#
Thirty-Year Anniversary of κ-(BEDT-TTF)_{2}Cu_{2}(CN)_{3}: Reconciling the Spin Gap in a Spin-Liquid Candidate

## Abstract

**:**

_{2}X; geometrical frustration; quantum spin liquid; valence bond solid; strongly correlated electron systems; Mott insulators; Hubbard model; bandwidth tuning; metal-insulator transitions

## 1. Introduction

## 2. Anatomy of the Magnetic Ground State of $\kappa $-(BEDT-TTF)${}_{2}$Cu${}_{2}$(CN)${}_{3}$

#### 2.1. Absence of Magnetic Order

#### 2.2. Evidence for a Spin Gap below ${T}^{\star}=6$ K

#### 2.2.1. Spin Susceptibility

#### 2.2.2. NMR Spin-Lattice Relaxation Rate

#### 2.2.3. Thermal Transport

#### 2.2.4. NMR Knight Shift and $\mu $-SR

#### 2.3. Structural Distortion at the ‘6K Anomaly’

#### 2.3.1. Thermodynamic Signatures

#### 2.3.2. Vibrational Spectroscopy and NQR

#### 2.4. B–T Phase Diagram of the Spin-Gapped Ground State

## 3. Conclusions and Outlook

#### Do Quantum Spin Liquids Exist in Solids?

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Anderson, P.W. Resonating valence bonds: A new kind of insulator? Mater. Res. Bull.
**1973**, 8, 153–160. [Google Scholar] [CrossRef] - Balents, L. Spin liquids in frustrated magnets. Nature
**2010**, 464, 199–208. [Google Scholar] [CrossRef] [PubMed] - Savary, L.; Balents, L. Quantum spin liquids: A review. Rep. Prog. Phys.
**2017**, 80, 016502. [Google Scholar] [CrossRef] [PubMed] - Zhou, Y.; Kanoda, K.; Ng, T.K. Quantum spin liquid states. Rev. Mod. Phys.
**2017**, 89, 25003. [Google Scholar] [CrossRef] - Broholm, C.; Cava, R.J.; Kivelson, S.A.; Nocera, D.G.; Norman, M.R.; Senthil, T. Quantum spin liquids. Science
**2020**, 367, eaay0668. [Google Scholar] [CrossRef] [Green Version] - Powell, B.J.; McKenzie, R.H. Quantum frustration in organic Mott insulators: From spin liquids to unconventional superconductors. Rep. Prog. Phys.
**2011**, 74, 56501. [Google Scholar] [CrossRef] - Kanoda, K.; Kato, R. Mott Physics in Organic Conductors with Triangular Lattices. Annu. Rev. Condens. Matter Phys.
**2011**, 2, 167–188. [Google Scholar] [CrossRef] - Shen, Y.; Li, Y.D.; Wo, H.; Li, Y.; Shen, S.; Pan, B.; Wang, Q.; Walker, H.C.; Steffens, P.; Boehm, M.; et al. Evidence for a spinon Fermi surface in a triangular-lattice quantum-spin-liquid candidate. Nature
**2016**, 540, 559–562. [Google Scholar] [CrossRef] [Green Version] - Xu, Y.; Zhang, J.; Li, Y.S.; Yu, Y.J.; Hong, X.C.; Zhang, Q.M.; Li, S.Y. Absence of Magnetic Thermal Conductivity in the Quantum Spin-Liquid Candidate YbMgGaO
_{4}. Phys. Rev. Lett.**2016**, 117, 267202. [Google Scholar] [CrossRef] [Green Version] - Li, Y.; Adroja, D.; Bewley, R.I.; Voneshen, D.; Tsirlin, A.A.; Gegenwart, P.; Zhang, Q. Crystalline Electric-Field Randomness in the Triangular Lattice Spin-Liquid YbMgGaO
_{4}. Phys. Rev. Lett.**2017**, 118, 107202. [Google Scholar] [CrossRef] [Green Version] - Norman, M.R. Colloquium: Herbertsmithite and the search for the quantum spin liquid. Rev. Mod. Phys.
**2016**, 88, 041002. [Google Scholar] [CrossRef] [Green Version] - Puphal, P.; Zoch, K.M.; Désor, J.; Bolte, M.; Krellner, C. Kagome quantum spin systems in the atacamite family. Phys. Rev. Mater.
**2018**, 2, 63402. [Google Scholar] [CrossRef] [Green Version] - Semeghini, G.; Levine, H.; Keesling, A.; Ebadi, S.; Wang, T.; Bluvstein, D.; Verresen, R.; Pichler, P.; Kalinowski, K.; Samajdar, R.; et al. Probing topological spin liquids on a programmable quantum simulator. Science
**2021**, 374, 1242–1247. [Google Scholar] [CrossRef] - Kitaev, A. Anyons in an exactly solved model and beyond. Ann. Phys.
**2006**, 321, 2–111. [Google Scholar] [CrossRef] [Green Version] - Singh, Y.; Gegenwart, P. Antiferromagnetic Mott insulating state in single crystals of the honeycomb lattice material Na
_{2}IrO_{3}. Phys. Rev. B**2010**, 82, 64412. [Google Scholar] [CrossRef] - Plumb, K.W.; Clancy, J.P.; Sandilands, L.J.; Shankar, V.V.; Hu, Y.F.; Burch, K.S.; Kee, H.Y.; Kim, Y.J. α-RuCl
_{3}: A spin-orbit assisted Mott insulator on a honeycomb lattice. Phys. Rev. B**2014**, 90, 41112. [Google Scholar] [CrossRef] [Green Version] - Johnson, R.D.; Williams, S.C.; Haghighirad, A.A.; Singleton, J.; Zapf, V.; Manuel, P.; Mazin, I.I.; Li, Y.; Jeschke, H.O.; Valentí, R.; et al. Monoclinic crystal structure of α-RuCl
_{3}and the zigzag antiferromagnetic ground state. Phys. Rev. B**2015**, 92, 235119. [Google Scholar] [CrossRef] [Green Version] - Banerjee, A.; Bridges, C.A.; Yan, J.Q.; Aczel, A.A.; Li, L.; Stone, M.B.; Granroth, G.E.; Lumsden, M.D.; Yiu, Y.; Knolle, J.; et al. Proximate Kitaev quantum spin liquid behaviour in a honeycomb magnet. Nat. Mater.
**2016**, 15, 733. [Google Scholar] [CrossRef] [Green Version] - Banerjee, A.; Yan, J.; Knolle, J.; Bridges, C.A.; Stone, M.B.; Lumsden, M.D.; Mandrus, D.G.; Tennant, D.A.; Moessner, R.; Nagler, S.E. Neutron scattering in the proximate quantum spin liquid α-RuCl
_{3}. Science**2017**, 356, 1055–1059. [Google Scholar] [CrossRef] [Green Version] - Takagi, H.; Takayama, T.; Jackeli, G.; Khaliullin, G.; Nagler, S.E. Concept and realization of Kitaev quantum spin liquids. Nat. Rev. Phys.
**2019**, 1, 264–280. [Google Scholar] [CrossRef] - Lee, S.S.; Lee, P.A. U(1) Gauge Theory of the Hubbard Model: Spin Liquid States and Possible Application to κ-(BEDT-TTF)
_{2}Cu_{2}(CN)_{3}. Phys. Rev. Lett.**2005**, 95, 36403. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Ng, T.K.; Lee, P.A. Power-Law Conductivity inside the Mott Gap: Application to κ-(BEDT-TTF)
_{2}Cu_{2}(CN)_{3}. Phys. Rev. Lett.**2007**, 99, 156402. [Google Scholar] [CrossRef] [Green Version] - Fu, M.; Imai, T.; Han, T.H.; Lee, Y.S. Evidence for a gapped spin-liquid ground state in a kagome Heisenberg antiferromagnet. Science
**2015**, 350, 655–658. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Khuntia, P.; Velazquez, M.; Barthélemy, Q.; Bert, F.; Kermarrec, E.; Legros, A.; Bernu, B.; Messio, L.; Zorko, A.; Mendels, P. Gapless ground state in the archetypal quantum kagome antiferromagnet ZnCu
_{3}(OH)_{6}Cl_{2}. Nat. Phys.**2020**, 16, 469–474. [Google Scholar] [CrossRef] - Wang, J.; Yuan, W.; Singer, P.M.; Smaha, R.W.; He, W.; Wen, J.; Lee, Y.S.; Imai, T. Emergence of spin singlets with inhomogeneous gaps in the kagome lattice Heisenberg antiferromagnets Zn-barlowite and herbertsmithite. Nat. Phys.
**2021**, 17, 1109–1113. [Google Scholar] [CrossRef] - Yamashita, S.; Nakazawa, Y.; Oguni, M.; Oshima, Y.; Nojiri, H.; Shimizu, Y.; Miyagawa, K.; Kanoda, K. Thermodynamic properties of a spin-1/2 spin-liquid state in a [kappa]-type organic salt. Nat. Phys.
**2008**, 4, 459–462. [Google Scholar] [CrossRef] [Green Version] - Miksch, B.; Pustogow, A.; Javaheri Rahim, M.; Bardin, A.A.; Kanoda, K.; Schlueter, J.A.; Hübner, R.; Scheffler, M.; Dressel, M. Gapped magnetic ground state in quantum spin liquid candidate κ-(BEDT-TTF)
_{2}Cu_{2}(CN)_{3}. Science**2021**, 372, 276–279. [Google Scholar] [CrossRef] - Dressel, M.; Tomić, S. Molecular quantum materials: Electronic phases and charge dynamics in two-dimensional organic solids. Adv. Phys.
**2020**, 69, 1–120. [Google Scholar] [CrossRef] - Geiser, U.; Wang, H.H.; Carlson, K.D.; Williams, J.M.; Charlier, H.A.; Heindl, J.E.; Yaconi, G.A.; Love, B.J.; Lathrop, M.W.; Schirber, J.E.; et al. Superconductivity at 2.8 K and 1.5 kbar in κ-(BEDT-TTF)
_{2}Cu_{2}(CN)_{3}: The first organic superconductor containing a polymeric copper cyanide anion. Inorg. Chem.**1991**, 30, 2586–2588. [Google Scholar] [CrossRef] - Komatsu, T.; Matsukawa, N.; Inoue, T.; Saito, G. Realization of Superconductivity at Ambient Pressure by Band-Filling Control in κ-(BEDT-TTF)
_{2}Cu_{2}(CN)_{3}. J. Phys. Soc. Jpn.**1996**, 65, 1340–1354. [Google Scholar] [CrossRef] - Kurosaki, Y.; Shimizu, Y.; Miyagawa, K.; Kanoda, K.; Saito, G. Mott Transition from a Spin Liquid to a Fermi Liquid in the Spin-Frustrated Organic Conductor κ-(ET)
_{2}Cu_{2}(CN)_{3}. Phys. Rev. Lett.**2005**, 95, 177001. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Shimizu, Y.; Kasahara, H.; Furuta, T.; Miyagawa, K.; Kanoda, K.; Maesato, M.; Saito, G. Pressure-induced superconductivity and Mott transition in spin-liquid κ-(ET)
_{2}Cu_{2}(CN)_{3}probed by^{13}C NMR. Phys. Rev. B**2010**, 81, 224508. [Google Scholar] [CrossRef] - Shimizu, Y.; Maesato, M.; Saito, G. Uniaxial Strain Effects on Mott and Superconducting Transitions in κ-(ET)
_{2}Cu_{2}(CN)_{3}. J. Phys. Soc. Jpn.**2011**, 80, 74702. [Google Scholar] [CrossRef] - Shimizu, Y.; Miyagawa, K.; Kanoda, K.; Maesato, M.; Saito, G. Spin Liquid State in an Organic Mott Insulator with a Triangular Lattice. Phys. Rev. Lett.
**2003**, 91, 107001. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Kandpal, H.C.; Opahle, I.; Zhang, Y.Z.; Jeschke, H.O.; Valentí, R. Revision of Model Parameters for κ-Type Charge Transfer Salts: An Ab Initio Study. Phys. Rev. Lett.
**2009**, 103, 67004. [Google Scholar] [CrossRef] [Green Version] - Itou, T.; Oyamada, A.; Maegawa, S.; Tamura, M.; Kato, R. Quantum spin liquid in the spin-1/2 triangular antiferromagnet EtMe
_{3}Sb[Pd_{(dmit)2}]_{2}. Phys. Rev. B**2008**, 77, 104413. [Google Scholar] [CrossRef] - Itou, T.; Oyamada, A.; Maegawa, S.; Kato, R. Instability of a quantum spin liquid in an organic triangular-lattice antiferromagnet. Nat. Phys.
**2010**, 6, 673–676. [Google Scholar] [CrossRef] - Yamashita, S.; Yamamoto, T.; Nakazawa, Y.; Tamura, M.; Kato, R. Gapless spin liquid of an organic triangular compound evidenced by thermodynamic measurements. Nat. Commun.
**2011**, 2, 275. [Google Scholar] [CrossRef] - Shimizu, Y.; Hiramatsu, T.; Maesato, M.; Otsuka, A.; Yamochi, H.; Ono, A.; Itoh, M.; Yoshida, M.; Takigawa, M.; Yoshida, Y.; et al. Pressure-Tuned Exchange Coupling of a Quantum Spin Liquid in the Molecular Triangular Lattice κ-(ET)
_{2}Ag_{2}(CN)_{3}. Phys. Rev. Lett.**2016**, 117, 107203. [Google Scholar] [CrossRef] [Green Version] - Hiramatsu, T.; Yoshida, Y.; Saito, G.; Otsuka, A.; Yamochi, H.; Maesato, M.; Shimizu, Y.; Ito, H.; Nakamura, Y.; Kishida, H.; et al. Design and Preparation of a Quantum Spin Liquid Candidate κ-(ET)
_{2}Ag_{2}(CN)_{3}Having a Nearby Superconductivity. Bull. Chem. Soc. Jpn.**2017**, 90, 1073–1082. [Google Scholar] [CrossRef] - Isono, T.; Kamo, H.; Ueda, A.; Takahashi, K.; Kimata, M.; Tajima, H.; Tsuchiya, S.; Terashima, T.; Uji, S.; Mori, H. Gapless Quantum Spin Liquid in an Organic Spin-1/2 Triangular-Lattice κ-H
_{3}(Cat-EDT-TTF)_{2}. Phys. Rev. Lett.**2014**, 112, 177201. [Google Scholar] [CrossRef] [PubMed] - Yamashita, S.; Nakazawa, Y.; Ueda, A.; Mori, H. Thermodynamics of the quantum spin liquid state of the single-component dimer Mott system κ-H
_{3}(Cat-EDT-TTF)_{2}. Phys. Rev. B**2017**, 95, 184425. [Google Scholar] [CrossRef] [Green Version] - Furukawa, T.; Miyagawa, K.; Taniguchi, H.; Kato, R.; Kanoda, K. Quantum criticality of Mott transition in organic materials. Nat. Phys.
**2015**, 11, 221–224. [Google Scholar] [CrossRef] [Green Version] - Pustogow, A.; Rösslhuber, R.; Tan, Y.; Uykur, E.; Böhme, A.; Wenzel, M.; Saito, Y.; Löhle, A.; Hübner, R.; Schlueter, J.A.; et al. Low-temperature dielectric anomaly arising from electronic phase separation at the Mott insulator-metal transition. NPJ Quantum Mater.
**2021**, 6, 9. [Google Scholar] [CrossRef] - Pustogow, A.; Bories, M.; Löhle, A.; Rösslhuber, R.; Zhukova, E.; Gorshunov, B.; Tomić, S.; Schlueter, J.A.; Hübner, R.; Hiramatsu, T.; et al. Quantum spin liquids unveil the genuine Mott state. Nat. Mater.
**2018**, 17, 773–777. [Google Scholar] [CrossRef] - Pustogow, A.; Saito, Y.; Löhle, A.; Sanz Alonso, M.; Kawamoto, A.; Dobrosavljević, V.; Dressel, M.; Fratini, S. Rise and fall of Landau’s quasiparticles while approaching the Mott transition. Nat. Commun.
**2021**, 12, 1571. [Google Scholar] [CrossRef] - Keimer, B.; Kivelson, S.A.; Norman, M.R.; Uchida, S.; Zaanen, J. From quantum matter to high-temperature superconductivity in copper oxides. Nature
**2015**, 518, 179–186. [Google Scholar] [CrossRef] - Hansmann, P.; Toschi, A.; Sangiovanni, G.; Saha-Dasgupta, T.; Lupi, S.; Marsi, M.; Held, K. Mott–Hubbard transition in V2O3 revisited. Phys. Status Solidi B
**2013**, 250, 1251–1264. [Google Scholar] [CrossRef] [Green Version] - Georges, A.; Kotliar, G.; Krauth, W.; Rozenberg, M.J. Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions. Rev. Mod. Phys.
**1996**, 68, 13–125. [Google Scholar] [CrossRef] [Green Version] - Vollhardt, D. Dynamical mean-field theory for correlated electrons. Ann. Phys. (Berl.)
**2012**, 524, 1–19. [Google Scholar] [CrossRef] - Limelette, P.; Georges, A.; Jérome, D.; Wzietek, P.; Metcalf, P.; Honig, J.M. Universality and Critical Behavior at the Mott Transition. Science
**2003**, 302, 89–92. [Google Scholar] [CrossRef] [Green Version] - Limelette, P.; Wzietek, P.; Florens, S.; Georges, A.; Costi, T.A.; Pasquier, C.; Jérome, D.; Mézière, C.; Batail, P. Mott Transition and Transport Crossovers in the Organic Compound κ-(BEDT-TTF)
_{2}Cu[N(CN)_{2}]Cl. Phys. Rev. Lett.**2003**, 91, 16401. [Google Scholar] [CrossRef] [PubMed] [Green Version] - McLeod, A.S.; van Heumen, E.; Ramirez, J.G.; Wang, S.; Saerbeck, T.; Guenon, S.; Goldflam, M.; Anderegg, L.; Kelly, P.; Mueller, A.; et al. Nanotextured phase coexistence in the correlated insulator V
_{2}O_{3}. Nat. Phys.**2016**, 13, 80–86. [Google Scholar] [CrossRef] [Green Version] - Terletska, H.; Vučičević, J.; Tanasković, D.; Dobrosavljević, V. Quantum Critical Transport near the Mott Transition. Phys. Rev. Lett.
**2011**, 107, 26401. [Google Scholar] [CrossRef] [PubMed] - Vučičević, J.; Terletska, H.; Tanasković, D.; Dobrosavljević, V. Finite-temperature crossover and the quantum Widom line near the Mott transition. Phys. Rev. B
**2013**, 88, 75143. [Google Scholar] [CrossRef] [Green Version] - Pinterić, M.; Čulo, M.; Milat, O.; Basletić, M.; Korin-Hamzić, B.; Tafra, E.; Hamzić, A.; Ivek, T.; Peterseim, T.; Miyagawa, K.; et al. Anisotropic charge dynamics in the quantum spin-liquid candidate κ-(BEDT-TTF)
_{2}Cu_{2}(CN)_{3}. Phys. Rev. B**2014**, 90, 195139. [Google Scholar] [CrossRef] [Green Version] - Dressel, M.; Lazić, P.; Pustogow, A.; Zhukova, E.; Gorshunov, B.; Schlueter, J.A.; Milat, O.; Gumhalter, B.; Tomić, S. Lattice vibrations of the charge-transfer salt κ-(BEDT-TTF)
_{2}Cu_{2}(CN)_{3}: Comprehensive explanation of the electrodynamic response in a spin-liquid compound. Phys. Rev. B**2016**, 93, 081201. [Google Scholar] [CrossRef] [Green Version] - Padmalekha, K.G.; Blankenhorn, M.; Ivek, T.; Bogani, L.; Schlueter, J.A.; Dressel, M. ESR studies on the spin-liquid candidate κ-(BEDT-TTF)2Cu2(CN)3: Anomalous response below T = 8 K. Phys. B Condens. Matter
**2015**, 460, 211–213. [Google Scholar] [CrossRef] [Green Version] - Abdel-Jawad, M.; Terasaki, I.; Sasaki, T.; Yoneyama, N.; Kobayashi, N.; Uesu, Y.; Hotta, C. Anomalous dielectric response in the dimer Mott insulator κ-(BEDT-TTF)
_{2}Cu_{2}(CN)_{3}. Phys. Rev. B**2010**, 82, 125119. [Google Scholar] [CrossRef] [Green Version] - Hotta, C. Quantum electric dipoles in spin-liquid dimer Mott insulator κ-ET
_{2}Cu_{2}(CN)_{3}. Phys. Rev. B**2010**, 82, 241104. [Google Scholar] [CrossRef] [Green Version] - Lunkenheimer, P.; Müller, J.; Krohns, S.; Schrettle, F.; Loidl, A.; Hartmann, B.; Rommel, R.; de Souza, M.; Hotta, C.; Schlueter, J.A.; et al. Multiferroicity in an organic charge-transfer salt that is suggestive of electric-dipole-driven magnetism. Nat. Mater.
**2012**, 11, 755. [Google Scholar] [CrossRef] [PubMed] - Gati, E.; Winter, S.M.; Schlueter, J.A.; Schubert, H.; Müller, J.; Lang, M. Insights from experiment and ab initio calculations into the glasslike transition in the molecular conductor κ-(BEDT–TTF)
_{2}Hg(SCN)_{2}Cl. Phys. Rev. B**2018**, 97, 75115. [Google Scholar] [CrossRef] [Green Version] - Fukuyama, H.; Kishine, J.I.; Ogata, M. Energy Landscape of Charge Excitations in the Boundary Region between Dimer–Mott and Charge Ordered States in Molecular Solids. J. Phys. Soc. Jpn.
**2017**, 86, 123706. [Google Scholar] [CrossRef] - Furukawa, T.; Kobashi, K.; Kurosaki, Y.; Miyagawa, K.; Kanoda, K. Quasi-continuous transition from a Fermi liquid to a spin liquid in κ-(ET)
_{2}Cu_{2}(CN)_{3}. Nat. Commun.**2018**, 9, 307. [Google Scholar] [CrossRef] [Green Version] - Miyagawa, K.; Kawamoto, A.; Nakazawa, Y.; Kanoda, K. Antiferromagnetic Ordering and Spin Structure in the Organic Conductor, κ–(BEDT-TTF)
_{2}Cu[N(CN)_{2}]Cl. Phys. Rev. Lett.**1995**, 75, 1174–1177. [Google Scholar] [CrossRef] - Lefebvre, S.; Wzietek, P.; Brown, S.; Bourbonnais, C.; Jérome, D.; Mézière, C.; Fourmigué, M.; Batail, P. Mott transition, antiferromagnetism, and unconventional superconductivity in layered organic superconductors. Phys. Rev. Lett.
**2000**, 85, 5420–5423. [Google Scholar] [CrossRef] [Green Version] - Shimizu, Y.; Miyagawa, K.; Kanoda, K.; Maesato, M.; Saito, G. Emergence of inhomogeneous moments from spin liquid in the triangular-lattice Mott insulator κ-(ET)
_{2}Cu_{2}(CN)_{3}. Phys. Rev. B**2006**, 73, 140407. [Google Scholar] [CrossRef] [Green Version] - Tamura, M.; Nakao, A.; Kato, R. Frustration-Induced Valence-Bond Ordering in a New Quantum Triangular Antiferromagnet Based on [Pd(dmit)
_{2}]. J. Phys. Soc. Jpn.**2006**, 75, 93701. [Google Scholar] [CrossRef] [Green Version] - Shimizu, Y.; Akimoto, H.; Tsujii, H.; Tajima, A.; Kato, R. Mott Transition in a Valence-Bond Solid Insulator with a Triangular Lattice. Phys. Rev. Lett.
**2007**, 99, 256403. [Google Scholar] [CrossRef] [Green Version] - Itou, T.; Oyamada, A.; Maegawa, S.; Kubo, K.; Yamamoto, H.M.; Kato, R. Superconductivity on the border of a spin-gapped Mott insulator: NMR studies of the quasi-two-dimensional organic system EtMe
_{3}P[Pd_{(dmit)2}]_{2}. Phys. Rev. B**2009**, 79, 174517. [Google Scholar] [CrossRef] - Manna, R.S.; de Souza, M.; Kato, R.; Lang, M. Lattice effects in the quasi-two-dimensional valence-bond-solid Mott insulator EtMe
_{3}P[Pd(dmit)_{2}]_{2}. Phys. Rev. B**2014**, 89, 45113. [Google Scholar] [CrossRef] [Green Version] - Yoshida, Y.; Ito, H.; Maesato, M.; Shimizu, Y.; Hayama, H.; Hiramatsu, T.; Nakamura, Y.; Kishida, H.; Koretsune, T.; Hotta, C.; et al. Spin-disordered quantum phases in a quasi-one-dimensional triangular lattice. Nat. Phys.
**2015**, 11, 679–683. [Google Scholar] [CrossRef] - Shimizu, Y.; Maesato, M.; Yoshida, M.; Takigawa, M.; Itoh, M.; Otsuka, A.; Yamochi, H.; Yoshida, Y.; Kawaguchi, G.; Graf, D.; et al. Magnetic field driven transition between valence bond solid and antiferromagnetic order in a distorted triangular lattice. Phys. Rev. Res.
**2021**, 3, 23145. [Google Scholar] [CrossRef] - Richardson, R.C. The Pomeranchuk effect. Rev. Mod. Phys.
**1997**, 69, 683–690. [Google Scholar] [CrossRef] - Rozen, A.; Park, J.M.; Zondiner, U.; Cao, Y.; Rodan-Legrain, D.; Taniguchi, T.; Watanabe, K.; Oreg, Y.; Stern, A.; Berg, E.; et al. Entropic evidence for a Pomeranchuk effect in magic-angle graphene. Nature
**2021**, 592, 214–219. [Google Scholar] [CrossRef] - Nakazawa, Y.; Yamashita, S. Thermodynamics of a Liquid-like Spin State in Molecule-based Magnets with Geometric Frustrations. Chem. Lett.
**2013**, 42, 1446–1454. [Google Scholar] [CrossRef] - Mizukoshi, K.; Nakamura, Y.; Yoshida, Y.; Saito, G.; Kishida, H. Optical Evaluation of Electronic Anisotropy in a Triangular Lattice System κ-(BEDT-TTF)
_{2}B(CN)_{4}. J. Phys. Soc. Jpn.**2018**, 87, 104708. [Google Scholar] [CrossRef] - Dumm, M.; Loidl, A.; Alavi, B.; Starkey, K.P.; Montgomery, L.K.; Dressel, M. Comprehensive ESR study of the antiferromagnetic ground states in the one-dimensional spin systems (TMTSF)
_{2}PF_{6},(TMTSF)_{2}AsF_{6}, and (TMTTF)_{2}Br. Phys. Rev. B**2000**, 62, 6512–6520. [Google Scholar] [CrossRef] - Hase, M.; Terasaki, I.; Uchinokura, K. Observation of the spin-Peierls transition in linear Cu
^{2+}(spin-1/2) chains in an inorganic compound CuGeO_{3}. Phys. Rev. Lett.**1993**, 70, 3651–3654. [Google Scholar] [CrossRef] - Isobe, M.; Ueda, Y. Magnetic Susceptibility of Quasi-One-Dimensional Compound α’- NaV2O5 –Possible Spin-Peierls Compound with High Critical Temperature of 34 K–. J. Phys. Soc. Jpn.
**1996**, 65, 1178–1181. [Google Scholar] [CrossRef] - Manna, R.S.; de Souza, M.; Brühl, A.; Schlueter, J.A.; Lang, M. Lattice Effects and Entropy Release at the Low-Temperature Phase Transition in the Spin-Liquid Candidate κ-(BEDT-TTF)
_{2}Cu_{2}(CN)_{3}. Phys. Rev. Lett.**2010**, 104, 16403. [Google Scholar] [CrossRef] [PubMed] - Pustogow, A.; Le, T.; Wang, H.H.; Luo, Y.; Gati, E.; Schubert, H.; Lang, M.; Brown, S.E. Impurity moments conceal low-energy relaxation of quantum spin liquids. Phys. Rev. B
**2020**, 101, 140401. [Google Scholar] [CrossRef] [Green Version] - Saito, Y.; Minamidate, T.; Kawamoto, A.; Matsunaga, N.; Nomura, K. Site-specific
^{13}C NMR study on the locally distorted triangular lattice of the organic conductor κ-(BEDT-TTF)_{2}Cu_{2}(CN)_{3}. Phys. Rev. B**2018**, 98, 205141. [Google Scholar] [CrossRef] [Green Version] - Olariu, A.; Mendels, P.; Bert, F.; Duc, F.; Trombe, J.C.; de Vries, M.A.; Harrison, A.
^{17}O NMR Study of the Intrinsic Magnetic Susceptibility and Spin Dynamics of the Quantum Kagome Antiferromagnet ZnCu_{3}(OH)_{6}Cl_{2}. Phys. Rev. Lett.**2008**, 100, 87202. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Freedman, D.E.; Han, T.H.; Prodi, A.; Müller, P.; Huang, Q.Z.; Chen, Y.S.; Webb, S.M.; Lee, Y.S.; McQueen, T.M.; Nocera, D.G. Site Specific X-ray Anomalous Dispersion of the Geometrically Frustrated Kagomé Magnet, Herbertsmithite, ZnCu
_{3}(OH)_{6}Cl_{2}. J. Am. Chem. Soc.**2010**, 132, 16185–16190. [Google Scholar] [CrossRef] - Zhu, Z.; Maksimov, P.A.; White, S.R.; Chernyshev, A.L. Disorder-Induced Mimicry of a Spin Liquid in YbMgGaO
_{4}. Phys. Rev. Lett.**2017**, 119, 157201. [Google Scholar] [CrossRef] [Green Version] - Kimchi, I.; Nahum, A.; Senthil, T. Valence Bonds in Random Quantum Magnets: Theory and Application to YbMgGaO
_{4}. Phys. Rev. X**2018**, 8, 31028. [Google Scholar] [CrossRef] [Green Version] - Itou, T.; Watanabe, E.; Maegawa, S.; Tajima, A.; Tajima, N.; Kubo, K.; Kato, R.; Kanoda, K. Slow dynamics of electrons at a metal-Mott insulator boundary in an organic system with disorder. Sci. Adv.
**2017**, 3, e1601594. [Google Scholar] [CrossRef] [Green Version] - Riedl, K.; Valentí, R.; Winter, S.M. Critical spin liquid versus valence-bond glass in a triangular-lattice organic antiferromagnet. Nat. Commun.
**2019**, 10, 2561. [Google Scholar] [CrossRef] [Green Version] - Isono, T.; Terashima, T.; Miyagawa, K.; Kanoda, K.; Uji, S. Quantum criticality in an organic spin-liquid insulator κ-(BEDT-TTF)
_{2}Cu_{2}(CN)_{3}. Nat. Commun.**2016**, 7, 13494. [Google Scholar] [CrossRef] [Green Version] - Abragam, A. Principles of Nuclear Magnetism; Oxford University Press: Hong Kong, China, 1983. [Google Scholar]
- Pratt, F.L.; Baker, P.J.; Blundell, S.J.; Lancaster, T.; Ohira-Kawamura, S.; Baines, C.; Shimizu, Y.; Kanoda, K.; Watanabe, I.; Saito, G. Magnetic and non-magnetic phases of a quantum spin liquid. Nature
**2011**, 471, 612. [Google Scholar] [CrossRef] [PubMed] - Yamashita, M.; Nakata, N.; Kasahara, Y.; Sasaki, T.; Yoneyama, N.; Kobayashi, N.; Fujimoto, S.; Shibauchi, T.; Matsuda, Y. Thermal-transport measurements in a quantum spin-liquid state of the frustrated triangular magnet nphys1134-m6gif1601313-(BEDT-TTF)
_{2}Cu_{2}(CN)_{3}. Nat. Phys.**2009**, 5, 44–47. [Google Scholar] [CrossRef] [Green Version] - Yamashita, M.; Nakata, N.; Senshu, Y.; Nagata, M.; Yamamoto, H.M.; Kato, R.; Shibauchi, T.; Matsuda, Y. Highly Mobile Gapless Excitations in a Two-Dimensional Candidate Quantum Spin Liquid. Science
**2010**, 328, 1246–1248. [Google Scholar] [CrossRef] [PubMed] - Bourgeois-Hope, P.; Laliberté, F.; Lefrançois, E.; Grissonnanche, G.; de Cotret, S.R.; Gordon, R.; Kitou, S.; Sawa, H.; Cui, H.; Kato, R.; et al. Thermal Conductivity of the Quantum Spin Liquid Candidate EtMe
_{3}Sb[Pd(dmit)_{2}]_{2}: No Evidence of Mobile Gapless Excitations. Phys. Rev. X**2019**, 9, 41051. [Google Scholar] [CrossRef] [Green Version] - Ni, J.M.; Pan, B.L.; Song, B.Q.; Huang, Y.Y.; Zeng, J.Y.; Yu, Y.J.; Cheng, E.J.; Wang, L.S.; Dai, D.Z.; Kato, R.; et al. Absence of Magnetic Thermal Conductivity in the Quantum Spin Liquid Candidate EtMe
_{3}Sb[Pd(dmit)_{2}]_{2}. Phys. Rev. Lett.**2019**, 123, 247204. [Google Scholar] [CrossRef] [Green Version] - Yamashita, M.; Sato, Y.; Tominaga, T.; Kasahara, Y.; Kasahara, S.; Cui, H.; Kato, R.; Shibauchi, T.; Matsuda, Y. Presence and absence of itinerant gapless excitations in the quantum spin liquid candidate EtMe
_{3}Sb[Pd_{( dmit )2}]_{2}. Phys. Rev. B**2020**, 101, 140407. [Google Scholar] [CrossRef] [Green Version] - Kato, R.; Uebe, M.; Fujiyama, S.; Hengbo, C. A Discrepancy in thermal conductivity measurement data of quantum spin liquid β
^{′}-EtMe_{3}Sb[Pd(dmit)_{2}]_{2}(dmit = 1,3-dithiol-2-thione-4,5-dithiolate). Crystals**2022**, 12, 102. [Google Scholar] [CrossRef] - Ando, Y.; Takeya, J.; Sisson, D.L.; Doettinger, S.G.; Tanaka, I.; Feigelson, R.S.; Kapitulnik, A. Thermal conductivity of the spin-Peierls compound CuGeO
_{3}. Phys. Rev. B**1998**, 58, R2913–R2916. [Google Scholar] [CrossRef] [Green Version] - Vasil’ev, A.N.; Pryadun, V.V.; Khomskii, D.I.; Dhalenne, G.; Revcolevschi, A.; Isobe, M.; Ueda, Y. Anomalous Thermal Conductivity of NaV
_{2}O_{5}as Compared to Conventional Spin-Peierls System CuGeO_{3}. Phys. Rev. Lett.**1998**, 81, 1949–1952. [Google Scholar] [CrossRef] [Green Version] - Gregor, K.; Motrunich, O.I. Nonmagnetic impurities in a S = 12 frustrated triangular antiferromagnet: Broadening of
^{13}C NMR lines in κ-(ET)_{2}Cu_{2}(CN)_{3}. Phys. Rev. B**2009**, 79, 24421. [Google Scholar] [CrossRef] [Green Version] - Itoh, K.; Itoh, H.; Naka, M.; Saito, S.; Hosako, I.; Yoneyama, N.; Ishihara, S.; Sasaki, T.; Iwai, S. Collective Excitation of an Electric Dipole on a Molecular Dimer in an Organic Dimer-Mott Insulator. Phys. Rev. Lett.
**2013**, 110, 106401. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Kobayashi, T.; Ding, Q.P.; Taniguchi, H.; Satoh, K.; Kawamoto, A.; Furukawa, Y. Charge disproportionation in the spin-liquid candidate κ-(ET)
_{2}Cu_{2}(CN)_{3}at 6 K revealed by^{63}Cu NQR measurements. Phys. Rev. Res.**2020**, 2, 42023. [Google Scholar] [CrossRef] - de Souza, M.; Brühl, A.; Müller, J.; Foury-Leylekian, P.; Moradpour, A.; Pouget, J.P.; Lang, M. Thermodynamic studies at the charge-ordering and spin-Peierls transitions in (TMTTF)2X. Phys. B Condens. Matter
**2009**, 404, 494–498. [Google Scholar] [CrossRef] - Winkelmann, H.; Gamper, E.; Büchner, B.; Braden, M.; Revcolevschi, A.; Dhalenne, G. Giant anomalies of the thermal expansion at the spin-Peierls transition in CuGeO
_{3}. Phys. Rev. B**1995**, 51, 12884–12887. [Google Scholar] [CrossRef] - Ramirez, A.P. A flood or a trickle? Nat. Phys.
**2008**, 4, 442–443. [Google Scholar] [CrossRef] - Sedlmeier, K.; Elsässer, S.; Neubauer, D.; Beyer, R.; Wu, D.; Ivek, T.; Tomić, S.; Schlueter, J.A.; Dressel, M. Absence of charge order in the dimerized κ-phase BEDT-TTF salts. Phys. Rev. B
**2012**, 86, 245103. [Google Scholar] [CrossRef] [Green Version] - Yakushi, K.; Yamamoto, K.; Yamamoto, T.; Saito, Y.; Kawamoto, A. Raman Spectroscopy Study of Charge Fluctuation in the Spin-Liquid Candidate κ-(BEDT-TTF)
_{2}Cu_{2}(CN)_{3}. J. Phys. Soc. Jpn.**2015**, 84, 84711. [Google Scholar] [CrossRef] - Drichko, N.; Beyer, R.; Rose, E.; Dressel, M.; Schlueter, J.A.; Turunova, S.A.; Zhilyaeva, E.I.; Lyubovskaya, R.N. Metallic state and charge-order metal-insulator transition in the quasi-two-dimensional conductor κ-(BEDT-TTF)
_{2}Hg(SCN)_{2}Cl. Phys. Rev. B**2014**, 89, 75133. [Google Scholar] [CrossRef] - Jeschke, H.O.; de Souza, M.; Valentí, R.; Manna, R.S.; Lang, M.; Schlueter, J.A. Temperature dependence of structural and electronic properties of the spin-liquid candidate κ-(BEDT-TTF)
_{2}Cu_{2}(CN)_{3}. Phys. Rev. B**2012**, 85, 35125. [Google Scholar] [CrossRef] [Green Version] - Manna, R.S.; Hartmann, S.; Gati, E.; Schlueter, J.A.; De Souza, M.; Lang, M. Low-Temperature Lattice Effects in the Spin-Liquid Candidate κ-(BEDT-TTF)
_{2}Cu_{2}(CN)_{3}. Crystals**2018**, 8, 87. [Google Scholar] [CrossRef] [Green Version] - Poirier, M.; de Lafontaine, M.; Miyagawa, K.; Kanoda, K.; Shimizu, Y. Ultrasonic investigation of the transition at 6 K in the spin-liquid candidate κ-(BEDT-TTF)
_{2}Cu_{2}(CN)_{3}. Phys. Rev. B**2014**, 89, 45138. [Google Scholar] [CrossRef] - Poirier, M.; Parent, S.; Côté, A.; Miyagawa, K.; Kanoda, K.; Shimizu, Y. Magnetodielectric effects and spin-charge coupling in the spin-liquid candidate κ-(BEDT-TTF)
_{2}Cu_{2}(CN)_{3}. Phys. Rev. B**2012**, 85, 134444. [Google Scholar] [CrossRef] - Cross, M.C.; Fisher, D.S. A new theory of the spin-Peierls transition with special relevance to the experiments on TTFCuBDT. Phys. Rev. B
**1979**, 19, 402–419. [Google Scholar] [CrossRef] [Green Version] - Cross, M.C. Effect of magnetic fields on a spin-Peierls transition. Phys. Rev. B
**1979**, 20, 4606–4611. [Google Scholar] [CrossRef] [Green Version] - Zeman, J.; Martinez, G.; van Loosdrecht, P.H.M.; Dhalenne, G.; Revcolevschi, A. Scaling of the H- T Phase Diagram of CuGeO
_{3}. Phys. Rev. Lett.**1999**, 83, 2648–2651. [Google Scholar] [CrossRef] [Green Version] - Langlois, A.; Poirier, M.; Bourbonnais, C.; Foury-Leylekian, P.; Moradpour, A.; Pouget, J.P. Microwave dielectric study of spin-Peierls and charge-ordering transitions in (TMTTF)
_{2}PF_{6}salts. Phys. Rev. B**2010**, 81, 125101. [Google Scholar] [CrossRef] [Green Version] - Hassan, N.; Cunningham, S.; Mourigal, M.; Zhilyaeva, E.I.; Torunova, S.A.; Lyubovskaya, R.N.; Schlueter, J.A.; Drichko, N. Evidence for a quantum dipole liquid state in an organic quasi–two-dimensional material. Science
**2018**, 360, 1101–1104. [Google Scholar] [CrossRef] [Green Version] - Hartmann, S.; Gati, E.; Yoshida, Y.; Saito, G.; Lang, M. Thermal Expansion Studies on the Spin-Liquid-Candidate System κ-(BEDT-TTF)
_{2}Ag_{2}(CN)_{3}. Phys. Status Solidi B**2019**, 256, 1800640. [Google Scholar] [CrossRef] - Poirier, M.; Proulx, M.O.; Kato, R. Ultrasonic investigation of the organic spin-liquid compound EtMe
_{3}Sb[Pd(dmit)_{2}]_{2}. Phys. Rev. B**2014**, 90, 45147. [Google Scholar] [CrossRef] - Huang, Y.Y.; Xu, Y.; Wang, L.; Zhao, C.C.; Tu, C.P.; Ni, J.M.; Wang, L.S.; Pan, B.L.; Fu, Y.; Hao, Z.; et al. Heat Transport in Herbertsmithite: Can a Quantum Spin Liquid Survive Disorder? Phys. Rev. Lett.
**2021**, 127, 267202. [Google Scholar] [CrossRef] - Sushkov, A.B.; Jenkins, G.S.; Han, T.H.; Lee, Y.S.; Drew, D.H. Infrared phonons as a probe of spin-liquid states in herbertsmithite ZnCu
_{3}(OH)_{6}Cl_{2}. J. Phys. Condens. Matter**2017**, 29, 95802. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Li, Y.; Pustogow, A.; Bories, M.; Puphal, P.; Krellner, C.; Dressel, M.; Valentí, R. Lattice dynamics in the spin-12 frustrated kagome compound herbertsmithite. Phys. Rev. B
**2020**, 101, 161115. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Crystal structure of $\kappa $-(BEDT-TTF)${}_{2}$Cu${}_{2}$(CN)${}_{3}$ [29]. (

**a**) Layers of organic BEDT-TTF${}^{+0.5}$ donor molecules are separated by polymeric Cu${}_{2}$(CN)${}_{3}^{-}$ anion sheets. (

**b**) Within the planes, (BEDT-TTF)${}_{2}$ dimers carrying one electronic charge and spin $S=1/2$ form a nearly ideal triangular lattice ($t/{t}^{\prime}=0.83$ [35]), yielding a $genuine$ Mott-insulating state [6,45] and pronounced effects of geometrical frustration [7,34]. (

**c**) Sources of disorder in the anion network are the randomly oriented C/N atoms in the bridging CN-groups [56,57] and Cu${}^{2+}$ impurities forming charged $S=1/2$ defects [27,30,58].

**Figure 2.**(

**a**) Unified phase diagram of genuine Mott insulators [44,45,46] based upon pressure-dependent measurements on EtMe${}_{3}$Sb[Pd(dmit)${}_{2}$]${}_{2}$, $\kappa $-(BEDT-TTF)${}_{2}$Ag${}_{2}$(CN)${}_{3}$ and $\kappa $-(BEDT-TTF)${}_{2}$Cu${}_{2}$(CN)${}_{3}$ [39,43,44,46]. Dotted rectangles indicate experimentally covered T–p ranges starting from ambient conditions ($p=0$, $T=300$ K marked by solid circles in respective color). Above the critical endpoint ${T}_{crit}$, the charge gap closes at the quantum-critical crossover (quantum Widom line QWL [43,54,55]) from a Mott insulator to incoherent semiconducting and bad metal states [45]. For $T<{T}_{crit}$ the MIT is first-order type. At low T the spin degrees of freedom are decisive for the magnetic ground state and the transition to superconducting and Fermi-liquid (FL) phases [46,64]. (

**b**) Near $T=0$ AFM interactions usually result in long-range order, seen in oxides (cuprates, V${}_{2}$O${}_{3}$ [47,48]) and organic systems such as $\kappa $-(BEDT-TTF)${}_{2}$Cu[N(CN)${}_{2}$]Cl [65,66] or based on [Pd(dmit)${}_{2}$]${}_{2}$ [7]. (

**c**) Due to the lack of AFM order down to a few millikelvin [34,37,39,67], the frustrated triangular-lattice compounds (${t}^{\prime}/t\approx 1$) from (

**a**) have been intensely discussed as QSL candidates [2,3,4,5,7]. (

**d**) EtMe${}_{3}$P[Pd(dmit)${}_{2}$]${}_{2}$ [68,69,70,71] and $\kappa $-(BEDT-TTF)${}_{2}$B(CN)${}_{4}$ [72,73] exhibit a VBS state below ${T}^{\star}$. Due to the Clausius–Clapeyron relation, the large spin entropy of the paramagnetic Mott state yields a positive slope of the insulator–metal boundary in the T–p phase diagram [6,7,45], whereas $d{T}_{\mathrm{MI}}/dp<0$ for AFM [51,66] or VBS [69,70].

**Figure 3.**Magnetic susceptibility $\chi $ of $\kappa $-(BEDT-TTF)${}_{2}$Cu${}_{2}$(CN)${}_{3}$. (

**a**) A weak temperature dependence at elevated temperatures is followed by a rapid drop of $\chi $ at ${T}^{\star}$ due to the opening of a spin gap $\Delta $ [27]. The reduction comes to a halt once ${\chi}_{s}$ becomes smaller than the contribution of impurity spins, which follows a Curie-like behavior ${\chi}_{imp}\propto \mathrm{C}/T$ until ${\mu}_{B}B\ge {k}_{B}T$ (C is the Curie constant). (

**b**–

**e**) The drop at ${T}^{\star}$ is seen prominently in ESR experiments on single crystals [27,30] and, less pronounced, in SQUID measurements on polycrystals [34,83]. The upturn for $T<{T}^{\star}$, seen most clearly in (

**b**), exhibits strong sample dependence due to a varying defect density. Miksch et al. [27] accomplished to identify and distinguish the intrinsic contribution from the impurity signal in the ESR spectra (

**d**), establishing that the ground state of $\kappa $-(BEDT-TTF)${}_{2}$Cu${}_{2}$(CN)${}_{3}$ has a spin gap.

**Figure 4.**Influence of paramagnetic impurities on NMR spin-lattice relaxation rate. (

**a**) ${T}_{1}^{-1}$ exhibits a rapid drop as a spin gap opens. In this nonmagnetic environment, paramagnetic impurities contribute a BPP-type relaxation (Equation (1)) forming a local maximum of ${T}_{1}^{-1}$ at lower temperatures. (

**b**) The ${}^{1}$H (blue; from Ref. [34]) and ${}^{13}$C (red; from Ref. [34]) relaxation rates of $\kappa $-(BEDT-TTF)${}_{2}$Cu${}_{2}$(CN)${}_{3}$ show similar behavior for $T\ge {T}^{\star}$. As the spin gap opens (dotted line indicates $\Delta =12$ K) the drop of ${T}_{1}^{-1}$ comes to a halt and a broad maximum appears. ${}^{1}$${T}_{1}^{-1}$ is smaller by a factor 80; note the different vertical scales. (

**c**) The grey highlighted area indicates the impurity-dominated region in a double-logarithmic plot. (

**d**–

**f**) Inhomogeneous relaxation from the impurities brings about deviations from single-exponential behavior, with a stretching exponent $\beta <1$ for $T<{T}^{\star}$.

**Figure 5.**(

**a**) The thermal conductivity $\kappa $ upon the transition to a VBS state is sketched based on the observations made in spin-Peierls compounds such as CuGeO${}_{3}$ and NaV${}_{2}$O${}_{5}$ [99,100]. Electronic heat transport ${\kappa}_{e}$ vanishes upon the opening of a spin gap at $T<{T}^{\star}$, yielding a drop of $\kappa ={\kappa}_{e}+{\kappa}_{ph}$ with cooling. At lower temperatures, the mean free path of phonons increases as the scattering off spin excitations is quenched in the singlet state. The associated enhancement of phononic thermal conductivity ${\kappa}_{ph}$ results in the formation of a maximum of $\kappa $ for $T<{T}^{\star}$. (

**b**) Strikingly similar behavior is seen for the thermal conductivity of $\kappa $-(BEDT-TTF)${}_{2}$Cu${}_{2}$(CN)${}_{3}$ reported by Yamashita et al. [93].

**Figure 7.**(

**a**) Field-temperature phase diagram of $\kappa $-(BEDT-TTF)${}_{2}$Cu${}_{2}$(CN)${}_{3}$ extracted from Ref. [113]. At small fields, the transition follows a BCS-type behavior ${B}^{\star}\left(T\right)={B}_{0}{(1-{T}^{\star}\left(B\right)/{T}^{\star}\left(0\right))}^{1/2}$ with ${B}_{0}\approx 60$ T. The sample ($\#1$–$\#3$) and frequency (13.0, 16.5 GHz) dependences of ${T}^{\star}(B=0)$ were assigned to different strain and microwave power conditions [113]. Light blue open diamonds indicate the transition in $\kappa $-(BEDT-TTF)${}_{2}$B(CN)${}_{4}$ (from NMR measurements in Ref. [73]), for which the magnetic field values have been multiplied by a factor 4. It is suggested that the VBS state is suppressed by a critical field of 30–60 T. (

**b**) Upon normalizing to ${T}^{\star}(B=0)$ and ${B}_{0}$, the low-field data of the two $\kappa $-(BEDT-TTF)${}_{2}X$ compounds [73,113] collapse with those of the spin-Peierls transitions in TMTTF${}_{2}$PF${}_{6}$ and CuGeO${}_{3}$ [116,117]. Deviations from mean-field-like BCS behavior set in when ${T}^{\star}\left(B\right)$ has reduced below 80% of ${T}^{\star}(B=0)$, which is indicated in (a) by the dashed extrapolated lines. (

**c**) Despite its comparatively low transition temperature, $\kappa $-(BEDT-TTF)${}_{2}$Cu${}_{2}$(CN)${}_{3}$ yields the largest ${B}_{0}$.

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Pustogow, A.
Thirty-Year Anniversary of *κ*-(BEDT-TTF)_{2}Cu_{2}(CN)_{3}: Reconciling the Spin Gap in a Spin-Liquid Candidate. *Solids* **2022**, *3*, 93-110.
https://doi.org/10.3390/solids3010007

**AMA Style**

Pustogow A.
Thirty-Year Anniversary of *κ*-(BEDT-TTF)_{2}Cu_{2}(CN)_{3}: Reconciling the Spin Gap in a Spin-Liquid Candidate. *Solids*. 2022; 3(1):93-110.
https://doi.org/10.3390/solids3010007

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Pustogow, Andrej.
2022. "Thirty-Year Anniversary of *κ*-(BEDT-TTF)_{2}Cu_{2}(CN)_{3}: Reconciling the Spin Gap in a Spin-Liquid Candidate" *Solids* 3, no. 1: 93-110.
https://doi.org/10.3390/solids3010007