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Properties of the frustrated spin 1/2 kagome material Cu3(HOTP)2

F. L. Pratt1,*, D. López-Alcalá2, V. Garcia-Lopez2, M. Clemente-León2, J. J. Baldoví2, and E. Coronado2,†

  • *Contact author: francis.pratt@stfc.ac.uk
  • †Contact author: eugenio.coronado@uv.es
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Phys. Rev. Research 7, 023007 – Published 2 April, 2025

DOI: https://doi.org/10.1103/PhysRevResearch.7.023007

Abstract

The metal-organic-framework compound Cu3(HOTP)2, also known as Cu3(HHTP)2, is a small-gap semiconductor containing a kagome lattice of antiferromagnetically coupled S=1/2CuII spins with intralayer nearest-neighbor exchange coupling J∼ 2 K. The intralayer J value obtained from calculations using density functional theory is shown to match with the experimental value for reasonable values of the Hubbard U parameter. Muon spin relaxation confirms no magnetic ordering down to 50 mK and sees spin fluctuations diffusing on a two-dimensional lattice. These properties are consistent with the hypothesis of a quantum spin-liquid ground state being present within highly decoupled kagome layers. Reduction of the spin-diffusion rate on cooling from the paramagnetic region to the low-temperature region reflects quantum entanglement, one of the key properties of a quantum spin liquid. It is also found that the layers become more strongly decoupled in the low-temperature region. Comparison of results for the spin diffusion, magnetic susceptibility, and specific heat in the low-temperature region suggests close proximity to a quantum critical point and a large density of low-energy spinless electronic excitations. A Z2-linear Dirac model for the low-energy spin excitations of the putative quantum spin-liquid ground state is found to provide the best match with experiment.

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References (71)

  1. Y. Zhou, K. Kanoda, and T.-K. Ng, Quantum spin liquid states, Rev. Mod. Phys. 89, 025003 (2017).
  2. L. Savary and L. Balents, Quantum spin liquids: A review, Rep. Prog. Phys. 80, 016502 (2017).
  3. Muon Spectroscopy: An Introduction, edited by S. J. Blundell, R. De Renzi, T. Lancaster, and F. L. Pratt (Oxford University Press, Oxford, 2021).
  4. Y. Misumi, A. Yamaguchi, Z. Zhang, T. Matsushita, N. Wada, M. Tsuchiizu, and K. Awaga, Quantum spin liquid state in a two-dimensional semiconductive metal-organic framework, J. Am. Chem. Soc. 142, 16513 (2020).
  5. Ch. Waldtmann, H.-U. Everts, B. Bernu, C. Lhuillier, P. Sindzingre, P. Lecheminant, and L. Pierre, First excitations of the spin 1/2 Heisenberg antiferromagnet on the kagomé lattice, Eur. Phys. J. B 2, 501 (1998).
  6. H. Nakano and T. Sakai, Density matrix renormalization group numerical study of the kagome antiferromagnet, J. Phys. Soc. Jpn. 80, 053704 (2011).
  7. H. C. Jiang, Z. Y. Weng, and D. N. Sheng, Numerical-diagonalization study of spin gap issue of the kagome lattice Heisenberg antiferromagnet, Phys. Rev. Lett. 101, 117203 (2008).
  8. S. Yan, D. A. Huse, and S. R. White, Spin-liquid ground state of the S = 1/2 kagome Heisenberg antiferromagnet, Science 332, 1173 (2011).
  9. S. Depenbrock, I. P. McCulloch, and U. Schollwöck, Nature of the spin-liquid ground state of the S = 1/2 Heisenberg model on the kagome lattice, Phys. Rev. Lett. 109, 067201 (2012).
  10. S. Nishimoto, N. Shibata, and C. Hotta, Controlling frustrated liquids and solids with an applied field in a kagome Heisenberg antiferromagnet, Nat. Commun. 4, 2287 (2013).
  11. Y.-C. He, M. P. Zaletel, M. Oshikawa, and F. Pollmann, Signatures of Dirac cones in a DMRG study of the kagome Heisenberg model, Phys. Rev. X 7, 031020 (2017).
  12. W. Zhu, S.-S. Gong, and D. N. Shen, Identifying spinon excitations from dynamic structure factor of spin-1/2 Heisenberg antiferromagnet on the kagome lattice, Proc. Natl. Acad. Sci. USA 116, 5437 (2019).
  13. R.-Y. Sun, H.-K. Jin, H.-H. Tu, and Y. Zhou, Possible chiral spin liquid state in the S = 1/2 kagome Heisenberg model, npj Quantum Mater. 9, 16 (2024).
  14. Y. Ran, M. Hermele, P. A. Lee, and X.-G. Wen, Projected-wave-function study of the spin-1/2 Heisenberg model on the kagomé lattice, Phys. Rev. Lett. 98, 117205 (2007).
  15. M. Hermele, Y. Ran, P. A. Lee, and X.-G. Wen, Properties of an algebraic spin liquid on the kagome lattice, Phys. Rev. B 77, 224413 (2008).
  16. Y. Iqbal, F. Becca, S. Sorella, and D. Poilblanc, Gapless spin-liquid phase in the kagome spin-1/2 Heisenberg antiferromagnet, Phys. Rev. B 87, 060405(R) (2013).
  17. Y. Iqbal, D. Poilblanc, and F. Becca, Vanishing spin gap in a competing spin-liquid phase in the kagome Heisenberg antiferromagnet, Phys. Rev. B 89, 020407(R) (2014).
  18. C. Zhang and T. Li, Variational study of the ground state and spin dynamics of the spin-1/2 kagome antiferromagnetic Heisenberg model and its implication for herbertsmithite ZnCu3(OH)6Cl2, Phys. Rev. B 102, 195106 (2020).
  19. H. J. Liao, Z. Y. Xie, J. Chen, Z. Y. Liu, H. D. Xie, R. Z. Huang, B. Normand, and T. Xiang, Gapless spin-liquid ground state in the S=1/2 kagome antiferromagnet, Phys. Rev. Lett. 118, 137202 (2017).
  20. J.-W. Mei, J.-Y. Chen, H. He, and X.-G. Wen, Gapped spin liquid with Z2 topological order for the kagome Heisenberg model, Phys. Rev. B 95, 235107 (2017).
  21. M. Hering, J. Sonnenschein, Y. Iqbal, and J. Reuther, Characterization of quantum spin liquids and their spinon band structures via functional renormalization, Phys. Rev. B 99, 100405(R) (2019).
  22. M. P. Shores, E. A. Nytko, B. M. Bartlett, and D. G. Nocera, A structurally perfect S = 1/2 kagomé antiferromagnet, J. Am. Chem. Soc. 127, 13462 (2005).
  23. M. R. Norman, Colloquium: Herbertsmithite and the search for the quantum spin liquid, Rev. Mod. Phys. 88, 041002 (2016).
  24. T.-H. Han, J. S. Helton, J. S. Chu, D. G. Nocera, J. A. Rodriguez-Rivera, C. Broholm, and Y. S. Lee, Fractionalized excitations in the spin-liquid state of a kagome-lattice antiferromagnet, Nature (London) 492, 406 (2012).
  25. M. Fu, T. Imai, T. H. Han, and Y. S. Lee, Evidence for a gapped spin-liquid ground state in a kagome Heisenberg antiferromagnet, Science 350, 655 (2015).
  26. P. Khuntia, M. Velazquez, Q. Barthélemy, F. Bert, E. Kermarrec, A. Legros, B. Bernu, L. Messio, A. Zorko, and P. Mendels, Gapless ground state in the archetypal quantum kagome antiferromagnet ZnCu3(OH)6Cl2, Nat. Phys. 16, 469 (2020).
  27. M. Hmadeh, Z. Lu, Z. Liu, F. Gándara, H. Furukawa, S. Wan, V. Augustyn, R. Chang, L. Liao, F. Zhou, E. Perre, V. Ozolins, K. Suenaga, X. Duan, B. Dunn, Y. Yamamoto, O. Terasaki, and O. M. Yaghi, New porous crystals of extended metal-catecholates, Chem. Mater. 24, 3511 (2012).
  28. L. Yang, X. He, and M. Dincǎ, Triphenylene-bridged trinuclear complexes of Cu: Models for spin interactions in two-dimensional electrically conductive metal-organic frameworks, J. Am. Chem. Soc. 141, 10475 (2019).
  29. The same material is called Cu3(HHTP)2 in some of the literature. We, however, prefer to use Cu3(HOTP)2, which reflects the deprotonated molecular structure of the HOTP molecule in the compound, in contrast to the original fully protonated HHTP molecule.
  30. V. Rubio-Giménez, M. Galbiati, J. Castells-Gil, N. Almora-Barrios, J. Navarro-Sánchez, G. Escorcia-Ariza, M. Mattera, T. Arnold, J. Rawle, S. Tatay, E. Coronado, and C. Martí-Gastaldo, Bottom-up fabrication of semiconductive metal-organic framework ultrathin films, Adv. Mater. 30, 1704291 (2018).
  31. N. Elstner and A. P. Young, Spin-1/2 Heisenberg antiferromagnet on the kagome lattice: High-temperature expansion and exact-diagonalization studies, Phys. Rev. B 50, 6871 (1994).
  32. P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I. Dabo, A. Dal Corso, S. de Gironcoli, S. Fabris, G. Fratesi, R. Gebauer, U. Gerstmann, C. Gougoussis, A. Kokalj, M. Lazzeri, L. Martin-Samos, N. Marzari et al., P. Umari and QUANTUM ESPRESSO: A modular and open-source software project for quantum simulations of materials, J. Phys.: Condens. Matter 21, 395502 (2009).
  33. J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77, 3865 (1996).
  34. G. Prandini, A. Marrazzo, I. E. Castelli, N. Mounet, and N. Marzari, Precision and efficiency in solid-state pseudopotential calculations, npj Comput. Mater. 4, 72 (2018).
  35. S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys, and A. P. Sutton, Electron-energy-loss spectra and the structural stability of nickel oxide: An LSDA+U study, Phys. Rev. B 57, 1505 (1998).
  36. S. Grimme, Semiempirical GGA-type density functional constructed with a long-range dispersion correction, J. Comput. Chem. 27, 1787 (2006).
  37. H. J. Monkhorst and J. D. Pack, Special points for Brillouin-zone integrations, Phys. Rev. B 13, 5188 (1976).
  38. E. Coronado et al., The ISIS Neutron and Muon Source Data Journal (2023), doi:10.5286/ISIS.E.RB2010775.
  39. M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, G. A. Petersson, H. Nakatsuji, X. Li, M. Caricato, A. V. Marenich, J. Bloino, B. G. Janesko, R. Gomperts, B. Mennucci, H. P. Hratchian, J. V. Ortiz, A. F. Izmaylov et al., Computer code Gaussian 16, Rev. B.01 (Gaussian, Inc., Wallingford, CT, 2016).
  40. V. I. Ovcharenko, E. V. Gorelik, S. V. Fokin, G. V. Romanenko, V. N. Ikorskii, A. V. Krashilina, V. K. Cherkasov, and G. A. Abakumov, Ligand effects on the ferro- to antiferromagnetic exchange ratio in Bis(o-Semiquinonato)Copper(II), J. Am. Chem. Soc. 129, 10512 (2007).
  41. Z. Meng, C. G. Jones, S. Farid, I. U. Khan, H. M. Nelson, and K. A. Mirica, Unraveling the electrical and magnetic properties of layered conductive metal-organic framework with atomic precision, Ang. Chem. Intl. Ed. 61, e202113569 (2022).
  42. J. Yoon and E. I. Solomon, Ground-state electronic and magnetic properties of a μ3-oxo-bridged trinuclear Cu(II) complex: Correlation to the native intermediate of the multicopper oxidases, Inorg. Chem. 44, 8076 (2005).
  43. T. Kawakami, R. Takeda, S. Nishihara, T. Saito, M. Shoji, S. Yamada, S. Yamanaka, Y. Kitagawa, M. Okumura, and K. Yamaguchi, Symmetry and broken-symmetry in molecular orbital descriptions of unstable molecules. 3 The nature of chemical bonds of spin frustrated systems, J. Phys. Chem. A 113, 15281 (2009).
  44. E. T. Spielberg, M. Fittipaldi, D. Geibig, D. Gatteschi, and W. Plass, Electronic and magnetic structure of a triacetylphlorogucinol-bridged C3-symmetric trinuclear copper complex: Magnetic characterization, ESR spectroscopy, and DFT calculations, Inorg. Chim. Acta 363, 4269 (2010).
  45. E. T. Spielberg, A. Gilb, D. Plaul, D. Geibig, D. Hornig, D. Schuch, A. Buchholz, A. Ardavan, and W. A. Plass, Spin-frustrated trinuclear copper complex based on triaminoguanidine with an energetically well-separated degenerate ground state, Inorg. Chem. 54, 3432 (2015).
  46. G. V. Baryshnikov, B. F. Minaev, A. T. Baryshnikova, and H. A. Ågren, Computational study of structural and magnetic properties of Bi- and trinuclear Cu(II) complexes with extremely long Cu-Cu distances, Chem. Phys. 491, 48 (2017).
  47. S. Chen, J. Dai, and X. C. Zeng, Metal-organic kagome lattices M3(2, 3, 6, 7, 10, 11-Hexaiminotriphenylene)2 (M = Ni and Cu): From semiconducting to metallic by metal substitution, Phys. Chem. Chem. Phys. 17, 5954 (2015).
  48. M. E. Foster, K. Sohlberg, C. D. Spataru, and M. D. Allendorf, Proposed modification of the graphene analogue Ni3(HITP)2 To yield a semiconducting material, J. Phys. Chem. C 120, 15001 (2016).
  49. M. E. Foster, K. Sohlberg, M. D. Allendorf, and A. A. Talin, Unraveling the semiconducting/metallic discrepancy in Ni3(HITP)2, J. Phys. Chem. Lett. 9, 481 (2018).
  50. J. L. Mancuso, A. M. Mroz, K. N. Le, and C. H. Hendon, Electronic structure modeling of metal-organic frameworks, Chem. Rev. 120, 8641 (2020).
  51. F. L. Pratt, S. J. Blundell, T. Lancaster, C. Baines, and S. Takagi, Low-temperature spin diffusion in a highly ideal S = 1/2 Heisenberg antiferromagnetic chain studied by muon spin relaxation, Phys. Rev. Lett. 96, 247203 (2006).
  52. F. Xiao, J. S. Moller, T. Lancaster, R. C. Williams, F. L. Pratt, S. J. Blundell, D. Ceresoli, A. M. Barton, and J. L. Manson, Spin diffusion in the low-dimensional molecular quantum Heisenberg antiferromagnet Cu(pyz)(NO3)2 detected with implanted muons, Phys. Rev. B 91, 144417 (2015).
  53. T. Lancaster, F. Xiao, B. M. Huddart, R. C. Williams, F. L. Pratt, S. J. Blundell, S. J. Clark, R. Scheuermann, T. Goko, S. Ward, J. L. Manson, Ch. Rüegg, and K. W. Krämer, Quantum magnetism in molecular spin ladders probed with muon-spin spectroscopy, New J. Phys. 20, 103002 (2018).
  54. S. Mañas-Valero, B. M. Huddart, T. Lancaster, E. Coronado, and F. L. Pratt, Quantum phases and spin liquid properties of 1T-TaS2, npj Quantum Mater. 6, 69 (2021).
  55. F. L. Pratt, F. Lang, W. Steinhardt, S. Haravifard, and S. J. Blundell, Spin dynamics, entanglement, and the nature of the spin liquid state in YbZnGaO4, Phys. Rev. B 106, L060401 (2022).
  56. M. A. Butler, L. R. Walker, and Z. G. Soos, Dimensionality of spin fluctuations in highly anisotropic TCNQ salts, J. Chem. Phys. 64, 3592 (1976).
  57. F. Verstraete, M. Popp, and J. I. Cirac, Entanglement versus correlations in spin systems, Phys. Rev. Lett. 92, 027901 (2004).
  58. P. Hauke, M. Heyl, L. Tagliacozzo, and P. Zoller, Measuring multipartite entanglement through dynamic susceptibilities, Nat. Phys. 12, 778 (2016).
  59. P. Laurell, A. Scheie, C. J. Mukherjee, M. M. Koza, M. Enderle, Z. Tylczynski, S. Okamoto, R. Coldea, D. A. Tennant, and G. Alvarez, Quantifying and controlling entanglement in the quantum magnet Cs2CoCl4, Phys. Rev. Lett. 127, 037201 (2021).
  60. J. J. Binney, N. J. Dowrick, A. J. Fisher, and M. E. J. Newman, The Theory of Critical Phenomena: An Introduction to the Renormalization Group (Oxford University Press, Oxford, 1992).
  61. S. Sachdev, Quantum Phase Transitions, 2nd ed. (Cambridge University Press, Cambridge, 2011).
  62. C. Xu and S. Sachdev, Global phase diagrams of frustrated quantum antiferromagnets in two dimensions: Doubled Chern-Simons theory, Phys. Rev. B 79, 064405 (2009).
  63. Y. Qi, C. Xu, and S. Sachdev, Dynamics and transport of the Z2 spin liquid: Application to κ−ET2Cu2(CN)3, Phys. Rev. Lett. 102, 176401 (2009).
  64. G. De Polsi, I. Balog, M. Tissier, and N. Wschebor, Precision calculation of critical exponents in the O(N) universality classes with the nonperturbative renormalization group, Phys. Rev. E 101, 042113 (2020).
  65. F. L. Pratt, P. J. Baker, S. J. Blundell, T. Lancaster, S. Ohira-Kawamura, C. Baines, Y. Shimizu, K. Kanoda, I. Watanabe, and G. Saito, Magnetic and non-magnetic phases of a quantum spin liquid, Nature (London) 471, 612 (2011).
  66. X. G. Wen, Mean-field theory of spin-liquid states with finite energy gap and topological orders, Phys. Rev. B 44, 2664 (1991).
  67. S. E. Sebastian, N. Harrison, C. D. Batista, L. Balicas, M. Jaime, P. A. Shrma, N. Kawashima, and I. R. Fisher, Dimensional reduction at a quantum critical point, Nature (London) 441, 617 (2006).
  68. R. Bag, S. Xu, N. E. Sherman, L. Yadav, A. I. Kolesnikov, A. A. Podlesnyak, E. S. Choi, I. da Silva, J. E. Moore, and S. Haravifard, Evidence of Dirac quantum spin liquid in YbZn2GaO5, Phys. Rev. Lett. 133, 266703 (2024).
  69. M. Klanjšek, A. Zorko, R. Žitko, J. Mravlje, Z. Jaglicic, P. K. Biswas, P. Prelovžek, D. Mihailovic, and D. Arcon, A high-temperature quantum spin liquid with polaron spins, Nat. Phys. 13, 1130 (2017).
  70. Z. Zeng, X. Ma, S. Wu, H.-F. Li, Z. Tao, X. Lu, X.-h. Chen, J.-X. Mi, S.-J. Song, G.-H. Cao, G. Che, K. Li, G. Li, H. Luo, Z. Y. Meng, and S. Li, Possible Dirac quantum spin liquid in the kagome quantum antiferromagnet YCu3(OH)6Br2[Brx(OH)1−x], Phys. Rev. B 105, L121109 (2022).
  71. Q. Barthelemy, A. Demuer, C. Marcenat, T. Klein, B. Bernu, L. Messio, M. Velázquez, E. Kermarrec, F. Bert, and P. Mendels, Specific heat of the kagome antiferromagnet herbertsmithite in high magnetic fields, Phys. Rev. X 12, 011014 (2022).

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Phys. Rev. Research 7, 023007– Published 2 April, 2025
  • Received 28 November 2024
  • Accepted 11 March 2025
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