Design of Spin-Frustrated Monomer-Type C₆₀^•− Mott Insulator

Abstract

Spin-frustrated monomer-type Mott insulator C₆₀^•− solids are discussed in this review article. For the C₆₀^•− solids, the interfullerene center-to-center distance (r) is the key parameter that controls the competition between covalent bond-formation, itinerancy, and spin frustration. Eight C₆₀^•− salts with various compositions and dimensionalities are reviewed. In all of these C₆₀^•− salts except one, neither bond-formation nor long-range magnetic ordering was observed down to low temperatures. A plot of Weiss temperature (|Θ_CW|) against r shows that |Θ_CW| grows rapidly below r = 10.0 Å.

1. Introduction: Quantum Spin Liquid State

Strong geometrical spin frustration suppresses the classical long-range magnetic ordering of the Néel state, and allows the novel quantum states such as the quantum spin liquid (QSL) state for two-dimensional (2D) S = 1/2 antiferromagnets (AFs), as proposed by Anderson [1,2]. The QSL phase is a quantum-disordered insulating phase, which has been theoretically predicted to have a ground state with many degenerate states [3], and, hence, to exhibit large spin entropy, even at 0 K. To obtain spin-frustrated materials, the geometries of spin lattice subject to contradictory constraints are crucial. The spin lattices—triangle (Figure 1a), kagome (Figure 1b), zigzag (Figure 1c), tetrahedron (Figure 1d), honeycomb (Figure 1e), and hyperkagome (Figure 1f)—have been discussed with the basic building block being the triangle [4]. The magnetic exchange interaction J of spins is proportional to the square of transfer interaction t (Equation (1)), where U is the on-site Coulomb repulsion energy,

J ~ 4t²/U

(1)

A small distance between two spins (r) induces a large t and eventually, a metallic band with itinerant electrons is formed. Equation (1) suggests that the QSL state may be adjacent to such an itinerant (metallic or superconducting (SC)) phase. When the distance between the spin sites in the triangular spin lattice becomes extremely large, all of the spins may act as Curie spins. With decreasing distance between the spins, spins feel frustration to each other and form a QSL or a compromised spin configurations which do not have spin frustration with geometry of spin lattice of 120° (triangle, Figure 1g), 109° (tetrahedron, Figure 1h), or collinear AF (AFC) (Figure 1i) structures [5,6,7,8,9,10,11]. When the triangular spin lattice is distorted, the AF phase may get preferentially stabilized. Therefore, the QSL, AF, 120° structure (Figure 1g), AFC, and metallic (or SC) phases compete with each other in a triangular spin lattice based on the parameters r, t, W, U, J, outer stimuli (temperature T, pressure P, magnetic field H, etc.), and the geometry of the spin lattice, where W is the bandwidth.

The Curie-Weiss temperature Θ_CW given by,

Θ_CW = 2zS(S + 1)J/3k_B

(2)

is a parameter showing the easiness to access the QSL state, where, z, S, and k_B are the number of the nearest neighbour sites, spin quantum number, and Boltzmann constant, respectively. The frustration index f defined by Equation (3),

f = − Θ_CW/T_m

(3)

was proposed by Ramirez as a measure of the spin frustration [13,14]. Strong spin-frustrated systems are thought to be those associated with f > 10. In Equation (3), T_m is the temperature at which magnetic ordering occurs.

The easiest way to design spin-frustrated systems is to arrange magnetic transition metals in a triangular or kagome geometry. Numerous inorganic compounds have been examined based on this idea. A few examples are VX₂ (X = Cl, Br) [15], ABO₂ [16,17,18,19,20,21] (A = monocation, B = trivalent transition metal ions: CuFeO₂ [16,17,18], LiNiO₂ [19], NaTiO₂, [19], LiCrO₂ [20,21,22]), ABX₃ [23,24] (X = halogenide ions: CsCoCl₃ [23], CsMnBr₃ [24]), and large clusters (Mo₇₂Fe₃₀, Mo₇₂V₂₀) [25] for the triangular lattice, and KFe₃(SO₄)₂(OH)₆ [26,27,28] and Rb₂SnCu₃F₁₂ [29] for the kagome lattice. Some typical magnetically frustrated inorganic systems of transition metal oxides studied prior 2000 have been summarized by Ramirez [13] and Greedan [4].

However, QSL systems are scarce in materials with spin quantum numbers S > 1/2, even in the triangular spin lattices and kagome lattices with f ≥ 10² [30,31,32]. Even for the triangular S = 1/2 spin systems with large |Θ_CW| and f, no QSL materials have been prepared due to the difficulty in maintaining the precise geometry of spin-frustrated lattices at low temperatures [33,34,35].

The first real QSL candidate was a charge-transfer (CT) salt of a dimer-type Mott insulator κ-(ET)₂Cu₂(CN)₃ [36], where ET is bis(ethylenedithio)tetrathiafulvalene (chemicals in this review are shown in Figure 2) and [Cu₂(CN)₃⁻]_∞ is a diamagnetic polymeric anion. The planar tridentate coordination of diamagnetic Cu(I) ions in [Cu₂(CN)₃⁻]_∞ is the main driving force for the two-dimensional (2D) triangular magnetic lattice that is composed of partially charged (ET)₂^•+, since [Cu₂(CN)₃⁻]_∞ has openings, and the arrangement of the anion openings is triangular due to the planar tridentate coordination of Cu(I) ions. The geometrical fit between a spin-site (ET)₂^•+ and the anion opening results in a triangular magnetic lattice (Figure 1j) according to a key-keyhole relation, where the key is the spin-site, (ET)₂^•+, and the keyhole is the anion opening.

The features of the Mott insulator κ-(ET)₂Cu₂(CN)₃ are: (1) it has a nearly equilateral triangular lattice (t′/t = 1.09) with very strong electron correlation (U/W = 0.93), (2) the QSL state is experimentally confirmed down to 20 mK [36,45,46], (3) the anisotropic SC state resides directly next to the QSL state without passing through the spin ordered AF state under pressure [47,48,49], (4) the transition from the QSL state to the metallic state shows positive pressure dependence, indicating that the residual spin entropy is in the QSL state [47,48,49,50], and (5) ¹³C NMR measurements under hydrostatic pressure on the salt of ET with ¹³C enriched at the central C=C bond (¹³C-ET) indicate d-wave SC symmetry [51]. Thus, a competition among the localized (and frustrated), itinerant, and exotic pairing of spins is manifested in this salt [36,45,46,47,48,49,50,51,52,53]. We have proposed designing principles for a QSL candidate residing next to the SC state based on the crystal, electronic, and spin structures for selected κ-(ET)₂X (X = anion) including X = Cu₂(CN)₃ as follows [54].

The requirements for a QSL state next to an itinerant state for κ-(ET)₂X are:

1)

the system has a low spin state (S = 1/2),

1)

the system should be a Mott insulator in ambient conditions,

1)

its Mott insulating state has both a partial CT state close to the itinerant region and a small Mott gap,

1)

the spin lattice should have a geometry that affords a strong geometrical frustration, i.e., t′/t ~ 1 for a triangular spin lattice,

1)

a high |Θ_CW| or high |J| value to observe the QSL state at the experimentally available temperatures, and

1)

the material must maintain weak energy dispersion along the weakest direction for the magnetic interactions of the 2D system, i.e., negligibly weak magnetic interaction perpendicular to the 2D magnetic layer in order to keep the geometry of spin-frustrated spin lattice down to low temperatures.

Since the discovery of the QSL state in κ-(ET)₂Cu₂(CN)₃ (|Θ_CW| = 375 K, |J|/k_B = 250 K, f > 1.3 × 10⁴), several materials based on the triangular, kagome, honeycomb, and hyperkagome spin lattices have been reported to have such a spin state [55,56,57]. Some organic and inorganic compounds with sufficiently large |Θ_CW|, |J|, and f examined on single crystals include κ-H₃(Cat-EDT-TTF)₂ (triangle, H₂Cat-EDT-TTF: catechol-fused ethylenedithio-TTF, |Θ_CW| = 120–150 K, |J|/k_B = 80–100 K, f > 2.4 × 10³) [58], a CT solid of (ethyltrimethylantimonate)[Pd(dmit)₂]₂ (triangle, dmit: 4,5-dimercapto-1,3-dithiole-2-thione, |Θ_CW| = 325–375 K, |J|/k_B = 220–250 K, f > 1.6 × 10⁴) [59], ZnCu₃(OH)₆Cl₂ (kagome, |Θ_CW| = 314 K, |J|/k_B = 180 K, f > 5.1 × 10³) [60,61,62], [NH₄]₂(C₇H₁₄N)[V₇O₆F₁₈] (kagome, |Θ_CW|= 81 K, f ≥2.0 × 10³) [63] where (C₇H₁₄N) is quinuclidinium ion, and [(C₂H₅)₃NH]₂Cu₂(oxalate)₃ (hyperhoneycomb, |Θ_CW|= 180 K, f > 3 × 10³) [64].

Another organic QSL system of a dimer-type Mott insulator κ-(ET)₂Ag₂(CN)₃ (t′/t = 0.97) was very recently developed by us (|Θ_CW| = 263 K, |J|/k_B = 175 K, f > 2.1 × 10³) having a different key-keyhole relation from that of κ-(ET)₂Cu₂(CN)₃ resulting in both a more robust QSL state and a higher critical temperature (T_c) of SC than the X = Cu salt [54,65], with similar relaxor ferroelectric response [66,67]. Among these QSL systems, only dimer-type ET salts κ-(ET)₂Cu₂(CN)₃ and κ-(ET)₂Ag₂(CN)₃ manifest competition among the itinerancy of electrons (metal), pairing of two electrons (SC), and localization (Mott insulator and QSL).

In order to have such competition between localized and itinerant spins in a monomer-type Mott insulator, the best candidate will be the C₆₀ CT materials having triangular or hexagonal packing of C₆₀^•− molecules (Figure 1k) among several kinds of such monomer-type Mott insulators based on TTF, TSF, ET, TCNQ, etc. [67,68,69], where TTF, TSF, and TCNQ are well known donor or acceptor molecules in the CT salts [70]. In these monomer-type Mott insulators, the upper-HOMO band for the CT solid of D^•+X⁻ (D: TTF, TSF, ET) and the lower-LUMO band for the CT solid of M⁺A^•− (A: TCNQ) are completely filled, where HOMO and LUMO are the highest occupied and lowest unoccupied molecular orbitals, respectively. The Mott gap being large makes it difficult for the competition between localization and itinerancy in the solids, thus requiring partial CT state or dimer-type Mott insulating state to satisfy the requirement 3 above mentioned. While for the C₆₀ CT solids, a partial CT state such as ET^1/2+ is not necessary and the completely ionized C₆₀^•− molecules are able to afford the itinerant state owing to the triply degenerate LUMO t_1u orbital, e.g., the compound CsC₆₀ quenched in liquid N₂ is reported to exhibit metallic behavior down to low temperatures [71,72].

In this review, the preparation, crystal and electronic structures, and the physical properties of spin-frustrated monomer-type Mott insulators of C₆₀^•− CT solids, namely (MDABCO⁺)(C₆₀^•−) (1), (Ph₃MeP⁺)(C₆₀^•−) (2), (TPC⁰)(MDABCO⁺)(C₆₀^•−) (3), (TPC⁰)(MQ⁺)(C₆₀^•−) (4), (PhCN⁰)(TMP⁺)(C₆₀^•−) (5), (PhCN⁰)(Ph₃MeP⁺)(C₆₀^•−) (6), (PhCl₂⁰){(Ph₃P)₃Au⁺)}₂(C₆₀^•−)₂(C₆₀) (7), and (DMI⁺)₃(C₆₀^•−)(I⁻)₂ (8), and the design of the QSL systems will be discussed.

2. Characteristic Features of C₆₀: Superconductors and Other Functions for C₆₀ Charge-Transfer Materials

Superconductivity (SC) is one of the most remarkable features of C₆₀ CT materials. An icosahedral C₆₀ molecule with I_h symmetry has triply degenerate LUMO and LUMO + 1 orbitals with t_1u and t_1g symmetries, respectively. Such multiple degeneracy (N) contributes to the relaxation of the Mott criterion [73,74], i.e., U/W ~ √N or an upper limit of U/W ~ 2.5, and the enhancement of the density of states at the Fermi level (D(ε_F)) inducing high T_c for SC [75], when C₆₀ is placed in the highly symmetric, i.e., cubic crystal field. Typical SC materials are represented as A₃C₆₀ (A: alkali metal), e.g., Rb₃C₆₀ (T_c = 29 K [76]), Rb₂CsC₆₀ (T_c = 31 K [77]), and RbCs₂C₆₀ (T_c = 33 K [77]), with a face-centered cubic (fcc) structure. The structural and physical properties of A₃C₆₀ and the related fullerene compounds were reviewed [73,78,79,80]. The critical temperature T_c varies monotonously with the lattice constant, independent of the type of the alkali dopant [77,81]. Thus far, about 40 SC materials have been synthesized with the highest T_c of 33 K (RbCs₂C₆₀) at normal pressure and 38 K (A15 or body-centered cubic (bcc) Cs₃C₆₀) under a pressure of approximately 0.7 Gpa [82]. The fcc phase of Cs₃C₆₀ also shows SC (T_c = 35 K) under an applied hydrostatic pressure of approximately 0.8 Gpa [83].

The critical temperature T_c decreases as the valence state (n) of C₆₀ deviates from n = −3 or lowering the symmetry of the crystal (non-cubic) such as Yb_2.75C₆₀ (n = −5.5, orthorhombic, T_c = 6 K) [84], Sm_2.75C₆₀ (n = −5.5, orthorhombic, T_c = 8 K) [85], Ba₄C₆₀ (n = −8, body-centered orthorhombic (bco), T_c = 6.7 K) [86,87], Sr₄C₆₀ (n = −8, bco, T_c = 4.4 K) [86] and K₃Ba₃C₆₀ (n = −9, bcc, T_c = 5.6 K) [88]. Eu₆C₆₀ with bcc packing undergoes a ferromagnetic transition at 12 K, owing from Eu²⁺ cations with S = 7/2 spin [89]. Ce_xC₆₀ shows the coexistence of SC and ferromagnetism below 13.5 K, although its crystal structure and composition are currently not clear [90].

A₃C₆₀ SCs show a dome-shaped curve of normalized T_c versus lattice volume [82,83,91], which may be a hallmark of the competition between electron-phonon attractive and electron-electron repulsive interactions. The observation of a Hebel-Slichter peak in the relaxation rate just below T_c in NMR and μSR indicate a BCS-type isotopic gap [92,93].

Besides SCs, (TDAE⁺)(C₆₀^•−) is a soft ferromagnet with a Curie temperature of 16.1 K, in which only spins on C₆₀^•− contribute to the ferromagnetism, where TDAE is tetrakis(dimethylamino)ethylene [94].

3. Requirements for Spin-Frustrated Spin Lattice of C₆₀^•−

3.1. Competition among Bond-Formation, Itinerancy, Localization, and Frustration in Fulleride Solids

It can be easily seen that the close packing of C₆₀^•− leads to triangular (Figure 1k) or honeycomb (Figure 1e) spin lattices. In the C₆₀ system, there is another factor, namely, the bond-formation between C₆₀^•− molecules, which leads to competition among itinerancy, localization, and frustration. Figure 3a,b summarize the electronic competition for bond-formation, such as dimers and polymers (Figure 2b) (top panel of Figure 3a), itinerancy (monomeric C₆₀ metallic solids, middle panel in Figure 3a), and localization (monomeric AF solids, bottom panel in Figure 3a) for 27 situations in C₆₀ CT solids as a function of center-to-center interfullerene distance r, as reported in the literatures [37,38,44,79,95,96,97,98]. The van der Waals (vdW) diameter [99] of C₆₀ is 10.18 Å indicated by green dotted line in Figure 3a,b.

When the interfullerene distance of the C₆₀ solids is small (r < 9.4 Å, which is indicated by blue dotted line in Figure 3a,b), the C₆₀^•− molecules tend to form dimers (9 and 10 in Figure 2b) [37,38] or polymers (11–15 in Figure 2b) by single or double bonds. Polymer 11 with K⁺ is a 3D metal [39], while those with Rb⁺ and Cs⁺ are one-dimensional (1D) metals and become spin-density-wave insulators at low temperatures [40,100]. Single carbon-carbon bonded polymer 12 in Na₂CsC₆₀ seems to be an SC with T_c about 3 K lower than that for the starting non-polymeric phase. Analogous lowering of T_c at the formation of polymer 12 in Na₂RbC₆₀ leads to very low T_c or the absence of SC [41,101].

The compound 13 is an insulator with high ionic conductivity [42], 14 is a diamagnetic semiconducting polymer [43], and 15 is a highly correlated metal [43].

Even when r > 9.4 Å, C₆₀^•− bond-formation proceeds for the cation species, which are not able to prevent bond-formation. The C₆₀^•− molecules in [K(DB18C6)]₄(C₆₀)₅12THF, where DB18C6 is dibenzo-18-crown-6 ether and THF is tetrahydrofuran, dimerize at r = 9.89 Å (at 225 K), but remain as a monomer at r = 10.1 Å [102]. For (MQ⁺)(Co^IIOEP)(C₆₀^•−)(PhCl₂), where OEP is octaethylporphyrin, no bond-formation between C₆₀^•− molecules (r = 9.88 Å at 100 K) is observed owing to the steric protection by the coordination of C₆₀^•− with Co^II, that gives rise to a diamagnetic state [103]. However, for {(TMP⁺)₂M^IITPP}(C₆₀^•−)₂(PhCN)₂(PhCl₂)₂ (M = Zn, Mn, TPP: tetraphenylporphyrin), where M^IITPP is weakly coordinated with TMP⁺, but, no coordination of C₆₀^•− with M^II, with r = 10.04 Å (M = Zn at 250 K) and r = 10.08 Å (M = Mn at 270 K), singly bonded (C₆₀⁻)₂ dimers form at 100 K [104]. Structural analysis of (Ph₃MeP⁺)(C₆₀^•−)(PhCN) indicates that the lowest value of r is 9.92 Å at room temperature (RT) for monomer state, and singly bonded (C₆₀⁻)₂ dimers appear when the temperature is reduced to 120 K for r = 9.28 Å [105]. (DMI⁺)₂(C₆₀^•−){Cd(diethyldithiocarbamate)I⁻} (r = 10.03 Å at 250 K) shows reduced-temperature dependent bond-formation, such as (1) monomeric C₆₀^•− upon instant quenching below 95 K, (2) a mixture of dimerized (C₆₀⁻)₂ and monomeric C₆₀^•−, and (3) stable singly bonded (C₆₀⁻)₂ only upon slow cooling for 6 h [106].

As can be seen in Figure 3a, C₆₀ compounds having an interfullerene distance in the range 9.4 Å < r < 10.0 Å at RT have not been extensively explored. It is highly plausible that the shorter interfullerene distance that is less than 9.7 Å instantaneously leads to bond-formation resulting in r ≤ 9.4 Å.

Above results indicate that, even at r ~ 9.9–10.1 Å, C₆₀^•− anion molecules dimerize, when the C₆₀^•− molecules are not properly protected against bond-formation.

An isolated fulleride anion structure has been observed for the interfullerene distance r ≥ 12 Å by using both bulky cation molecules (Ph₄P⁺, PPN⁺, [Ru(BPY)₃]²⁺) [107,108,109,110,111,112,113,114,115,116], and fulleride anions with a charge of −2 or −3 and alkali metal cations, which coordinate with THF, crown ether or cryptand molecules [113,114,115,116]. The preparative methods of these compounds were reviewed [95].

When r is greater than 9.73 Å and less than 12 Å, the “bond-formation”, “monomer metal”, and “monomer AF” in Figure 3a compete with each other. The “monomer metal” is observed for many C₆₀ⁿ⁻ solids, e.g., K₃Ba₃C₆₀ (bcc, r = 9.74 Å) [88], Li₃CsC₆₀ (fcc, r = 9.98 Å) [117], Na₂RbC₆₀ (fcc, r = 9.96 Å) [118], K₃C₆₀ (fcc, r = 10.07 Å) [119], and RbCs₂C₆₀ (fcc, r = 10.19 Å) [77]. The “monomer AF” state is observed for a wide range of r. The A15 Cs₃C₆₀ (r = 10.20 Å) solid shows AF ordering below T_N = 46 K (Θ_CW = −68 K) [120], and the fcc phase (r = 10.44 Å) also shows AF ordering at 2.2 K (Θ_CW = −105 K) at normal pressure [83,121,122], where T_N is Néel temperature. The intercalation of NH₃ molecules (e.g., (NH₃)K₃C₆₀, r = 10.57 Å at RT) results in a phase transition from a cubic to orthorhombic lattice structure accompanied by the appearance of AF ordering instead of SC [123] and exhibits an SC state under pressure (T_c = 28 K at 15 kbar) [124].

Figure 3 will be a guiding map for the search for C₆₀ functional materials. A phase diagram of C₆₀³⁻ solids indicates that the SC state is in the vicinity of spin ordered AF states while the critical temperature T_c decreases as approaching to the AF phase [82,120,125]. Therefore, the C₆₀ system is a potential candidate for the QSL state to be close to the metallic, SC, and bond-formed states.

The C₆₀^•− molecules in both the A15 and fcc phases of Cs₃C₆₀ are not in the high-spin state (S = 3/2), which prefers spin-ordering, but are in the low-spin states (S = 1/2) due to the splitting of the t_1u-orbitals by the Jahn-Teller effect. The triangular spin lattices of fcc have strong spin frustration with strong AF interactions characterized by its frustration index f ~ 48 for Cs₃C₆₀. However, the Cs₃C₆₀ solid in the fcc structure prefers the AF state as compared to the QSL state. A large frustration index f and a low-spin state are necessary for QSL state formation, however, the geometry of the triangular spin lattice is also important. For example, κ-(ET)₂X with a nearly equilateral triangular spin lattice (t′/t = 0.97–1.09 for X = Ag₂(CN)₃ and Cu₂(CN)₃) exhibits the QSL state [36,54], while the distorted triangular spin lattice (e.g., t′/t = 0.715 for X = Cu[N(CN)₂]Cl [126,127], t′/t = 1.79 for X = CF₃SO₃) [128] exhibits an AF ordered spin state. Similarly a spin-frustrated system in which t′/t deviates from unity shows AF ordering, i.e., Cs₂CuCl₄ (t′/t = 1.71) with T_N = 0.69 K [129].

The fundamental and simple prerequisites for the formation of QSL C₆₀ system are:

(I) 

existence of C₆₀^•− or low-spin state of C₆₀³⁻ (S = 1/2),

(II) 

no polymerization between C₆₀ molecules,

(III) 

triangular or hexagonal packing of C₆₀ with equal interfullerene distance r, or t′/t ~ 1, and

(IV) 

strong AF interactions.

It is believed that smaller r values lead to stronger AF interactions. The critical r between the “itinerant”, “localized spin”, and “bond-formation” has not been elucidated yet, and the competition strongly depends on the environment of the C₆₀ molecules, pattern, and dimensionality of packing of C₆₀ molecules, charge of C₆₀ molecules, etc.

3.2. Charged State of C₆₀ and Effective On-Site Coulomb Repulsion

The C₆₀ⁿ⁻ species (n = 0, 1, 2, 3) are well discriminated by IR and UV-Vis-NIR spectra [78,79,80,95]. The on-site Coulomb energy of free C₆₀ molecule (U₀) was calculated by DFT as 2.7–3.1 eV [130,131,132]. The effective U (U_eff = U₀ − V, V is nearest neighbour Coulomb repulsion energy) values in solids are found to be 0.8–1.3 eV [130,131,132], when an electron is added to a C₆₀ molecule surrounded by other C₆₀ molecules in the fcc lattice, which is due to polarization by the charged C₆₀ molecules. While, Auger spectroscopy yields U_eff in the range 1.4–1.6 eV [133,134]. The first CT band of non-metallic M⁺(C₆₀^•−) solid corresponds to U_eff. With organic cation or supramolecular cation in this paper, the IR and NIR absorption spectra of CT solids in KBr show CT band peaks at 0.25 and 0.68 eV for (TPC⁰)(MDABCO⁺)(C₆₀^•−) [12], 0.69 eV for (TDAE⁺)(C₆₀^•−) [12], 0.71 eV for (TPC⁰)(MQ⁺)(C₆₀^•−) [12], 0.74 eV for (PhCN⁰)(Ph³MeP⁺)(C₆₀^•−) [105], and 0.77 eV for (PhCN⁰)(TMP⁺)(C₆₀^•−) [12]. The lowest band of 0.25 eV in (TPC⁰)(MDABCO⁺)(C₆₀^•−) is ascribed to the intraband absorption due to the metallic nature. The estimated U_eff (U_eff = 0.68–0.77 eV) should be smaller than those for (alkali metal⁺)(C₆₀^•−) owing to the high polarizability of organic cation molecules, which is similar to that proposed for TCNQ CT solids [135].

The effect of orbital degeneracy on the Mott-Hubbard criterion leads to a conclusion that the Mott transition takes place at U/W = √3 or an upper limit of U/W ~ 2.5, attributed to triple degeneracy or negligible splitting of t_1u orbitals [73,94]. Therefore, this relaxed Mott criterion is effective as long as the splitting of t_1u orbitals by Jahn-Teller distortion is not large enough, and forms one LUMO band. The upper limit of U/W requires W to be 0.27 eV, by considering the lowest value of U_eff (calculated value—0.68 eV), which is about 2–3 times the calculated W value (0.10–0.15 eV for (TPC⁰)(MDABCO⁺)(C₆₀^•−)) by the AM1 method [12]. A preliminary DFT calculation for this CT solid indicates a total W of about 0.48 eV at 160 K [12], which is still smaller than the estimated U_eff and is not able to account for the metallic nature. Since the calculated W values in these systems are not reliable at present, only the ratio of overlap integrals or transfer interactions will be discussed in the following.

3.3. Packing of C₆₀ and Magnetic Interactions in Fulleride Solids

So far, no structure-magnetic property relationship, especially concerning the geometrical spin frustration and the QSL state has been studied for the triangular or hexagonally packed C₆₀^•− solids. Hexagonal packing of C₆₀^•− has been suggested for (tetramethylammonium)(C₆₀^•−)∙1.5THF polycrystals based on the postulated structure, from both the calculation of total energy for various arrangements of the component molecules and the observed powder diffraction pattern [136]. The estimated r was 10.13 Å. The conductivity of the pellet sample was 10⁻² S·cm⁻¹ and the effective magnetic moment of the complex was ~1.75 μ_B in good agreement with the value for the system containing one S = 1/2 spin per formula unit. However, the susceptibility did not indicate AF interactions. (Na⁺)(C₆₀^•−)(THF)₅ polycrystals [137] showed a Curie-Weiss behavior above 200 K (μ = 1.70 μ_B, Θ_CW = −58 K) suggesting triangular or hexagonal packing of C₆₀^•−. The χ (magnetic susceptibility) value decreased sharply below 180 K (μ = ca. 0.8 μ_B), probably owing to the dimerization of C₆₀^•−, though no crystal structure above 200 K was found.

Recent work on needle-like single crystal (up to 6 mm) of (K⁺)(C₆₀^•−)(THF)₅(THF)₂ revealed that K⁺ cation coordinate with five THF molecules (by K∙∙∙O contacts) to form bulky cationic building units of [K(THF)₅]⁺, which separates C₆₀^•− corrugated layers with a distorted square arrangement [138]. At RT, C₆₀^•− molecules display rotational disorder, and ordering takes place below 240 K with a simultaneous appearance of a Dysonian EPR line down to 40 K. In the non-conductive region in the 240–295 K range, Curie-Weiss behavior (Θ_CW = −54 K) was observed with r = ~10.01 Å at 260 K. A metallic nature in the conductive region between 240 K and 40 K was suggested by the observation of the Dysonian EPR line shape, though the Dysonian line shape has been observed not only in metallic materials, but also in semiconductors with very high conductivity [139]. It should be emphasized that “localization” or “itinerancy” of spins in C₆₀ solids is strongly associated with the rotational disorder and ordering of C₆₀ molecules.

Crystals of [K([2.2.2]crypt)]₂(C₆₀²⁻)∙4(toluene) exhibit a 2D distorted hexagonal layer of C₆₀²⁻ dianion molecules separated by layers of [K([2.2.2]crypt)]⁺ cations [115]. The shortest interfullerene distance of 13.77 Å is too long owing to the large size of cation to have appropriate intermolecular interactions for the spin-frustrated system, regardless of the magnitude of effective spin of the dianion system.

3.4. Requirements for Triangular or Hexagonal Packing of C₆₀^•− by Key-Keyhole Relation: 2D or 3D Polycationic Template

The donor molecules that can reduce fullerene moieties are sparse because of the weak electron-accepting ability of fullerene moiety [140]. Simple use of very strong donor species, such as alkali metal has not led to 2D triangular or hexagonal packing of C₆₀^•−. A strategy to obtain new versatile ionic complexes, including polymerized fullerenes has been developed by Konarev, et al. [12,37,38,95,103,104,105,106,141,142,143,144,145,146,147,148,149], which can lead to multi-component CT complexes composed of small-sized strong donor molecules including alkali metals (D₂), which are able to ionize fullerene, and a structure-forming neutral molecule including solvent molecules (D₁), e.g., {[(D₁⁰)(D₂⁺)](fullerene⁻)}. A variety of single crystals of [(D₁⁰)(D₂⁺)](C₆₀^•−) with versatile packing patterns of C₆₀^•− molecules have been prepared according to this concept, which satisfies requirement (I) described in Section 3.1.

The distance between the cation species in the supramolecule formed between D₁ and D₂ molecules [(D₁⁰)(D₂⁺)] is the essential parameter to control both the bond-formation and the AF interaction among C₆₀^•− molecules. Therefore, the r value is the key to satisfy requirements (II) and (IV). When the supramolecular assemblies of D₁ and D₂ molecules have a periodic cationic site with sufficient space to hold one C₆₀^•− molecule, C₆₀^•− molecules (spin site = key) will be arranged according to the pattern of the cationic parts of supramolecular assemblies [(D₁⁰)(D₂⁺)] (=keyhole). This is the key-keyhole relation between C₆₀^•− molecules and polycationic template [(D₁⁰)(D₂⁺)]. To construct triangular or hexagonal packing of C₆₀^•−, the template [(D₁⁰)(D₂⁺)] should have a triangular or hexagonal pattern of cationic sites. It was observed that the threefold symmetry of D₁ and D₂ molecules satisfies requirement (III) [12,146]. Higher than threefold symmetry would be satisfactory. However, the design principle to satisfy the requirement for t′/t ~ 1 has not been developed yet for multi-component solid [(D₁⁰)(D₂⁺)](C₆₀^•−) and two-component solid (D⁺)(C₆₀^•−).

A good chemical choice of D₁ and/or D₂ concerning the donor strength, size, shape, and symmetry satisfies all of the requirements, (I)–(III). Among the multi-component complexes {[(D₁⁰)(D₂⁺)](C₆₀^•−)}, C₆₀^•− molecules form 2D hexagonal packing (triangular spin lattice, Figure 1k) in (TPC⁰)(MDABCO⁺)(C₆₀^•−) (3) and (TPC⁰)(MQ⁺)(C₆₀^•−) (4) and three-dimensional (3D) one in (DMI⁺)₃(C₆₀^•−)(I⁻)₂ (8) [12,145,146]. The coexistence of monomer metallic and monomer AF phases was observed in 3 (r = 10.07 Å at RT). Insulators 4 and 8 are AF insulators with 2D and 3D packing of C₆₀^•−, respectively, with r = 10.12–10.18 Å at 250 K and 10.06–10.12 Å at 100 K for 4 and 11.05 and 13.36 Å at 100 K for 8. Even in two-component materials, (D⁺)(C₆₀^•−), the bond-formation sometimes is prevented when r is large enough and the spins on C₆₀^•− interact antiferromagnetically. In (MDABCO⁺)(C₆₀^•−) (1), 3D close packing of C₆₀^•− and high |Θ_CW| were realized and the geometry of the spin lattice is 3D distorted bipyramid with r = 10.01–10.11 Å at 250 K and 9.91–10.12 Å at 100 K [147]. The solid (Ph₃MeP⁺)(C₆₀^•−) (2) has double chains of C₆₀^•− with zigzag spin lattice having weak interchain interactions with r = 10.08–10.10 Å at 100 K [105].

4. Key-Keyhole Relations in C₆₀^•− Charge-Transfer Solids

4.1. Three-Component Materials

4.1.1. Coexistence of Itinerant and Frustrated Spins in 2D Hexagonal Packing of C₆₀^•− in (TPC⁰)(MDABCO⁺)(C₆₀^•−)

Single crystals of (TPC⁰)(MDABCO⁺)(C₆₀^•−) (3) were obtained by the diffusion method. C₆₀, reductant CH₃CH₂SNa, and MDABCO∙I were stirred in a PhCl₂/PhCN mixture. TPC was dissolved in the obtained solution and n-hexane was layered. The diffusion was carried out over a period of two months to give black hexagonal prisms on the walls of the tube of sizes up to 0.5 × 2 × 2 mm³ (Figure 4c) [146].

The formation of 3 can be well interpreted by two kinds of key-keyhole relation (Figure 4). In the first step, three TPC molecules (D₁) surround MDABCO⁺ molecule (D₂⁺), both molecules have threefold symmetry, to form cationic supramolecular unit (Supramolecule 1: [(TPC⁰)₃(MDABCO⁺)]) as shown in Figure 4(a-1).

A C₆₀^•− molecule fit into the concave of the supramolecular unit to form a unit of {Supramolecule 2: [(TPC⁰)₃(MDABCO⁺)](C₆₀^•−)} according to the second key-keyhole relation (Figure 4(b-1)). The Supramolecule 2 units assemble to form 2D layer of hexagonally packed C₆₀^•− in the ab plane ([(TPC⁰)(MDABCO⁺)(C₆₀^•−)]_∞, Figure 4(b-2)). In another view, C₆₀^•− molecules were assembled according to the 2D sheet of the polycationic template [(TPC⁰)(MDABCO⁺)]_∞ shown in Figure 4(a-2) to form crystal 3. Such packing of C₆₀^•− was generated on one side of the polycationic template and is denoted as Layer A.

Figure 5 shows the size of MDABCO⁺ (Figure 5a,b), the height of TPC (Figure 5c), and some parts of Supramolecule 1 (Figure 5d) at 200 K. The neighboring TPC molecules are separated by 9.82 Å, so that it is able to prevent the bond-formation between C₆₀^•− molecules when they pack on the template in a hexagonal lattice structure, where the intermolecular distance of MDABCO⁺ molecules corresponds to N···N (blue points in Figure 4(a-2)) with separation distance of 9.99 Å. The MDABCO⁺ molecule is thicker (7.27 Å) than the thickness of the holder, which is composed of three TPC molecules (7.05 Å). Consequently, one side of Supramolecule 1 has a concave shape where N atoms are centered and the Me of N-Me⁺ group of MDABCO⁺ extrudes from the holder on the opposite side. Therefore, the C₆₀^•− hexagonal layer on the top of the polycationic template in Figure 4(a-2), namely Layer A (Figure 6a,b) has different steric and electronic environment than the C₆₀^•− hexagonal layer at the bottom of the polycationic template (Layer B, Figure 6a,c).

Figure 7 demonstrates the key-keyhole relation for layer B. Three TPC⁰ and three MDABCO⁺ molecules constitute Supramolecule 3, [TPC⁰]₃[MDABCO⁺]₃ (Figure 7a), according to the first key-keyhole relation. Here, six methyl groups of six MDABCO⁺ molecules, three from the top (TPC⁰)(MDABCO⁺) layer and three from the bottom layer, constitute an octopore to hold one C₆₀^•− molecule to afford {(TPC⁰)₆(MDABCO⁺)₆(C₆₀^•−)}, according to the second key-keyhole relation.

Figure 7b shows one Supramolecule 4, (TPC⁰)₃(MDABCO⁺)₃(C₆₀^•−). The 2D assembly of Supramolecule 4 generates the hexagonally packed C₆₀^•− Layer B (Figure 7c), which corresponds to Figure 6c.

At 300 K, C₆₀^•− molecules are ordered in Layer A (r = 10.07 Å, overlap integral s = 1.91 × 10⁻³, Figure 8a), while C₆₀^•− molecules in Layer B are disordered. At the same temperature, half of the MDABCO⁺ cations are disordered between three orientations that are linked by their rotation about the lattice threefold axis. On Layer A, the r value decreases monotonously to 9.97 Å at 183 K at which temperature a transition from rhombohedral to triclinic occurs. Assuming linear shrinkage of the interfullerene distance along the a axis below 160 K, r = 9.54 Å is evaluated at around 4 K, where no dimerization was experimentally detected. The calculated Fermi surfaces of both C₆₀^•− layers have a closed 2D pocket at Γ point and suggest 2D metallic nature in the ab plane (Figure 8b).

With decreasing temperature, ordering of the orientations of C₆₀^•− and MDABCO⁺ started below 200 K. A complete ordering of all three of the component molecules was found in the crystal structure at 160 K (Figure 6). These observations suggest that the orientation disorder of C₆₀^•− is closely linked with that of MDABCO⁺. The calculated overlap integrals at 160 K are s = 2.57 × 10⁻³ (//a), 2.03 × 10⁻³ (//b), and 2.76 × 10⁻³ (//a + b) for Layer A and 2.45 × 10⁻³ (//a), 2.21 × 10⁻³ (//b), and 1.61 × 10⁻³ (//a + b) for Layer B. The calculated bandwidth W is 0.103 eV at 300 K for Layer A, 0.150 eV and 0.133 eV for Layers A and B at 160 K, respectively. The calculated anisotropy of the transfer interactions t_a:t_b:t_a+b = 1:1:1 for Layer A above 183 K changed to t_a:t_b:t_a+b = 1.27:1:1.36 for Layer A and t_a:t_b:t_a+b = 1.52:1.37:1 for Layer B at 160 K. The ratio of the triangular spin lattice is defined as 2t_a/(t_b + t_a+b), 2t_b/(t_a + t_a+b), and 2t_a+b/(t_a + t_b). However, the last two definitions provide inadequate t′/t values of 0.60–0.76 for layer A and 0.54–0.69 for Layer B at 160 K that suggests a much enhanced 2D nature than that at RT. So using the ratio t′/t = 2t_a/(t_b + t_a+b), the calculated anisotropy is 1.00 (300 K) and 1.07 (160 K) for Layer A and 0.99 (185 K) and 1.28 (160 K) for Layer B. The spin frustration of this spin lattice is comparable to that of κ-(ET)₂Cu₂(CN)₃ (t′/t = 1.09 at RT, 1.07 at 100 K) [54] and stronger than a Mott insulator κ-(ET)₂B(CN)₄ [150] (t′/t = 1.42 at RT, 1.61 at 100 K, ground state is valence-bond solid). However, the increase of W together with the ordering of C₆₀^•− molecules in Layer B below 200 K gave rise to a superior itinerancy (metallic state) than localization (QSL state) for this salt.

The resistivity measurements carried out using the four probe method indicate that salt 3 is metallic within the ab plane from 360 (1.8 S·cm⁻¹) to 200 K (14 S·cm⁻¹), after which a rapid enhancement of the metallic nature occurs from 200 K to 185 K (33 S·cm⁻¹) (Figure 9a).

The temperature range of this anomaly between 200 K and 185 K, indicated by two red lines, coincides well with that of an ordering of C₆₀^•− in Layer B, showing that the ordered C₆₀^•− radical anions in Layer B start to participate in the metallic transport below 200 K. The resistivity could not be measured correctly below 70 K because of a large increase in the contact resistance, due to its air-sensitivity, even though it was measured in an inert atmosphere. The contactless microwave and optical measurements revealed that the conductivity increased down to 100–25 K and the metallic state is preserved down to 5 K (Figure 9b) [12]. The optical conductivity is nearly flat below 150 K down to 5 K.

The asymmetry ratio of the Dysonian EPR line shape between the maximum and minimum of the absorption derivative (A/B, Figure 9c inset) of 3 is compared with that of β-(ET)₂I₃ (Figure 9c) [151], which is highly metallic and shows SC with T_c = 1.5 K [152]. For β-(ET)₂I₃, the EPR signal is Lorentzian (A/B = 1) at RT and the peak ratio A/B increases below 130 K to about A/B = 3.0 (at approximately 50 K) followed by a gradual decrease down to 5 K. In comparison, the A/B values of 3 are considerably large even at RT (A/B = 2.4). It exhibits an abrupt increase below 200 K reaching a maximum of 4.2 at 183 K, which coincides well with the rapid conductivity increase, then falls to 2.3–2.4 below 183 K. The A/B ratio slowly decreases below 100 K, but the Dysonian shape is observed even at 4 K (A/B = 1.64), thus confirming the existence of a highly conducting state down to 4 K.

At 230–330 K, molar magnetic susceptibility (χ_M) can be fitted by a combination of the Pauli and Curie-Weiss terms: χ_M = χ₀ + C/(T − Θ_CW) with a constant χ₀ = 6.5 × 10⁻⁴ emu∙mol⁻¹, C = 0.160 emu∙K∙mol⁻¹, and Θ_CW = −31 K (blue curve in Figure 9d). C of 0.160 emu∙K∙mol⁻¹ corresponds to the contribution of about 43% of the spins from the total amount of C₆₀ (C = 0.374 emu∙K∙mol⁻¹ for 100% of spins). Consequently, the spins in one layer (Layer B) are treated as localized ones and they interact antiferromagnetically with Θ_CW of −31 K. A reversible decrease in χ_M is observed at 200–230 K. Below 200 K, the temperature-independent susceptibility (χ′₀) of about 10.0 × 10⁻⁴ emu∙mol⁻¹ is attributed to the Pauli paramagnetic contribution, implying a metallic state down to 1.9 K. The scenario is that ordering of both C₆₀^•− in layer B and MDABCO⁺ triggered a transition from a non-metallic and antiferromagnetically frustrated state to a metallic state for spins of C₆₀^•− in Layer B, while ordered C₆₀^•− in Layer A kept its 2D itinerancy over the entire temperature range. The strong coupling between the ordering of C₆₀ and physical properties is intriguing and was previously observed in some fullerene salts [153,154,155,156].

Salt 3 is the first 2D monomer-type C₆₀^•− organic metal composed of only light elements (C, H, N). Even though rapidly cooled AC₆₀^•− (A = Cs and Rb) were reported to be monomer-type metals below 150 K and 125 K, respectively, definitive information is needed concerning the stoichiometry, metallic behavior, dimensionality, and crystal structure to confirm a monomer-type metal [71,72,157].

Summarizing the information concerning the geometry of spin lattice of 3, the C₆₀^•− molecules form hexagonal stacking according to the geometry of cationic template (TPC⁰)(MDABCO⁺), both of the component molecules have threefold symmetry, by key-keyhole relation. Layer B has t′/t = 0.99 (185 K), which indicates strong spin frustration and is close to those of QSL candidates κ-(ET)₂M₂(CN)₃ (M = Cu; t′/t = 1.09, Ag; t′/t = 0.97), |Θ_CW| = 31 K is estimated in the range of 260−300 K, and r = 10.07 Å (RT). No dimerization of C₆₀^•− occurred down to 1.9 K.

4.1.2. Only Frustrated Spins in 2D Hexagonal Packing of C₆₀^•− in (TPC⁰)(MQ⁺)(C₆₀^•−)

By a using MQ⁺ instead of MDABCO⁺, where MQ⁺ is N-methylquinuclidinium cation, both of which have threefold symmetry, an AF insulator (TPC⁰)(MQ⁺)(C₆₀^•−) (4) was obtained [12]. The C₆₀ molecule, a 10-fold molar excess of CH₃CH₂SNa, and a 5-fold molar excess of MQ∙I were reacted in a PhCl₂/PhCN mixture. Into a filtered solution, TPC was dissolved and filtered. n-Hexane was layered over the obtained solution. Black hexagonal prisms up to 0.2 × 0.5 × 0.5 mm³ were harvested after two months.

A major difference between MQ⁺ (Figure 10a–c) and MDABCO⁺ is that the vertical size of MQ⁺ (7.87 Å) is larger than that of MDABCO⁺ cation (7.27 Å) since the carbon atom with hydrogen in MQ⁺ instead of uncharged nitrogen atom in MDABCO⁺ that is caused some kind of distortion in the layered packing of C₆₀ molecules. The size of MQ⁺ is 5.86 Å × 6.40 Å (Figure 10c), which is very close to that of MDABCO⁺. Figure 10d,e show the sizes of the fundamental units at 250 K. Similar to the MDABCO⁺ salt, three TCP and one MQ⁺ molecules form Supramolecule 5; (TPC⁰)₃(MQ⁺), with a periodicity of TPC molecules of an average of 10.15 Å (Figure 10d) and periodicity of N-site in MQ⁺ molecules of an average of 10.04 Å (Figure 10e).

Crystal 4 has lower symmetry (triclinic unit cell) than 3 at RT. Similar to 3, crystal 4 has layered packing in which hexagonal fullerene layers alternate with the (TPC⁰)(MQ⁺) layers along the с axis (Figure 11a) in the sequence of (TPC⁰)(MQ⁺) layer/C₆₀^•− Layer A/(TPC⁰)(MQ⁺) layer/C₆₀^•− Layer B. Figure 11b,c show the view along the c axis for Layer A and Layer B, respectively.

The key-keyhole relation between C₆₀ in Layer A and (TPC⁰)(MQ⁺) is presented in Figure 12. Three TPC molecules and one MQ⁺ form Supramolecule 5, (TPC⁰)₃(MQ⁺), via the first key-keyhole relation. Then, C₆₀^•− molecule fits into the concave of Supramolecule 5 to give Supramolecule 6, [(TPC⁰)₃(MQ⁺)](C₆₀^•−), similar to the case of {[(TPC⁰)₃(MDABCO⁺)](C₆₀^•−)} (Figure 4(b-1)) by the second key-keyhole relation. The Supramolecule 6 units assembled to form (Supramolecule 6)_∞ generating Layer A of (TPC⁰)(MQ⁺)(C₆₀^•−) (Figure 11b and Figure 12d).

From a different point of view, C₆₀^•− molecules fit into the concaves in the polycationic template of layered unit of [(TPC⁰)(MQ⁺)]_∞ (Figure 12c) to form Layer A in 4 (Figure 12d). Figure 12e,f illustrate how one C₆₀^•− molecule in Layer A is embedded between two layers of (TPC⁰)(MQ⁺) where the TPC molecules in the upper and lower layers are shown in different colors. C₆₀^•− molecules in Layer A are well fitted in the TPC hole formed by the six TPC molecules in the upper and lower (TPC⁰)(MQ⁺) layers (Figure 12e). In Supramolecule 5 [(TPC⁰)₃(MQ⁺)] (Figure 10d), the extra hydrogen atom (red circle in Figure 10a) in MQ⁺ prevents the MQ⁺ cation from arranging vertically relative to the fullerene layers resulting in a lowered crystal symmetry. Figure 13g,h show the calculated Fermi surfaces by the tight-binding method combined with the semiempirical (AM1) molecular orbital calculations based on crystal structures at 250 K and 100 K, respectively. The band calculation of Layer A at 250 K indicates that the salt has 2D Fermi surfaces. However, due to the doubling of the unit cell along the b axis, a semi-metallic Fermi surface was estimated at 100 K.

Surprisingly, Layer B also has a hexagonal packing of C₆₀^•− molecules in spite of the use of longer cation molecule MQ⁺ by 0.60 Å than MDABCO⁺. Figure 13 shows the formation of Layer B similar to that in Figure 7. Figure 13a shows Supramolecule 7 by the first key-keyhole relation made of three TPC and three MQ⁺ molecules. The C₆₀ molecule fit into the hollow site where three MQ⁺ molecules formed that corresponds to the crossing points of black lines in Figure 13c to form Supramolecule 8 [(TPC⁰)₃(MQ⁺)₃(C₆₀^•−)] (Figure 13b). This is the second key-keyhole relation.

The 2D assembly of supramolecular units [(TPC⁰)₃(MQ⁺)₃(C₆₀^•−)] leads to Layer B on the (TPC⁰)(MQ⁺) layer (Figure 13d). From another view, the C₆₀^•− molecules assemble using polycationic template of 2D assembly of the first supramolecular units (Figure 13c) and leads to Layer B on the (TPC⁰)(MQ⁺) layer (Figure 13d).

In Layer B, C₆₀^•− molecules are sandwiched between 2D layer of (TPC⁰)(MQ⁺) formed by Supramolecule 7 (Figure 13e). Here, the C₆₀^•− molecules are more in contact with MQ⁺ than with the TPC molecules. The intermolecular interactions between C₆₀^•− and MQ⁺ are dominant factors that determine the packing of (TPC⁰)(MQ⁺) and C₆₀^•−. C₆₀^•− molecules in Layer B, which are allocated at each hollow site where the six MQ⁺ cation molecules combine: three MQ⁺ molecules in one (TPC⁰)(MQ⁺) layer and three MQ⁺ in the neighboring (TPC⁰)(MQ⁺) layer constitute an octopore for C₆₀^•− molecule in Layer B (Figure 13e,f). The fusion of the assemblies of (TPC⁰)(MQ⁺) and C₆₀^•− molecules while maintaining the relation of octopores in Figure 13e,f leads to the packing pattern of C₆₀^•− in Layer B in the ab plane (Figure 13d). Similar to Layer A, the band calculation of Layer B indicates that the salt has 2D Fermi surfaces at 250 K and a semi-metallic Fermi surface at 100 K (Figure 13g,h).

The MQ⁺ cations are disordered between two orientations at 250 K and even 100 K. In spite of the disorder of MQ⁺, the fullerene anions are ordered in 4 at both temperatures. The center-to-center interfullerene distances of 10.12, 10.16, and 10.18 Å at 250 K (Figure 14a,b) are noticeably larger than the distance of 10.07 Å in 3 at 300 K. Owing to the increased interfullerene distance in 4, the average overlap integrals are smaller than those of Layer A in 3, giving rise to localized nature of spins (Mott insulator) in 4 with distorted triangular spin lattice.

The calculated overlap integrals at 100 K are s = 0.78 × 10⁻³ (//a), 1.82 × 10⁻³ (//b), and 2.24 × 10⁻³ (//a + b) for Layer A and 2.81 × 10⁻³ (//a), 1.97 × 10⁻³ (//b), and 1.51 × 10⁻³ (//a + b) for Layer B. The calculated bandwidths are 0.103 (0.112) and 0.097 (0.113) eV for Layer A and Layer B at 250 K (100 K), respectively. Similar to 3, the ratios 2t_b/(t_a + t_a+b) and 2(t_a + t_b)/(t_a + t_b) are 0.85 and 1.17 for Layer A and 0.91 and 0.64 for Layer B at 100 K. Since the calculated Fermi surface shows 1D properties, it is more appropriate to use 2t_a/(t_a + t_a+b) instead of 2t_b/(t_a + t_a+b) and 2t_a+b/(t_a + t_b). The calculated anisotropy of the transfer interactions at 250 K is t_a:t_b:t_a+b = 1.04:1:1 (t′/t = 1.04) and 1:1.40:1.23 (t′/t = 0.76) for Layer A and Layer B, respectively. The anisotropy changed to 1:0.90:1.11 (t′/t = 0.99) for Layer A and 1:0.70:0.54 (t′/t = 1.61) for Layer B at 100 K. The anisotropy of Layer A is close to that of κ-(ET)₂Cu₂(CN)₃, and the geometrical spin frustration is comparable to that of 3.

The relatively large distances between C₆₀^•− prevent their dimerization but allow for the manifestation of a magnetic interaction between them. Reciprocal molar magnetic susceptibility is described well by the Curie-Weiss law in the 30–300 K range with negative Weiss temperature of Θ_CW = −27 K (Figure 14c), indicating AF interaction of spins in the fullerene layers. The |Θ_CW| is small, owing to the weaker AF interactions than that in 3 because of the larger interfullerene distance. In spite of the strong AF interaction of spins, magnetic ordering is not observed down to 1.9 K in this distorted triangular spin lattice system (f > 14). The resistivity measurements were impossible owing to small size of the crystals.

In summary, concerning the geometry of spin lattice of 4, the C₆₀^•− molecules form hexagonal stacking according to the geometry of cationic template (TPC⁰)(MQ⁺) by key-keyhole relation similar to that for 3. There are two fulleride layers and both have distorted hexagonal arrangement of C₆₀^•− with t′/t = 0.99 for Layer A and t′/t = 1.61 for Layer B. Owing to large r (10.12–10.18 Å at 250 K), small |Θ_CW| (27 K) is estimated in the range of 30–300 K. No dimerization of C₆₀^•− occurred down to 1.9 K (f > 14).

4.1.3. Distorted Hexagonal Packing of C₆₀^•− in (PhCN⁰)(TMP⁺)(C₆₀^•−), (PhCN⁰)(Ph₃MeP⁺)(C₆₀^•−), and (PhCl₂⁰)[(Ph₃P)₃Au⁺]₂(C₆₀^•−)₂(C₆₀)

(PhCN⁰)(TMP⁺)(C₆₀^•−)

Single crystals of the three-component salt (PhCN⁰)(TMP⁺)(C₆₀^•−) (5) were unintentionally obtained, where TMP⁺ is N,N,N′-trimethylpiperazinium cation (D₂⁺) and solvent molecule PhCN is neutral (D₁⁰) [148]. The crystal structure was solved at 120 K and 90 K. The C₆₀ molecules are disordered at 120 K and ordered at 90 K with trebling of the unit cell, which made the calculation of band parameters difficult.

The 2D C₆₀^•− hexagonal layer in 5 is sandwiched between the layers composed of (PhCN⁰)(TMP⁺) (Figure 15a). Since the components molecules D₁ and D₂ do not have threefold symmetry, the C₆₀^•− layer shows distorted hexagonal packing (Figure 15b). Further, the TMP⁺ cations form pairs and are deeply embedded in the C₆₀^•− layers to deform the C₆₀^•− packing and are found near the center of the hexagonally arranged C₆₀^•− molecules. Methyl groups of TMP⁺ and PhCN molecules work to prevent the close approach of C₆₀^•− molecules and no bond-formation between C₆₀^•− was detected down to 2 K. The shortest r value along the interlayer direction is 10.39 Å at 120 K, indicating weak interlayer interactions, where the spin lattice should be 2D.

Only three types of interfullerene interactions with different r (9.90, 10.20, and 10.43 Å at 120 K) are essential within the C₆₀ layer and the ratio of the transfer interactions are t₁:t₂:t₃ = 0.9:1:0.5 for the major C₆₀ orientation. In spite of the hexagonal environment of the fullerenes in the layers, vdW C∙∙∙C contacts are formed with only three fullerene neighbors. Therefore, the geometry of model spin lattice is not triangular owing to weak magnetic interaction shown by dashed green lines (for r = 10.43 Å, s₃ = 0.66 × 10⁻³) in Figure 15c. Blue (for r = 9.90 Å, s₁ = 1.25 × 10⁻³) and red (for r = 10.20 Å, s₂ = 1.41 × 10⁻³) lines represent the main interactions J₁:J₂:J₃ = 0.79:1:0.22. Even though the r value for blue line is much shorter than that of the red line, s₂ is larger than s₁ owing to more favorable orientation of C₆₀ for s₂. The main magnetic interactions indicated by red lines extend along the b axis, and such 1D zigzag magnetic chains are connected by magnetic interactions along the c axis by blue lines. The unit of the spin lattice is edge-shared hexagonal, which is composed of four red lines and two blue lines (Figure 15c) and forms a 2D layer.

The temperature dependence of the reciprocal magnetic susceptibility for salt 5 is linear in the 70–300 K range with Θ_CW = −11 K (f = 5.5, no dimerization of C₆₀^•− occurred). The resistivity at RT is ρ = 7 × 10⁷ Ω·cm.

(PhCN⁰)(Ph₃MeP⁺)(C₆₀^•−)

Insoluble precipitates obtained by the reduction of C₆₀ with sodium fluorenone ketyl in PhCl₂ in the presence of Ph₃PMeBr were dissolved by the addition of PhCN. n-Hexane was layered on the filtered solution to grow single crystals of (PhCN⁰)(Ph₃MeP⁺)(C₆₀^•−) (6) where Ph₃MeP⁺ has a threefold symmetry [105]. The crystal structure at 250 K (Figure 16) shows hexagonal packing of C₆₀^•− and supramolecules (PhCN⁰)(Ph₃PMe⁺) are located in the centers of fullerene hexagon.

Hence, C₆₀^•− has only three negatively charged fullerene neighbors similar to that observed in 5 (Figure 15b). The somewhat low value of r = 9.92, 9.96, and 10.07 Å may result in higher spin frustration than for 5. The lowest interlayer r is 10.15 Å suggesting strong 2D nature within the C₆₀ layer in Figure 16. The C₆₀^•− molecules a₁-a₄ in Figure 16 form a flat layer, where the overlap integrals are s(a₁−a₂) = 2.14 × 10⁻³, s(a₁−a₃) = 2.29 × 10⁻³, s(a₁−a₄) = 0.70 × 10⁻³, and s(a₂−a₃) ~ s(a₃–a₄) ~ s(a₂–a₄) ~ s(a₁–a_1′) = 0. The spin lattice geometry (Figure 16b) is approximated as the 1D nonuniform zigzag chain along the b axis with alternating red and blue lines and the lines are connected by weak magnetic interactions by green lines (ratio of J values = 1:0.87:0.09). Salt 6 shows much stronger 1D properties than those of salt 5.

The EPR intensity decreases from 295–220 K, smoothly followed by a rapid decrease due to reversible dimerization of C₆₀^•− below 220 K. Upon cooling down to 120 K, the C₆₀^•− radical anion pairs, which has r = 9.92 Å at 250 K, form singly bonded (C₆₀⁻)₂ dimers with r = 9.28 Å. Therefore, no Θ_CW value is determined in this system. The steric protection to avoid the bond-formation is not sufficient in this solid. The solvent free crystal (Ph₃MeP⁺)(C₆₀^•−) (2) exhibits completely different structural and physical properties (vide infra) without the dimerization down to 1.9 K.

(PhCl₂⁰){(Ph₃P)₃Au⁺}₂(C₆₀^•−)₂(C₆₀)

By using a very bulky cation (Ph₃P)₃Au⁺ with C_3v symmetry, single crystals of (PhCl₂⁰){(Ph₃P)₃Au⁺}₂(C₆₀^•−)₂(C₆₀) (7) with a highly symmetric trigonal lattice were obtained, in which the supramolecule {(PhCl₂⁰)[(Ph₃P)₃Au⁺]₂} is a cationic template that accommodates C₆₀ molecules hexagonally [149]. The crystal structure was solved for a crystal slowly cooled down to 100 K.

Hexagonal corrugated C₆₀ layer is sandwiched between the layers of {(PhCl₂⁰)[(Ph₃P)₃Au⁺]₂} along the c axis (Figure 17a).

Fullerenes and PhCl₂ molecules located on the C_3v symmetry axes are statistically disordered between three orientations. The (Ph₃P)₃Au⁺ cations are ordered and located on the C_3v symmetry axis. The (Ph₃P)₃Au⁺ cations are too large in size to fit into the size of a C₆₀ molecule, but the size of supramolecule {(PhCl₂⁰)[(Ph₃P)₃Au⁺]₂} approximately corresponds to that of three C₆₀ molecules. The C₆₀ layer and {(PhCl₂⁰)[(Ph₃P)₃Au⁺]₂} layer, which is indicated by black arrows in Figure 17b, alternate along the c axis. C₆₀ molecules move from the layers toward planar PhCl₂ molecules to form strongly corrugated C₆₀ layers.

The fullerene layer consists of different charged C₆₀ molecules with −1 and 0 denoted as I and II, respectively, in Figure 17a–c). Interestingly, C₆₀^•− molecules are sandwiched between a (Ph₃P)₃Au⁺ cation molecule and a PhCl₂ molecule while C₆₀⁰ molecules are sandwiched between two (Ph₃P)₃Au⁺ molecules along the c axis, as shown in Figure 17b. Negatively charged and neutral C₆₀ molecules are closely packed within hexagonal layers with r (I∙∙∙II) = 10.02 Å, while between C₆₀^•−, it is long with r (I∙∙∙I) = 10.37 Å due to corrugation. The magnetic interactions between C₆₀^•− molecules in the neighboring fullerene layers are expected to be small based on its r value (r ~ 13.9 Å). As a result, the magnetic interactions are 2D. Each C₆₀^•− has only three negatively charged fullerene neighbors within a fullerene layer, namely C₆₀^•− molecule a₁ is surrounded by C₆₀^•− molecules a₂–a₄ in Figure 17c. They form distorted tetrahedral spin lattice composed of C₆₀^•− molecules a₁-a₄ (red lines in Figure 17d). The a₁ molecule projects out of the a₂–a₄ plane in Figure 17d by only 3.03 Å. The tetrahedral units are arranged in the 2D plane by apex-sharing. The overlap integrals have not been obtained due to severe disorder of fullerene molecules. Owing to the very large center-to-center distance between the C₆₀^•− molecules, the AF interaction is weak (Θ_CW = −5 K, f = 2.6). In order to enhance the magnetic interactions, smaller sized cationic supramolecules than [(PhCl₂⁰){(Ph₃P)₃Au⁺}₂] well matched with two C₆₀ molecules would be preferable. The resistivity at RT is approximately ρ = 4 × 10⁵ Ω·cm. EPR measurements confirmed no dimerization of C₆₀^•− down to 4.2 K.

4.1.4. Frustrated Spins in 3D Hexagonal Packing of C₆₀^•− in (DMI⁺)₃(C₆₀^•−)(I⁻)₂

Hexagonal packing of C₆₀^•− molecules is also formed in (DMI⁺)₃(C₆₀^•−)(I⁻)₂ (8) [145], where cationic template is delivered by a three-dimensional (3D) network of [(DMI⁺)₃(I⁻)₂] (Figure 18), where DMI⁺ is N,N’-dimethylimidazolium cation with no threefold symmetry. This system is not within the (D₁⁰)(D₂⁺)(C₆₀^•−) scheme, but the supramolecular cationic template [(DMI⁺)₃(I⁻)₂] has threefold symmetry.

The single crystals were obtained by a diffusion method. C₆₀, an excess of DMI∙I and reductant CH₃CH₂SNa were stirred in a PhCl₂/PhCN mixture. The mixture was cooled and n-hexane was layered over the solution. The diffusion was carried out during one month to give the single crystals on the wall of the tube with the size of 1 × 1 × 0.5 mm³.

The 3D network of [(DMI⁺)₃(I⁻)₂] is held together by the hydrogen (H)-bonds between the H atoms of DMI⁺ cation (Figure 18a) and I⁻ anion. The crystal structures at 100 K are shown in Figure 18b,c. The H atoms at 3- and 4-positions of DMI⁺ have H-bonds with one kind of I⁻ anion, denoted as I_B⁻. One I_B⁻ anion is surrounded by six DMI⁺ molecules; namely, I_B⁻ anion in red color in Figure 18b,c has H-bonds with six H atoms at 4-position (three up and three down along the c axis, see Figure 18c). The supramolecular unit composed of three DMI⁺ molecules and two I_B⁻ anion molecules; Supramolecule 9 [(DMI⁺)₃(I_B⁻)₂], which corresponds to the unit encircled by magenta dotted circle in Figure 18b,c, has a similar threefold symmetry like TPC molecule as shown in Figure 18d. Three Supramolecular 9 units encircle one C₆₀^•− molecules to form Supramolecule 11 [(DMI⁺)₃(I_B⁻)₂]₃(C₆₀^•−) which is schematically seen in Figure 18f where magenta figure Y represents Supramolecule 9 and B represents an I_B⁻ anion.

H atoms at 1-position have contacts with different kind of I⁻ anions (I_A⁻), and three DMI⁺ molecules and one I_A⁻ anion molecule form another supramolecular unit [(DMI⁺)₃(I_A⁻)] in the ab plane encircled by blue circle (Supramolecule 10, Figure 18e) in Figure 18b,c, which has also threefold symmetry like TPC. Similar to Supramolecule 11, one C₆₀^•− molecule is surrounded by three Supramolecular 10 units [(DMI⁺)₃(I_A⁻)] in the ab plane to form Supramolecule 12, where blue figure Y represents Supramolecule 10 and A represents an I_A⁻ anion (Figure 18g).

The top and bottom of Supramolecule 11 are capped by Supramolecule 10 along the c axis, yielding Supramolecule 13, [(Supramolecule 9)₃(Supramolecule 10)(C₆₀^•−)], as schematically shown in Figure 18(h-1), where C₆₀^•− molecules labeled 1–7 form one layer (Layer 1). Similar capping occurs for Supramolecule 12 by Supramolecule 9 to form Supramolecule 14, [(Supramolecule 9)(Supramolecule 10)₃(C₆₀^•−)], in Figure 18(h-2), where C₆₀^•− molecules labeled a-c form one layer (Layer a). Two kinds of Supramolecule units 13 and 14 stack alternately along the c axis to give the single crystal of 8. Figure 18(h-3) schematically illustrates the molecular packing. In the crystal viewed along the c axis (Figure 18(h-4,h-5)), four C₆₀^•− molecules, namely three molecules from Layer 1 (drawn in black in Figure 18(h-5)) and one molecule from Layer a (drawn in blue in Figure 18(h-5)) form tetrahedral [(1,2,7,a), (3,4,7,b) and (5,6,7,c)] structure. The labeling of C₆₀^•− molecules is the same as that in Figure 18(h-1–h-4)) Center-to-center distances between C₆₀^•− of 13.36 Å are marked by blue dotted lines and those of 11.05 Å (between blue and black circles) are marked by red lines in Figure 18(h-5).

As a result, geometry of the unit of model spin lattice is not triangular, but distorted tetragonal, i.e., tetrahedron with t′/t ~ 0. The AF spin configuration is expected within a 1D Mott insulating C₆₀^•− chain of black-blue-black-blue- fullerenes along the c axis, while within a layer of blue or black fullerenes in the ab plane parallel spin configurations are expected. Such spin units form a column along the c axis and the columns are arranged in the bc plane (Figure 19a). The other spin unit composed of C₆₀^•− molecules [(3,4,7,b) and (5,6,7,c)] has the same distorted tetrahedral geometry and form similar 2D packing to that in Figure 19a. These 2D spin sheets are connected, for example, through 7 and 7′ to form 3D network of magnetic interactions. It is emphasized here that according to the 3D template network composed of DMI⁺ and I⁻ molecules, the C₆₀^•− radical anions provide the 2H-hexagonal 3D packing by a 3D key-keyhole relation between cation [(DMI⁺)₃(I⁻)₂]⁺ and anion C₆₀^•− molecules.

The geometry of the spin lattice of 8 is expected to be 3D apex sharing triangular bipyramid based on the r values. We calculated overlap integrals based on the two main orientations of C₆₀ molecules at 100 K and the mean square of each is s = 0.12 × 10⁻³. Based on the overlap integrals, the model spin lattice is shown in Figure 19b, which are arranged in the ab plane to form 3D apex sharing triangular bipyramid.

The effective magnetic moment of 8 is 1.64 μ_B, slightly smaller than the value of 1.73 μ_B for the system containing one S = 1/2 spin per formula unit. The Curie-Weiss temperature of −9.6 K in the 40–300 K range indicates AF coupling of spins (Figure 19d). Effective magnetic moment also decreases below about 50 K (Figure 19e) due to AF coupling of spins. Because of the large interfullerene distance, 11.05 Å at 100 K, and small overlap integrals, the AF interaction is not strong. The long-range magnetic ordering is not observed down to 1.9 K.

In summary, concerning the geometry of the spin lattice of 8, the cationic supramolecule [(DMI⁺)₃(I⁻)₂] forms threefold assemblies that act as a template for the C₆₀^•− molecules to achieve hexagonal stacking of (C₆₀^•−). The overlap integrals lead to a 3D apex sharing triangular bipyramid with r = 11.05 Å at 100 K. Owing to a large r, a small |Θ_CW| (9.6 K) is estimated in the range of 50–300 K. No dimerization of C₆₀^•− takes place down to 1.9 K (f ~ 5).

4.2. Two-Component Materials

4.2.1. Frustrated Spins in 3D Corrugated Packing of C₆₀^•− in (MDABCO⁺)(C₆₀^•−) 3D Spin Lattice

3D close packing of C₆₀^•− and high |Θ_CW| are realized in the two-component CT solid of (MDABCO⁺)(C₆₀^•−) (1) where cation molecules MDABCO⁺ show threefold symmetry [147]. Black block single crystals of 1 (~0.4 × 0.4 × 0.5 mm³) were obtained by the reduction of C₆₀ with slight excess of strong reductant sodium fluorenone ketyl in PhCl₂ in the presence of stoichiometric amount of MDABCO∙I. Then, n-hexane was layered to precipitate CT solids. The crystal structures were solved at 250 K and 100 K. At 250 K, there is one independent C₆₀^•− and one ordered MDABCO⁺ molecules. The ordering of C₆₀^•− is observed below 160 K, with a trebling of the unit cell b axis.

The C₆₀^•− radical anions form square corrugated fullerene layers in the bc plane and the C₆₀ layers alternate with the MDABCO⁺ cations along the a direction (Figure 20a at 100 K). 2D C₆₀ layer in the bc plane is corrugated along the c axis (Figure 20b at 100 K). The corrugation of C₆₀ layer is clearly shown in Figure 20c (at 100 K) and C₆₀ columns extend along the b axis.

A C₆₀^•− molecule (drawn in red in Figure 20d at 250 K, labeled as Ⓐ) is surrounded by eight neighboring C₆₀^•− molecules labeled as ①, ②, and ③). There are three kinds of small r values: 10.01 Å with molecule ③, 10.08 Å with molecule ②, and 10.11 Å with molecule ① at 250 K. Four C₆₀^•− molecules, all of them are molecules ② among the eight neighbors, are located in the bc plane to form corrugated C₆₀^•− layer with vdW C∙∙∙C contacts in the 3.17–3.30 Å. Four other C₆₀^•− molecules, two ① and two ③, are located in the adjacent fullerene layers without short vdW C∙∙∙C contacts, though the r between molecules Ⓐ and ③ is shorter than that between molecules Ⓐ and ②.

The ordering of fullerenes is observed below 160 K with trebling of the unit cell b axis. As a result, the packing of C₆₀^•− molecules becomes denser and strongly anisotropic; namely r changes from Ⓐ−② = 10.08 Å at 250 K to (10.01, 10.04, 10.05 Å) at 100 K in the bc plane. Similarly, Ⓐ−③ and ①−② reduce to 9.91 and 9.96 Å, but Ⓐ−① becomes a little longer, 10.12 Å. Additional vdW interfullerene C∙∙∙C contacts are formed for each C₆₀^•− within the bc plane and between fullerene layers to account for the total number of such contacts as 18 for each C₆₀^•− at 100 K.

Since the structure at 100 K is complicated by the appearance of one and a half independent molecules both for C₆₀^•− and MDABCO⁺ molecules, at first the schematic of a possible spin lattice is examined based on the crystal structure at 250 K. Figure 20(e-1) shows the packing of C₆₀^•− viewed along the a axis, that corresponds to Figure 20a at 100 K. The square corrugated C₆₀^•− packing composed of C₆₀^•− molecules (one Ⓐ and four ②) form a pyramidal shape with r(Ⓐ−②) = 10.08 Å, which is schematically shown in Figure 20(e-2). The other combination of C₆₀^•− molecules ① and ③ form the equivalent pyramid and these two pyramids have short contacts with r values of 10.01 Å for 1−2 and 10.11 Å for ①−Ⓐ. As a consequence, the unit of intermolecular interactions is approximated as a distorted bipyramid, as shown in Figure 20(e-3). The plane of ②−②′−②′′−②′′′ bisects the bond ①−Ⓐ by 6.91 Å and 3.20 Å. The distorted bipyramids are connected to each other by sharing edge in the bc plane to form a 2D sheet. Figure 20(e-4) shows the part of the sheet extending along the c axis. The other unit of bipyramidal spin lattice composed of C₆₀^•− molecules ①, Ⓐ, and ③ forms equivalent 2D layer with different orientation (Figure 20(e-5)). The model geometry of the possible spin lattice of 1 is obtained by sharing apexes of the layers in Figure 20(e-4) and Figure 20(e-5) along the a axis to form 3D distorted bipyramidal spin lattice in which corrugated layers alternate as Layer Ⓐ②/Layer ①③ along the a axis (Figure 20(e-6)).

At low temperatures, the C₆₀^•− environment is strongly anisotropic in terms of r and overlap integrals s. The r values inside corrugated Layer Ⓐ② or Layer ①③, namely bonds Ⓐ−② and ①−, show shrinkage by 0.3–0.7%, while those bonds that are connecting neighboring corrugated layers, namely bonds ①−② and Ⓐ−③, show larger shrinkage by 0.5–1.0%. Therefore, though the corrugated square nature shown by Figure 20(e-2) is important to account for the spin interactions, the spin interactions between corrugated layers along the a axis becomes more significant at low temperatures. The model spin lattice geometry keeps the distorted bipyramidal one down to low temperatures. No dimerization was detected down to 1.9 K.

The temperature dependence of the molar magnetic susceptibility χ_M of 1 showed a maximum at 46 K, followed by a decrease but then an increase below 10 K owing to the Curie impurity of about 2.7% of total amount of C₆₀^•−. Figure 21 shows the temperature dependence of χ_M and 1/χ_M after the correction of Curie impurity. The magnetic susceptibility clearly indicates a characteristic peak near 50 K. Such peak in χ_M has been usually detected in the low-D (1D-2D) Mott insulators with strong spin frustration, such as κ-(ET)₂X (X = Cu₂(CN)₃, Ag₂(CN)₃, B(CN)₄, CF₃SO₃) [36,45,46,47,48,49,50,51,54,65,66,128,150,158]. The Θ_CW temperature of −118 K was derived in the 70–300 K range (Figure 21b). The temperature dependence of χ_M was fitted by the Heisenberg model for square 2D AF coupling of spins [159] to give J/k_B = −25.3 K. The long-range magnetic ordering is not observed down to 1.9 K (f > 62).

The overlap integrals between C₆₀^•− radical anions based on the real crystal structures are helpful to understand the situation of intermolecular interaction. At 250 K, 1 contains only one crystallographically independent C₆₀^•− and one MDABCO⁺, however the C₆₀^•− radical anions are disordered between three orientations with 49.4/25.3/25.3 (%) occupancies. On the other hand, 1 contains one and a half crystallographically independent C₆₀^•− and MDABCO⁺ at 100 K. Half of C₆₀^•− is well ordered at 100 K, while another C₆₀^•− is rotationally disordered between two orientations with 91.3/8.7 (%) occupancies. Although it is not reasonable to discuss the overlap integrals at 250 K due to the severe orientational disorder, we can calculate the overlap integrals in 1 at 100 K, assuming that all of the C₆₀^•− are well ordered ignoring the minor orientation (8.7%) in one of the two kinds of C₆₀^•−.

Figure 22 shows the network between the two kinds of C₆₀^•− radical anions at 100 K that are connected by several magnitudes of overlap integrals. The dominant interaction (red line) is 4.74 × 10⁻³ between the molecules 1 with 9.96 Å distance that forms a pair of C₆₀^•− between the adjacent layers. The second largest interaction (purple line) is 2.66 × 10⁻³ between the molecules 2 with 10.14 Å uniform distance that extends linearly along the a axis. The third one (black line) is 1.96 × 10⁻³ between the molecules 1 with 10.05 Å distance which forms zig-zag path ways along the c axis. Note that the shortest distance of 9.91 Å (orange line) between the different kinds of molecules 1 and 2 resulted in only the fifth largest interaction of 1.25 × 10⁻³. As demonstrated also in 2 and 5, the magnitude of overlap integrals between C₆₀^•− anion radicals does not necessarily scale with the closeness between them. The relative orientation of the molecular orbitals in the nearest neighbors as well as the center-to-center distances between them plays an important role to characterize the molecular interactions in the crystal.

The small size of the MDABCO⁺ cations with threefold symmetry and the absence of solvent molecules induced a densely packed 3D bipyramidal C₆₀^•− packing in 1 resulting in strong AF interactions. However, the strong anisotropic packing of C₆₀^•− may reduce the AF interaction considerably. The key-keyhole relation between MDABCO⁺ and C₆₀^•− is not clear in 1, however, it should be emphasized that MDABCO⁺ cations work to prevent the bond-formation even though the r values became small (①-② = 9.91 Å, Ⓐ-③ = 9.96 Å at 100 K).

4.2.2. Frustrated Spins in Double Chains of Triangles from C₆₀^•− in (Ph₃MeP⁺)(C₆₀^•−): Weakly Coupled Zigzag Chains

Single crystals of (Ph₃MeP⁺)(C₆₀^•−) (2), where Ph₃MeP⁺ is a triphenylmethylphosphonium cation with threefold symmetry, were prepared by the reduction of C₆₀ by (Ph₃MeP⁺)(vanadyl(IV) phthalocyanine) in PhCl₂ and slow mixing of the obtained PhCl₂ solution in n-hexane [105].

The crystal structure determined at 100 K indicates nearly ordered state of C₆₀^•− molecules. Crystal 2 involves a nearly isolated double chain with nearly equivalent fullerene triangles with small r values 10.08 Å, 10.103 Å, and 10.103 Å at 100 K (Figure 23a). The overlap integral is s₂ = 1.19 × 10⁻³ for r = 10.079 Å while that with longer r = 10.103 Å has larger s₁ = 1.79 × 10⁻³ owing to favorable orientation of C₆₀^•− molecules. Even though the double chain can be characterized as a zigzag chain, the chains are not isolated but are coupled weakly with a separation of r = 10.309 Å (s₃ = 0.58 × 10⁻³) within the bc plane (Figure 23b). Therefore, the possible spin lattice of 2 consists of pseudo 3D weakly coupled zigzag chains (Figure 23c) with t′/t = 1.50 and J₁:J₂:J₃ = 1:0.44:0.10.

Several zigzag-chain spin systems, which are the simplest frustrated magnet and treated by J₁-J₂ model or zigzag chain model, have been developed, such as CaV₂O₄ (S = 1, V³⁺, T_N = 69 K) [160], Cu[2-(2-aminomethyl)pyridine]Br₂ (S = 1/2, Cu²⁺) [161,162], (VO)(μ₃-MoO₄)(BPY) (S = 1/2, V⁴⁺) [163,164], and F₂PIMNH [165]. A theoretical study predicted that the S = 1/2 zigzag chain has a gapless phase for J₁/J₂ < 0.241 [166,167], as exemplified for Cu[2-(2-aminomethyl)pyridine]Br₂ (J₁/J₂ = 0.2, J₁/k_B = 8.5 K) [162] and (VO)(μ₃-MoO₄)(BPY) (J′/J = 0.2, J₁/k_B = 51 K) [164]. It is known that the interchain magnetic interactions are critical for the spin-ladder system either to manifest a Néel ordered or disordered spin-frustrated state, and the critical value is reported to be J/J′ = 0.11, where J and J′ are intraladder and interladder interactions, respectively [168]. Even though the actual J₃ values for the above zigzag systems were not estimated in these reports, it is likely that the J₃ values are very small according to their crystal structures. If we are able to expand the separation between the zigzag chains for 2 using more bulky cation molecules than Ph₃MeP⁺, we may have a real zigzag system of C₆₀^•−.

The temperature dependence of the molar magnetic susceptibility χ_M shows an increase below 10 K, owing to the Curie impurity of about 1.2% of total amount of C₆₀^•−. Figure 23d showed the temperature dependence of χ_M and 1/χ_M after the correction of Curie impurity. No peak of χ_M was observed even though 2 has strong spin frustration. The magnetic behavior is described well by the Curie-Weiss law in the 30–300 K range with Θ_CW = −60 K, and no AF ordering was observed down to 1.9 K (f > 30). Between C₆₀^•− molecules arranged along the a axis, Me groups of Ph₃MeP⁺ cation molecules penetrate and prevent the dimerization even at r = 10.08–10.10 Å.

5. Summary

Geometrical spin frustration is discussed for monomer-type Mott insulators of C₆₀ CT solids. When compared with the ET QSL system, bond-formation between C₆₀ molecules and disorder of C₆₀ molecule additionally participate in the competition among the itinerancy, localization, and spin frustration. The donor ability, size, shape, and symmetry of donor molecules in multi-component approach provide suitable geometrical space and spatial regulation for C₆₀^•− molecules by forming versatile supramolecules through the key-keyhole relation. A hexagonal packing of C₆₀^•− is achieved by the multi-component concept using cations (MDABCO⁺ or MQ⁺) and structure defining molecule (TPC) with threefold symmetry. C₆₀^•− molecules are packed according to the pattern of the polycationic supramolecular template of [(TPC⁰)(MDABCO⁺)] or [(TPC⁰)(MQ⁺)]. (TPC⁰)(MDABCO⁺)(C₆₀^•−) has uniform close packed hexagonal layers of two types, with an ordered C₆₀^•− layer (Layer A) and disordered C₆₀^•− layer (Layer B). The Layer A with t′/t = 1.00 at 300 K shows 2D metallic conductivity, whereas AF interaction of spins is observed in nonmetallic Layer B above 200 K. This AF layer has monomer-type Mott insulating state with t′/t = 0.99 (at 185 K) and Θ_CW = −31 K. The disordered layer becomes metallic below 200 K through the ordering of C₆₀^•−. (TPC⁰)(MQ⁺)(C₆₀^•−) also has a 2D hexagonal packing of C₆₀^•− and shows relatively longer interfullerene distances (10.12–10.18 Å at 250 K) than that in (TPC⁰)(MDABCO⁺)(C₆₀^•−) (10.07 Å at 300 K). (TPC⁰)(MQ⁺)(C₆₀^•−) has a monomer-type Mott insulating state with t′/t = 0.99 (Layer A at 100 K), t′/t = 1.61 (Layer B at 100 K), and Θ_CW = −27 K (f ~ 14). Solvent molecules participate in the formation of three-component C₆₀ solids to achieve C₆₀ hexagonal packing in (PhCN⁰)(TMP⁺)(C₆₀^•−) and (PhCN⁰)(Ph₃MeP⁺)(C₆₀^•−). The former has a distorted edge-shared honeycomb 2D spin lattice with J₁:J₂:J₃ = 0.79:1:0.22 and Θ_CW = −11 K (f ~ 5), and the latter has a non-uniform 1D zigzag chain of C₆₀^•− as an effective spin geometry and forms singly bonded dimers that are below 220 K. (PhCl₂⁰)[(Ph₃P)₃Au⁺]₂(C₆₀^•−)₂(C₆₀⁰) has a charge-disproportionated hexagonal 2D layer and the C₆₀^•− molecules form an apex-sharing tetrahedral spin lattice with large r = 10.37 Å at 100 K and small Θ_CW = −5 (f ~ 3). 3D hexagonal packing is realized in (DMI⁺)₃(C₆₀^•−)(I⁻)₂ in which H-bonds between DMI⁺ and I⁻ formed a polycationic template [(DMI⁺)₃(I⁻)₂] with threefold symmetry. A possible geometry of spin lattice is an apex-sharing bipyramid (t′/t = 0) and the long interfullerene distance of 11.05 Å at 100 K resulted in weak magnetic interaction Θ_CW = −9.6 K (f ~ 5). For two-component systems, a triangular unit of C₆₀^•− is observed in (MDABCO⁺)(C₆₀^•−) and (Ph₃MeP⁺)(C₆₀^•−) both cation molecules have threefold symmetry. 3D close packing of C₆₀^•− was observed in (MDABCO⁺)(C₆₀^•−) with a possible geometry of spin lattice of deformed 3D edge-shared bipyramid. The AF interaction with high Θ_CW of −118 K (f ~ 62) was observed though the bipyramidal geometry becomes distorted at low temperatures. (Ph₃MeP⁺)(C₆₀^•−) has double chains that are composed of a triangular arrangement of C₆₀^•−, resulting in weakly coupled zigzag chains with t′/t = 1.50, J₁:J₂:J₃ = 1:0.44:0.10, and Θ_CW = −60 K (f > 30).

Even though the overlap integrals between C₆₀ molecules depend on the molecular orientation of C₆₀, the center-to-center distance between C₆₀^•− molecules r is the key parameter that determines the competition among the bond-formation, itinerancy, and spin frustration. Figure 24 shows the relation between |Θ_CW| values and interfullerene distances r in this study. In the following, the summary and the perspective developed are presented.

• |Θ_CW| seems to increase rapidly when r < 10 Å and magnetic dimensionality is 3D. Such low values of r were realized for two-component CT solids with cation molecules of small size with threefold symmetry. However, it is difficult to find a good key-keyhole relation to provide uniform triangular or hexagonal packing of C₆₀^•− for the two-component case. Furthermore, single crystals of CT solids were not always obtainable. For example, a quinuclidinium cation, which is smaller than MDABCO⁺ and MQ⁺, gave no CT solids so far.
• The cationic supramolecular template with threefold symmetry leads to uniform triangular or hexagonal packing of C₆₀^•− for three-component case based on the key-keyhole relation. For three-component case, it is critical to decrease the r value and increase the magnetic dimensionality.

In all of these monomer-type C₆₀^•− Mott insulator except 6, no long-range magnetic ordering nor bond-formation were observed down to low temperatures. However, their |Θ_CW| values in the range 10–118 K, are not large enough to detect QSL state at experimentally available temperatures. This is apparent from the comparison with the |Θ_CW| values of the QSL systems obtained; |Θ_CW| = 180, 263, 314, 375, and 325-375 K for [(C₂H₅)₃NH]₂Cu₂(oxalate)₃, κ-(ET)₂Ag₂(CN)₃, ZnCu₃(OH)₆Cl₂, κ-(ET)₂Cu₂(CN)₃, and EtMe₃Sb[Pd(dmit)₂], respectively [36,59,60,61,62,64,65]. It is important to explore a C₆₀^•− Mott insulator with r = 9.4–10 Å to detect a QSL state neighboring metallic and SC states, as a decrease in r increases |t|, |J|, and |Θ_CW| values drastically according to their relations (Equations (1) and (2)).