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Bi RH, Su Y, Wang Y, Sun L, Dou W. Spin-lattice relaxation with non-linear couplings: Comparison between Fermi's golden rule and extended dissipaton equation of motion. J Chem Phys 2024; 161:024105. [PMID: 38984964 DOI: 10.1063/5.0212870] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/07/2024] [Accepted: 06/24/2024] [Indexed: 07/11/2024] Open
Abstract
Fermi's golden rule (FGR) offers an empirical framework for understanding the dynamics of spin-lattice relaxation in magnetic molecules, encompassing mechanisms like direct (one-phonon) and Raman (two-phonon) processes. These principles effectively model experimental longitudinal relaxation rates, denoted as T1-1. However, under scenarios of increased coupling strength and nonlinear spin-lattice interactions, FGR's applicability may diminish. This paper numerically evaluates the exact spin-lattice relaxation rate kernels, employing the extended dissipaton equation of motion formalism. Our calculations reveal that when quadratic spin-lattice coupling is considered, the rate kernels exhibit a free induction decay-like feature, and the damping rates depend on the interaction strength. We observe that the temperature dependence predicted by FGR significantly deviates from the exact results since FGR ignores the higher order effects and the non-Markovian nature of spin-lattice relaxation. Our methods can be easily extended to study other systems with nonlinear spin-lattice interactions and provide valuable insights into the temperature dependence of T1 in molecular qubits when the coupling is strong.
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Affiliation(s)
- Rui-Hao Bi
- Department of Chemistry, School of Science and Research Center for Industries of the Future, Westlake University, Hangzhou, Zhejiang 310024, China
| | - Yu Su
- Hefei National Research Center for Physical Sciences at the Microscale, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Yao Wang
- Hefei National Research Center for Physical Sciences at the Microscale, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Lei Sun
- Department of Chemistry, School of Science and Research Center for Industries of the Future, Westlake University, Hangzhou, Zhejiang 310024, China
- Institute of Natural Sciences, Westlake Institute for Advanced Study, Hangzhou, Zhejiang 310024, China
- Key Laboratory for Quantum Materials of Zhejiang Province, Department of Physics, School of Science and Research Center for Industries of the Future, Westlake University, Hangzhou, Zhejiang 310024, China
| | - Wenjie Dou
- Department of Chemistry, School of Science and Research Center for Industries of the Future, Westlake University, Hangzhou, Zhejiang 310024, China
- Institute of Natural Sciences, Westlake Institute for Advanced Study, Hangzhou, Zhejiang 310024, China
- Key Laboratory for Quantum Materials of Zhejiang Province, Department of Physics, School of Science and Research Center for Industries of the Future, Westlake University, Hangzhou, Zhejiang 310024, China
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2
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Dey S, Sharma T, Rajaraman G. Unravelling the role of spin-vibrational coupling in designing high-performance pentagonal bipyramidal Dy(iii) single ion magnets. Chem Sci 2024; 15:6465-6477. [PMID: 38699254 PMCID: PMC11062094 DOI: 10.1039/d4sc00823e] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/02/2024] [Accepted: 03/22/2024] [Indexed: 05/05/2024] Open
Abstract
At the cutting edge of high-performance single-molecule magnets (SMMs) lie lanthanide-based complexes, renowned for their potent magnetic anisotropy. SMMs containing one metal centre are defined as single-ion magnets (SIMs). The performance of SMMs is measured generally via the barrier height for magnetisation reversal (Ueff) and blocking temperature (TB), below which the magnetisation is fully frozen. To enhance the Ueff and TB values in lanthanide-based SMMs, the static crystal field splitting of mJ levels has been effectively adjusted through ligand design, leveraging the oblate/prolate ground state 4f electron density shape. However, the maximum fine-tuning achievable through ligand design, known as the axial limit, has already been reached in this class of compounds. This necessitates new design principles to enhance SMM characteristics to better suit end-user applications. Among other avenues that can be explored to improve SMM characteristics, a deeper understanding of spin-phonon coupling is critical to advancing TB values. However, there are only a handful of examples where this has been deciphered. In this work, using a combination of DFT and ab initio CASSCF calculations, we have performed spin-phonon calculations on five classes of pentagonal bipyramidal Dy(iii) SIMs exhibiting TB values in the range of 4.5 K to 36 K ([Dy(bbpen)Br] (1, H2bbpen = N,N'-bis(2-hydroxybenzyl)-N,N'-bis(2-methylpyridyl)ethylenediamine), [Dy(OCMe3)Br(THF)5][BPh4] (2) [Dy(OSiMe3)Br(THF)5] [BPh4] (3), [Dy(LN5)(Ph3SiO)2](BPh4)·CH2Cl2 (4) and [L2Dy(H2O)5][I]3·L2·H2O (5, L = tBuPO(NHiPr)2)). Unlike the method employed elsewhere for the calculation of spin-phonon coupling, in this work, we have employed a set of criteria and intuitively selected vibrational modes to perform the spin-phonon coupling analysis. The approach provided here not only reduces the computational cost significantly but also suggests chemical intuition to improve the performance of this class of compounds. Our calculations reveal that low-energy vibrational modes govern the magnetisation relaxation in these SIMs. A flexible first coordination sphere found on some of the complexes was found to be responsible for low-energy vibrations that flip the magnetisation, reducing the TB values drastically (complexes 2 and 3). On the other hand, a rigid first coordination sphere and a stiff ligand framework move the spin-vibrational coupling that causes the relaxation to lie beyond the secondary coordination sphere, resulting in an increase in TB values. Our calculations also reveal that not only the atoms in the first coordination sphere but also those in the secondary coordination sphere affect the performance of the SMMs. Learning from this exercise, we have undertaken several in silico models based on these vibrations to improve the TB values. Some of these predictions were correlated with literature precedents, offering confidence in the methodology employed. To this end, our comprehensive investigation, involving twenty-three molecules/models and five sets of geometries for pentagonal bipyramidal Dy(iii) single-ion magnets (SIMs), unveils a treasure trove of chemically sound design clues, poised to enhance the TB values in this fascinating molecular realm.
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Affiliation(s)
- Sourav Dey
- Department of Chemistry, Indian Institute of Technology Bombay Powai 400076 Mumbai India
| | - Tanu Sharma
- Department of Chemistry, Indian Institute of Technology Bombay Powai 400076 Mumbai India
| | - Gopalan Rajaraman
- Department of Chemistry, Indian Institute of Technology Bombay Powai 400076 Mumbai India
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3
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Gálico DA, Rodrigues EM, Halimi I, Toivola J, Zhao H, Xu J, Moilanen JO, Liu X, Hemmer E, Murugesu M. Confining single Er 3+ ions in sub-3 nm NaYF 4 nanoparticles to induce slow relaxation of the magnetisation. Nat Commun 2024; 15:3498. [PMID: 38664382 PMCID: PMC11045814 DOI: 10.1038/s41467-024-47682-x] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/26/2022] [Accepted: 04/09/2024] [Indexed: 04/28/2024] Open
Abstract
Molecular systems known as single-molecule magnets (SMMs) exhibit magnet-like behaviour of slow relaxation of the magnetisation and magnetic hysteresis and have potential application in high-density memory storage or quantum computing. Often, their intrinsic magnetic properties are plagued by low-energy molecular vibrations that lead to phonon-induced relaxation processes, however, there is no straightforward synthetic approach for molecular systems that would lead to a small amount of low-energy vibrations and low phonon density of states at the spin-resonance energies. In this work, we apply knowledge accumulated over the last decade in molecular magnetism to nanoparticles, incorporating Er3+ ions in an ultrasmall sub-3 nm diamagnetic NaYF4 nanoparticle (NP) and probing the slow relaxation dynamics intrinsic to the Er3+ ion. Furthermore, by increasing the doping concentration, we also investigate the role of intraparticle interactions within the NP. The knowledge gained from this study is anticipated to enable better design of magnetically high-performance molecular and bulk magnets for a wide variety of applications, such as molecular electronics.
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Affiliation(s)
- Diogo A Gálico
- Department of Chemistry and Biomolecular Sciences, University of Ottawa, Ottawa, ON, K1N 6N5, Canada
| | - Emille M Rodrigues
- Department of Chemistry and Biomolecular Sciences, University of Ottawa, Ottawa, ON, K1N 6N5, Canada
| | - Ilias Halimi
- Department of Chemistry and Biomolecular Sciences, University of Ottawa, Ottawa, ON, K1N 6N5, Canada
| | - Juho Toivola
- Department of Chemistry, Nanoscience Centre, University of Jyväskylä, P.O. Box 35, FI-40014, Jyväskylä, Finland
| | - He Zhao
- Department of Chemistry, National University of Singapore, 3 Science Drive 3, Singapore, 117543, Singapore
| | - Jiahui Xu
- Department of Chemistry, National University of Singapore, 3 Science Drive 3, Singapore, 117543, Singapore
| | - Jani O Moilanen
- Department of Chemistry, Nanoscience Centre, University of Jyväskylä, P.O. Box 35, FI-40014, Jyväskylä, Finland.
| | - Xiaogang Liu
- Department of Chemistry, National University of Singapore, 3 Science Drive 3, Singapore, 117543, Singapore
| | - Eva Hemmer
- Department of Chemistry and Biomolecular Sciences, University of Ottawa, Ottawa, ON, K1N 6N5, Canada.
- Centre for Advanced Materials Research (CAMaR), University of Ottawa, Ottawa, ON, K1N 6N5, Canada.
| | - Muralee Murugesu
- Department of Chemistry and Biomolecular Sciences, University of Ottawa, Ottawa, ON, K1N 6N5, Canada.
- Centre for Advanced Materials Research (CAMaR), University of Ottawa, Ottawa, ON, K1N 6N5, Canada.
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4
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Staab J, Chilton N. Correction to Analytic Linear Vibronic Coupling Method for First-Principles Spin-Dynamics Calculations in Single-Molecule Magnets. J Chem Theory Comput 2024; 20:2969-2970. [PMID: 38552184 PMCID: PMC11008092 DOI: 10.1021/acs.jctc.4c00239] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/24/2024] [Indexed: 04/10/2024]
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Emerson-King J, Gransbury GK, Whitehead GFS, Vitorica-Yrezabal IJ, Rouzières M, Clérac R, Chilton NF, Mills DP. Isolation of a Bent Dysprosium Bis(amide) Single-Molecule Magnet. J Am Chem Soc 2024; 146:3331-3342. [PMID: 38282511 PMCID: PMC10859956 DOI: 10.1021/jacs.3c12427] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/07/2023] [Revised: 01/09/2024] [Accepted: 01/10/2024] [Indexed: 01/30/2024]
Abstract
The isolation of formally two-coordinate lanthanide (Ln) complexes is synthetically challenging, due to predominantly ionic Ln bonding regimes favoring high coordination numbers. In 2015, it was predicted that a near-linear dysprosium bis(amide) cation [Dy{N(SiiPr3)2}2]+ could provide a single-molecule magnet (SMM) with an energy barrier to magnetic reversal (Ueff) of up to 2600 K, a 3-fold increase of the record Ueff for a Dy SMM at the time; this work showed a potential route to SMMs that can provide high-density data storage at higher temperatures. However, synthetic routes to a Dy complex containing only two monodentate ligands have not previously been realized. Here, we report the synthesis of the target bent dysprosium bis(amide) complex, [Dy{N(SiiPr3)2}2][Al{OC(CF3)3}4] (1-Dy), together with the diamagnetic yttrium analogue. We find Ueff = 950 ± 30 K for 1-Dy, which is much lower than the predicted values for idealized linear two-coordinate Dy(III) cations. Ab initio calculations of the static electronic structure disagree with the experimentally determined height of the Ueff barrier, thus magnetic relaxation is faster than expected based on magnetic anisotropy alone. We propose that this is due to enhanced spin-phonon coupling arising from the flexibility of the Dy coordination sphere, in accord with ligand vibrations being of equal importance to magnetic anisotropy in the design of high-temperature SMMs.
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Affiliation(s)
- Jack Emerson-King
- Department
of Chemistry, The University of Manchester, Oxford Road, Manchester M13 9PL, U.K.
| | - Gemma K. Gransbury
- Department
of Chemistry, The University of Manchester, Oxford Road, Manchester M13 9PL, U.K.
| | - George F. S. Whitehead
- Department
of Chemistry, The University of Manchester, Oxford Road, Manchester M13 9PL, U.K.
| | | | | | | | - Nicholas F. Chilton
- Department
of Chemistry, The University of Manchester, Oxford Road, Manchester M13 9PL, U.K.
- Research
School of Chemistry, The Australian National
University, Sullivans
Creek Road, Canberra, ACT 2601, Australia
| | - David P. Mills
- Department
of Chemistry, The University of Manchester, Oxford Road, Manchester M13 9PL, U.K.
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6
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Mattioni A, Staab JK, Blackmore WJA, Reta D, Iles-Smith J, Nazir A, Chilton NF. Vibronic effects on the quantum tunnelling of magnetisation in Kramers single-molecule magnets. Nat Commun 2024; 15:485. [PMID: 38212305 PMCID: PMC10784566 DOI: 10.1038/s41467-023-44486-3] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/13/2023] [Accepted: 12/13/2023] [Indexed: 01/13/2024] Open
Abstract
Single-molecule magnets are among the most promising platforms for achieving molecular-scale data storage and processing. Their magnetisation dynamics are determined by the interplay between electronic and vibrational degrees of freedom, which can couple coherently, leading to complex vibronic dynamics. Building on an ab initio description of the electronic and vibrational Hamiltonians, we formulate a non-perturbative vibronic model of the low-energy magnetic degrees of freedom in monometallic single-molecule magnets. Describing their low-temperature magnetism in terms of magnetic polarons, we are able to quantify the vibronic contribution to the quantum tunnelling of the magnetisation, a process that is commonly assumed to be independent of spin-phonon coupling. We find that the formation of magnetic polarons lowers the tunnelling probability in both amorphous and crystalline systems by stabilising the low-lying spin states. This work, thus, shows that spin-phonon coupling subtly influences magnetic relaxation in single-molecule magnets even at extremely low temperatures where no vibrational excitations are present.
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Affiliation(s)
- Andrea Mattioni
- Department of Chemistry, School of Natural Sciences, The University of Manchester, Oxford Road, Manchester, M13 9PL, UK.
| | - Jakob K Staab
- Department of Chemistry, School of Natural Sciences, The University of Manchester, Oxford Road, Manchester, M13 9PL, UK
| | - William J A Blackmore
- Department of Chemistry, School of Natural Sciences, The University of Manchester, Oxford Road, Manchester, M13 9PL, UK
| | - Daniel Reta
- Department of Chemistry, School of Natural Sciences, The University of Manchester, Oxford Road, Manchester, M13 9PL, UK
- Faculty of Chemistry, The University of the Basque Country UPV/EHU, Donostia, 20018, Spain
- Donostia International Physics Center (DIPC), Donostia, 20018, Spain
- IKERBASQUE, Basque Foundation for Science, Bilbao, 48013, Spain
| | - Jake Iles-Smith
- Department of Physics and Astronomy, School of Natural Sciences, The University of Manchester, Oxford Road, Manchester, M13 9PL, UK
| | - Ahsan Nazir
- Department of Physics and Astronomy, School of Natural Sciences, The University of Manchester, Oxford Road, Manchester, M13 9PL, UK
| | - Nicholas F Chilton
- Department of Chemistry, School of Natural Sciences, The University of Manchester, Oxford Road, Manchester, M13 9PL, UK.
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