# Ingredients for Generalized Models of κ-Phase Organic Charge-Transfer Salts: A Review

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## Abstract

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## 1. Introduction

#### 1.1. Crystal and Electronic Structure

#### 1.2. Phase Diagram

**Figure 3.**Experimental temperature–pressure phase diagrams of $\kappa $-phase charge-transfer salts: (

**a**) $\kappa $-(BEDT-TTF)${}_{2}$Cu[N(CN)${}_{2}$]Cl (‘$\kappa $-Cl’) [1,3,4,9,16,21,22] and (

**b**) $\kappa $-(BEDT-TTF)${}_{2}$Cu${}_{2}$(CN)${}_{3}$ (‘$\kappa $-CuCN’) [7,10,23,34]. At ambient pressure and low-enough temperatures, both compounds exhibit a Mott insulating ground state (i.e., a moderate to large correlation strength, $U/W$). Upon applying pressure, $U/W$ is reduced and eventually a first-order Mott metal-insulator transition is induced (red line) which ends in a second-order critical endpoint (red circle). On both sides of the Mott transition, intriguing electronic orders emerge at lowest temperatures. On the Mott insulating side, $\kappa $-Cl orders antiferromagnetically [35], which is well-understood given the anisotropy of its triangular lattice (${t}^{\prime}<t$). The antiferromagnetic (AFM) order is believed to be accompanied by the emergence of long-range ferroelectric order, making this compound multiferroic [16]. In contrast, $\kappa $-CuCN is characterized by an almost isotropic triangular lattice with ${t}^{\prime}\phantom{\rule{0.166667em}{0ex}}\sim \phantom{\rule{0.166667em}{0ex}}t$. For a long time, it has, therefore, been considered as a candidate for quantum spin-liquid (QSL) behavior [10]. However, there exist recent results which argue in favor of the formation of a valence bond solid (VBS) [12,13,34,36,37,38]. On the metallic side of the Mott transition, both compounds exhibit superconductivity (SC). A region of percolative superconductivity SC${}_{\mathrm{perc}}$ can also be found in the Mott insulating state close to the metal-insulator boundary.

#### 1.3. Outline of This Review

## 2. Magnetic Exchange beyond Heisenberg

#### 2.1. Magnetic Bilinear Anisotropic Interactions

#### 2.2. Four-Spin Ring Exchange

## 3. New Physics in the Extended Molecule-Based Model

#### 3.1. Ferroelectricity in the Mott Insulating Ground State

#### 3.2. Superconductivity in Extended Molecule-Based Models

## 4. Coupling of Correlated Electrons to the Crystal Lattice

#### 4.1. Critical Elasticity around the Mott Critical Endpoint

#### 4.2. Phonon Anomalies Probed by Inelastic Neutron Scattering

## 5. Role of Disorder

#### 5.1. Experimental Study of Phenomena Close to the Mott Transition under Controlled Disorder

#### 5.2. Disorder in the Magnetic Phase: Scenario of a Valence Bond Solid Host with Orphan Spins

## 6. Conclusions and Outlook

- As discussed in the review, ab-initio extracted magnetic models for triangular $\kappa $-phase charge-transfer salts indicate the importance of four-spin interaction terms with spin-orbit coupling effects, as well as, in the presence of a magnetic field, possible products of an odd number of spin operators, such as the scalar spin chirality. What (quantum) phases are to be expected arising from these interactions that are relevant for these materials?
- How do intra-dimer charge and spin degrees of freedom conspire in the ground-state properties of frustrated Mott insulating $\kappa $-phase charge-transfer salts? In particular, what impact do the intra-dimer charge degrees of freedom have in promoting (impeding) the formation of long-range spin order? Can these effects be theoretically described with models containing both, spin and charge degrees of freedom?
- What is the role of magnetoelastic coupling in the formation of novel states of matter, such as putative spin-liquid states in $\kappa $-(BEDT-TTF)${}_{2}X$? Can we develop accurate models to capture these effects?
- Can we quantitatively describe the impact of disorder on the properties of these charge-transfer salts close to the Mott metal–insulator transition experimentally and theoretically? To this end, how can we accurately quantify the level of disorder in real materials and determine the nature of the disorder and their spatial distribution? Which models and methods allow to theoretically describe the interaction between disorder and bulk properties properly?
- What novel phases may be realized under non-equilibrium conditions [176]?

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) Layered structure of organic electron donor and inorganic electron acceptor layer, shown for the example $\kappa $-(BEDT-TTF)${}_{2}$Cu${}_{2}$(CN)${}_{3}$; (

**b**) structure of BEDT-TTF and BETS molecules; and (

**c**) $\kappa $ packing motif of the organic layer of $\kappa $ phase charge-transfer salts from the top view.

**Figure 2.**Center: Illustration of the mapping of the organic layer of $\kappa $ phase charge-transfer salts to an anisotropic triangular lattice. The various ingredients for generalized models beyond the strongly dimerized one-band Hubbard picture discussed in this work are illustrated in the boxes.

**Figure 4.**Magnetic exchange in $\kappa $-phase charge-transfer salts. (

**a**) Heisenberg exchange terms J, ${J}^{\prime}$, ${J}^{\u2033}$, ${J}^{\u2034}$ and four-spin ring-exchange K and ${K}^{\prime}$ on the two distinct four-site plaquettes on the anisotropic triangular lattice. (

**b**) The SOC-induced DM vector $\mathbf{D}$ is oriented approximately along the long side of the organic molecule, here illustrated for the example $\kappa $-(BEDT-TTF)${}_{2}$Cu${}_{2}$(CN)${}_{3}$. Figures reprinted from Refs. [51,64].

**Figure 5.**(

**a**) Classical phase diagram for the four-spin ring-exchange on the anisotropic triangular lattice. Indicated are the locations for $\kappa $-(BEDT-TTF)${}_{2}$Cu[N(CN)${}_{2}$]Cl ($\kappa $-Cl), $\kappa $-(BEDT-TTF)${}_{2}$Cu${}_{2}$(CN)${}_{3}$ ($\kappa $-CuCN), and $\kappa $-(BETS)${}_{2}$Mn[N(CN)${}_{2}$]${}_{3}$ ($\kappa $-Mn), based on ab-initio calculations in Refs. [12,51,64,67] (see Table 1). The depicted phases include the two-sublattice (2SL) Néel order, non-coplanar chiral (NCC) order, and spin-vortex crystal (SVC) order. For this phase diagram, the constraints ${J}^{\prime}/J={J}^{\u2034}/{J}^{\u2033}={K}^{\prime}/K$, $K/{J}^{\u2033}=2$, and $\left|\mathbf{D}\right|=|\mathbf{\Gamma}|=0$ were enforced. (

**b**) ${}^{13}$C NMR spectra measured in Ref. [74] (“Exp.”) and simulated for hypothetical SVC and 2SL phases [51]. Figures adapted from Ref. [51].

**Figure 6.**Role of intra-dimer charge degrees of freedom revealed by dielectric measurements: (

**a**–

**c**) Frequency-dependent dielectric constant ${\u03f5}^{\prime}$ on three $\kappa $-phase organic charge-transfer salts: (

**b**) $\kappa $-(BEDT-TTF)${}_{2}$Cu[N(CN)${}_{2}$]Cl ($\kappa $-Cl) [16,84,88], (

**c**) $\kappa $-(BEDT-TTF)${}_{2}$Cu${}_{2}$(CN)${}_{3}$ ($\kappa $-CuCN) [85] and (

**d**) $\kappa $-(BEDT-TTF)${}_{2}$Hg(SCN)${}_{2}$Cl [86]. Symbols represent the measured data, solid lines are guide to the eyes. The red dashed lines represent Curie–Weiss fits of the data in (

**b**,

**d**), whereas they show the frequency dependence of the peak position in (

**c**). These three materials differ by the degree of dimerization (${t}_{1}/{t}^{\prime}$), as well as the frustration strength (${t}^{\prime}/t$, defined in the effective dimer model). Values for ${t}_{1}/{t}^{\prime}$ and ${t}^{\prime}/t$ for each of the materials are included below the figures and taken from Refs. [18,42,86]; (

**d**) Schematic view of charge distribution on the dimerized $\kappa $-phase structure. For temperatures above the ferroelectric transition temperature (i.e., T > ${T}_{\mathrm{FE}}$), charge is equally distributed within a dimer. For T < ${T}_{\mathrm{FE}}$, intra-dimer charge order sets in, giving rise to a macroscopic polarization.

**Figure 7.**Theoretical results for $\kappa $-(BEDT-TTF)${}_{2}$Cu[N(CN)${}_{2}$]Br, calculated by an RPA spin-fluctuation approach using hopping parameters extracted from ab-initio-based density functional theory calculations in the extended molecular model: (

**a**) Eight-node mixed-symmetry superconducting gap $\Delta $, (

**b**) $|\Delta |$ as a function of the angle $\varphi $ with respect to ${k}_{x}$, (

**c**) quasi-particle DOS ${\rho}_{\mathrm{qp}}$ in the superconducting state. The three coherence peaks A, B, C are consistent with scanning tunneling spectroscopy results [60]. Figures adapted from Ref. [60].

**Figure 8.**Coupling of correlated $\pi $-electrons with the crystal lattice in $\kappa $-(BEDT-TTF)${}_{2}$Cu[N(CN)${}_{2}$]Cl. (

**a**) Strain–stress relationships across the pressure-induced Mott metal–insulator transition [9]: The relative length-change data on the left as a function of hydrostatic pressure indicate a wide temperature region above the critical endpoint located at ${T}_{\mathrm{cr}}$∼ 37 K, in which strongly non-linear strain–stress relationships are observed. This result implies that the crystal lattice itself becomes soft as the electronic degrees of freedom become critical. This effect has been termed “critical elasticity” [131,132] and prevails in an extended region around the endpoint (right panel); (

**b**) ambient-pressure study of the damping factor of a low-lying optical phonon, likely related to the BEDT-TTF breathing mode, as a function of temperature [129]. The damping factor is inversely proportional to the phonon lifetime. Three characteristic regimes were identified in the data: (i) low damping at high temperatures ($T\phantom{\rule{0.166667em}{0ex}}>\phantom{\rule{0.166667em}{0ex}}{T}_{\mathrm{ins}}$) before the charge gap opens, (ii) high damping when the charges localize on the dimer (${T}_{\mathrm{N}}\phantom{\rule{0.166667em}{0ex}}<\phantom{\rule{0.166667em}{0ex}}T\phantom{\rule{0.166667em}{0ex}}<\phantom{\rule{0.166667em}{0ex}}{T}_{\mathrm{ins}}$) and (iii) low damping below the onset of spin and charge order ($T\phantom{\rule{0.166667em}{0ex}}<\phantom{\rule{0.166667em}{0ex}}{T}_{\mathrm{N}}$). Figures adapted from Refs. [9,129].

**Figure 9.**Experimental studies of the impact of weak disorder in $\kappa $-phase charge-transfer salts. (

**a**) Experimentally, disorder can be deliberately introduced either by (i) creating molecular defects through X-ray irradiation [139] or (ii) by controlling the ethylene endgroup conformation via the cooling rate through the glass-forming temperature ${T}_{\mathrm{glass}}$ [137]; (

**b**) temperature–pressure phase diagram for $\kappa $-(BEDT-TTF)${}_{2}$Cu[N(CN)${}_{2}$]Cl, subjected to different irradiation times [140]. Symbols represent the discontinuous Mott metal–insulator transition. (

**b**) adapted from Ref. [140].

**Figure 10.**(

**a**) Disorder scenarios in a valence-bond solid state, discussed for $\kappa $-(BEDT-TTF)${}_{2}$Cu${}_{2}$(CN)${}_{3}$. Local spin 1/2, or so-called “orphan spins”, may be caused by (i) vacancies in the anion layer, emphasized by the red circle or (ii) specific domain wall patterns, a possible result of randomness in the ethylene endgroup conformation. (

**b**) Magnetic-torque setup, with crystal axes a and c (with ${a}^{*}\perp bc$, ${c}^{*}\perp ab$), magnetic field $\mathbf{H}$ and the angle $\theta $ between sample and field. (

**c**) Theoretical magnetic torque $\tau $ dependence on field angle $\theta $, with indication of the angle shift ${\theta}_{0}$ and resolution of impurity (${\tau}_{\mathrm{I}}$) and bulk (${\tau}_{\mathrm{B}}$) contributions. Figures adapted from Ref. [12].

**Table 1.**Representative exchange parameters in K for indicated materials $\kappa $-(BEDT-TTF)${}_{2}$Cu[N(CN)${}_{2}$]Cl ($\kappa $-Cl), $\kappa $-(BEDT-TTF)${}_{2}$Ag${}_{2}$(CN)${}_{3}$ ($\kappa $-AgCN), $\kappa $-(BEDT-TTF)${}_{2}$B(CN)${}_{4}$ ($\kappa $-BCN), $\kappa $-(BETS)${}_{2}$Mn[N(CN)${}_{2}$]${}_{3}$ ($\kappa $-Mn): Bilinear exchange J, ${J}^{\prime}$, $\mathbf{D}$ (defined in Equation (3) and Figure 4) and averaged four-spin ring exchange K, ${K}^{\prime}$ (defined in Equation (5) and Figure 4). The DM vectors for the $Pnma$ salts are given in the coordinate system $(a,\text{}b,\text{}c)$, while the $P21/c$ values are indicated with $\left(\right)$ * and given with respect to $(a,\text{}b,\text{}c$*). The ratio ${J}^{\prime}/J$ indicates the deviation from the isotropic triangular limit ${J}^{\prime}/J=1$ and $K/J$ indicates the significance of the four-spin ring exchange. This table is not intended to be a complete representation of available results, but serves as orientation with selected values.

Material | $\mathit{J};\phantom{\rule{0.166667em}{0ex}}{\mathit{J}}^{\prime};\phantom{\rule{0.166667em}{0ex}}\mathit{D}$ | $\mathit{K};\phantom{\rule{0.166667em}{0ex}}{\mathit{K}}^{\prime}$ | ${\mathit{J}}^{\prime}/\mathit{J}$ | $\mathit{K}/\mathit{J}$ |
---|---|---|---|---|

$\kappa $-Cl | 482; 165; $(-3.6,-3.6,-0.2)$ [64] | 62; 21 [67] | 0.34 | 0.13 |

$\kappa $-AgCN | 250; 158; $(-2.9,-0.9,-2.9)$ * [64] | 20; 13 [67] | 0.63 | 0.08 |

$\kappa $-CuCN | 228; 268; $(+3.3,+0.9,+1.0)$ * [64] | 18; 18 [12] | 1.18 | 0.08 |

$\kappa $-BCN | 131; 366; $(+1.0,+4.2,-0.1)$ [64] | 05; 15 [67] | 2.79 | 0.04 |

$\kappa $-Mn | 260; 531; $(+22.6,-1.9,+8.8)$ * [51] | 16; 39 [51] | 2.04 | 0.06 |

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**MDPI and ACS Style**

Riedl, K.; Gati, E.; Valentí, R.
Ingredients for Generalized Models of *κ*-Phase Organic Charge-Transfer Salts: A Review. *Crystals* **2022**, *12*, 1689.
https://doi.org/10.3390/cryst12121689

**AMA Style**

Riedl K, Gati E, Valentí R.
Ingredients for Generalized Models of *κ*-Phase Organic Charge-Transfer Salts: A Review. *Crystals*. 2022; 12(12):1689.
https://doi.org/10.3390/cryst12121689

**Chicago/Turabian Style**

Riedl, Kira, Elena Gati, and Roser Valentí.
2022. "Ingredients for Generalized Models of *κ*-Phase Organic Charge-Transfer Salts: A Review" *Crystals* 12, no. 12: 1689.
https://doi.org/10.3390/cryst12121689