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8
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Advance of Mechanically Controllable Break Junction for Molecular Electronics. Top Curr Chem (Cham) 2017; 375:61. [DOI: 10.1007/s41061-017-0149-0] [Citation(s) in RCA: 24] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/06/2016] [Accepted: 05/16/2017] [Indexed: 10/19/2022]
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9
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Sýkora R, Novotný T. Graph-theoretical evaluation of the inelastic propensity rules for molecules with destructive quantum interference. J Chem Phys 2017; 146:174114. [DOI: 10.1063/1.4981916] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/29/2022] Open
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10
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Hsu LY, Rabitz H. Theory of molecular conductance using a modular approach. J Chem Phys 2016; 145:234702. [DOI: 10.1063/1.4972131] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Liang-Yan Hsu
- Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA
| | - Herschel Rabitz
- Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA
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11
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Lykkebo J, Romano G, Gagliardi A, Pecchia A, Solomon GC. Single-molecule electronics: Cooling individual vibrational modes by the tunneling current. J Chem Phys 2016; 144:114310. [DOI: 10.1063/1.4943578] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/30/2022] Open
Affiliation(s)
- Jacob Lykkebo
- Nano-Science Center and Department of Chemistry, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen Ø, Denmark
| | - Giuseppe Romano
- Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA
| | - Alessio Gagliardi
- Technische Universität München, Electrical Engineering and Information Technology, Arcisstr. 21, 80333 München, Germany
| | - Alessandro Pecchia
- Consiglio Nazionale delle Ricerche, ISMN, Via Salaria Km 29.6, 00017 Monterotondo, Rome, Italy
| | - Gemma C. Solomon
- Nano-Science Center and Department of Chemistry, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen Ø, Denmark
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12
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Xiang D, Wang X, Jia C, Lee T, Guo X. Molecular-Scale Electronics: From Concept to Function. Chem Rev 2016; 116:4318-440. [DOI: 10.1021/acs.chemrev.5b00680] [Citation(s) in RCA: 816] [Impact Index Per Article: 102.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/26/2023]
Affiliation(s)
- Dong Xiang
- Beijing
National Laboratory for Molecular Sciences, State Key Laboratory for
Structural Chemistry of Unstable and Stable Species, College of Chemistry
and Molecular Engineering, Peking University, Beijing 100871, China
- Key
Laboratory of Optical Information Science and Technology, Institute
of Modern Optics, College of Electronic Information and Optical Engineering, Nankai University, Tianjin 300071, China
| | - Xiaolong Wang
- Beijing
National Laboratory for Molecular Sciences, State Key Laboratory for
Structural Chemistry of Unstable and Stable Species, College of Chemistry
and Molecular Engineering, Peking University, Beijing 100871, China
| | - Chuancheng Jia
- Beijing
National Laboratory for Molecular Sciences, State Key Laboratory for
Structural Chemistry of Unstable and Stable Species, College of Chemistry
and Molecular Engineering, Peking University, Beijing 100871, China
| | - Takhee Lee
- Department
of Physics and Astronomy, and Institute of Applied Physics, Seoul National University, Seoul 08826, Korea
| | - Xuefeng Guo
- Beijing
National Laboratory for Molecular Sciences, State Key Laboratory for
Structural Chemistry of Unstable and Stable Species, College of Chemistry
and Molecular Engineering, Peking University, Beijing 100871, China
- Department
of Materials Science and Engineering, College of Engineering, Peking University, Beijing 100871, China
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13
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Bessis C, Della Rocca ML, Barraud C, Martin P, Lacroix JC, Markussen T, Lafarge P. Probing electron-phonon excitations in molecular junctions by quantum interference. Sci Rep 2016; 6:20899. [PMID: 26864735 PMCID: PMC4750039 DOI: 10.1038/srep20899] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/27/2015] [Accepted: 01/12/2016] [Indexed: 12/31/2022] Open
Abstract
Electron-phonon coupling is a fundamental inelastic interaction in condensed matter and in molecules. Here we probe phonon excitations using quantum interference in electron transport occurring in short chains of anthraquinone based molecular junctions. By studying the dependence of molecular junction’s conductance as a function of bias voltage and temperature, we show that inelastic scattering of electrons by phonons can be detected as features in conductance resulting from quenching of quantum interference. Our results are in agreement with density functional theory calculations and are well described by a generic two-site model in the framework of non-equilibrium Green’s functions formalism. The importance of the observed inelastic contribution to the current opens up new ways for exploring coherent electron transport through molecular devices.
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Affiliation(s)
- C Bessis
- Université Paris Diderot, Sorbonne Paris Cité, MPQ, UMR 7162, CNRS, 75205 Paris Cedex 13, France
| | - M L Della Rocca
- Université Paris Diderot, Sorbonne Paris Cité, MPQ, UMR 7162, CNRS, 75205 Paris Cedex 13, France
| | - C Barraud
- Université Paris Diderot, Sorbonne Paris Cité, MPQ, UMR 7162, CNRS, 75205 Paris Cedex 13, France
| | - P Martin
- Université Paris Diderot, Sorbonne Paris Cité, ITODYS, UMR 7086, CNRS, 15 rue J.-A. de Baïf, 75205 Paris Cedex 13, France
| | - J C Lacroix
- Université Paris Diderot, Sorbonne Paris Cité, ITODYS, UMR 7086, CNRS, 15 rue J.-A. de Baïf, 75205 Paris Cedex 13, France
| | - T Markussen
- QuantumWise A/S, Fruebjergvej 3, Box 4, DK-2100 Copenhagen, Denmark
| | - P Lafarge
- Université Paris Diderot, Sorbonne Paris Cité, MPQ, UMR 7162, CNRS, 75205 Paris Cedex 13, France
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14
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Lu SY, Lin JS. A nano Ag 5 cluster tip probing the vertical transfer of CO (ads) adsorbed on Ag(110) with simulated inelastic electron tunneling spectroscopy. RSC Adv 2015. [DOI: 10.1039/c5ra06728f] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022] Open
Abstract
A nano Ag5 tip probing the transfer of CO(ads) on Ag(110) surface is investigated with simulated inelastic electron tunneling spectroscopy generated by combining DFT-based MD simulations with a FT-ACF of the derivative of local density of states.
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Affiliation(s)
- Shao-Yu Lu
- Department of Chemistry
- Tamkang University
- Tamsui
- Taiwan 25137
| | - Jyh-Shing Lin
- Department of Chemistry
- Tamkang University
- Tamsui
- Taiwan 25137
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15
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Ding Z, Sun Z, Li G, Meng F, Wu M, Ma Y, Chen X. The inelastic electron tunneling spectroscopy of curved finite-sized graphene nanoribbon based molecular devices. RSC Adv 2015. [DOI: 10.1039/c5ra09727d] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022] Open
Abstract
The inelastic electron scattering properties of the molecular devices of curved finite-sized graphene nanoribbon (GNR) slices have been studied by combining the density functional theory and Green's function method.
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Affiliation(s)
- Zongling Ding
- School of Physics and Material Science
- Anhui University
- Hefei, China
- Co-operative Innovation Research Center for Weak Signal-Detecting Materials and Devices Integration
- Anhui University
| | - Zhaoqi Sun
- School of Physics and Material Science
- Anhui University
- Hefei, China
| | - Guang Li
- School of Physics and Material Science
- Anhui University
- Hefei, China
| | - Fanming Meng
- School of Physics and Material Science
- Anhui University
- Hefei, China
| | - Mingzai Wu
- School of Physics and Material Science
- Anhui University
- Hefei, China
| | - Yongqing Ma
- School of Physics and Material Science
- Anhui University
- Hefei, China
| | - Xiaoshuang Chen
- National Lab. of Infrared Physics
- Shanghai Institute for Technical Physics
- Chinese Academy of Sciences
- Shanghai, China
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16
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Reuter MG, Hansen T. Communication: Finding destructive interference features in molecular transport junctions. J Chem Phys 2014; 141:181103. [DOI: 10.1063/1.4901722] [Citation(s) in RCA: 27] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Affiliation(s)
- Matthew G. Reuter
- Department of Chemistry, Northwestern University, Evanston, Illinois 60208, USA
| | - Thorsten Hansen
- Department of Chemistry, H. C. Ørsted Institute, University of Copenhagen, DK 2100 Copenhagen, Denmark
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17
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Hsu LY, Xie D, Rabitz H. Light-driven electron transport through a molecular junction based on cross-conjugated systems. J Chem Phys 2014; 141:124703. [DOI: 10.1063/1.4895963] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/29/2023] Open
Affiliation(s)
- Liang-Yan Hsu
- Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA
| | - Dan Xie
- Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA
| | - Herschel Rabitz
- Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA
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18
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Lykkebo J, Gagliardi A, Pecchia A, Solomon GC. IETS and quantum interference: Propensity rules in the presence of an interference feature. J Chem Phys 2014; 141:124119. [DOI: 10.1063/1.4896234] [Citation(s) in RCA: 19] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Jacob Lykkebo
- Nano-Science Center and Department of Chemistry, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen Ø, Denmark
| | - Alessio Gagliardi
- Technische Universität München, Electrical Engineering and Information Tech., Arcisstr. 21, 80333 München, Germany
| | - Alessandro Pecchia
- Consiglio Nazionale delle Ricerche, ISMN, Via Salaria km 29.6, 00017 Monterotondo (Rome), Italy
| | - Gemma C. Solomon
- Nano-Science Center and Department of Chemistry, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen Ø, Denmark
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19
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Venkatramani R, Wierzbinski E, Waldeck DH, Beratan DN. Breaking the simple proportionality between molecular conductances and charge transfer rates. Faraday Discuss 2014; 174:57-78. [DOI: 10.1039/c4fd00106k] [Citation(s) in RCA: 40] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/19/2022]
Abstract
A theoretical framework is presented to describe and to understand the observed relationship between molecular conductances and charge transfer rates across molecular bridges as a function of length, structure, and charge transfer mechanism. The approach uses a reduced density matrix formulation with a phenomenological treatment of system–bath couplings to describe charge transfer kinetics and a Green's function based Landauer–Buttiker method to describe steady-state currents. Application of the framework is independent of the transport regime and includes bath-induced decoherence effects. This model shows that the relationship between molecular conductances and charge transfer rates follows a power-law. The nonlinear rate–conductance relationship is shown to arise from differences in the charge transport barrier heights and from differences in environmental decoherence rates for the two experiments. This model explains otherwise puzzling correlations between molecular conductances and electrochemical kinetics.
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Affiliation(s)
- Ravindra Venkatramani
- Department of Chemical Sciences
- Tata Institute of Fundamental Research
- Mumbai 400 005, India
- Department of Chemistry
- Duke University
| | | | | | - David N. Beratan
- Department of Chemistry
- Duke University
- Durham, USA
- Departments of Biochemistry and Physics
- Duke University
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