Update 1 of: Calorimetric Investigation of Phase Transitions Occurring

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Update 1 of: Calorimetric Investigation of Phase Transitions Occurring in Molecule-Based Magnets† Michio Sorai,*,‡ Yasuhiro Nakazawa,‡,§ Motohiro Nakano,∥ and Yuji Miyazaki‡ ‡

Research Center for Structural Thermodynamics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan Department of Chemistry, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan ∥ Department of Applied Chemistry, Graduate School of Engineering, Osaka University, Suita, Osaka 565-0871, Japan §

This is a Chemical Reviews Perennial Review. The root paper of this title was published in Chem. Rev., 2006, 106 (3), 976−1031, DOI: 10.1021/cr960049g; Published (Web): February 21, 2006. Updates to the text appear in red type. 6.1. Dodecanuclear Manganese Clusters (Mn12) 6.2. Octanuclear Iron Clusters (Fe8) 6.3. Other Single-Molecule Magnets 6.4. Single-Chain Magnets (SCMs) 6.5. Antiferromagnetic Spin Rings 6.6. Other 0D Metal Complex Systems 7. Spin Crossover Phenomena 7.1. Entropy-Driven Phenomena: Coupling with Phonon 7.2. Phase Transition without Change in Crystal Symmetry 7.3. Cooperativeness of Spin Crossover Phenomena 7.4. Two-Step Spin-State Conversion 7.5. Thermal Hysteresis: Bistable States 7.6. Mechanochemical Effect in Spin Crossover Complex 8. Intramolecular Electron Transfer in Mixed-Valence Compounds 8.1. Trinuclear Mixed-Valence Compounds 8.2. Binuclear Mixed-Valence Compounds 9. Thermochromic Complexes 9.1. Thermochromism Due to a Change in the Coordination Number 9.2. Thermochromism Due to Molecular Motion of the Ligands 9.3. Enthalpy Change Due to Thermochromism 10. Spin-Peierls Transitions 10.1. Organic Spin-Peierls Compounds 10.2. Inorganic Spin-Peierls Compounds 11. Organic Conductors 11.1. Donor−Acceptor Type Charge Transfer Salts 11.2. (BEDT-TTF)2X Salts 11.3. π−d Interacting Systems 11.4. Organic Spin Liquid Systems 12. Concluding Remarks Author Information Corresponding Author Notes Biographies Acknowledgments

CONTENTS 1. Introduction 1.1. Scope 1.2. Characteristic Aspects of Molecular Thermodynamics 1.3. Heat-Capacity Calorimetry 2. Organic Free Radicals 2.1. Nitronyl Nitroxide Radicals 2.2. TEMPO Radicals 2.3. Verdazyl Radicals 2.4. Thiazyl Radicals 2.5. Other Radicals 3. One-Dimensional Magnets 3.1. Metallocenium Salts of Radical Anions 3.2. Bimetallic Chain Complexes 3.3. Helical Chain with Competing Interactions 3.4. Lithium Phthalocyanine 3.5. Quasi-1D Inorganic Complex 3.6. Mixed-Valence MMX Type Complexes 3.7. MnIII-Porphyrin−TCNE and M(TCNQ)2 3.8. Spin-Gapped Systems 4. Two-Dimensional Magnets 4.1. Assembled-Metal Complexes 4.2. Mixed-Valence Assembled-Metal Complexes 4.3. Inorganic Layered Complex 4.4. Multilayer Systems 5. Three-Dimensional Magnets 5.1. Cyanide-Bridged Bimetallic Complexes: Prussian Blue Analogues 5.2. Dicyanamide-Bridged Bimetallic Complexes 5.3. Dicyanoargentate-Bridged Complexes 6. Single-Molecule Magnets © XXXX American Chemical Society

B B B D D D H I I K M M O P R R R U V W W AA AC AD AD AD AG AI AI

AK AL AL AM AM AN AN AN AO AP AQ AT AU AV AV AY AZ AZ BA BB BC BC BE BF BF BH BM BP BR BR BR BR BR BS

Received: April 16, 2012

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dx.doi.org/10.1021/cr300156s | Chem. Rev. XXXX, XXX, XXX−XXX

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“molecular thermodynamics”. Among various thermodynamic measurements, heat-capacity calorimetry is an extremely useful tool to investigate thermal properties of materials at low temperatures. In most thermodynamic textbooks and monographs, interpretation of the thermodynamic laws and derivation of thermodynamic equations occupy many parts of the pages. However, descriptions about how to interpret experimental results are scarce or limited to very classical examples, if any. Therefore, a brief description will be made, at first, about what kinds of microscopic and macroscopic information are derived from actual calorimetry.9−14 Heat capacity is usually measured under constant pressure and designated as Cp. From the definition Cp ≡ (∂H/∂T)p, enthalpy is determined by integration of Cp with respect to temperature T:H = ∫ Cp dT. Because Cp is alternatively defined as Cp ≡ T(∂S/∂T)p, entropy S is also obtainable by integration of Cp with respect to ln T:S = ∫ Cp d lnT. Condensed states of matter are basically controlled by the interplay among molecular structures, intermolecular interactions, and molecular motions. Functionalities of materials manifest themselves as the result of a concerted effect among these three factors. When a delicate balance of these three factors is broken, the condensed state faces a catastrophe and is transformed into another phase. This is the so-called phase transition. Therefore, phase transition is a good probe for elucidation of the interplay among these three factors. Most phase transitions observed in molecule-based magnets are not first-order but second-order or higher-order phase transitions. Although the first-order phase transition takes place isothermally with a latent heat, the higher-order phase transition exhibits a sharp peak at a critical temperature Tc with long heatcapacity tails below and above Tc. The extent of order lost in the higher-order phase transition is described in terms of an “order parameter”. The heat-capacity tail above Tc reflects the short-range order still remaining in the system. In the vicinity of the critical temperature Tc, the heat capacity C is often reproduced by the following equations: C ∝ ε−α (T > Tc) and C ∝ |−ε|−α′ (T < Tc), where ε ≡ (T − Tc)/Tc and α and α′ are critical exponents, from which one can get information such as the lattice structure and the interaction involved in the system. The entropy gain at the phase transition plays a diagnostic role for the interpretation of the mechanisms governing phase transitions. A clue to correlate the entropy with the microscopic aspect is the Boltzmann principle given by S = kNA ln W = R ln W, where k and NA are the Boltzmann and the Avogadro constants, R is the gas constant, and W stands for the number of energetically equivalent microscopic states. If one applies this principle to the transition entropy, the following relationship is easily derived because ΔS is the entropy difference between the high- (HT) and low-temperature (LT) phases: ΔS = R ln(WH/ WL), where WH and WL mean the number of microscopic states in the HT and LT phases, respectively. In many cases, the LT phase corresponds to an ordered state and hence WL = 1. Therefore, on the basis of experimental ΔS value, one can determine the value of WH. Magnetic interactions in insulating molecule-based magnets are usually approximated by the localized spin model and described in terms of the following basic spin Hamiltonian:

BT BT

1. INTRODUCTION 1.1. Scope

Magnetism is one of the important functionalities of materials and has been a basic research subject to understand materials. Magnetism has been explored mainly in the field of solid-state physics for inorganic substances consisting of atoms or ions of transition elements, in which electron spins on atomic orbitals mainly contribute to the magnetism. In this sense, traditional magnetism may be called “atom-based magnetism”.1 However, the situation in this field of science changed dramatically after the discovery of bulk ferromagnetism in molecular complexes such as decamethylferrocenium tetracyanoethenide [FeIII{C5(CH3)5}2][TCNE] (the Curie temperature TC = 4.8 K) reported in 1987, MnIICuII(obbz)·H2O [obbz = N,N′oxamidobis(benzoato)] (TC = 14 K) reported in 1989, and even in the pure organic compound p-NPNN (p-nitrophenylnitronyl-nitroxide) (TC = 0.60 K) reported in 1991. Because spins of unpaired electrons on delocalized molecular orbitals are responsible for the magnetism, this new magnetism has been called “molecule-based magnetism”.1−8 Magnetic functionality has widely been used for various practical applications, and this need will be further increased in the future. One of the great advantages of the use of moleculebased magnets lies in the various possibilities of chemical modification onto a molecule; thereby, the magnetic property is finely adjusted by the chemical tuning. In the last two decades, various kinds of molecule-based magnets have been synthesized, and their chemical and physical properties have been elucidated from microscopic and macroscopic standpoints by use of various experimental methods. Among them, thermodynamic methods are unique in the sense that the energetic and entropic aspects inherent in materials can be directly observed. However, the number of scientific papers concerning thermodynamic research is not so high in comparison to other spectroscopic and structural studies. This is partly based on the misunderstanding that thermodynamics is a classical and old-fashioned science. The aim of this paper is to survey calorimetric studies on molecule-based magnets published in the last two decades and to reveal the important roles played by calorimetric studies. In addition to updating the root paper, we added two topics in this perennial review: One is chapter 10 “Spin-Peierls Transitions” and the other is chapter 11 “Organic Conductors”. It should be remarked here that in many papers referred to in this review article, the use of physical quantities “magnetic field strength” (symbol, H; unit, A m−1) and “magnetic flux density” (symbol, B; unit, T) has often been confused. However, in view of a review article, we shall follow descriptions adopted in the original papers without correction. 1.2. Characteristic Aspects of Molecular Thermodynamics

Physical quantities obtained from thermodynamic measurements reflect macroscopic aspects of materials in the sense that they are derived as the ensemble averages in a given system. However, because those quantities are closely related to the microscopic energy schemes of all kinds of molecular degrees of freedom in a statistical manner, one can gain detailed knowledge on the microscopic level on the basis of precise calorimetry. This field of study has been recognized as

The sum is taken over all pairwise interactions of ith and jth spins in a lattice. The quantity Si,k (k = x, y, z) is the kB


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component of the ith spin operator Si, and Jij indicates the effective exchange integral between spins. When the sign of Jij is positive, the spin−spin interaction assumes a ferromagnetic character, while it becomes antiferromagnetic for negative value. According to the magnitude of Jij,k, the spin system is classified into the following models: (i) Jij,x = Jij,y = Jij,z (isotropic Heisenberg model) (ii) Jij,x ≠ Jij,y ≠ Jij,z (anisotropic Heisenberg model) (iii) Jij,x = Jij,y or Jij,x ≠ Jij,y and Jij,z = 0 (isotropic or anisotropic XY model) (iv) Jij,x = Jij,y = 0 and Jij,z ≠ 0 (Ising model) Because the molecule-based magnets consist of molecular parts as the building blocks, their structures are anisotropic; thus, physical properties inevitably become structurally anisotropic. In many cases, this leads to low-dimensional magnets in which one- (1D) or two-dimensional (2D) interaction is dominant.15−17 Short-range and long-range orders formed by spins crucially depend on the magnetic lattice structure. These features are very sensitively reflected in heat capacity. When paramagnetic species form clusters magnetically isolated from one another, the magnetic clusters can be regarded as being of zero-dimension (0D), whose spin-energy scheme consists of a bundle of energy levels. In such a case, there exists no phase transition and the magnetic heat capacity exhibits a broad anomaly characteristic of the cluster geometry. For a linear-chain structure (1D lattice), no phase transition is theoretically expected because fluctuation of the spin orientation is extremely large. As shown in Figure 1 for the

In the case of a 2D lattice, a phase transition showing a remarkable short-range order effect takes place when the interaction is of the Ising type, while there is no phase transition for the Heisenberg type. Contrary to this, a threedimensional structure (3D) gives rise to a phase transition with a minor short-range order effect, independent of the type of spin−spin interaction. In actual magnetic materials, interaction paths, through which exchange spin−spin interaction takes place, are often much more complicated than being classified into a homogeneous single dimension. As a result, it happens that apparent dimensionality seems to change with temperature. This feature is called “dimensional crossover”. Figure 2 shows

Figure 2. Dimensional crossover between 1D and 2D lattices for the spin S = 1/2 ferromagnetic Ising model with the intrachain interaction parameter J and the interchain interaction parameter J′. Dashed and dot−dashed curves correspond to 1D (J′ = 0) and 2D (J = J′) models, respectively. A solid curve indicates the heat capacity of a rectangular lattice (J′ = 0.01J), showing the dimensional crossover between 1D and 2D lattices. Reprinted with permission from ref 18. Copyright 1960 Taylor & Francis Group.

the dimensional crossover between 1D and 2D lattices for the spin S = 1/2 ferromagnetic Ising model with the intrachain interaction parameter J and the interchain interaction parameter J′.18 The dashed curve corresponds to 1D (J′ = 0) and the dot−dashed curve is the heat capacity of a square planar 2D lattice (J = J′) corresponding to the exact solution by Onsager.19 The solid curve indicates the heat capacity of a rectangular lattice (J′ = 0.01J), which shows the dimensional crossover between 1D and 2D lattices. At high temperatures, the heat-capacity curve asymptotically approaches the 1D curve, while at low temperatures the existent 2D interchain interaction, although weak, diminishes the fluctuation of the system and leads to a phase transition. When a system consists of a finite number of energy levels, its heat capacity exhibits a broad anomaly, which asymptotically approaches zero at high temperatures. This type of anomaly is designated as the Schottky anomaly and is encountered in paramagnetic clusters and in systems involving Zeeman splitting, tunnel splitting, zero-field splitting, and so on. Spin-wave (magnon) theory has been proven to be a good approximation to describe the LT properties of magnetic substances. The limiting LT behavior of heat capacity due to spin-wave excitation CSW is conveniently given by the formula,15 CSW ∝ Td/n, where d is the dimensionality and n is defined as the exponent in the dispersion relation. For antiferromagnetic magnons, n is equal to 1, while for

Figure 1. Theoretical heat capacities of magnetic chains with spin S = 1/2: (a) ferro- or antiferromagnetic Ising model, (b) ferro- or antiferromagnetic XY model, (c) antiferromagnetic Heisenberg model, and (d) ferromagnetic Heisenberg model. Reprinted with permission from ref 15. Copyright 1974 Taylor & Francis Group.

1D S = 1/2 case,15 only a broad heat-capacity anomaly characteristic of 1D structure is observed. Heat capacities of the 1D magnetic systems characterized by Ising and XY type interactions are identical for ferromagnetic and antiferromagnetic interactions (curves a and b). Contrary to this, the system characterized by Heisenberg type interaction provides quite different heat-capacity curves between antiferromagnetic (curve c) and ferromagnetic chains (curve d). By comparing experimental heat capacity with these theoretical curves, one can easily estimate the type of spin−spin interaction operating in the material. C


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ferromagnetic magnons n = 2. Thus, for example, the spin-wave heat capacity of a 3D ferromagnet is proportional to T3/2 and that of a 2D antiferromagnet is proportional to T2. However, because actual low-dimensional magnets contain, more or less, weak 3D interactions, the dimensionality sensed by spin at extremely low temperatures is d = 3. One of the most characteristic aspects inherent in thermodynamics is that there exists no selection rule. In other words, any degrees of freedom originating in atoms, molecules, electrons, and spins can contribute to thermodynamic quantities. For example, the heat capacity of a solid measured at constant pressure, Cp, consists of contributions from lattice vibrations, Clat, and magnetic freedom, Cmag. Therefore, if one discusses the magnetic event, one should separate the magnetic part from the observed value. Except for scarce materials with simple crystal structures, it is usually very difficult to estimate Clat on the basis of the first principle of lattice dynamics. To avoid this difficulty, various approximate methods for separation have been proposed. Usually, the lattice heat capacity (or normal heat capacity) has been approximated by a polynomial function of temperature in a narrow interval or a combination of Debye and/or Einstein heat-capacity functions. In particular, the polynomial function of odd power of temperature, Clat = aT3 + bT5 + cT7 + ···, is often a good approximation at low temperatures. At moderate and high temperatures, the effective frequency-distribution method proposed by Sorai and Seki20 is a useful method. In this method, an effective frequency-distribution spectrum of lattice vibrations is first determined in the temperature region(s), where a relevant event has no contribution, and then, the hypothetical lattice heat capacity in the event temperature region is estimated on the basis of the obtained spectrum.

(iii) In the relaxation method, the sample is heated with a short pulse of energy and the resulting temperature rise and fall (due to the environment) is measured. The time constant of the exponential decay of temperature gives the heat capacity, provided the thermal resistance is known by a calibration measurement. In addition to home-built apparatuses, one can use commercially available calorimeters. The relaxation method usually deals with samples ranging between 0.5 g and 10 mg. The accuracy of the commercially available relaxation calorimeter (Quantum Design, Physical Property Measurement System; PPMS) is critically examined.21c,d The difficulty encountered in extraction of accurate heat capacity by conventional relaxation calorimetry at firstorder or very sharp second-order phase transition is overcome by applying the so-called scanning method.21e (iv) DSC is one of the methods belonging to thermal analysis, in which the physical property of a sample is monitored under a programmed temperature control and the results are recorded together with the temperature. Many kinds of DSC are commercially available. The amount of sample used for this method is a few milligrams to ∼10 mg. The great merit is a simple operation supported by an equipped computer.

2. ORGANIC FREE RADICALS The study of genuine organic ferromagnets is one of the research subjects attracting many organic and physical chemists, because the carrier of spin is neither d- nor f-electron but pelectron and also because one can easily design a variety of molecular and crystal structures in comparison to traditional magnets such as metals and metal oxides. Furthermore, because most organic molecules are magnetically isotropic, they can be regarded as ideal Heisenberg spin systems, which lead to interesting quantum spin systems. Despite tiny magnetic anisotropy of the spins, however, most organic free radical crystals exhibit low-dimensional magnetic properties owing to structural anisotropy and relative arrangement of their constituent molecules. In 1991, Kinoshita and his collaborators22 discovered that βphase crystal of p-nitrophenyl nitronyl nitroxide (p-NPNN) becomes a bulk ferromagnet. Since then, several bulk ferromagnetic organic radical crystals have successfully been synthesized as follows: nitronyl nitroxide derivatives,23−27 2,2,6,6-tetramethylpiperidin-1-oxyl (TEMPO)-derivatives,28−33 verdazyl derivatives,34 the α-phase crystal of N,N′-dioxy-1,3,5,7tetramethyl-2,6-diazaadamantane,35 and a complex consisting of tetrakis(dimethylamino)ethylene (TDAE) and C60.36

1.3. Heat-Capacity Calorimetry

Roughly classifying the methods of determination of heat capacity, there are four experimental methods: (i) adiabatic calorimetry, (ii) a.c. calorimetry, (iii) relaxation calorimetry, and (iv) differential scanning calorimetry (DSC).21a (i) Adiabatic calorimetry is a static method to determine precisely absolute values of heat capacity in thermal equilibrium. Because this calorimetry has mainly been used for the purposes of basic research, commercially available apparatuses are scarce. Consequently, researchers construct their own calorimeters suitable for the specific purposes. The adiabatic principle can be employed for heat-capacity measurements from cryogenic temperatures to ∼1000 K. A shortcoming of this method is that it requires a large amount of specimen, usually several grams to ∼100 mg. (ii) The other three calorimetries belong to dynamic methods. In a.c. calorimetry, the sample is heated periodically by chopped light or sinusoidal Joule heating. The amplitude of the temperature response at the same frequency is inversely proportional to the heat capacity. Because the absolute values of heat capacity of a sample cannot directly be obtained by this method, it is necessary to adjust the scale of the experimental values by use of known heat-capacity data determined by another method. The amount of sample needed for the a.c. method is as small as several micrograms. Therefore, this method enables us to measure the heat capacity of a tiny single crystal. An apparatus workable under high pressure is also developed.21b

2.1. Nitronyl Nitroxide Radicals

Crystalline p-NPNN (Figure 3) forms four crystalline polymorphs, monoclinic α-phase, 37 orthorhombic βphase,38,39 monoclinic βh-phase,37,40 and triclinic γ-phases,41 which all have ferromagnetic intermolecular coupling. In 1991, Kinoshita et al.42,43 discovered on the basis of magnetic and heat-capacity measurements that γ-phase of p-NPNN exhibits a ferromagnetic phase transition at TC = 0.65 K (see Figure 4) and the heat-capacity hump above TC is well-approximated by the S = 1/2 1D ferromagnetic Heisenberg model with J/k = 4.3 K. In the same year, they also discovered that β-phase of pNPNN shows a ferromagnetic phase transition at TC = 0.60 K (Figure 4).22 Nakazawa et al.40 reported that the magnetic phase transition of the γ-phase of p-NPNN at TC = 0.65 K is D


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net. The magnetic entropy of the β-phase rose very steeply at TC as encountered in ordinary 3D ferromagnets and reached the value R ln 2 expected for S = 1/2 spin systems, while that of the γ-phase increased gradually above TN due to the short-range order effect characteristic of 1D magnetic systems and approached to R ln 2. The magnetic field dependence of the heat capacity of the γ-phase was analyzed in the framework of a mean field approximation to provide the interchain interactions J′/k = 0.224 K and J″/k = −0.176 K.44 On the other hand, the 3D magnetic interaction of the β-phase was estimated to be J/k = 0.6 K. The critical temperature and the magnetic properties of all the radical crystals described in this chapter are summarized in Table 1. Takeda et al. 45 investigated the mechanism of the intermolecular magnetic interactions of the β-phase of pNPNN by experiments under pressure and magnetic field. From the pressure variable heat-capacity and a.c. magnetic susceptibility measurements, a drastic depression of the Curie temperature and a pressure-induced ferromagnetic-to-antiferromagnetic transition at Pc = (0.65 ± 0.05) GPa were observed. This transition was also detected by the pressure dependence of the lattice parameters. At ambient pressure, the effective intermolecular magnetic interaction was estimated to be zJ/k = 2.3 K by the magnetic heat capacity under magnetic field. Furthermore, the magnetic heat capacity turned to be 2D ferromagnetic under a high pressure such as P = 0.72 GPa. Sugawara et al.24,46 designed a new nitronyl nitroxide derivative carrying a hydroquinone moiety, 2-(2′,5′-dihydroxyphenyl)-4,4,5,5-tetramethyl-4,5-dihydro-1H-imidazolyl-1-oxy3- oxide (HQNN; Figure 3), to control the molecular assembly and thus the magnetic property by the intra- and intermolecular hydrogen bonding. HQNN crystallizes into either of two polymorphs: monoclinic α- and β-phases. The temperature dependence of the magnetic susceptibility of α-HQNN was reproduced well by the singlet−triplet model with the ferromagnetic intradimer exchange interaction of J/k = 0.93 K with a positive Weiss constant of θ = +0.46 K. The a.c. magnetic susceptibility increased rapidly around 0.5 K, suggesting a ferromagnetic phase transition around this temperature. The temperature dependence of the heat capacity of α-HQNN exhibited a λ-shaped anomaly with a peak due to the ferromagnetic phase transition at TC = 0.42 K. The associated entropy change was evaluated to be 5.4 J K−1 mol−1, which is in accord with the theoretical value R ln 2. As seen in Table 2a,46 the ratio of TC and θ, and relative ratio of the entropy gained below TC (designated as SC) and that above TC [designated as (S∞ − SC)] are close to the 3D Heisenberg model for a simple cubic lattice. These facts suggest that αHQNN is a 3D ferromagnet. On the other hand, the temperature dependence of the magnetic susceptibility of βHQNN was fitted well by the singlet−triplet model with the ferromagnetic intradimer exchange interaction of J/k = 5.0 K assuming the interdimer interaction as a negative Weiss constant of θ = −0.32 K. They also synthesized an analogous organic radical 2-(3′,5′-dihydroxyphenyl)-4,4,5,5-tetramethyl4,5-dihydro-1H-imidazolyl-1-oxy-3-oxide (RSNN; Figure 3) whose crystal structure resembles that of β-HQNN. The temperature dependence of the magnetic susceptibility was also reproduced well by the singlet−triplet model with the ferromagnetic intradimer exchange interaction of J/k = 10 K with a negative Weiss constant of θ = −4.0 K. Although the analysis using the Bleaney−Bowers equation (a magnetic susceptibility equation proposed for isolated spin-dimer

Figure 3. Molecular structures of nitronyl nitroxide radicals.

Figure 4. Molar heat capacities of β- and γ-phases of p-NPNN. The solid curves for these phases at the LT side represent the spin-wave approximation for 3D ferromagnetic and 3D antiferromagnetic systems, respectively. Reprinted with permission from ref 40. Copyright 1992 American Physical Society.

not ferromagnetic but antiferromagnetic from the magnetic field variable a.c. magnetic susceptibility and heat-capacity data. It was found by differential thermal analysis (DTA) measurement that the β-phase is more stable than the γ-phase, and the γ-phase sample reported previously42,43 was contaminated by a small amount of the β-phase. Spin-wave analyses for the magnetic heat capacities of both the β- and γ-phases below their magnetic phase transition temperatures revealed that the magnetic heat-capacity data of the β-phase are proportional to T 3/2, corresponding to a 3D ferromagnet, while those of the γ-phase are proportional to T3, suggesting a 3D antiferromagE


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Table 1. Critical Temperature and Magnetic Properties of Organic Radical Crystals dimensionality compound

Tc (K)

below Tc

p-NPNN (β-phase) p-NPNN (γ-phase) HQNN (α-phase) HQNN (β-phase) RSNN o-FPNN 2,5-DFPNN F5PNN ImNN BImNN F4BImNN DNPNN MOTMP MATMP CATMP BATMP p-CDTV p-BDTV p-CDpOV DPTOV p-NTpV NDpV NDpMV p-NDpMV TOV p-CyDOV p-NCC6F4CNSSN (α-phase) p-NCC6F4CNSSN (β-phase) BDTA BBDTA·GaBr4 DMTzNC-TCNQ m-MPYNN·BF4 DCPTTCPPA PNNNO (biradical) F2PNNNO (biradical) PIMNO (biradical) PNNBNO (triradical) BImtBN BImPhtBN ClBImtBN Me2BImtBN Tris-NO (triradical) (N-Me-2,6-di-Me-Pz)(TCNQ)2 NNDPP·FeBr4 (biradical)

0.60 0.65 0.42

3D-F 3D-AF 3D-F

0.3 0.45

3D-F

0.72 1.1 0.14 0.15 0.28 0.19 0.68

3D-AF 3D-F 3D-AF 3D-AF 3D-F 3D-F 3D-AF

0.21 5.45 1.16

3D-F 3D-AF 3D-AF

4.8 0.135 8 35.5 11 0.42 1.49

3D-AF 3D-AF 3D-AF 3D-AF 3D-AF 3D-F 3D-AF

0.40 1.1

3D-AF 3D-AF

2.5 0.28 1.7

3D-AF 3D-F 3D-AF

0.74 6.7

above Tc 3D-F 1D-F F-dimer F-dimer F-dimer 1D-F 1D-F 1D-AF AF-dimer 1D-F 1D-F 1D-F 1D-F 1D-F 2D-F 1D-F 1D-F 1D-AF 1D-F 1D-AF 1D-F 1D-AF

interaction parameter

θ (K)

J·k−1 (K)

refs

0.6 (3D) 4.3 (1D); 0.224, −0.176 (inter-1D) 0.93 (intradimer) 5.0 (intradimer) 10 (intradimer) 0.6 (1D) 0.70 (1D) −3.1, −1.24 (alternate-1D) −88 (intradimer) 22(1D) 22 (1D); −0.12 (inter-1D) 5.6 (1D); 0.9 (inter-1D) 0.45 (1D); 0.03, −0.02 (inter-1D) 0.70 (1D); −0.045 (inter-1D) 0.42 (2D); 0.024 (inter-2D) 0.95 (1D); 0.026 (inter-1D) 6.0 (1D) −20.6 (1D) 5.5 (1D); 3 × 10−2 (inter-1D) 7.25 (1D) 3.5 (1D); −1.5 (inter-1D) −5.8 (1D)

22, 40 40, 42, 43 46 46 46 23 26, 47 50 53 53b, 54 55 27 58, 60, 61 62, 64 31, 65 28, 66 73, 77, 79b 73, 77 34 70 71 71 71 71 79 80a, 81 87a, 88 87b, 88 89 90 91 92 96 97 97 97 99 100, 102 101, 103 101 101 104 105 106

1.2 2 0.46 −0.32 −4.0 0.48 0.66

8.2 11 0.16 0.69 0.7 3.0 −25 3.35 −7.5 3.1 −7.3 −18

1D-AF 2D-AF SP

−9.9 −25 −102 −93

0.75 SL F-dimer

−56.5, −22.5 (alternate-1D) −4.5 (2D) −42 (1D) −83 (1D) −99, 4.4 (1D); 0.15 (inter-1D) −32.5 (1D) −1.6 (2D) 14.0 (1D); 0.02, −0.01 (inter-1D) 319 (intraradical); −7.3 (1D) 204 (intraradical); −34, −3.7 (2D) 108 (intraradical); −6 (1D) 430, −108 (intraradical); −0.3 (1D) −1.2 (2D); −1.9 (inter-2D)

1D-F + 1D-AF 1D-AF 2D-AF 1D-F 1D-AF 2D-AF 1D-AF SL BL SL or BL AF-dimer AF-dimer

3D-AF 3D-F

Weiss constant

−22 (intradimer) −24 (intradimer) 3.0, −0.68 (interradical) −14 (1D); −560 (inter-1D) ≥700 (intraradical)

ferromagnetic Heisenberg model with J/k = (0.6 ± 0.1) K. The temperature dependence of the a.c. magnetic susceptibility below 0.25 K also suggested that this magnetic transition is ferromagnetic. On the other hand, among four difluorophenylα-nitronyl nitroxides, only the 2,5-derivative (2,5-DFPNN; Figure 3) had ferromagnetic interactions.26 a.c. magnetic susceptibility and heat-capacity measurements revealed that 2,5-DFPNN exhibits a ferromagnetic phase transition at TC = 0.45 K at ambient pressure.26,47 As shown in Figure 5, when pressure was applied, the heat-capacity anomaly was linearly shifted toward HT side up to 0.57 K at 1.52 GPa.47,48 Furthermore, application of pressure made the shape of the heat-capacity anomaly above TC changed to that characteristic of the S = 1/2 2D Heisenberg ferromagnet of a square lattice

system) modified with an enhancement factor T/(T − θ) is widely applied to weakly coupled spin-dimer systems, it is not obvious for these compounds possessing no remarkable shortrange order to be modeled by such a singlet−triplet system. Nakatsuji et al. prepared some halogen-substituted nitronyl nitroxide derivatives, 2-(2′-halophenyl)-α-nitronyl nitroxides23 and difluorophenyl-α-nitronyl nitroxides.26 Among four kinds of 2-(2′-halophenyl)-α-nitronyl nitroxides, only 2-(2′-fluorophenyl)-α-nitronyl nitroxide (o-FPNN; Figure 3) indicated a positive Weiss constant θ = +0.48 K.23 a.c. magnetic susceptibility and heat-capacity measurements revealed a magnetic phase transition around 0.3 K. A broad heat-capacity hump observed at the high temperature side of the phase transition peak was well reproduced by the S = 1/2 1D F


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interaction ratio α = |J2/J1| = 0.4 (J1/k = −3.1 K) giving rise to a spin-gap system. Mito et al.50 performed heat-capacity measurement of F5PNN crystal under pressure up to 0.78 GPa to confirm the pressure-induced crossover from the alternating bond system to the uniform one. The exponential temperature dependence of heat capacity at low temperature and at ambient pressure was changed gradually and continuously to a linear one with increasing pressure, and eventually the alternating magnetic interaction was transformed into the uniform one (α = 1, J1/k = −4.5 K) at 0.65 GPa. On the other hand, Yoshida et al.51 investigated the magnetic-field dependence of the heat capacity of F5PNN crystal. Two critical magnetic fields were found at HC1 (= 3 T) and HC2 (= 6.5 T). In the H ≤ HC1 magnetic field region (spin-gapped phase; Figure 6a), a heatcapacity hump around 2 K due to the short-range ordering in 1D magnets was suppressed and broadened with an increasing magnetic field. Moreover, the exponential behavior changed gradually to an almost linear T dependence, indicating the suppression of the energy gap, which was estimated to be ΔE/k = 4.0 K in zero magnetic field.51b However, the magnetic heat capacity at 2.5 T was not monotonously decreased to zero when the temperature was approached to 0 K. In the HC1 ≤ H ≤ HC2 magnetic field region (gapless Tomonaga−Luttinger liquid phase; Figure 6b,c), an upturn or a cusp-like anomaly51a due to the field-induced magnetic ordering (FIMO) was observed for T ≤ 0.2 K, which was enhanced with increasing magnetic field up to 5 T and suppressed in the higher magnetic field. Furthermore, another hump emerged around 0.6 K at 4.5 T. With increasing magnetic field, this hump was shifted to the lower temperature, and then merged with the sharp anomaly due to the FIMO at 5.5 T. This hump was not finally observed at 6 T, while a shoulder appeared around 0.7 K at 6.5 T. In the H ≥ HC2 magnetic field region (spin-polarized phase; Figure 6d), the exponential decay was seen again at low temperatures as in the case for H ≤ 2.5 T. However, the magnetic heat capacity at 7 T did not approach zero monotonously down to 0 K as is the case of 2.5 T. The shoulder found at 6.5 K was also seen and developed with increasing magnetic field. These features were qualitatively in good agreement with the theoretical calculations on the S = 1/2 two-leg spin ladder model with the intra- and interchain interactions, J///k = −1 K and J⊥/k = −5.28 K, respectively52 (see the insets in Figure 6). As a new strategy to control intermolecular packing and magnetic behavior, Yoshioka et al.53 introduced an imidazole moiety into nitronyl nitroxide to form hydrogen-bonded NH···ON chains in the solid state. They synthesized 2(imidazole-2-yl)-4,4,5,5-tetramethylimidazoline-1-oxyl-3-oxide (ImNN; Figure 3) and 2-(benzimidazole-2-yl)-4,4,5,5-tetramethylimidazoline-1-oxyl-3-oxide (BImNN; Figure 3). Magnetic susceptibility measurements revealed that ImNN with S = 1/2 makes dimerization with the very strong antiferromagnetic interaction Jdimer/k = −88 K,53 while BImNN forms S = 1/2 ferromagnetic Heisenberg chain with the strong ferromagnetic interaction Jchain/k = +22 K.53b,54 Recently, Lahti et al.55 synthesized a derivative of BImNN, 2-(4,5,6,7-tetrafluorobenzimidazol-2-yl)-4,4,5,5-tetramethy-4,5-dihydro-1H-imidazole-3oxide-1-oxyl (F4BImNN; Figure 3). Magnetic measurements55 revealed that F4BImNN behaves as an S = 1/2 1D ferromagnetic spin system with Jchain/k = +22 K and forms a bulk antiferromagnet at TN ≈ 0.7 K. Heat capacities of F4BImNN under magnetic fields 55b also indicated an antiferromagnetic phase transition at TN = 0.72 K and a hump characteristic of 1D spin systems above TN (see Figure

Table 2. (a) Ratio of TC/θ and Entropy of Spin System in Lower Temperature Region than TC (SC) and That in Higher Temperature Region than TC (S∞ − SC),a,b (b) Results for Nonlinear Least-Squares Fits to Cmag(BImPhtBN) versus T Data at Zero Fieldc (a) model or crystal

TC/θ

SC (%)

(S∞ − SC) (%)

mean field Ising sc Ising diamond Heisenberg fcc Heisenberg bcc Heisenberg sc α-HQNN46

1 0.75 0.67 0.67 0.63 0.56 0.56

100 81 74 67 65 62 60

0 19 26 33 35 38 40

(b) model

J/k

variance (nσ2)

ST 1DCH 2DSQ SQ bilayer

−22.1 K (JST) −16.8 K (J1DCH) −12.8 K (J2DSQ) −2.8 K (JSQ) −21.8 K (Jinter) −4.0 K (JCH) −21.7 K (Jinter)

0.576 20.7 2.33 0.502

spin ladder

0.501

a

Reprinted with permission from ref 46. Copyright 1997 American Chemical Society. bSC, entropy change between 0 K and TC; S∞, entropy change of magnetic phase transition. cAll fits were done using the data between 8.2 and 101.5 K.

Figure 5. Pressure dependence of the magnetic heat capacity of 2,5DFPNN in the hydrostatic pressure region up to 1.52 GPa. The dotted and solid curves express the theoretical curves of S = 1/2 1D Heisenberg ferromagnetic system with intrachain interaction J1D/k = 0.70 K and S = 1/2 2D Heisenberg ferromagnetic system with intraplane interaction J2D/k = 0.70 K, respectively. Reprinted with permission from ref 47. Copyright 2003 American Physical Society.

with J2D/k = 0.70 K. The enhancement of TC was considered to originate mainly from the development of the 2D ferromagnetic interaction on the ac plane. The ferromagnetic signal of the a.c. magnetic susceptibility, however, was decreased by applying pressure and almost disappeared at P ≥ 5.0 GPa, which would arise from the decrease of the ferromagnetic interaction along the b-axis. One of the halogen-substituted nitronyl nitroxide derivatives, pentafluorophenyl nitronyl nitroxide (F5PNN; Figure 3), was synthesized by Hosokoshi et al.49 This is an S = 1/2 1D Heisenberg antiferromagnet with the alternating magnetic G


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Figure 6. Temperature dependence of the magnetic heat capacity Cm in magnetic fields. (a) 0 ≤ H/T ≤ 2.5 (spin-gapped phase), (b,c) 3 ≤ H/T ≤ 6.5 (gapless phase), and (d) 7 ≤ H/T ≤ 8 (spin-polarized phase). A solid line in (b) shows linear T dependence of Cm. Down arrows in (b) and (c) indicate a hump in Cm. Up arrows in (b) and (c) and down arrows in (d) indicate a shoulder Cm. Inserts show theoretical calculations of the strongly coupled two-leg ladder system. Reprinted with permission from ref 51c. Copyright 2005 American Physical Society.

7). From the magnetic field dependence of TN, the critical magnetic field was estimated to be Hc ≈ 0.18 T, from which the interchain magnetic interaction is zJinter/k = (−)0.12 K.

2.2. TEMPO Radicals

Most TEMPO radical crystals possess antiferromagnetic interactions.33,57−59 Kamachi et al.58−64 prepared some TEMPO derivatives (Figure 9a) and investigated their magnetic properties. They found ferromagnetic interactions in 4methacryloyloxy-TEMPO (MOTMP) and 4-acryloyloxyTEMPO (MATMP)crystals.58−60 Ohmae et al. carried out heat-capacity measurements of MOTMP58,60 and MATMP.62 As shown in Figure 10, MOTMP exhibited a magnetic phase transition at Tc = 0.14 K together with a broad hump arising from a short-range ordering characteristic of low-dimensional magnets. This broad anomaly was well-reproduced by the ferromagnetic 1D Heisenberg model with the exchange interaction parameter of J/k = 0.45 K (curve 2). As far as the crystal structure of this compound is concerned, there seems to be no indication of any 1D structure concerning the packing of molecules. This finding of 1D character was afterward confirmed by magnetic susceptibility measurements done below 1 K.62 Since the lattice heat capacity (curve 1) was extremely small in this temperature region, the separation of the magnetic contribution from the observed value was accurately done. The entropy gain due to the phase transition and the hump amounted to 5.8 J K−1 mol−1, which is substantially the same as the expected spin entropy of R ln 2 (= 5.76 J K−1 mol−1). A similar result has been obtained for MATMP. Although the spin-wave analysis (line 3) suggested a ferromagnetic 3D long-range order, magnetic measurements below Tc revealed that the magnetic state is not ferromagnetic but antiferromagnetic.61,63,64 Nogami et al.28−33 synthesized many TEMPO radicals and discovered bulk ferromagnetic behaviors in some TEMPO crystals (Figure 9a). Among them, Miyazaki et al. measured heat capacities of CATMP65 and 4-benzylideneamino-TEMPO (BATMP)66 crystals and found ferromagnetic phase transitions at TC = 0.28 and 0.19 K, respectively. Figure 11 illustrates the excess heat capacity of CATMP plotted on logarithmic and normal scales. As shown by the solid curves, the heat-capacity anomaly appearing above TC is well-reproduced by the 2D ferromagnetic Heisenberg model with the intralayer magnetic interaction of J/k = 0.42 K. The spin-wave analysis (broken curves in Figure 11) suggested 3D ferromagnetic state in the long-range ordered phase with an interlayer magnetic interaction of J′/k = 0.024 K. X-ray structural analyses28,31 revealed that both crystals have 2D sheet structures. This result

Figure 7. Experimental heat capacity for F4BImNN versus temperature at several external magnetic fields. Reprinted with permission from ref 55b. Copyright 2008 American Chemical Society.

Very recently, Shiomi et al.27 found that the achiral 3,5dinitrophenyl derivative of nitronyl nitroxide radical (DNPNN; Figure 3)56 crystallizes in two enantiomorphs with chiral space groups of P43 and P41. The P43 form (L form) has left-handed stacking of the molecules, giving the helical chirality in a crystalline solid. In the other form of P41 (R form), the righthanded stacking corresponds to a mirror image of the L form. Magnetic susceptibility measurement suggested that the L form of DNPNN undergoes a ferromagnetic phase transition at TC = 1.1 K and is an S = 1/2 1D ferromagnet with the intrachain exchange interaction 2J/k = +12.0 K above TC, which is the same as those of the R form. Heat-capacity measurements under magnetic fields (Figure 8a) suggested that the L form of DNPNN gives rise to a sharp heat-capacity peak due to a ferromagnetic phase transition at TC = 1.1 K accompanying a heat-capacity hump which is well reproduced by the S = 1/2 1D ferromagnetic Heisenberg model with the intrachain exchange interaction 2J/k = +11.2 K. Experimental magnetic entropy (Figure 8b) is consistent with R ln 2 = 5.76 J K−1 mol−1 as expected for S = 1/2 spins. H


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Figure 8. (a) Magnetic heat capacity Cmag of DNPNN (L form). The static magnetic fields from 0 T up to 9 T applied to the polycrystalline sample are shown by different colors. Red solid line represents the theoretical Cmag value calculated by the ferromagnetic chain model with 2J/k = +11.2 K, as fitted to the observed Cmag under 0 T. (b) Magnetic entropy Smag of DNPNN (L form) obtained by numerical integration of the Cmag values. Black circles indicate the experimental values of Smag. Blue and red lines represent the theoretical values of magnetic entropy calculated from the spin-wave theory and the S = 1/2 1D ferromagnetic Heisenberg model (2J/k = +11.2 K), respectively. The horizontal line corresponds to the entropy Smag = R ln 2 for S = 1/2 spins. Reprinted with permission from ref 27. Copyright 2011 American Chemical Society.

consists of the observation of the heat-capacity anomaly above TC. Mito et al.67 performed magnetic and heat-capacity measurements under pressure to investigate the pressure dependence of the ferromagnetic phase transition of CATMP crystal. They found three pressure-induced phenomena: (i) down−up variation with a period of about 0.4 GPa for P < 0.9 GPa and monotonous and slight increase for P ≥ 0.9 GPa of the magnetic transition temperature, (ii) ferromagnetic-to-antiferromagnetic transition around 0.5 GPa, and (iii) a reduction of the magnetic lattice dimensionality from 2D Heisenberg ferromagnet to 1D one, which can be understood by the pressure-induced rotation of the methyl moiety relevant to the interaction mechanism through C−H···O−N contacts.

NDpMV), and a 2D antiferromagnet 1,3,5-triphenyl-6oxoverdazyl (TOV) crystal.79 Recently, this research group studied the doping effect of 3(4-cyanophenyl)-1,5-dimethyl-6-thioxoverdazyl (p-CyDTV) on 3-(4-cyanophenyl)-1,5-dimethyl-6-oxoverdazyl (p-CyDOV), which shows spin-Peierls transition at TSP = 15.0 K,80 by heat capacity measurements of (p-CyDOV)1−x(p-CyDTV)x crystals.81 Antiferromagnetic phase transitions were observed at TN = (0.135 ± 0.02) K, (0.290 ± 0.02) K, and (0.164 ± 0.02) K for the crystals with x = 0, 0.01, and 0.07, respectively. The coexistence of antiferromagnetic long-range order and the spinPeierls state was confirmed in the systems with x = 0 and 0.01, while in the system with x = 0.07 only a single antiferromagnetic phase appeared below TN = 0.164 K.

2.3. Verdazyl Radicals

2.4. Thiazyl Radicals

Verdazyls provide a family of stable organic radicals (Figure 9b). Ferromagnetic interactions were first demonstrated in 3(4-nitrophenyl)-1,5,6-triphenylverdazyl (p-NTpV), which indicated a positive Weiss constant of θ = 1.6 K.68 Several verdazyl derivatives have so far been reported to exhibit magnetic phase transitions.34,69−72 Mukai et al. measured magnetic susceptibility73 and heat capacity69,74−77 measurements of 3-(4-chlorophenyl)-1,5-dimethyl-6-thioxoverdazyl (p-CDTV) and 3-(4-bromophenyl)1,5-dimethyl-6-thioxoverdazyl (p-BDTV) crystals. Takeda and his collaborators34,78 investigated magnetic-field dependence of the heat capacity of 3-(4-chlorophenyl)-1,5-diphenyl-6-oxoverdazyl (p-CDpOV) crystal. They observed a sharp peak due to a ferromagnetic phase transition at TC = 0.21 K and a wide plateau around 1−4 K, characteristic of 1D quantum ferromagnets (Figure 12). This sharp peak was easily affected by the small magnetic field, and another round hump appeared, increased in height and shifted toward the high temperature region as the magnetic field was increased. They also investigated the magnetic and thermal properties of four 1D antiferromagnets,71 p-NTpV, 3-nitro-1,5-diphenylverdazyl (NDpV), 6-methyl-3-nitro-1,5-diphenylverdazyl (NDpMV), and 6-methyl-3-(4-nitrophenyl)-1,5-diphenylverdazyl (p-

Thiazyl radicals (Figure 9c) are candidates of HT bulk magnets because they are characterized by strong antiferromagnetic exchange interactions between radical centers.82 Their unique intermolecular interactions have resulted in unusual solid-state properties.83−86 The dithiadiazolyl radical p-NCC6F4CNSSN87 (Figure 9c) has two crystal polymorphs: triclinic α-phase and orthorhombic β-phase. The α-phase becomes antiferromagnetic below TN = 8 K.87a Although the β-phase is also an antiferromagnet, it gives rise to a spontaneous magnetization below TN = 36 K. This implies noncolinear antiferromagnetism.87b Palacio et al.88 performed heat-capacity measurement of the β-phase and observed a peak due to the noncolinear antiferromagnetic phase transition at TN = 35.5 K, above which a broad shoulder characteristic of low-dimensional magnets was found. Fujita et al.89 studied the magnetic and thermal properties of 1,3,2-benzodithiazolyl (BDTA, Figure 9c). Temperature dependence of the magnetic susceptibility of BDTA is shown in Figure 13. The as-prepared sample is in a diamagnetic state (DS) in the range from 4.4 to 360 K, at which it is suddenly transformed into a paramagnetic state (PS). On cooling down to 4.2 K, the sample remains in PS. However, when the temperature is increased, the PS state is irreversibly transformed to the DS around 270 K. On the basis of this result, I


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Figure 9. Molecular structures of (a) TEMPO radicals, (b) verdazyl radicals, (c) thiazyl radicals, and (d) other radicals.

Shimizu et al.90 synthesized the organic radical cation salt BBDTA·GaIIIBr4 (BBDTA = benzo[1,2-d:4,5-d′]bis[1,3,2]dithiazole; Figure 9d) and carried out magnetic susceptibility and heat-capacity measurements. This salt consists of antiferromagnetic chains and ferromagnetic chains. Because the antiferromagnetic chains do not contribute to the magnetic

together with the result of the calorimetric study, one can get a Gibbs energy phase diagram shown in Figure 13. Namely, DS is the stable phase and PS is a metastable phase below 346 K, above which the stabilities of both phases are reversed. Heatcapacity measurement of the supercooled PS phase indicated an antiferromagnetic phase transition at TN = 11 K. J


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Figure 10. Molar heat capacity of MOTMP. Curve 1, lattice heat capacity; curve 2, ferromagnetic 1D Heisenberg model (J/k = 0.45 K), and line 3, spin-wave contribution. Reprinted with permission from ref 60. Copyright 1993 Taylor & Francis Group.

Figure 12. Field dependence of the magnetic heat capacity of pCDpOV crystal. Solid curves are the theoretical results for the S = 1/2 1D ferromagnetic system. Reprinted with permission from ref 34. Copyright 1995 The Physical Society of Japan.

Figure 13. Temperature dependence of the paramagnetic susceptibility χp for the virginal sample of BDTA radical crystal upon heating (gray circles) and upon cooling (closed circles). Open circles show the dependence for the sample after the thermal cycles upon heating. The inset shows the expected Gibbs energy phase diagram. Reprinted with permission from ref 89. Copyright 2002 Elsevier Ltd. Figure 11. Magnetic heat capacities of CATMP free radical crystal on (a) logarithmic and (b) normal scales. Solid curves indicate the heat capacities calculated from the HT series expansion for the S = 1/2 2D ferromagnetic Heisenberg model of square lattice with J/k = 0.42 K. Broken curves show the heat capacities derived from the spin-wave theory for 3D ferromagnets. Reprinted with permission from ref 65. Copyright 2000 The Chemical Society of Japan.

0.42 K. The interchain interaction between the ferromagnetic chains was estimated to be 2zJ′/k = 0.29 K. 2.5. Other Radicals

Takagi et al.91 reported magnetic susceptibility and heat capacity of an organic radical salt, [3,3′-dimethyl-2,2′thiazolinocyanine]-[7,7,8,8-tetracyanoquinodimethane] (DMTzNC-TCNQ) (Figure 9d). The heat capacity showed a sharp peak due to an antiferromagnetic phase transition at TN = 1.49 K, below which it gave rise to T n dependence expected for a gapless spin-wave excitation.

property at low temperatures because of a large intrachain interaction 2JAF/k = −197 K, the ferromagnetic chains with 2JF/k = 8.7 K show a 3D ferromagnetic ordering alone at TC = K


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Figure 14. (a) Heat capacity of F2PNNNO for the magnetic field parallel to the a axis. (b) phase diagram of F2PNNNO determined by the heat capacity measurements. The line is guide to the eye. Reprinted with permission from ref 98. Copyright 2007 Elsevier Ltd.

magnetic field Hc2 = 15.4 T (see Figure 14a). The positions of the heat-capacity peaks give the phase diagram as shown in Figure 14b. The phase boundary of the ordered phase is symmetric with respect to the horizontal line of H = 12.4 T. Hosokoshi et al. also designed a novel triradical, 2-[3′,5′-bis(Ntert-butylaminoxyl)phenyl]-4,4,5,5-tetramethyl-4,5-dihydro-1Himidazol-1-oxyl 3-oxide (PNNBNO) (Figure 9d) to obtain an organic ferrimagnet. 99 The magnetic susceptibility of PNNBNO was well-reproduced by the calculation with 2JF/k = 860 K, 2JAF/k = −216 K, and 2J′AF/k = −0.6 K. In the heat capacity of PNNBNO, a λ-shaped peak due to a ferromagnetic phase transition was observed at TC = 0.28 K. The total magnetic entropy gain reached 91% of R ln 2 at 3.3 K, which suggested that the effective S = 1/2 species undergo a magnetic phase transition at TC = 0.28 K. One of the promising strategies for controlling packing of organic free radicals in the solid state is to use hydrogen bonding. Recently, Lahti et al.100,101 synthesized new organic radicals hydrogen-bonded benzimidazole-based tert-butylnitroxides, benzimidazole-2-tert-butylnitroxide (BImtBN), 4-(1Hbenzimidazol-2-yl-phenyl)-tert-butylnitroxide (BImPhtBN), 5(6)-chloro-BImtBN (ClBImtBN), and 5,6-dimethyl-BImtBN (Me2BImtBN) (Figure 9d). The single-crystal X-ray crystallography100,101 of BImtBN showed that bilayers made up of two sheets parallel to the ab plane are formed, whereas the magnetic susceptibility measurement100 revealed that its magnetism can be well expressed by the S = 1/2 square planar 2D antiferromagnetic Heisenberg model with the intralayer exchange interaction of J/k = −1.6 K. The magnetic structure of BImPhtBN resembled that of BImtBN, that is, the S = 1/2 square planar 2D antiferromagnetic Heisenberg model with the intralayer exchange interaction of J/k = −15 K,101 while ClBImtBN and Me2BImtBN obeyed the singlet−triplet models with the intradimer exchange interactions J/k = −22 and −24 K, respectively.101 Miyazaki et al.102 performed heat-capacity measurement of BImtBN. An antiferromagnetic phase transition was observed at TN = 1.7 K. As compared in Figure 15, the heat capacity hump observed above TN was well-reproduced by the S = 1/2 antiferromagnetic bilayer Heisenberg model with the intralayer interaction J1/k = −1.2 K and the interlayer interaction J2/k = −1.9 K. The magnetic entropy was determined to be 5.34 J K−1 mol−1, which is in good agreement with the theoretical value for the magnetic ordering of S = 1/2 spin systems, R ln 2 = 5.76 J

Wada et al.92 studied the magnetic property of an organic radical salt, m-N-methylpyridinium α-nitronyl nitroxide (mMPYNN)-BF4 (Figure 9d), which is known to be an S = 1 2D Kagomé Heisenberg antiferromagnet.93 The heat-capacity maximum due to a magnetic short-range order was observed at 1.4 K, which was about half of the antiferromagnetic interaction 2|J|/k = 3.1 K in the Kagomé lattice. However, Uekusa and Oguchi94 suggested that m-MPYNN·BF4 is not the S = 1 Kagomé antiferromagnet but a magnetic hexagonalnetwork system with 2J1/k = 20.1 K, 2J2/k = −3.1 K, and 2J3/k = −2.0 K. Subsequent heat-capacity measurement at very low temperatures95 pointed out the existence of another heatcapacity peak at 0.12 K, which does not depend on magnetic fields. As this anomaly is not due to magnetism, it is not attributed to the spin entropy. Teki et al.96 investigated the magnetic and thermal properties of N-[(2,4-dichlorophenyl)thio]-2,4,6-tris(4-chlorophenyl)phenylaminyl (DCPTTCPPA) (Figure 9d). The heat capacity indicated a λ-type anomaly due to an antiferromagnetic phase transition at TN = 0.40 K. The interchain exchange interactions were estimated to be J′/k = 0.02 K and J″/k = −0.01 K. Hosokoshi et al. synthesized stable biradicals 2-[4′-(N-tertbutyl-N-oxyamino)phenyl]-4,4,5,5-tetramethyl-4,5-dihydro-1Himidazol-1-oxyl3-oxide (PNNNO), 2-[2′,6′-difluoro-4′-(N-tertbutyl-N-oxyamino)phenyl]-4,4,5,5-tetramethyl-4,5-dihydro-1Himidazol-1-oxyl 3-oxide (F2PNNNO), and 2-[4′-(N-tert-butylN-oxyamino)phenyl]-4,4,5,5-tetramethyl-4,5-dihydro-1H-imidazol-1-oxyl (PIMNO) (Figure 9d).97 From the magnetic measurements of PNNNO and PIMNO, both biradicals were well-understood by 1D antiferromagnetic chain models of ferromagnetic spin pairs with 2JF/k = 638 K and 2JAF/k = −14.5 K for PNNNO and with 2JF/k = 216 K and 2JAF/k = −12 K for PIMNO. The heat capacities of PNNNO and PIMNO showed λ-shaped peaks due to antiferromagnetic phase transitions at TN = 1.1 and 2.5 K, respectively. On the other hand, F2PNNNO was thought to be a 2D Heisenberg system, in which the ferromagnetic spin pairs with 2JF/k = 407 K were connected by two types of antiferromagnetic interactions with 2J′AF/k = −67 K and 2JAF/k = −7.4 K. Recent heat-capacity measurements under high magnetic fields98 revealed that F2PNNNO does not exhibit long-range magnetic order and has an energy gap Δ/k = 12.6 K below the lower critical magnetic field Hc1 = 9.5 T and that a λ-shaped heat-capacity peak due to an antiferromagnetic phase transition appears between Hc1 and the upper critical L


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Figure 16. Zero-field magnetic heat capacity versus temperature for BImPhtBN with fits to several antiferromagnetic Heisenberg models. Exchange interaction parameters estimated for the models are listed in Table 2b. Reprinted with permission from ref 103. Copyright 2008 American Chemical Society.

Radváková et al.105 measured magnetic susceptibility and heat capacity of the genuine organic radical anion salt (N-Me2,6-di-Me-Pz)(TCNQ)2 (N-Me-2,6-di-Me-Pz = N-methyl-2,6dimetylpyrazine; Figure 9d). Both magnetic susceptibility and heat-capacity data were well-reproduced by the S = 1/2 two-leg Heisenberg spin-ladder model with 2Jrung/k = −28 K, Jrung/Jleg = 40, and the external magnetic field h/k = 50. A novel radical-substitute radical cation, 2-nitronyl-nitroxide5,10-dihydro-5,10-diphenylphenazine (NNDPP; Figure 9d), was synthesized by Okada et al.,106a which behaves as an S = 1 spin system at low temperatures because of a very large intramolecular ferromagnetic interaction Jintra/k ≥ +700 K. By using NNDPP, they prepared a new radical-substitute radical cation salt NNDPP·FeIIIBr4.106b Magnetic and heat-capacity measurements (see Figure 17) elucidated that this salt is a 3D long-range-ordered ferrimagnet with Tc = 6.7 K. The acquired magnetic entropy was 23.8 J K−1 mol−1. This value coincides well with the value expected for a spin system composed of S = 1 (NNDPP) and S = 5/2 [Fe(III)], which is given by R ln (3 × 6) = 24.0 J K−1 mol−1.

Figure 15. Magnetic heat capacity of BImtBN crystal on (a) logarithmic and (b) normal scales. The broken curve drawn in plot a shows the heat capacity obtained from the spin-wave theory for 3D antiferromagnets. In (b), curves A and B indicate the heat capacities calculated on the basis of the HT series expansion for the S = 1/2 square planar 2D Heisenberg model with J/k = −1.6 K and for the bilayer model with the intralayer interaction J1/k = −1.2 K and the interlayer interaction J2/k = −1.9 K, respectively. Curve C shows the heat capacity calculated by the dimer model with Jd/k = −2.8 K. Curve D represents the heat capacity calculated by the bilayer model with J1/ k = −1.4 K and J2/k = −1.3 K derived from the magnetic data. Reprinted with permission from ref 102. Copyright 2002 American Chemical Society.

K−1 mol−1. On the other hand, a hump in the magnetic heat capacity of BImPhtBN103 was observed with a maximum at about 15 K in zero magnetic field, which did not shift at magnetic fields up to 9 T. As shown in Figure 16, this hump was fitted to several antiferromagnetic Heisenberg models. The best fits were obtained using spin ladder and coupled spin bilayer models (see Table 2b). The experimental magnetic entropy agreed well with the expected value for S = 1/2 spin systems, R ln 2 = 5.76 J K−1 mol−1. Takeda et al.104a investigated magnetic properties of N,N,Ntris[p-(N-oxyl-tetra-butyamino)phenyl]amine (Tris-NO; Figure 9d)104b by magnetic susceptibility and heat-capacity measurements. The magnetic susceptibility measurements indicated that tris-NO is in the ground doublet state below 60 K, that is, Tris-NO at low temperatures behaves as a system of interacting S = 1/2 spins. An antiferromagnetic phase transition occurred at TN = 0.74 K. The magnetic entropy attained to more than 95% of R ln 2 for S = 1/2 spins at TN, indicating more evidence of the ground-state doublet within a molecule. The analyses of the magnetic field dependence of heat capacity and magnetization revealed that the intermolecular ferromagnetic and antiferromagnetic interactions are working with the respective values 2zfJf/k = 6.0 K and 2zafJaf/k = −1.35 K.

3. ONE-DIMENSIONAL MAGNETS Because superexchange interactions are principally confined within nearest neighbors, the lattice dimensionality of molecule-based magnets is basically influenced by the anisotropy of the constituent molecule carrying spins and by the relative arrangement of the molecules. This situation leads to low-dimensional magnets in which 1D or 2D interactions are dominant. 3.1. Metallocenium Salts of Radical Anions

As a strategy to realize molecule-based bulk ferromagnets, Miller et al.107,108 proposed charge-transfer complexes, for example, decamethylferrocenium tetracyanoethenide [DMFc][TCNE]. As shown in Figure 18,109 the solid-state structure of [DMFc][TCNE] consists of parallel stacks of alternating [DMFc]+ and [TCNE] − radical ions, although disorder in the anion positions108 has prevented refinement of the orthorhombic structure. The crystal structure includes neighboring stacks that are in and out of registry. Magnetic ordering in this complex was intensively studied by magnetization and magnetic susceptibility measurements, 57Fe Mössbauer spectroscopy, and neutron diffraction. Below 4.8 M


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Figure 17. Heat capacities Cp and magnetic heat capacities Cmag of NNDPP·FeIIIBr4 as functions of temperature under various magnetic fields. (a) Cp with an estimated curve of the heat capacity Clat due to lattice contribution (solid line). Inset: magnetic field dependence of Cp. (b) Temperature dependence of Cmag = Cp − Clat. Inset: magnetic field dependence of Cmag. In the insets, the y axis shows only a relative scale to compare the peak positions under the magnetic fields. Reprinted with permission from ref 106b. Copyright 2009 American Chemical Society.

Figure 19. Magnetic heat capacity of [DMFc][TCNE]. (a) 1D S = 1/ 2 ferromagnetic Heisenberg model (J⊥/k = J||/k = 13 K), (b) 1D anisotropic Heisenberg model (J||/k = 25 K; J⊥/J|| = 0.5), and (c) Ising model (J||/k = 19 K; J⊥/k = 0 K). Reprinted with permission from ref 111. Copyright 1990 Elsevier Ltd. Figure 18. Example of one type of stacking arrangement of donor and acceptor molecules for out of registry chains of the compound [DMFc][TCNE].

strong short-range order characteristic of the 1D stacking structure in this crystal. The temperature region of this hump coincides with the anomalous region of the 57Fe Mössbauer spectra. The total entropy gain due to the phase transition at 4.74 K and the noncooperative anomaly around 8.5 K amounted to ΔS = (12 ± 1) J K−1 mol−1 and is wellapproximated by 2R ln 2 = 11.53 J K−1 mol−1. This fact confirms that the charge transfer from the donor to the acceptor is complete and that the present complex consists of a spin 1/2 cation and a spin 1/2 anion. As can be seen in Figure 19, the hump cannot be reproduced by a usual 1D S = 1/2 isotropic Heisenberg model (curve a: J⊥/ k = J||/k = 13 K). This anomaly is favorably accounted for in terms of anisotropic Ising character (curve c: J||/k = 19 K; J⊥/k = 0 K). The fit was obtained for an anisotropic Heisenberg model (curve b: J||/k = 25 K; J⊥/J|| = 0.5). The origin of this Ising anisotropy can be attributed to the anisotropic g-tensor of the [DMFc]+ cation radical. For the superexchange interaction

K, the charge-transfer complex displays the onset of spontaneous magnetization in zero-applied magnetic field, consistent with a 3D ferromagnetic ground state. However, 57 Fe Mössbauer spectroscopy exhibited zero-field Zeeman split spectra even up to 20 K. This was a great mystery, and at that time, there had been reported no conclusion about the question, whether the Mössbauer line-width broadening is due to single ion relaxation or to a cooperative effect involving solitons in a 1D magnet. Heat capacities of [DMFc][TCNE] were reported by Chakraborty et al.110 and by Nakano and Sorai.111 Figure 19 represents the magnetic heat capacity of [DMFc][TCNE].111 The phase transition expected for the ferromagnetic ordering was actually observed at 4.74 K. Another feature is a remarkable hump centered around 8.5 K. This hump corresponds to the N


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between two magnetic ions A and B, the anisotropy of magnetic interaction is known to be proportional to the product of the gvalues, J⊥/J|| ≈ (g⊥Ag⊥B)/(g||Ag||B). The present cation [DMFc]+ has anisotropic g-values (g|| = 4.0 and g⊥ = 1.3), whereas the anion [TCNE]− is isotropic (g⊥ = g|| = 2.0). Consequently, the magnitude of the interaction anisotropy is expected to be J⊥/J|| = 0.33. This value agrees well with J⊥/J|| = 0.30 derived from the magnetic susceptibility for single crystals.112 The calorimetric result suggesting a very strong Ising character of the magnetic interaction, J⊥/J|| < 0.5, supports the conclusion obtained from the single-crystal susceptibility. Close examination of the magnetic elementary excitations in the ordered phase provides more evidence for the strong Ising character. The excess heat capacity obviously exhibits the exponential temperature dependence characteristic of excitation over an energy gap rather than the power-law dependence of spin-wave excitation. The energy gap was estimated to be ΔE/k = 28 K from the data in the range of 1−3 K. The dominant thermal excitation in [DMFc][TCNE] is inferred to be the bound-state type inherent in the Ising magnet. Decamethylferrocenium 7,7,8,8-tetracyano-p-quinodimethanikde [DMFc][TCNQ] is isomorphous with [DMFc][TCNE]. Miller et al.113 have reported the metamagnetic behavior of [DMFc][TCNQ], in which a dominant interaction is intracolumnar ferromagnetism with a weak intercolumnar antiferromagetism. Heat-capacity measurements of [DMFc][TCNQ]114 exhibited an antiferromagnetic phase transition at TN = 2.54 K with the entropy gain of ΔS = 11.2 J K−1 mol−1. The magnetic lattice anisotropy of the [TCNQ] complex was smaller than that of the [TCNE] complex, as exemplified by a smaller energy gap of ΔE/k = 6.5 K.

Figure 20. Excess heat capacities of MnCu(obbz)·5H2O around the antiferromagnetic phase transition temperature TN. The solid line indicates the theoretical heat-capacity curve estimated by the HT series expansion for S = 2 1D ferromagnetic Heisenberg model with J/k = 0.75 K. The broken straight line shows the heat capacity due to the spin-wave excitation. Reprinted with permission from ref 116a. Copyright 1999 The Chemical Society of Japan.

heat-capacity hump centered around 4 K is due to the shortrange order effect characteristic of a low-dimensional magnet. The entropy arising from these two anomalies was 12.1 J K−1 mol−1. MnCu(obbz)·5H2O has a 2D or 3D network structure formed by assembly of 1D chains, in which Mn2+ and Cu2+ ions are alternately bridged by two different kinds of groups: oxamido and carboxylato. The magnetic susceptibility measurement115 indicated that this complex can be regarded as a 1D ferrimagnet composed of two different spins S = 5/2 of Mn2+ and S = 1/2 of Cu2+ above TN. The magnetic structure of this 1D ferrimagnetic chain is schematically drawn in Figure 21. The exchange interaction parameters for the oxamido and carboxylato bridges between Mn2+ and Cu2+ have been determined to be J1/k = −21 K and J2 = −3.6 K, respectively.

3.2. Bimetallic Chain Complexes

Kahn and his collaborators115 proposed a strategy to design molecule-based compounds exhibiting a spontaneous magnetization by assembling ordered bimetallic chains within the crystal lattice in a ferromagnetic fashion. The novel ferromagnet that they prepared was MnIICuII(obbz)·H2O [obbz = N,N′oxamidobis(benzoato)] with a high Curie temperature, which is obtained by partial dehydration of the molecule-based antiferromagnet MnIICuII(obbz)·5H2O. On the basis of powder X-ray diffraction (XRD) patterns and XANES and EXAFS spectra at both Mn and Cu edges for MnCu(obbz)·5H2O and MnCu(obbz)·H2O, it has been shown that these complexes have very similar structures and that Mn2+ and Cu2+ ions are in octahedral and elongated tetragonal surroundings, respectively. These structural results imply that Mn2+ and Cu2+ ions are bridged alternately by oxamido and carboxylato groups to form 1D chain structures, which are two- or three-dimensionally connected through Mn2+−O(oxamido) bonds. In MnCu(obbz)·5H2O, only one water molecule is coordinated to the Mn2+ ion and the remaining four water molecules are noncoordinated, consistent with the easy dehydration. Magnetic measurements115 indicate that MnCu(obbz)·H2O exhibits a 3D ferromagnetic order below Tc = 14 K while MnCu(obbz)·5H2O shows a 3D antiferromagnetic order below Tc = 2.3 K. Heat capacities of MnCu(obbz)·5H 2 O and MnCu(obbz)·H2O were measured by Sorai and his collaborators116 at temperatures from 0.1 to 300 K by means of adiabatic calorimetry. Figure 20 shows the excess heat capacities ΔCp below 20 K. 116a A phase transition peak due to 3D antiferromagnetic ordering was observed at TN = 2.18 K. A

Figure 21. (a) Schematic magnetic structure of the 1D ferrimagnetic chain in MnCu(obbz)·5H2O. Open and filled circles stand for Mn2+ and Cu2+ ions, respectively. Long and short arrows indicate spins 5/2 and 1/2, respectively. Antiferromagnetic interaction J1 is much stronger than J2. (b) One-dimensional ferromagnetic chain consisting of the resultant S = 2 spins expected for the antiferromagnetically strongly coupled Mn2+ and Cu2+ spin pair. Double circle stands for the Mn2+−Cu2+ pair.116a. O


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Evangelisti et al. reported heat capacities of bimetallic compounds containing 4f-3d electrons. They involve (i) Gd 2 (ox)[Cu(pba)] 3 [Cu(H 2 O) 5 ]·20H 2 O and Gd 2 [Cu(opba)] 3 ·6DMSO·4H 2 O [opba = o-phenylenebis(oxamato)],120 (ii) Ln2[Cu(opba)]3·S (S = solvated molecules; Ln = Tb, Dy, Ho);121a and (iii) L2[M(opba)]3·xDMSO·yH2O (L = La, Gd, Tb, Dy, Ho; M = Cu, Zn; DMSO = dimethyl sulfoxide; hereafter abbreviated by L2M3).121b As to (i), both compounds undergo 3D ferromagnetic ordering with TC = 1.05 K and TC = 1.78 K, respectively. For the compounds of (ii), three compounds undergo 3D magnetic ordering, with Tc = 0.81 K and Tc = 0.74 K for the Tb- and Dy-containing compounds, respectively. For the Ho-containing compound, only the HT tail of the λ-type anomaly was detected by the heat-capacity measurement. These results are compared with the magnetic properties of the isostructural compound Gd2[Cu(opba)]3·S, which is a ferromagnet. The results for L2[M(opba)]3·xDMSO·yH2O showed that the Cu-containing complexes (with the exception of La2Cu3) undergo long-range magnetic order at temperatures below 2 K and that the magnetic ordering is ferromagnetic for Gd2Cu3, whereas for Tb2Cu3 and Dy2Cu3, it is probably antiferromagnetic. The authors120,121 have claimed that these compounds belong to the spin-ladder system owing to their ladder-shaped molecular packing in the crystal lattice. However, their heat capacities do not show the quantum effects at low temperatures characteristic of the spin-ladder system, which is one of the spin-gapped systems. Therefore, as far as the magnetic heat capacities are concerned, these complexes are a family of lowdimensional magnets showing long-range order.

This strong alternation shows that the 1D ferrimagnetic chain can be regarded as a 1D pseudoferromagnetic chain made up of S = 2 resultant spins arising from pairing between the spins of Mn2+ and Cu2+. This approximation is reasonable from the fact that the exchange interaction for the oxamido bridge is much stronger than that for the carboxylato bridge. The total entropy gain theoretically expected for the spin multiplicity of Mn2+ and Cu2+ amounts to R ln(6 × 2) (= 20.7 J K−1 mol−1). However, because this ferrimagnetic chain can be regarded as an S = 2 pseudoferromagnetic chain for the reason described above, only R ln 5 (= 13.4 J K−1 mol−1) contributes to the entropy at low temperatures. Because the experimental entropy gain 12.1 J K−1 mol−1 agrees rather well with this value, one may conclude that the pseudoferromagnetic chain model is a good approximation. The remaining entropy of R ln(12/5) (= 7.3 J K−1 mol−1) would be acquired at higher temperatures by thermal excitation to the S = 3 excited spin level. As seen in Figure 20, the validity of the present pseudoferromagnetic chain approximation is given by the fact that the heat-capacity tail above TN is well-reproduced by this model with J/k = 0.75 K. The effective value for the exchange parameter J between the adjacent resultant spins with S = 2 is derived as J/k = 0.70 K from the experimental values (J1/k = −21 K and J2 = −3.6 K) on the basis of the spin-vector projection method.117 In the case of MnCu(obbz)·H2O, a broad phase transition due to 3D ferromagnetic ordering was observed around 17 K.116b Above the transition temperature, a heat-capacity tail arising from the short-range order characteristic of a lowdimensional magnet was found. As in the case of MnCu(obbz)·5H2O, the entropy gained by the phase transition (12.7 J K−1 mol−1) was close to R ln 5 (= 13.4 J K−1 mol−1) expected for the resultant spin model with the S = 2 ground state. Magnetization measurements116b for MnCu(obbz)·H2O revealed a bend of the hysteresis loop above the transition temperature and a magnetic relaxation below the transition temperature. These results lead to the conclusion that the broadness of the phase transition may be attributed to superparamagnetism caused by the fact that the present complex is prepared only in a fine powder. Coronado et al.118 reported magnetic and thermal properties of bimetallic chain complexes MnIIMII(EDTA)·6H2O (M = Co, Ni, Cu; EDTA = hexadentate ligand ethylenediamineN,N,N′,N′-tetraacetate) in the 1.4−30 K temperature region. Their structure consists of infinite zigzag chains, built up from two alternating [Mn,M] sites. The degree of J-alternation of these compounds varies in the order [Mn,Ni] < [Mn,Cu] ≪ [Mn,Co]. A qualitative explanation of this trend is given on the basis of the structural features and the electronic ground state of the interacting ions. The magnetic behavior exhibits the characteristic features of 1D ferrimagnets, but interchain interactions lead at very low temperature to a long-range antiferromagnetic ordering as outlined by the λ-type heatcapacity peak. Drillon and his collaborators119 reported a theoretical model for ferrimagnetic Heisenberg bimetallic chains [1/2 − S] (S = 1 to 5/2) and provided heat-capacity curves as a function of temperature. It was shown that the main features for the infinite chain are conveniently given by extrapolating the results obtained for spin rings consisting of several pairs of spins. Because the model chains have been characterized by uniform interaction parameter J, application of this model to the systems with strongly alternating interaction should be careful.

3.3. Helical Chain with Competing Interactions

One-dimensional localized magnets have served as suitable systems by which theoretical predictions are tested. Harada122 proposed that the competition between nearest-neighbor (nn) and next-nearest-neighbor (nnn) exchange can give rise to a helical phase, whose 2-fold chiral degeneracy (clockwise and counterclockwise turns of spins along a chain) leads to the excitation of topologically stable chiral domain walls, separating two domains of opposite chirality. Quasi-1D molecular magnets GdIII(hfac)3NITR (hfac = hexafluoro-acetylacetonate, NITR = 2-R-4,4,5,5-tetramethyl4,5-dihydro-1H-imidazolyl-1-oxyl 3-oxide, and R = alkyl or phenyl groups)123 were regarded as good candidates to test Harada’s prediction. Their magnetic properties are determined by Gd3+ rare-earth ions with spin S = 7/2 and the NITR organic radicals with spin s = 1/2, alternating along the chain and interacting via competing nn and nnn exchange. It should be noted here that a capital S and a lower case s are used for the spin quantum number to discriminate the two kinds of spins in a chain. The chains are well-separated one from the other by the presence of the bulky ligands, and only weak dipolar interactions are supposed between chains. The spin Hamiltonian describing the exchange interaction in a single chain is written as follows: N /2

H = −J1

N /2

∑ (S2n− 1·s2n + s2n·S2n+ 1) − J2 ∑ S2n− 1·S2n+ 1 n=1 N /2

− J2′

∑ s 2n · s 2n + 2 n=1

P

n=1

(2)


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gave rise to very small heat-capacity peaks at Tpeak ≈ 2 K, the experimental heat-capacity curves are drastically different from the previous cases. In the case of Gd(hfac)3NITEt, because the total entropy was ΔS = 22.7 J K−1 mol−1, all of the magnetic contributions have been observed. The initial slope of the heat capacity near 0 K was almost linear with respect to T, suggesting a 1D antiferromagnetic state even below the peak temperature. On the basis of these results, Bartolomé et al.125 assumed that the peaks observed for Gd(hfac)3NITEt and Gd(hfac)3NITi-Pr are not due to 3D ordering but to a different mechanism, and they hypothesized that the peaks at Tpeak ≈ 2 K could be due to the excitation of topologically stable chiral domain walls, separating two domains of opposite chirality, first predicted by Harada.122 To confirm this interpretation, Bartolomé et al. performed numerical transfer matrix calculations of the magnetic heat capacity for the classical planar model and obtained a justification in a qualitative way. For the compounds with lower frustration (R = Me, Ph), this model provided a broad heat-capacity peak with no indication for the excitation of chiral domain walls. For the compounds with a higher degree of frustration (R = Et, i-Pr), a peak showing the signature of the excitation of chiral domain walls appeared. Affronte et al.123 further studied the LT heat capacity C(T) of Gd(hfac)3NITi-Pr. The anomaly observed in C(T) was reproducibly found at T0 = 2.09 K, although the C(T) curves measured on different samples showed clear evidence for aging effects. An interpretation, capable of explaining the C(T) data both in the LT and in the critical region, calls for the onset, below T0, of nonconventional chiral long-range order due to interchain interactions. The chiral phase can be described as a collection of parallel corkscrews, all turning clockwise (or all counterclockwise), whereas their phases are random. An estimation of T0 according to this model and using the exchange constants evaluated from the linear C-vs-T slope supports this interpretation. The chiral order is compatible with the breaking of the chain into finite segments, due to aging effects, while the phase coherence necessary for the onset of 3D helical long-range order is destroyed. Lascialfari et al.126a measured the heat capacity of Gd(hfac)3NITi-Pr under magnetic field and zero-field muon spin resonance (μ+SR). The heat capacity measured in zero magnetic field exhibited a λ-type peak at T0 = (2.08 ± 0.01) K that disappeared upon the application of a 5 T magnetic field (Figure 23). Conversely, the μ+SR data do not present any anomaly at T ≈ 2 K, proving the lack of divergence of the twospin correlation function as required for usual 3D long-range helical order. Moreover, no muon spin precession can be evinced from the μ+SR asymmetry curves, thus excluding the presence of a long-range-ordered magnetic lattice. These results provide indications for a LT phase where chiral order is established in the absence of long-range helical order. Cinti et al.126b,c reinvestigated the magnetic properties of the quasi-1D helimagnets Gd(hfac)3NITR (R = Et, i-Pr, Ph) on the basis of magnetic susceptibility, zero-field muon spin resonance, and heat capacity measurements. For Gd(hfac)3NITi-Pr,126b the results can be coherently explained only in terms of the Villain’s conjecture: a chiral order at intermediate temperatures and helical order at low temperatures. In the case of weakly frustrated molecular magnetic chain compound Gd(hfac)3NITPh,126b new heat capacity measurements show the presence of a phase transition to the 3D helical order at T ≈ 0.63 K. In the temperature range 25−250 mK, the experimental

where the index n refers to the spin position along the chain, N is the number of spins along the chain, J1 > 0 is the ferromagnetic nn exchange constant, and J2 < 0 and J2′ < 0 are the antiferromagnetic nnn exchange constants. Under an appropriate condition, the ground state of this isolated chain is characterized by 2-fold degeneracy: clockwise turn of the spins and counterclockwise turn shown in Figure 22a,b, respectively.123

Figure 22. Helical configurations of a spin chain with easy-plane anisotropy. (a) Helical ground state with clockwise turn of the spins (+chirality); (b) helical ground state with counterclockwise turn of the spin (−chirality); and (c) dotted line, domain wall separating two regions with opposite chirality. The simplest case of S = s and of a localized domain wall was represented in the figures. Reprinted with permission from ref 123. Copyright 1999 American Physical Society.

Gatteschi and his collaborators studied thermal properties of Gd(hfac)3NITR, where R = isopropyl (= i-Pr), ethyl (= Et), methyl (= Me), and phenyl (= Ph), at low temperature.123−126 As a result, it was found that the thermal behavior is classified into two groups: One is that shown by the compounds with R = Me and Ph, and the other is that given by the compounds with R = i-Pr and Et. In the case of Gd(hfac)3NITMe and Gd(hfac)3NITPh, they exhibited a neat λ-type heat-capacity peak at Tc = 0.68 and 0.6 K, respectively.125 The total entropy gain due to the phase transition was 22.9 and 22.0 J K−1 mol−1, respectively. Because these values are close to the expected spin entropy due to Gd3+ and the radical spin, ΔS = R ln(8 × 2) = 23.0 J K−1 mol−1, the phase transitions observed in these compounds arise from the magnetic origin. On the LT side of the peak, both compounds showed a tendency to a T3/2 temperature dependence, suggesting a 3D ferromagnetic spinwave contribution. Therefore, the λ-type peak corresponds to the onset of 3D long-range order and a dimensional crossover takes place from 1D at high temperature to 3D at low temperature. In contrast, Gd(hfac)3NITEt and Gd(hfac)3NITi-Pr are strongly frustrated compounds. Although these two compounds Q


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Figure 23. Molar heat capacities (normalized to the gas constant R) against temperature of Gd(hfac)3NITiPr, measured in zero field (solid circles), in a field of 0.5 T (open circles) and 5 T (open squares). The solid line is a guide for the eye. Reprinted with permission from ref 126a. Copyright 2003 American Physical Society.

Figure 24. Molar heat capacity of [CuII(NO3)2(pyr)] vs temperature at constant magnetic field parallel to b. Solid curves are a fit to an exact diagonalization model. The dashed line is the phonon contribution determined from the fit. Reprinted with permission from ref 129. Copyright 1999 American Physical Society.

heat capacity data are reproduced very well by a simple spin wave theory. For Gd(hfac)3NITEt126c the heat capacity presents two anomalies at T0 = 2.19 K and TN = 1.88 K, while susceptibility and zero-field muon spin resonance show anomalies only at TN = 1.88 K. The results suggest an experimental validation of Villain’s conjecture of a two-step magnetic ordering in quasi-1D XY helimagnets: The paramagnetic phase and the helical spin solid phases are separated by a chiral spin liquid, where translational invariance is broken without violation of rotational invariance.

[Cu(NO3)2(pyr)] for a number of magnetic fields H parallel to b-axis. As H is increased, the broad feature observed in zerofield is suppressed, and the maximum gradually shifts to lower T. The solid curves are the results of a fit to a model based on exact diagonalization of short chains. The dashed line is the estimated phonon contribution determined from the fit. Because the results are in excellent agreement with numerical calculations based on the Bethe ansatz with no adjustable parameters, this material is an ideal 1D S = 1/2 Heisenberg antiferromagnet with nearest-neighbor exchange constant of J/k = 5.2 K (in the −2JSi·Sj scheme). This makes the material an excellent model system for exploring the T = 0 critical line that is expected in the H−T phase diagram of the 1D S = 1/2 Heisenberg antiferromagnet.

3.4. Lithium Phthalocyanine

The 1D molecular semiconductor lithium phthalocyanine (LiPc) has a half-filled conduction band and is expected to be an organic metal.127 However, because of the strong Coulomb repulsion, the system is a 1D Mott−Hubbard insulator with a Hubbard gap of 72 kJ mol−1 as inferred from optical measurements. LiPc has one unpaired electron on the inner ring of the macrocycle. These electrons are localized along the molecular stack and behave like an S = 1/2 antiferromagnetic spin chain. The heat capacity of LiPc has been measured by Dumm et al.127,128 in the temperature range of 1.5 ≤ T ≤ 300 K. The magnetic heat capacity shows a broad maximum at Tmax ≈ 18 K. This anomaly originates in the short-range order effect in the spin chains and is well-described by a model of an S = 1/2 antiferromagnetic Heisenberg chain. The intrachain exchange constant is given by J/k ≈ 40 K.

3.6. Mixed-Valence MMX Type Complexes

Halogen-bridged mixed-valence binuclear metal complexes, the so-called MMX chain compounds, have drawn attention as quasi-1D electronic systems characterized by strong electron− phonon, electron−electron, and magnetic interactions. There are following four extreme valence-ordering states: (a) averaged valence (AV) state: ··· −X−−M2.5+−M2.5+−X−−M2.5+−M2.5+− ··· (b) charge density wave (CDW) state: ··· −X−−M2+−M2+−X−−M3+−M3+− ··· (c) charge polarization (CP) state: ··· −X−−M2+−M3+−X−−M2+−M3+− ··· (d) alternate charge polarization (ACP) state: ··· −X−−M2+−M3+−X−−M3+−M2+− ··· Mode (a) is an AV state and has the possibility of either a Mott−Hubbard insulator (U > W ≈ 4t) or a 1D metal (U < W), where U, W, and t are the on-site Coulomb repulsion, bandwidth, and transfer integral, respectively. Mode (b) is a CDW insulating state, where doubling of the unit cell occurs. Modes (c) and (d) are a CP state and an ACP state, respectively. As an unpaired electron exists in the (−M2+− M3+−) unit, modes (c) and (d) are anticipated to be paramagnetic. Because no cell doubling occurs in the case of mode (c), either a Mott−Hubbard insulator or a 1D metal is expected. On the other hand, mode (d) is expected to be an insulator analogous to a spin-Peierls state. Platinum complexes [PtIIPtIII(RCS2)4I] are typical examples of this category, where R is n-alkyl-group. Two platinum atoms are bridged by four

3.5. Quasi-1D Inorganic Complex

To understand the intrinsic properties of quantum criticality, it is important to explore its phenomenology in simple and wellcontrolled model systems. One of the simplest quantum critical many-body systems is a linear chain of antiferromagnetically coupled spin 1/2 objects. Quantum criticality in this system is particularly interesting because it is possible to continuously tune the critical exponents by the application of a magnetic field. Hammar et al.129 explored the magnetic properties of a copper complex [CuII(NO3)2(pyr)] (pyz = pyrazine; C4H4N2) on the basis of high-field magnetization, field-dependent heatcapacity measurements, and zero-field inelastic magnetic neutron scattering. Heat capacities of [CuII(NO3)2(pyr)] were measured by use of a relaxation method in the range of 0.1−10 K up to an applied magnetic field of 9 T.129 Figure 24 shows the temperature dependence of the total heat capacity of R


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bidentate RCS2− ligands, and such binuclear complex units are linearly bridged by iodide ions to one another. Kitagawa et al.130−133 reported the first observation of a metallic property in [Pt2(dta)4I] (dta: dithioacetato, R = CH3−). The electrical conductivity and thermoelectric power measurements for a single crystal of [Pt2(dta)4I] parallel to the chain axis b revealed that this complex exhibits a metallic temperature dependence above a metal−semiconductor transition temperature TM−S = 300 K. This metal−semiconductor transition was also confirmed by magnetic susceptibility measurement for a polycrystalline sample.132,133 Furthermore, temperature dependence of the electrical conductivity and thermoelectric power measurements showed a first-order phase transition around 360 K. From single-crystal X-ray structure analyses,134 it was revealed that a first-order phase transition occurs at Ttrs = 365 K from the LT phase with space group C2/c to the HT phase with A2/m, in which the cell volume is a half of that of the LT phase. Moreover, the magnetic susceptibility shows a convexity at 80 K, which suggests the existence of a phase transition. From variabletemperature infrared and polarized Raman spectroscopies at room temperature and 129I Mössbauer spectroscopy132,133 at 16 and 80 K, together with the experimental results mentioned above, it is expected that the [Pt2(dta)4I] complex is in a 1D metallic phase of the AV state above TM−S = 300 K, in a semiconducting phase of the CP state between 90 and 300 K, and in an insulating phase of the ACP state below 80 K. To elucidate the relationship between various phases, heatcapacity measurements based on adiabatic calorimetry were made by Miyazaki et al.135 for a powder sample of this complex. Molar heat capacities of [Pt2(dta)4I] crystal are plotted in Figure 25. As described above, various experiments so far

heat-capacity measurements based on a.c. calorimetry were performed for a single crystal of [Pt2(dta)4I] in the 15−317 K range at a frequency of 2 Hz. However, no thermal anomaly was detected around 80 and 300 K. In general, the change of a crystal lattice involved in a Mott transition is extremely small. Therefore, the thermal anomaly due to the Mott transition at 300 K, if any, is too small to be detected by these calorimetric methods. On the other hand, because a Peierls or spin-Peierls transition gives rise to a change of crystal lattice due to dimerization, a thermal anomaly would be observed at 80 K if the transition actually occurs. In the case of [Pt2(dta)4I] crystal, however, as the spin in the crystal is still alive below 80 K, the change in the lattice would be very small. This might be the reason for no detection of a thermal anomaly around 80 K by these two calorimetries. In contrast, a heat-capacity peak due to a structural phase transition was observed at Ttrs = 373.4 K. An XRD study134 revealed that the plane formed by SCS atoms of a ligand is statically tilted at room temperature, while above about 370 K the ligand plane can randomly convert between two tilted directions available for right and left sides with equal probability. Therefore, this phase transition is of the order− disorder type concerning conformational change of the dta ligands (see Figure 26). The total entropy gain due to the phase

Figure 26. Part of a 1D arrangement of the halogen-bridged mixedvalence complex [Pt2(dta)4I]. In the HT phase, two conformational disorders of each dta-ligand are drawn with equal probability.134

transition was only ΔS = (5.25 ± 0.07) J K−1 mol−1. If the four dithioacetato ligands reorient independently between two positions, the total entropy gain expected for the order− disorder phase transition would be 4 × R ln 2 = 23.05 J K−1 mol−1. However, this is quite far from the observed value. The observed entropy is just 1/4 of the independent motion model. This fact clearly indicates that order−disorder motions of the four dithioacetato ligands might be synchronized. In that case, the theoretical entropy is only R ln 2 = 5.75 J K−1 mol−1. This value agrees well with the observed value. Therefore, one can safely conclude that the ligand motions in this complex are not independent but synchronized. Similar structural phase transitions have been found by adiabatic calorimetry for the homologous series of complexes with longer alkyl chains: [Pt2(n-C3H7CS2)4I],136 [Pt2(nC4H9CS2)4I],137 and [Pt2(n-C5H11CS2)4I].138 Table 3 lists the transition temperatures and the entropy gains at the phase transitions calorimetrically determined for the [Pt 2 (nCmH2m+1CS2)4I] (1 ≤ m ≤ 5) complexes. The phase transitions with a large entropy gain always involve a order−disorder mechanism of the conformational changes of the ligands. In the case of [Pt2(n-C4H9CS2)4I], for example, the heatcapacity measurement by Ikeuchi et al.137 revealed two large

Figure 25. Molar heat capacity of [Pt2(dta)4I] crystal. The broken curve implies normal heat capacity. Reprinted with permission from ref 135. Copyright 2002 American Chemical Society.

reported have suggested the metal−semiconductor transition at TM−S = 300 K, corresponding to a Mott transition from a quasi1D metallic phase of the AV state to a semiconducting phase of the CP state and a spin-Peierls-like phase transition at 80 K to the ACP state. However, the calorimetric results did not give rise to any thermal anomalies at 80 and 300 K. Very sensitive S


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Table 3. Transition Temperature and the Entropy Gain at the Phase Transitions in Halogen-Bridged Mixed-Valence Quasi-1D Complexes [Pt2(n-CmH2m+1CS2)4I] LT phase → RT phase Ttrs (K)

complexes

RT phase → HT phase

ΔtrsS (J K−1 mol−1)

a

[Pt2(CH3CS2)4I] [Pt2(C2H5CS2)4I]b [Pt2(n-C3H7CS2)4I]c [Pt2(n-C4H9CS2)4I]d [Pt2(n-C5H11CS2)4I]e [Pt2(n-C5H11CS2)4I]f model of disorderg [Ni2(C2H5CS2)4I] [Ni2(n-C3H7CS2)4I] [Ni2(n-C4H9CS2)4I]

∼180 and ∼230 209 213.5 207.4 220.5 none 205.6 260

0.21 and 0.13 14.6 20.09 49.1 52.4 15.36

Ttrs (K)


refs

373.4

5.25 ± 0.07

358.8 323.5 324

10 7.46 ∼0

135 147b 136 137 138 138

7.68 148a 148a 148b

19.7 ∼4

a

No thermal anomalies at TM−S = 300 K and T(spin Peierls-like transition) = 80 K.132,134 bElectrical properties show a metal-semiconductor transition at TM−S = 205 K.139,142 cStructural phase transition was reported at Ttrs = 359 K.140 dPhase transitions were reported at 210 and 324 K.141−143 The calorimetric study137 additionally detected a small phase transition at 114 K. eData for as-grown sample. The phase transition was reported at 210 K.143 fData for the sample once heated above the transition at 324 K and then supercooled. A very broad higher-order phase transition was detected around 170 K. gSee the text.

1D chain (the c-axis of tetragonal system) with an inversion center, implying tetragonal I4/m and the AV electronic state. ORTEP diagrams of the LT, RT, and HT phases of [Pt2(nC4H9CS2)4I] are shown in Figure 28.142 In the LT phase, no

phase transitions at 213.5 and 323.5 K (see Figure 27). These transition temperatures agree well with those found by the

Figure 28. ORTEP diagram (thermal ellipsoids set at the 50% probability level) of [Pt2(n-C4H9CS2)4I]: (a) LT phase at 167 K, (b) RT phase at 298 K, and (c) HT phase at 350 K. Reprinted with permission from ref 142. Copyright 2002 Wiley-VCH.

Figure 27. Molar heat capacity of [Pt2(n-C4H9CS2)4I] as a function of temperature. A small heat-capacity anomaly was also detected at 114 K. Reprinted with permission from ref 137. Copyright 2002 American Physical Society.

structural disorder has been detected. In contrast, the RT phase has a lattice periodicity of three in the Pt−Pt−I− repeating unit. There are two crystallographically independent Pt−Pt−I− units with the population ratio of 1:2. Whereas structural disorder does not exist in two units, in the remaining third unit, the dithiocarboxylato (CS2−) groups undergo a structural disordering between two tilted positions realized by clockwise and counterclockwise twist motions with a population ratio of 1:1; moreover, each butyl group has two conformations with a ratio of 1:1. On the other hand, the HT phase has a lattice periodicity of one, showing that all of the Pt−Pt−I− units are equivalent. Interestingly, although the CS2− groups show 2-fold disorder with respect to the twist motion in the HT phase, the butyl chains are seemingly ordered. Because the twist motions of the four CS2− groups in [Pt2(CH3CS2)4I] were synchronized,135 Ikeuchi et al.137 estimated possible structural entropies of each phase on the basis of various combinations of synchronized and independent motions of the CS2− and butyl groups and compared them with

transport, magnetic, and a single-crystal XRD studies by Mitsumi et al.142 These structural phase transitions were afterward confirmed by ESR143 and resistance measurements under uniaxial compression.143 The X-ray structural study142 revealed that the space group changes from tetragonal P4/n in the LT phase (below 213.5 K) to tetragonal I4/m in the roomtemperature (RT) phase persisting between 213.5 and 323.5 K. In both phases, the crystal consists of a neutral 1D chain with a Pt−Pt−I− repeating unit lying on the crystallographic 4-fold axis parallel to the c-axis. The periodicity of the crystal lattice in the 1D chain direction, however, changes from 2-fold with a Pt−Pt−I− period in the LT phase to 3-fold in the RT phase. As judged by Pt−Pt and Pt−I bond distances, the valence-ordering structure in the LT phase can be regarded as an ACP state, while that in the RT phase is close to the AV state. The HT phase appearing above 323.5 K has uniform structure along the T


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Alternative candidate of mixed valence MMX type complex is [NiIINiIII(RCS2)4I]. Heat capacities of the complex with R = C2H5, n-C3H7, and n-C4H9 have been measured by Ikeuchi et al.148 As shown in Table 3, the phase behavior of [M2(RCS2)4I] (M = Pt or Ni) is quite different between the Pt and Ni complexes, implying the presence of motional correlation in dynamics of RCS2 ligands.

the experimental values. As a result, good agreement was obtained for the independent motion model. When the entropy due to structural disorder is assumed to be zero in the LT phase, the configurational entropy of the RT phase is estimated as R/3·ln(24 × 24) (= 15.36 J K−1 mol−1), in which the factor 1/3 originates in the portion of the disorder unit in the 3-fold periodicity of Pt−Pt−I− units and the double 24 terms correspond to the number of conformations attained by four CS2− and four butyl groups in a Pt−Pt−I− unit. The entropy of the HT phase is simply R·ln24 (= 23.04 J K−1 mol−1), because only the four CS2− groups contribute to the entropy. Therefore, the entropy changes at the phase transitions amount to ΔS(LT → RT phase) = 15.36 J K−1 mol−1 and ΔS(RT → HT phase) = 7.68 J K−1 mol−1. As shown in Table 3, agreement between the experimental value and the disorder model is excellent for the RT−HT phase transition, whereas the experimental entropy gain at the LT−RT phase transition (20.09 J K−1 mol−1) is fairly large. The excess entropy beyond the present disorder model has been attributed to the contribution from the spinPeierls behavior inherent in this phase transition. This disorder model is also applicable to the phase transitions in [Pt2(nC3H7CS2)4I].136 In the case of [Pt2(n-C5H11CS2)4I],138 the LT−RT phase transition with a large entropy gain (49.1 J K−1 mol−1) occurs at 207.4 K. Although the RT−HT phase transition also takes place at 324 K, the entropy change is substantially zero and this phase transition is always supercooled. Therefore, the interpretation of the phase transitions is not so easy as compared with other homologous complexes. Saito et al.138 proposed a more general interpretation of the RT−HT phase transitions occurring in [Pt 2 (n-C 3 H 7 CS 2 ) 4 I], [Pt 2 (nC4H9CS2)4I], and [Pt2(n-C5H11CS2)4I] in terms of “alkyl group acting as entropy reservoir” and “entropy transfer between dithiocarboxylato and alkyl groups”. Because alkyl chains are very flexible and capable of doing labile motions in condensed states,144 they have a potential capacity of entropy amounting to about 10 J K−1 (mol of CH2) −1.145 Therefore, the alkyl chain attached to the ligand can serve as “entropy reservoir”. This concept and the “entropy transfer mechanism” have recently been proposed for the interpretation of phase transitions occurring in liquid crystalline systems by Saito et al.146,147a The “entropy transfer mechanism” can be seen in the fact that the entropy reserved in alkyl groups in the RT phase is transferred to the dithiocarboxylato groups when the RT−HT phase transition occurs (see Figure 29). It is of interest to remark here that the complex [Pt2(C2H5CS2)4I] does not give rise to any structural phase transition147b and that the synchronized twist motion of the four CS2− groups is observed only in [Pt2(CH3CS2)4I].

3.7. MnIII-Porphyrin−TCNE and M(TCNQ)2

An interesting family of molecule-based magnets is the chargetransfer complexes between manganese(III)-tetraphenylporphyrin and TCNE, [MnTPP][TCNE]·(solv), where solv implies the crystalline solvent molecule. They form a quasi1D-ferrimagnetic chain structure with the spin quantum numbers of 2 of [MnTPP]+ and 1/2 of [TCNE]−. The first example was reported by Miller et al.149 for [MnTPP][TCNE]·2PhMe, where PhMe is toluene. The real part of the a.c. magnetic susceptibility of this complex exhibits a maximum at the critical temperature Tc = 12.5 K.150 Because the magnetic properties of this complex are sensitive to the stoichiometry of the solvent molecule and thus the lattice defects and disorder, the magnetic properties are rather complicated. The imaginary part of the a.c. magnetic susceptibility shows a second, lower temperature peak, which is similar to those seen in “reentrant” spin glass or superparamagnetism.150 The compounds introduced here are analogous complexes [MnT(R)PP][TCNE]·(solv). Molecular structures of [MnT(R)PP] and TCNE are shown in Figure 30. The complexes

Figure 30. Molecular structures of [MnT(R)PP] and TCNE.

with (1, R = n-C12H25 and solv =2PhMe), (2, R = n-C14H29 and solv = MeOH), and (3, R = F and solv = 0.5MeOH). The complexes 1, 2, and 3 exhibit the phase transition evidenced by the spontaneous magnetization at the critical temperature Tc = 22,151 20.5,152 and 27 K,151 respectively. The a.c. magnetic susceptibility exhibits a very large peak at the critical temperatures. For the complex 1, the transition peak is especially sharp. In the sense that although the spontaneous magnetization appears below Tc, the real and imaginary parts of the a.c. magnetic susceptibility depend on the frequency of measurements, and there exists simultaneously both the ferrimagnetic order and the spin-glass property; the latter is

Figure 29. Schematic relationship of the entropies possessed by dithiocarboxylato ligands and alkyl chains for the RT−HT phase transitions in [Pt2(n-C3H7CS2)4I], [Pt2(n-C4H9CS2)4I], and [Pt2(nC5H11CS2)4I].138 U


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caused by random anisotropy due to the crystalline solvent molecules.151 One may naturally expect some heat-capacity anomalies around the critical temperatures. Figure 31 illustrates the molar

M ∝ [ND( ↑ ) − ND( ↓ )]·n = NA ·[ND( ↑ ) − ND( ↓ )]/[ND( ↑ ) + ND( ↓ )] (4)

Because the parameter n does not appear in the final form of the magnetization equation, the formation of domain does not influence the magnetization at all. On the other hand, when the magnetic domains are formed, the magnetic susceptibility χ is given by the following equation, χ = (NA /n) ·[(n·μeff )2 /3kT ] = n·[NA ·μeff 2 /3kT ]

where μeff is the effective magnetic moment of a single spin. From this equation, one can know that the magnetic susceptibility is enhanced in proportion to n. Consequently, the enhanced effect observed in the magnetic susceptibility is surely based on the formation of magnetic domains in the crystal. Dunbar and her collaborators155,156 prepared the homologous series M(TCNQ)2 [M = Mn(II), Fe(II), Co(II), and Ni(II)] and characterized magnetic properties by the tools of d.c. and a.c. magnetometry. The d.c. magnetic measurements reveal a spontaneous magnetization for the four materials at low temperatures with a weak field coercivity of 20, 750, 190, and 270 G at 2 K for Mn(TCNQ)2, Fe(TCNQ)2, Co(TCNQ)2, and Ni(TCNQ)2, respectively. At low temperatures, a.c. susceptibility measurements confirm the presence of a magnetically ordered phase at 44, 28, 7, and 24 K for Mn(TCNQ)2, Fe(TCNQ)2, Co(TCNQ)2, and Ni(TCNQ)2, respectively, but do not support the description of this system as a typical magnet. In the absence of the a.c. magnetic data, the behavior is indicative of ferri- or ferromagnetic ordering (depending on the metal), but in fact, a complete investigation of their physical properties revealed their true nature to be a glassy magnet. The glassiness, which is a high magnetic viscosity known to originate from randomness and frustration, is revealed by a frequency dependence of the a.c. susceptibility data and is further supported by a lack of a λ-type peak in the heat-capacity data. These results clearly demonstrate that molecule-based materials with a presumed magnetic ordering may not always be exhibiting truly cooperative behavior.

Figure 31. Molar heat capacity of [MnT(R)PP][TCNE]·(solv). Curve A: R = n-C14H29 and solv = MeOH. Curve B: R = F and solv =0.5 MeOH. Reprinted with permission from ref 153. Copyright 2005 Elsevier Ltd.

heat capacities of 2 and 3.153 Quite mysteriously, any heatcapacity anomalies due to the magnetic ordering have not been detected around 20.5 and 27 K. A clue to solve this mystery is hidden in the characteristic magnetic structure of these complexes. They consist of a quasi-1D-ferrimagnetic chain with a very strong antiferromagnetic interaction between the spin quantum number 2 of the manganese ion and the spin 1/2 of the TCNE radical. The intrachain spin−spin interaction parameters for 2 and 3 are as large as Jintra/k = −148152 and −236 K,151 respectively. This type of magnetic chain brings about a very broad heat capacity extending over a wide temperature region. A very large fraction of the magnetic entropy above Tc cannot contribute to the magnetic phase transition, because this entropy corresponds to the strong short-range order still remaining above Tc. The entropy available for the phase transition is an extremely small fraction below Tc. This is the reason for substantially no effect in the heat capacity. The heat-capacity measurement supporting this situation was recently reported by Nakazawa et al.154 for [MnT(X)PP][TCNE]·2PhMe, where X = Cl, Br and 2PhMe = toluene. However, there still remains a big mystery. Why is an enhanced effect possible for the magnetic susceptibility, while substantially no effect in the heat capacity? To solve this mystery, Sorai and his collaborators153 anticipated formation of magnetic domains in the actual crystal. They assumed the following model: (i) An antiferromagnetically coupled pair of spins (S = 2 and 1/2) forms a resultant spin (S = 3/2). (ii) The resultant spins have a tendency to form ferromagnetic shortrange order. A region of the short-range order is assumed to form a magnetic domain of uniform size. (iii) The number of spins in a domain is n. (iv) Under zero magnetic field, the net magnetic moments of domains fluctuate paramagnetically. The number of domains with up-spin and down-spin are designated as ND(↑) and ND(↓), respectively. These numbers are related to the Avogadro constant NA through the following equation, [ND( ↑ ) + ND( ↓ )]·n = NA

(5)

3.8. Spin-Gapped Systems

Haldane’s conjecture157 that 1D antiferromagnetic chains consisting of integer spins should have “spin gaps” in their excitation spectra, making a sharp contrast to gapless half-odd spin chains, has launched a lot of research both theoretically and experimentally on spin-gapped systems where the Néelorder ground states are suppressed by strong quantum fluctuations.158 The spin-gapped systems include Haldane systems,159,160 spin-ladder systems with even-numbered legs,161 bond alternation systems, and so on. Because their common feature is the existence of an energy gap above the singlet ground state (Figure 32), the magnetic heat capacity at low temperatures shows an exponential growth with temperature and allows quantitative estimation of spin gaps. Among molecule-based Haldane systems, calorimetric studies were reported on a series of antiferromagnetic S = 1 Heisenberg chain systems, [Ni(en)2(NO2)]ClO4 (NENP),162 NMe4 [Ni(NO2 ) 3 ] (TMNIN),163,164 [Ni(dmpn) 2N 3 ]ClO 4 (NDMAZ),165 and [Ni(dmpn)2N3]PF6 (NDMAP).166−169 Magnetic heat capacities deduced from temperature dependence of optical linear birefringence were also reported on NENP and [Ni(en)2(NO2)]PF6 (NENF).170 These studies

(3)

As the magnetization M is proportional to the effective number of spins orienting to an easy axis, it is expressed by the equation, V


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Figure 32. Schematic diagram of excitation energy and magnetization for Ni-based Haldane systems under magnetic field. Eg stands for the Haldane gap above the singlet ground state, and the shaded area means spin-wave continuum. Reprinted with permission from ref 160. Copyright 2000 Elsevier Ltd.

Figure 34. B−T phase diagram of NDMAP determined from the heat capacity (C) and magnetization (M) measurements. H, quantumdisordered Haldane phase; P, thermally disordered paramagnetic phase; and LRO, field-induced long-range ordered phase. Reprinted with permission from ref 167. Copyright 2001 American Physical Society.

clearly showed that the magnetic long-range order is absent under zero magnetic field and that the spin gaps are dominated by the exchange interaction J along the chains rather than the single-ion anisotropy D of spin bearer Ni(II). The effect of external magnetic field was thoroughly examined to bring about quantum phase transitions. As shown in Figure 33, the field-

dimensionality and special care should be taken at the choice of theoretical models.176 Ferromagnetic−antiferromagnetic alternating Heisenberg chains are also interesting spin-gapped systems, whose strong ferromagnetic-coupling limit coincides with an S = 1 Haldane system. On the basis of the calorimetric results, a spin gap ΔE/ k = 20.9 K was reported in (Me2CHNH3)CuCl3, exhibiting good agreement with the estimate from the magnetic susceptibilities.177 This compound showed λ-type heat-capacity anomalies indicating the occurrence of a field-induced longrange order above 9.0 T.

4. TWO-DIMENSIONAL MAGNETS To obtain spontaneous magnetization in metal-assembled complexes, 3D ferromagnetic or ferrimagnetic ordering over a whole crystal is required. One of the useful synthetic strategies was to bundle 1D ferro- or ferrimagnetic chains threedimensionally and to afford a net magnetization. This idea was applied first by Kahn and his collaborators178−180 to synthesize the ferromagnetic compound [MnIICuII(pbaOH)(H2O)3] [pbaOH = 2-hydroxy-1,3-propylene-bis(oxamato)]. They proposed an idea to bring about ferromagnetic interactions among ferrimagnetic chains. However, it owes much to accidental matching of overlap phase in neighboring chains. Avoiding the difficulty of crystal packing regulation of 1D element, O̅ kawa and his collaborators adopted another strategy to build a higher-dimensional structure of coordination polymer by a single reaction process.

Figure 33. Temperature dependence of magnetic heat capacity of NDMAP under magnetic field parallel to c-axis. Reprinted with permission from ref 167. Copyright 2001 American Physical Society.

induced magnetic phase transitions were clearly detected for NDMAP as sharp λ-type heat-capacity anomalies. The field temperature phase diagram was presented in Figure 34. It revealed that the critical fields are varied with the angle between the external field and the crystallographic axis, since the Ni(II) ions have magnetic anisotropy of easy-plane type. As a candidate of two-leg spin-ladder systems, the heat capacity of [Cu2(C5H12N2)Cl4] was reported.171−174 Round heat-capacity peaks characteristic of spin-gapped systems were observed. Field-induced magnetic order similar to the Haldane systems was also found under external fields from 7 to 13 T, where spin-gap behaviors were suppressed and alternatively λtype peaks appeared. However, it should be mentioned that a recent neutron diffraction study suggested another interpretation of the spin gap in this compound on the basis of a frustrated 3D spin-liquid model rather than a spin-ladder model.175 It has already been pointed out that the magnetism of weakly coupled spin dimers is not sensitive to the lattice

4.1. Assembled-Metal Complexes

O̅ kawa et al. prepared the ferromagnetic assembled-metal complexes NBu4[MIICrIII(ox)3] (Bu = n-C4H9, M = Mn, Fe, Co, Ni, Cu, Zn; H 2 ox = oxalic acid) 1 8 1 , 1 8 2 and NBu4[MIIFeIII(ox)3] (M = Fe, Ni).183 These complexes except for the Zn compound exhibit ferromagnetic phase transitions around 6−14 K. The network structure of {[MIICrIII(ox)3] −}∞ is depicted schematically in Figure 35. The hub unit with D3 point symmetry, [CrIII(ox)3]3−, combines three M2+ ions through its ox groups. This network structure can be embedded in two possible crystal structures, i.e., 2D honeycomb or 3D helical structures. A tris(chelate) complex can take left-handed W


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Unfortunately, however, because these complexes were prepared as fine powders, a single-crystal XRD study was unsuccessful. As heat capacity is sensitive to lattice dimensionality, the present authors and co-workers184,185 tried to determine the actual dimensionality on the basis of heatcapacity measurements for the complexes NBu4[CuIICrIII(ox)3] and PPh4[MnIICrIII(ox)3] (Ph = C6H5). For example, as shown in Figure 37, the mixed-metal NBu4[CuIICrIII(ox)3] complex

Figure 35. [MIICrIII(ox)3]− network structure (M = Mn, Fe, Co, Ni, Cu, Zn; H2ox = oxalic acid).

or right-handed helicity around a D3 axis, which are called Δ- or Λ-isomers, respectively. When the neighboring M2+ and Cr3+ coordination octahedra take a combination of (Δ + Λ), a 2D honeycomb lattice with alternating hetero spins is formed (Figure 36a), while the combination of either (Δ + Δ) or (Λ + Λ) provides a 3D structure, i.e., a complicated helix with 10membered rings (Figure 36b). Figure 37. Molar heat capacities of NBu4[CuIICrIII(ox)3]. Solid curves indicate the normal heat capacities. Reprinted with permission from ref 185. Copyright 2003 American Institute of Physics.

gave rise to a phase transition at 6.98 K. This phase transition is attributable to the ferromagnetic spin ordering because the entropy gain ΔS = 16.2 J K−1 mol−1 agrees well with the theoretical value R ln(2 × 4) = 17.28 J K−1 mol−1 expected for the spin system consisting of Cu2+ (spin 1/2) and Cr3+ (spin 3/ 2) and also because the transition temperature agrees well with the onset temperature of the spontaneous magnetization.182 Figure 38 represents the excess heat capacity due to the magnetic phase transition of this complex. A remarkable feature is the existence of a dominant short-range order manifested by the large tail of the magnetic heat capacity. This type of heatcapacity anomaly is characteristic of low-dimensional magnets. Because 1D magnetic interaction paths cannot be found in this complex, one may conclude that this series of mixed-metal complexes has 2D magnetic structures. Because the spin−spin interaction in this complex is of Heisenberg type, no phase transition is theoretically expected for 2D systems. However, the occurrence of the magnetic phase transition does not conflict with the existence of dominant 2D interactions. When a 3D interaction, even though weak, exists at low temperatures, dimensional crossover between 2D and 3D may happen and bring about a phase transition. In fact, the heat capacities of NBu4[CuIICrIII(ox)3] in the spin-wave excitation region are well-reproduced by T1.51, whose exponent is approximated by d/n = 3/2, implying that the ordered state at low temperatures is actually a 3D ferromagnet.15 On the basis of this fact, the authors concluded that the dominant magnetic interaction may be 2D. However, one should be very careful to draw a conclusion about the dimensionality when the number of the nearest neighbor

Figure 36. Possible schematic network structure of [MIICrIII(ox)3]−. (a) Two-dimensional honeycomb and (b) 3D helical structures. X


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3D structures by the HT series expansion up to the seventh cumulant with the Padé approximation186 and fitted the excess heat capacities above the transition temperature. Because the number of the nearest neighbors is extremely small, two formulas for 2D and 3D structures closely resemble each other. As a result, the short-range order effect was extremely large for both 2D and 3D structures. However, as can be seen in Figure 38, good agreement is obtained for the 2D structure, where the superexchange interaction parameter is J/k = 5.0 K. Therefore, from a thermodynamic viewpoint, it is concluded that the magnetic lattice structure might be 2D. It is of great interest, however, that the short-range order effect estimated for 3D structure is also unusually large as compared with those for sc, bcc, or fcc lattices. This fact clearly implies that in addition to the dimensionality the important factor governing the shortrange order effect due to magnetic fluctuation is the number of the nearest neighbors. It should be remarked here that X-ray structural study has been made for PPh4[MnIICrIII(ox)3] by Decurtins et al. 187 and for NBu 4 [Mn II Cr III (ox) 3 ] and NBu4[FeIICrIII(ox)3] by Ovanesyan et al.188 These complexes are crystallized in 2D honeycomb structures. The coincidence of the dimensionality derived from the HT series expansion method with the results of X-ray structural analysis would guarantee the usefulness of the calorimetry. Recently, Bhattacharjee et al. measured heat capacities of analogous assembled-metal complexes NBu 4 [Fe II Fe III (ox) 3 ], 189−191 NBu 4 [Zn II Fe III (ox) 3 ], 189,190 NPr4[FeIIFeIII(ox)3] (Pr = n-C3H7),192 NBu4[CoIIFeIII(ox)3],193 and NPe4[MnIIFeIII(ox)3] (Pe = nC5H11)194 and showed that these compounds exhibit similar behavioral characteristics of 2D structures. Antorrena et al.195 also reported that the magnetic heat capacities of PPh4[MnCr(ox)3] and PPh4[FeIICrIII(ox)3] indicate 2D characters. Except for NBu4[ZnIIFeIII(ox)3], all of the complexes exhibited the magnetic phase transitions at low temperatures. The critical temperatures due to the magnetic ordering Tc and the entropy gains at Tc are listed in Table 4. Many of the complexes showed a small subpeak below Tc, which would be characterized by a spin-glasslike behavior. In addition to the magnetic phase transition, these complexes exhibited, respectively, a heatcapacity peak above 200 K. These phase transitions seem to be attributable to an order−disorder mechanism of tetraalkylammonium cations. The structural transition temperatures Ttrs and the entropy gains are listed in Table 4. Day and his collaborators196 found that the mixed-valence assembled-metal complexes N(n-CnH2n+1)4[FeIIFeIII(ox)3] (n = 3−5) behave as ferrimagnets with Tc between 33 and 48 K and that they exhibit a crossover from positive to negative

Figure 38. Excess heat capacities due to the magnetic phase transition of NBu4[CuIICrIII(ox)3] on (a) logarithmic and (b) normal scales. Thick curves represent the 2D structure, while thin curves correspond to the 3D structure. Dashed curves at the lowest temperature region indicate the spin-wave contribution. Reprinted with permission from ref 185. Copyright 2003 American Institute of Physics.

paramagnetic ions z is extremely small. The value z = 3 for the present complexes is extremely small in comparison with z = 6 (sc), z = 8 (bcc), and z = 12 (fcc) of 3D lattices or z = 4 (square and Kagomé) and z = 6 (triangular) of 2D lattices. In many cases, the short-range order effect is regarded as a direct manifestation of low-dimensionality of spin−lattices. Exceptional cases rarely arise where a 3D magnet shows a large shortrange order effect comparable to low-dimensional magnets. In this point, the present series of compounds are very unique, because they have extremely small numbers of the nearest neighbor spins, z = 3, in either 2D honeycomb or 3D helical lattices. A small z may bring about a remarkable short-range order effect, and the discrimination between 2D and 3D structures becomes a delicate problem. To examine quantitatively the dimensionality of the magnetic structure of the present complex, Hashiguchi et al.185 formulated the magnetic heat capacities for possible 2D and

Table 4. Transition Temperature and Entropy of Transition in Assembled-Metal Complexes Bridged by Oxalato Ligands complexes N(n-C4H9)4[FeIIFeIII(ox)3] N(n-C4H9)4[ZnIIFeIII(ox)3] N(n-C3H7)4[FeIIFeIII(ox)3] N(n-C4H9)4[CoIIFeIII(ox)3] N(n-C5H11)4[MnIIFeIII(ox)3] P(C6H5)4[FeIICrIII(ox)3] N(n-C4H9)4[FeIICrIII(ox)3] N(n-C4H9)4[CuIICrIII(ox)3] P(C6H5)4[MnIICrIII(ox)3]

Tc (K) 16.3 and 43.3 paramagnetic until 10 and 38 12 and 29.7 23 and 27.1 10.4 12.0 6.98 5.59 5.85

ΔcS (J K−1 mol−1)

ΔcS(spin) (J K−1 mol−1)

4.89 + 22.65 (= 27.54) at least 5 K 31.98 25.5 33.22 5.4 23.3 16.2 25.1 23.2

28.27 14.90 28.27 26.42 29.80 24.90 24.90 17.28 26.42 26.42 Y

[=R ln(5 × 6)] [=R ln(1 × 6)] [=R ln(5 × 6)] [= R ln(4 × 6)] [=R ln(6 × 6)] [=R ln(5 × 4)] [=R ln(5 × 4)] [=R ln(2 × 4)] [=R ln(6 × 4)] [=R ln(6 × 4)]

Ttrs (K)


refs

217 227 none 210 226

11.47 13.32 28.5 13.07

235.5 226.9 71.3

23.5 11.0 1.48

189−191 189, 190 192 193 194 195 186 184, 185 184, 185 195


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magnetization near 30 K when cooled in a field of 10 mT. To get more insights into this rarely occurring negative magnetization phenomenon, Bhattacharjee et al. measured heat capacities of NBu4[FeIIFeIII(ox)3]189−191 and NPr4[FeIIFeIII(ox)3]192 under a nonmagnetic field. Figure 39

(3-methoxysalicylideneaminato)201 and a ferrimagnet [NEt4][Mn III (salen)] 2 [Fe III (CN) 6 ] (salen = N,N′-ethylenebis(salicylideneaminato).202 Their molecular assembly is schematically drawn in Figure 40. When these crystals were

Figure 40. (a) Molecular structure of 3-MeOsalen. (b) Schematic drawing of 2D assembly of K[MnIII(3-MeOsalen)]2[FeIII(CN−)6], where [MnIII(3-MeOsalen)]+ is simply expressed by Mn.

Figure 39. Magnetic heat capacities of NBu4[FeIIFeIII(ox)3]. The solid and broken curves represent the values estimated by assuming T−2 and T3 temperature dependences, respectively. Reprinted with permission from ref 189. Copyright 2000 The Physical Society of Japan.

mechanically perturbed by grinding or pressing, their heat capacities around the magnetic phase transition temperatures were decreased. A plausible explanation for this type of mechanochemical effect is that enhanced lattice defects and imperfections in the crystal lattice produced by mechanical perturbation would lead to an incomplete magnetic phase transition and consequently a part of the paramagnetic species characteristic of the HT phase would remain as the so-called residual paramagnetism even below the magnetic transition temperature. The mechanochemical effect has often been encountered in other soft matter. A dramatic mechanochemical effect was observed in the spin crossover complex [FeIII(3MeOsalenEt)2]PF6.11,203−206 Bhattacharjee et al.207 reported the heat capacity of a weak ferromagnet [MnIII (cyclam)][Fe III(CN) 6 ]·3H 2 O (cyclam =1,4,8,11-tetraazacyclotetradecane) under zero magnetic field in the 0.1−300 K range by adiabatic calorimetry. A heatcapacity anomaly corresponding to the magnetic phase transition was detected at 6.2 K. The observed entropy gain (ΔS = 19.29 J K−1 mol−1) is in good agreement with the theoretical magnetic entropy (R ln 5 × 2 = 19.14 J K−1 mol−1) considering the spin multiplicity of HS Mn3+ and low-spin (LS) Fe3+ ions. Heat-capacity studies under applied magnetic fields indicated an antiferromagnetic to ferromagnetic transition. A field dependence study of magnetization at different temperatures revealed a field-induced metamagnetic transition. The most intensively studied cyanide-bridged magnets have been Prussian blue analogues with 3D network structures. Recently, new cyanide-bridged molecule-based magnets having 2D structures were synthesized as follows: KI2MnII3(H2O)6[MoIII(CN)7]2·6H2O by Kahn et al.,208 (tetrenH5)0.8CuII4[MV(CN)8]4·7.2H2O (M = W, Mo; tetren = tetraethylenepentamine) by Podgajny et al.,209 and (dienH3){CuII3[MV(CN)8]3}·4H2O (M = W, Mo; dien = diethylenetriamine) by Korzeniak et al.210 Heat-capacity measurements have been made on a single-crystal sample of the double-layered coordination polymer {(tetrenH5)0.8CuII4[WV(CN)8]47.2H2O}n in the presence of the

shows the excess heat capacity observed under nonmagnetic field for NBu4[FeIIFeIII(ox)3]. Two heat-capacity anomalies were detected at T2 (= Tc) = 43.3 K and at T3 = 16.3 K. Because the entropy gain due to these two anomalies, 27.54 J K−1 mol−1, is well-approximated by the theoretical value 28.27 J K−1 mol−1 [= R ln(5 × 6)] expected for the spin multiplicity of high-spin (HS) Fe2+ (spin 2) and HS Fe3+ (spin 5/2), these two anomalies can be attributed to the magnetic origin. The anomaly at 16.3 K was attributed, without definite evidence, to the formation of a spin-glasslike state at low temperatures. This state was ascribed to the existence of a small fraction of metal vacancies, which would be produced to compensate electrically a partial oxidation FeII → FeIII occurring during sample preparation. No heat-capacity anomaly has been detected around 30 K in NBu4[FeIIFeIII(ox)3], as expected from the magnetic measurements.196 However, it should be noted here that the magnetic measurements were made with precooled samples under certain magnetic fields. Thus, Bhattacharjee et al.191 measured its heat capacities under various applied magnetic fields. The magnetic heat capacity was tentatively fitted with the relation Cmag = αHβ. The temperature variation of the parameter β exhibited two sharp minima below and above 30 K. This observation seems to be in good correlation with the negative magnetization phenomenon. Another family of 2D assembled-metal complex is the cyanide-bridged molecule-based magnets given by the formula Xk[A(L)]l[B(CN)6]m·nS (X, monovalent nonmagnetic ion; A, B, di- or trivalent transition metal ions; L, organic ligand; S, solvent molecule; k, l, m, n, numbers of stoichiometry). Various network structures can be realized depending on the choice of A, B, and L. On the basis of this strategy, many complexes have so far been synthesized and investigated.197−200 Miyazaki et al. measured heat capacities of an antiferromagnet K[MnIII(3MeOsalen)]2[FeIII(CN−)6] (3-MeOsalen = N,N′-ethylenebisZ


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external magnetic field (up to 5 T) directed perpendicular to the plane of the layers.211 The heat capacity determined at zero magnetic field exhibited a rather broad peak centered at Tc = 32.7 K arising from the ferromagnetic to paramagnetic phase transition. Ferromagnetism can be attributed to the interaction through the CN bridges of the unpaired electrons of CuII (spin S = 1/2) and WV (S = 1/2). The layered structure brings about the quasi-2D character of the magnetic coupling. In relation to the 2D character, the heat-capacity curve has a large tail above Tc. As the external magnetic field was increased, the peak temperature of the heat-capacity curve was lowered and the peak was suppressed. The entropy gain due to the magnetic anomaly was only 6.9 J K−1 mol−1, which is much less than the maximum value Smax = 8R ln 2 (= 46.1 J K−1 mol−1) expected for eight unpaired electrons. The entropy change at zero magnetic field was only 15% of Smax, indicating that there is a great deal of magnetic ordering at temperatures above TC. Bałanda et al.212a reinvestigated the magnetic properties of the quasi-2D coordination polymer {(tetrenH5)0.8CuII4[WV(CN)8]4·7.2H2O}n by use of a single crystal. The crystal is built of cyano-bridged CuII−WVanionic doublelayer sheets lying in the ac plane. The space between the double layers is filled with water molecule and (tetrenH5)5+ solvent molecules. It was found that the 3D magnetic ordering at Tc = 33 K gives rise to antiferromagnetic structure, which under relatively small magnetic field changes to ferromagnetic. There is strong easy-plane anisotropy confining the magnetic moments to the ac crystallographic plane. Detailed analysis of the scaling behavior of the d.c. susceptibility above Tc is performed. For the direction of the external magnetic field parallel to the ac crystallographic plane the ordering process involves one stage only, whereas for the direction parallel to the b crystallographic axis a two-stage process is revealed. The corresponding crossover at about 39 K is from the 2D shortrange order state to the 3D long-range one. The wellestablished short-range order above the transition is consistent with the small value of the entropy gained at the phase transition. Czapla et al.212b studied magnetic properties of the structurally similar compound CuII2+x{CuII4[WV(CN)8]4−2x[WIV(CN)8]2x}·4H2O, in which the free spaces between the double layers are filled with paramagnetic CuII ions leading to a unique magnetic network. This compound exhibits the transition to a magnetically ordered phase at Tc = 38.7 K. The critical behavior was investigated using complementary methods: a.c. magnetometry, relaxation calorimetry, and muon spin-rotation spectroscopy. As the result, the system is found to be close to 3D Heisenberg model. The shift of the heat capacity anomaly toward higher temperatures with increasing applied field indicates the presence of ferromagnetic interactions. Fukuoka et al.213 elucidated thermodynamic nature of the antiferromagnetic transitions occurring in chiral and racemic molecular magnets [W(CN) 8 ] 4 [Cu(pn)H 2 O] 4 [Cu(pn)]2·2.5H2O, where pn is 1,2-diaminopropane. Heat capacity anomalies were observed at 7.8 and 7.2 K for the chiral Senantiomer and the racemic compounds, respectively. In the case of the S-enantiomer, a shoulder structure appears just below the main peak in a Cp/T vs T plot. This shoulder is enhanced to a sharp anomaly with a first-order character by applying magnetic field of about 1 T in the crystallographic baxis direction. On the other hand, in the case of racemic compound, only a broadening of the peak is observed.

Appearance of the complicated peak structure including the first order character is ascribed to peculiar magnetic ordering derived from crystallographic chirality related to the Dzyaloshinskii−Moriya interaction. 4.2. Mixed-Valence Assembled-Metal Complexes

By replacing the symmetric oxalate ligand ox2− (= C2O42−) in the assembled-metal complex with an asymmetric dithiooxalato ligand dto2− (= C2O2S22−), Kojima et al.214 reported a novel mixed-valence assembled-metal complex N(nC3H7)4[FeIIFeIII(dto)3]. This complex consists of Fe(II) and Fe(III) ions, which are quasi-octahedrally coordinated either by six oxygen atoms, the [FeO6] site, or by six sulfur atoms, the [FeS6] site (see Figure 41) and is crystallized at room

Figure 41. Schematic drawing of the dto-bridged network structure in N(n-C3H7)4[FeIIFeIII(dto)3].

temperature in the tetragonal crystal system with the space group P63.215 Because the ligand field at the [FeS6] site is much stronger than that at the [FeO6] site, the iron ion surrounded by sulfur atoms is characterized by a LS state, while that surrounded by oxygen atoms shows a HS state. As shown in Figure 42,214 at RT iron ions sitting in the [FeS6] site are LS Fe(III) with spin quantum numbers S = 1/2, while those of the [FeO6] site are HS Fe(II) with S = 2. When the temperature is decreased, electron transfer occurs from the [FeO6] site to the [FeS6] site around 110 K and their oxidation states are

Figure 42. Schematic representation of the electron-transfer phenomenon in N(n-C3H7)4[FeIIFeIII(dto)3]. It should be remarked that symbol S has been used both for the spin quantum number and for the entropy due to the spin multiplicity. Reprinted with permission from ref 214. Copyright 2001 Elsevier Ltd. AA


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interchanged. As the result, iron ion at the [FeO6] site becomes HS Fe(III) with S = 5/2 and that at the [FeS6] site is LS Fe(II) with S = 0. On the other hand, when the temperature of the crystal is increased, the electron is reversely transferred around 120 K. The spin multiplicity at the HT phase is 2 × 5 = 10, while that at the LT phase is 1 × 6 = 6. Because the spin-state conversion takes place in the whole [FeO6][FeS6] system by the electron transfer, this phenomenon seems to be, at first glance, spin crossover. However, this is not the case, because the spin state at a given site always remains either HS or LS state when the electron transfer occurs. Nakamoto et al.216 measured the heat capacities of this complex under constant pressure (see Figure 43) and observed

Figure 43. Molar heat capacity of N(n-C3H7)4[FeIIFeIII(dto)3] crystal as a function of temperature. Reprinted with permission from ref 216. Copyright 2001 Wiley-VCH.

Figure 44. (a) Molar heat capacities of N(n-C3H7)4[FeIIFeIII(dto)3] crystal in the vicinity of the phase transition at 122.4 K. Open circles are the same data as shown in Figure 43, corresponding to the heat capacities of the specimen well-annealed. Filled marks indicate the data obtained after cooling to the temperatures as follows: 110 (●), 100 (■), and 90 K (▲). The solid curves are for an eye guide. The dotted curve stands for the normal heat capacity. (b) Excess molar heat capacities due to the phase transition. Reprinted with permission from ref 216. Copyright 2001 Wiley-VCH.

a sharp peak at 122.4 K, a broad heat-capacity anomaly centered at 253.5 K, and a very small anomaly due to the magnetic ordering around 7 K. The origin of the heat-capacity anomaly at 253.5 K can be attributed to an order−disorder type of phase transition of the N(n-C3H7)4+ cation, judging from the fact that similar heat-capacity anomalies have been observed in the analogous molecule-based oxalate complexes (see Table 4). The anomaly at 122.4 K obviously arises from the phase transition due to the charge transfer, because the transition temperature agrees well with the temperature at which the anomaly was observed in the magnetic susceptibilities.214 Although the phase transition at 122.4 K does not take place isothermally, this can be regarded as a first-order phase transition, because a supercooling phenomenon has been observed. As shown in Figure 44a, no anomaly was detected in the measurement done for the specimen cooled to 110 K. The sample cooled to 100 K exhibited a small anomaly. This anomaly became larger as the cool-down temperature was lowered. The heat capacities shown by the open circles in Figure 44a correspond to the values obtained for the specimen well-annealed around 90 K. The excess heat capacity due to the phase transition is shown in Figure 44b. The entropy arising from the charge-transfer phenomenon was ΔtrsS = 9.20 J K−1 mol−1, which is by 4.95 J K−1 mol−1 larger than the value expected for the change in the spin multiplicity R ln(10/6) = 4.25 J K−1 mol−1. To interpret the origin responsible for this excess entropy, two possibilities have been discussed as follows:216 One is the orbital angular momentum, and the other is the lattice vibrations. The orbital angular momentum depends on the ligand-field symmetry. In

the case of Oh symmetry, the ground terms of FeII(HS) and FeIII(LS) ions in the HT phase are 5T2g and 2T2g, respectively. Because the ground terms of FeIII(HS) and FeII(LS) ions in the LT phase are 6A1g and 1A1g, respectively, the contribution of entropy from the orbital degeneracy is R ln [(3 × 3)/(1 × 1)] = 18.27 J K−1 mol−1. This value is far beyond the experimental entropy. However, the orbital degeneracy is lifted by the lowsymmetry ligand field to give a nondegenerate or lessdegenerate ground state. When a trigonal distortion is encountered in the [FeO6] and [FeS6] sites, the ground terms 5T2g of FeII(HS) and 2T2g of FeIII(LS) in Oh ligand-field symmetry are altered to 5E and 2A, respectively. In that case, the orbital entropy is reduced to R ln [(2 × 1)/(1 × 1)] = 5.76 J K−1 mol−1. This value is fairly close to the excess entropy of 4.95 J K−1 mol−1. When the ligand-field symmetry is lower than trigonal symmetry, the orbital contribution to the entropy is further reduced and eventually the contribution to entropy becomes the so-called “spin only” value. In such a case, one should look for another cause for the excess entropy. The most plausible candidate is the change in the intramolecular vibrations, in particular the metal−ligand skeletal vibrations. The magnitude of vibrational entropy crucially depends on the bond lengths between the metal ion and the ligands. The bond length is basically affected either by the spin state or by the oxidation state of the metal ion. In the case of the present complex, the bond lengths are determined by the oxidation state of the metal ion. According to Shannon,217 typical ionic radii of LS iron ion with coordination number 6 are 75 and 69 pm for FeII(LS) and AB


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crystal structure219 of [(5CAP)2CuBr4] shows the existence of layers of distorted copper−bromide tetrahedra parallel to the ab plane, separated by the organic cations along the c-axis. Magnetic pathways are available through the bromide−bromide contacts within the layers and provide for moderate antiferromagnetic exchange. Magnetic susceptibility measurements219 of polycrystalline [(5CAP)2CuBr4] reveal a broad maximum around 7 K. This maximum is well-reproduced above 5.5 K by a curve based on a HT series expansion for the S = 1/2 square planar Heisenberg antiferromagnet with the exchange parameter of J/k = −4.3 K. The experimental and the theoretical values sharply diverge near 5.1 K, indicating the existence of a long-range magnetic ordering. Figure 46 shows the magnetic heat capacity of this compound below 20 K.220 A heat-capacity peak with a large

FeIII(LS), respectively. Namely, the bond length between the metal ion and the ligand is shorter when the oxidation state of the metal ion is high. As schematically shown in Figure 45, the

Figure 45. Schematic representation of the relationship between the electron transfer and the metal−ligand bond lengths.

metal−ligand bond lengths are elongated by 13.5 pm at the [FeO6] site at the phase transition while those at the [FeS6] site are contracted by 6 pm. As a result, when the phase transition due to the electron transfer occurs, the [FeO6] site remarkably contributes positively to the vibrational entropy, whereas the [FeS6] site has a negative contribution. If the oxidation and reduction sites consisted of identical ligands, these two conflicting contributions to entropy would cancel out each other and thus no vibrational entropy would be expected. In the present case, however, because the ligands are not identical and the relative atomic masses are different (O, 15.9994; S, 32.065), the imbalance part of the cancellation seems to be the origin of the small excess entropy beyond the contribution from the change in the spin multiplicity. The present mixed-valence complex surely provides an interesting system, in which the spin state of the whole system dramatically changes by virtue of the electron transfer. From a viewpoint of functionality, the present material is expected to have a great potentiality.

Figure 46. Magnetic heat capacities of [(5CAP)2CuBr4] crystal as a function of temperature on (a) logarithmic and (b) normal scales. Solid curves indicate theoretical heat capacities for the S = 1/2 square planar antiferromagnetic Heisenberg model with J/k = −4.3 K. The broken line shows the heat capacity in accordance with the spin-wave theory for 3D antiferromagnets. Reprinted with permission from ref 220. Copyright 2000 American Chemical Society.

4.3. Inorganic Layered Complex

It is well-known that 2D Ising spin systems give rise to a magnetic phase transition,19 while ideal 2D Heisenberg spin systems do not bring about any magnetic phase transitions.218 However, actual 2D magnetic substances exhibit a magnetic phase transition at low temperatures and lead to a 3D magnetically ordered state due to the presence of weak Ising anisotropy, the existence of weak interlayer magnetic interaction between the 2D magnetic layers, or both. Landee and his collaborators are endeavoring to expand the available examples of 2D quantum antiferromagnets through the application of the principles of molecule-based magnetism. They are focusing on layered S = 1/2 Heisenberg antiferromagnets with a formula [A2CuX4], where A = 5CAP (2-amino-5-chloropyridinium) or 5MAP (2-amino-5-methylpyridinium) and X = Cl or Br. The compound [(5CAP)2CuBr4] contains S = 1/2, Cu2+ ions related by C centering, yielding four equivalent nearest neighbors. The

tail was observed at 5.08 K. As the entropy gain due to this phase transition ΔS = 5.65 J K−1 mol−1 agrees well with the R ln 2 (= 5.76 J K−1 mol−1) expected for the S = 1/2 spin system, this phase transition is surely based on the magnetic origin. The large heat-capacity tail at the HT side of the peak due to a short-range order effect is well-accounted for in terms of the S = 1/2 square planar antiferromagnetic Heisenberg model with the intralayer exchange interaction of J/k = −4.3 K. This value is in excellent agreement with the value J/k = −4.3 K, obtained from the magnetic susceptibility analysis.219 The magnetic state in the ordered phase is estimated from the low-temperature heat capacity by applying the spin-wave analysis. As described in section 1.2, the spin-wave heat capacity CSW is proportional to Td/n. By fitting this relation to the experimental values in the 0.9−2.3 K range, the parameter has been determined as d/n = AC


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summarized in Table 5,222,225 the Curie temperatures TC for all of the hybrid compounds are roughly the same as those observed in the NBu4+ derivatives.182

2.98, which is approximated as 3/1. This fact suggests a 3D antiferromagnetic state. 4.4. Multilayer Systems

According to the concept “magnetic lattice engineering” introduced by de Jongh,221 Coronado and his collaborators222 have succeeded in preparing hybrid functional materials formed by two molecular networks: (i) Hybrid magnets constructed from combinations of an extended ferromagnetic or ferrimagnetic inorganic network with a molecular paramagnetic metal complex acting as a template. (ii) Hybrid organic−inorganic compounds combining an organic π-electron donor network that furnishes the pathway for electronic conductivity, with inorganic metal complexes that act as structural and/or magnetic components. The first is a family of organometallic−inorganic magnetic compounds of formula [metallocenium][MIIMIII(ox)3],222−224 formed by 2D bimetallic oxalato-bridged honeycomb net [MnIIMIII(ox)3]− (MII = Mn, Fe, Co, Cr, Ni, Cu; MIII = Cr, Fe) and a metallocenium cation (decamethylferrocenium [DMFe]+, decamethylcobaltocenium [DMCo]+, and decamethylmanganocenium [DMMn]+) (see Figure 47),223 where [DMFe]+ is paramagnetic while [DMCo]+ is nonmagnetic. Heat-capacity measurements224 revealed the ferromagnetic phase transitions for the complexes [DMFe][MnIICrIII(ox)3] (MII = Mn, Fe, Co, Cu). Interestingly, as far as the magnetic ordering is concerned, the paramagnetic [DMFe]+ cations are not directly involved in the long-range magnetic ordering. As

Table 5. Curie Temperatures of the A+[MIIMIII(ox)3]− Complexes222,225 TC (K) +

MII, MIII

A = [DMFe]+

+

A = [DMCo]+

A+ = [DMMn]+

A+ = NBu4+

A+ = (BEDT-TTF)3+

Mn, Cr Fe, Cr Co, Cr Cu, Cr Mn, Fe Fe, Fe

5.3 13.0 9.0 7.0 28.4 43.3

5.1 12.7 8.2 6.7 25.4 44.0

5.3 13.0 9.3 7.0 27.8 45.0

6 12 10 7 28 45

5.5

The second family is characterized by the coexistence of ferromagnetism and conductivity in molecule-based layered compounds. The presence of two cooperative properties in the same crystal lattice might result in new physical phenomena and novel applications. Coronado et al.222,225 synthesized a hybrid organic−inorganic compound (BEDTTTF)3[MnIICrIII(ox)3] consisting of organic π-electron donor cation bis(ethylenedithio)tetrathiafulvalene (BEDT-TTF) and the bimetallic oxalato complex anion [MnIICrIII(ox)3]−. This layered compound displays both ferromagnetism and metallic conductivity. As shown in Figure 48,225 the structure consists of organic layers of BEDT-TTF cations alternating with honeycomb layers of the bimetallic oxalate complex [MnIICrIII(ox)3]−. The BEDT-TTF cations are tilted with respect to the inorganic layer by an angle of 45°. The magnetic properties of this compound show that it is a ferromagnet below a Curie temperature of TC = 5.5 K. The magnetic features are identical to those already reported for [metallocenium][MIIMIII(ox)3]223,224 and NBu4[MIIMIII(ox)3].182 This fact implies that the magnetic ordering in these 2D phases occurs within the bimetallic layers. This compound exhibits a metallic behavior over the whole temperature region of 2−300 K studied. Although the thermal properties seem to be of interest, there are no reports on heatcapacity measurements.

5. THREE-DIMENSIONAL MAGNETS 5.1. Cyanide-Bridged Bimetallic Complexes: Prussian Blue Analogues

Bimetallic transition metal hexacyanides form 3D network structures similar to the Prussian blue.226,227 A very high Tc = 315 K was found for V[Cr(CN)6]0.86·2.8H2O.228 Ohkoshi et al.229,230a found an interesting temperatureinduced phase transition due to the metal-to-metal chargetransfer mechanism in rubidium manganese hexacyanoferrate, RbMn[Fe(CN)6], with a large thermal hysteresis. The product of magnetic susceptibility and temperature (χm·T) is decreased around 225 K on cooling and increased around 300 K on heating, a large thermal hysteresis of 75 K (see Figure 49). XRD revealed that the structure of the HT phase is a fcc system with a space group F4̅3m and that of the LT phase is a tetragonal system with I4m ̅ 2. X-ray photoelectron spectroscopy and IR spectroscopy indicated that the electronic and spin states in the HT and LT phases are assigned as MnII(t2g3eg2, 6 A 1g ; S = 5/2)-NC-Fe III (t 2g 5 , 2 T 2g ; S = 1/2) and

Figure 47. View of the structure of [decamethylmetallocenium][MIIMIII(ox)3] in the ab plane showing the honeycomb magnetic layers (a) and in the ac plane (b). Reprinted with permission from ref 223. Copyright 1997 The Royal Society of Chemistry. AD


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Figure 48. Structures of the hybrid material (BEDT-TTF)3[MnIICrIII(ox)3]. (a) View of the [MnIICrIII(ox)3]− bimetallic layers. Filled and open circles in the vertices of the hexagons represent the two types of metals. (b) Structure of the organic layer. (c) Representation of the hybrid structures along the c-axis, showing the alternating organic/inorganic layers. Reprinted with permission from ref 225. Copyright 2000 Nature Publishing Group.

Figure 49. Observed χm·T against T plot for RbMnII[FeIII(CN)6] on cooling (i) and on heating (ii). First (○), second (□), and third (▼) measurements. Reprinted with permission from ref 230a. Copyright 2004 American Chemical Society.

Figure 50. Relationship between the electronic states of Mn and Fe in the HT and LT phases and the charge transfer. Reprinted with permission from ref 230a. Copyright 2004 American Chemical Society.

MnIII(eg2b2g1a1g1, 5B1g; S = 2)-NC-FeII(b2g2eg4, 1A1g; S = 0), respectively. This phenomenon is caused by a metal-to-metal charge transfer between Mn and Fe ions and a Jahn−Teller distortion of MnIII ion in the LT phase (see Figure 50). Molar heat capacities determined by use of a relaxation method230a and DSC230b are compared in Figure 51. A phase

transition peak was detected in both measurements in the range of 290−300 K. However, the entropy change observed at the phase transition was quite different between them: ΔS(obsd) = ∼6 J K−1 mol−1 for the former while ΔS(obsd) = (61 ± 5) J K−1 mol−1 for the latter. Possible candidates contributing to the AE


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experimental value, ΔS(obsd) = ∼6 J K−1 mol−1, observed by a relaxation method,230a whereas far from the value, ΔS(obsd) = (61 ± 5) J K−1 mol−1, detected by DSC.230b At present there is no explanation for such a large entropy change. Tokoro et al.230 suggested a possible mechanism of the phase transition as follows: Prussian blue analogues belong to the class II mixed-valence compounds,231 which have the potential to exhibit a charge-transfer phase transition and are represented by two parabolic adiabatic potential-energy surfaces due to valence isomers in the nuclear coordinate of the vibronic coupling model.232 When these two vibronic states interact, the ground-state surface has two minima. In the present crystal, the MnIII−NC−FeII vibronic state becomes a ground state in the LT phase and MnIII ions show an elongation type Jahn−Teller distortion. When the temperature is increased, the Jahn−Teller distortion is diminished and the MnII−NC−FeIII vibronic state becomes a ground state in the HT phase via a phase transition. Because the electronic state in the LT phase is [MnIII(5B1 g) and FeII(1A1 g)], there still remains a spin multiplicity of five. Tokoro et al.233 measured its heat capacity at low temperatures and found a phase transition at Ttrs = 11.0 K arising from the spin ordering (Figure 52). The entropy of phase transition was

Figure 51. Molar heat capacity of RbI0.94MnII[FeIII(CN)6]0.98·0.2H2O in the 200−350 K temperature range. Large circles are the data obtained by relaxation calorimetry (ref 230a), leading to ΔS = ∼6 J K−1 mol−1 mol and small circles are derived from DSC measurements (ref 230b), leading to ΔS = (61 ± 5) J K−1 mol−1. Reprinted with permission from ref 230b. Copyright 2011 American Institute of Physics.

entropy gain at the phase transition are the change in (i) spinmanifold ΔS(spin), (ii) electron orbital degeneracy ΔS(orbital), and (iii) molecular vibration, the so-called phonon ΔS(phonon). Because the electronic states of the LT and HT phases are [MnIII(5B1 g) and FeII(1A1 g)] and [MnII(6A1 g) and FeIII(2T2 g)], the entropy expected for (i) and (ii) are ΔS(spin) = R ln(12/5) = 7.28 J K−1 mol−1 and ΔS(orbital) = R ln(3/1) = 9.13 J K−1 mol−1, respectively. However, because the extended X-ray absorption fine structure data of the HT phase shows a small distortion in the octahedral coordinates of the metal ions, 2 T2g of FeIII is reduced to 2B2g. The saturated value of χm·T at the HT phase also supports the absence of the orbital degeneracy. These facts indicate that the entropy gain due to the electronic origin is reduced only to spin, ΔS(spin) = R ln(12/5) = 7.28 J K−1 mol−1. As to (iii), the phonon entropy ΔS(phonon) is caused mainly by a change in the meta-ligand skeletal vibrations when the charge transfer occurs. The bond length between a metal ion and its ligands in a charge transfer system basically depends on the oxidation state of the metal ion before and after the charge transfer. The bond length r(M−L) between a metal ion (M) and its ligand (L) is long when the metal ion is in a low valence state, while short for a high valence state. As shown in Figure 50, the oxidation state of the Mn ion in the LT phase is three, while two in the HT phase. Thus the bond length r(Mn− N) between the central Mn and the ligand N atoms is longer in the HT than in the LT phase. Therefore, the vibrational entropy change at the phase transition, ΔS(phonon, Mn site), is positive. Contrary to this, the oxidation state of the Fe ion in the LT phase is two, while three in the HT phase. Consequently, the bond length r(Fe−C) between the central Fe and the ligand C atoms is shorter in the HT than in the LT phase. In this case, the vibrational entropy change at the phase transition, ΔS(phonon, Fe site), becomes negative. These opposite contributions to entropy at the Mn and Fe sites lead to a very small phonon entropy at the phase transition, ΔS(phonon). As a result, one can anticipate that the entropy gain at the charge transfer transition is close to the spin-only value, ΔS(spin) = 7.28 J K−1 mol−1. This value is close to the

Figure 52. Molar heat capacity of RbMnII[FeIII(CN)6] below 30 K and the magnetic phase transition. The solid curve corresponds to the lattice heat capacity. Reprinted with permission from ref 233. Copyright 2004 Elsevier Ltd.

estimated as (11.8 ± 0.9) J K−1 mol−1. This value is close to R ln(2S + 1) = 13.4 J K−1 mol−1, the value expected for the ordering of magnetic spins on the MnIII (S = 2) sites for RbIMnIII[FeII(CN)6]. On the basis of the spin-wave analysis and the comparison of the entropies gained below and above Ttrs,16 the ordered magnetic state was determined to be a 3D Heisenberg type ferromagnetic lattice of MnIII sites. Because the diamagnetic LS FeII sites are connected with the paramagnetic MnIII sites, simple application of the superexchange mechanism to the present ferromagnetic ordering is difficult. As a plausible mechanism, the authors233 assumed the valence delocalization mechanism, in which ferromagnetic coupling arises from the charge-transfer configuration.234 With respect to the conclusion of isotropic 3D long-range order, the authors noted that the absence of anisotropic behavior in the magnetic heat capacity was unexpected for tetragonally distorted Mn(III) ions. Materials showing a large thermal hysteresis loop are useful for applications such as memory device. Ohkoshi et al.235 succeeded in expanding the thermal hysteresis of 75 K for RbMnII[FeIII(CN)6] by tuning the chemical composition. They AF


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found unusually large thermal hysteresis for the phase transitions in a series of compounds RbxMnII[FeIII(CN)6](x+2)/3·zH2O, where x = 0.94 and z = 0.3 for (1); x = 0.85 and z = 0.8 for (2); and x = 0.73 and z = 1.4 for (3). As x is decreased, the transition temperatures detected both on heating and on cooling are lowered and the hysteresis loop width is increased as ΔT = 86, 94, and 116 K for 1, 2, and 3, respectively. The heat-capacity measurement has also been reported for [Mn2(H2O)5Mo(CN)7]·4H2O by Larionova et al.236 This is a 3D cyanide-bridged bimetallic complex consisting of ladders made of edge-sharing lozenge motifs (Mo−CN−Mn−N C)2 running along the a-direction. These ladders are linked further along the b- and c-directions. Thus, the structure of this complex is totally different from that of the Prussian blue analogues characterized by fcc. A λ-type heat-capacity peak was observed at TC = 50.5 K, corresponding to the transition between the ferromagnetic and the PSs. Magnetic susceptibility measurements revealed the presence of two ferromagnetically ordered states, and a transition between these two states occurred at 43 K. However, as the entropy change is very weak, no heat-capacity anomaly is detected around this temperature.

paramagnetic state as well as in the ordered state for [Mn{N(CN)2}2] is given in Figure 54.240 The structure consists of discrete MnN6 octahedra, which are axially elongated and successively tilted in the ab plane.

5.2. Dicyanamide-Bridged Bimetallic Complexes

Figure 54. Crystallographic unit cell in the paramagnetic regime as well as in the ordered state for [Mn{N(CN)2}2]. The structure consists of discrete MnN6 octahedra, which are axially elongated and successively tilted in the ab plane. Reprinted with permission from ref 240. Copyright 2000 American Physical Society.

Miller and his collaborators237 have investigated the coordination chemistry of dicyanamide, N(CN)2−. As shown in Figure 53, this bifurcated molecule has the potential of being

Two crystal polymorphs are known for [CoII{N(CN)2}2]:237 One is a pink crystal designated as α-[Co{N(CN)2}2] in which Co2+ ion is octahedrally coordinated by six N atoms while the other is blue β-[Co{N(CN)2}2] in which Co2+ ion is tetrahedrally coordinated by four N atoms. Heat capacities of [M{N(CN)2}2] have been measured for the complexes with M = Fe, α-Co, and Ni by Kurmoo and Kepert238 and for those with M = α-Co, β-Co, Ni, Cu, and Zn by Kmety et al.241 Figure 55 represents the molar heat capacities of α-[Co{N(CN)2}2], β-[Co{N(CN)2}2], and [Ni{N(CN)2}2].241 The essential features of the data are peaks followed by rather large heat-capacity tails at the HT side of the peak. The α-[Co{N(CN)2}2] and [Ni{N(CN)2}2] give rise to λ-type peaks at TC = 9.37 and 21.1 K, respectively. These anomalies are associated with the onset of long-range ferromagnetic order. The entropy gains due to these phase transitions were 0.91R ln 2 and R ln 3, respectively. This fact implies that the magnetic ordering originates from a groundstate doublet (J = 1/2) in the form of R ln(2J + 1) for the Co2+ ion in α-[Co{N(CN)2}2] and a ground-state triplet (J = 1) for the Ni2+ system. On the other hand, β-[Co{N(CN)2}2] showed a small peak at 2.45 K and a broader one at 8.35 K. These anomalies are assigned to a canted antiferromagnetic ordering at 8.35 K and then spin reorientation to a different canted antiferromagnetic state. Because the entropy gain (0.89R ln 2) is close to R ln 2, the ground state of the tetrahedral cobalt complex is also doublet. Because the Co2+ ion has a 3d7 electronic configuration, it gives rise to a 4F ground state. In an octahedral crystal field, the orbital ground state is 4T1g followed by 4T2g (first excited state) and 4A1g (second excited state). On the basis of the crystal structure at 1.6 K, the Co2+ ion resides at the core of an axially distorted octahedron. In the presence of the axial crystal field, the 4T1g level is split further into 4A2g followed by 4Eg. Through the combined effect of the axially distorted crystal field and spin−orbit coupling, the spin quartet

Figure 53. Bifurcated ligand molecule dicyanamide N(CN)2−. (1a) Terminal ligand, (1b) μ2-ligand, (1c) μ2-ligand, (1d) μ3-ligand, and (1e) μ4-ligand. Reprinted with permission from ref 237. Copyright 1998 American Chemical Society.

polydentate. N(CN)2− can act as a terminal ligand (1a) or as a μ2-ligand (1b,c), μ3-ligand (1d), or μ4-ligand (1e). The reaction of [MII(H2O)6](NO3)2 (M = Mn, Fe, Co, Ni, Cu) and N(CN)2− leads to formation of isostructural [M{N(CN)2}2]. Crystal structures are orthorhombic with a space group Pnnm.237,238−240 The N(CN)2− anion is triply coordinated through its three nitrogen atoms (1d). It bridges the metal ions to form infinite 3D metal−organic frameworks with a rutile type structure. The framework contains doubly bridged ···M− (NC−N−CN)2−M··· ribbons that link approximately orthogonally through the amide nitrogen atoms. As a representative one, the crystallographic unit cell in the AG


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Figure 55. Molar heat capacities of (a) α-[Co{N(CN)2}2], (b) β[Co{N(CN)2}2], and (c) [Ni{N(CN)2}2]. The solid curves represent the estimated lattice heat capacities. Reprinted with permission from ref 241. Copyright 2002 Elsevier Ltd.

Figure 56. Magnetic heat capacity (a) and magnetic entropy gain (b) of [Mn{N(CN)2}2] as a function of temperature. Reprinted with permission from ref 240. Copyright 2000 American Physical Society.

of the 4A2g level is split into two Kramers doublets (| ± 1/2⟩ and | ± 3/2⟩). The value of ΔS = R ln 2 indicates that the separation between the two doublets is large as compared with TC. As a result, at low temperature, only the lowest doublet is thermally populated and the system behaves as an effective J = 1/2 system. For a Co2+ ion in a tetrahedral crystal field encountered in β-[Co{N(CN)2}2], the orbital ground state is 4 A2 followed by 4T2 (first excited state) and 4T1 (second excited state). As in the case of α-[Co{N(CN)2}2], the spin quartet of 4 A2 is split into two Kramers doublets and the system behaves as an effective J = 1/2 system. [Cu{N(CN)2}2] is paramagnetic even at 1.7 K, the lowest temperature attainable by the calorimeter.241 The LT upturn observed in the heat capacity against temperature plot suggests that [Cu{N(CN)2}2] (J = 1/2) undergoes a phase transition to an ordered state at a temperature below 1.7 K. The application of a large magnetic field of 8 T dramatically shifted the ordering heat-capacity peak from below 1.7 K to about 7.4 K. Judging from the direction of the peak shift, the ordered state has been concluded as being ferromagnetic state. Heat-capacity measurements of [MnII{N(CN)2}2] have been reported by Batten et al.239 and by Kmety et al.240 Figure 56 shows the magnetic heat capacity and the entropy gain.240 A λtype anomaly is observed at TN = 15.62 K, and the entropy gain due to this phase transition amounted to R ln 6, the value expected from the spin multiplicity of the HS manganese in an octahedral crystal field. Magnetization measurements239 of [MnII{N(CN)2 }2] confirmed that this phase transition originates in a magnetic long-range ordering into a spin-canted antiferromagnetic state (weak ferromagnet). In addition to the λ-type peak, a weak shoulder appeared around 7 K. This anomaly was connected to the observed anomalous behavior of the crystal structure below 10 K, at which a sign reversal of the thermal expansion occurs. The fractional changes in length between 10 and 300 K and those between 4.6 and 10 K are Δa/ a = +0.71 and −0.05%, Δb/b = +0.79 and +0.04%, and Δc/c = −0.38 and +0.02%.240

Manson et al.242 have studied the magnetic properties of [Mn{N(CN)2}]2·(pyz), where pyz = pyrazine, in detail by using d.c. magnetization, a.c. susceptibility, heat capacity, and neutron diffraction. The material crystallizes in the monoclinic space group P21/n with Z = 4 at 1.5 K (Figure 57).242a The

Figure 57. Zero-field magnetic structure of [Mn{N(CN)2}]2·(pyz) at 1.5 K. Arrows denote the spin configuration of the ordered Mn2+ moments aligned along the short ac-diagonal. Reprinted with permission from ref 242b. Copyright 2003 Elsevier Ltd.

heat capacity of this complex exhibited a 3D antiferromagnetic ordering below TN = 2.53 K. Figure 58 presents a schematic illustration of a magnetic phase diagram for a weakly anisotropic antiferromagnet. At a specific temperature just below TN, Hsf and Hc intersect at the tricritical point, which indicates the coexistence of the antiferromagnetic, spin-flop, and paramagnetic phases. As shown in Figure 59, the spin-flop transition was detected by variable temperature d.c. magnetic AH


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Figure 61. Field-dependent heat capacity of [Mn{N(CN)2}]2·(pyz) obtained at 2 K. Reprinted with permission from ref 242a. Copyright 2001 American Chemical Society.

Figure 58. Schematic representation of the magnetic field vs temperature diagram for a weakly anisotropic antiferromagnet. Reprinted with permission from ref 242a. Copyright 2001 American Chemical Society.

5.3. Dicyanoargentate-Bridged Complexes

Č ernák et al.243 reported the crystal structure and magnetic properties of dicyanoargentate-bridged complex [Cu(en)2Ag2(CN)4] (en = ethylenediamine or 1,2-diaminoethane). This compound crystallizes in the orthorhombic space group Pnnm. The structure is formed of free linear [Ag(CN)2]− anions and infinite cationic chains [−Cu(en)2−NC−Ag− CN−]+ containing paramagnetic copper atoms bridged by a second kind of linear dicyanoargentate species. The coordination geometry of the copper atoms corresponds to an elongated tetragonal bipyramid with two chelating (en) molecules in the equatorial positions and N-bonded bridging cyano groups in the axial positions. The heat-capacity measurements in the 40 mK to 2 K temperature range detected only the HT part of a λ-like anomaly associated with the onset of a long-range order because the lowest available temperature of 40 mK is still too high for detailed study of magnetic correlations. However, the heat-capacity study revealed that despite the chain structure this compound may represent an S = 1/2 3D magnetic system characterized by a low value of the exchange coupling constant, J/k much less than 60 mK. Heat capacities of similar compounds [Cu(bpy)2Ag2(CN)4]·H2O (bpy = 2,2′-bipyridine)244 and [Cu(en)Ni(CN)4]245,246 measured in the temperature range from 80 mK to 8 K revealed significant deviations from the assumed 1D magnetic behavior. The behavior of short-range correlations was described by an S = 1/2 quadratic Heisenberg antiferromagnetic model with the exchange constant J/k = −60 and −180 mK, respectively. Intrachain covalent bonds are completed by bifurcated hydrogen bonds of the N···H···N(C)···H···N type connecting adjacent chains and creating a 2D net of exchange paths. The origin of the λ-like anomaly in the heat capacity observed at (81 ± 2) mK for [Cu(bpy)2Ag2(CN)4]·H2O and at (128 ± 2) mK for [Cu(en)Ni(CN)4] were ascribed to a 2D−3D crossover in magnetic lattice dimensionality activated by the interlayer dipolar interaction.

Figure 59. Variable-temperature magnetic susceptibility of [Mn{N(CN)2}]2·(pyz) at various external filed in the vicinity of TN. Reprinted with permission from ref 242a. Copyright 2001 American Chemical Society.

susceptibility measurement under magnetic field. Figure 60 is the magnetic susceptibility measured at 2 K as a function of

Figure 60. ac magnetic susceptibility of [Mn{N(CN)2}]2·(pyz) as a function of external magnetic field obtained at 2 K. Reprinted with permission from ref 242a. Copyright 2001 American Chemical Society.

applied magnetic field. The spin-flop and the critical fields were determined as Hsf = 0.43 T and Hc = 2.83 T, respectively. To confirm the type of field-induced magnetic phase transition found in the magnetic measurements, Manson et al.242a performed isothermal heat-capacity measurements at 2 K (Figure 61). The Cp(H) data yielded a single anomaly at 2.83 T, which is attributed to the spin-flop to paramagnetic phase transition. Although the spin-flop transition is first-order, it was difficult to detect using this technique.

6. SINGLE-MOLECULE MAGNETS Single-molecule magnets (SMMs) are a class of paramagnetic metal complex compounds, which show slow magnetization reversal at low temperatures as an isolated molecule property.247,248 In this meaning, they belong to 0D spin systems, whose spin Hamiltonian should be exactly solved in principle even though it is too laborious of a task. Most of them are polynuclear complexes where paramagnetic transition metal AI


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ions are magnetically coupled through some bridging ligands to provide a large resultant spin on a molecule. There is a requirement for the SMM behavior that a large molecular magnetic anisotropy of easy-axis type should impose a restriction on the molecular spin so as to fix magnetic poles with respect to the molecular shape. Figure 62 depicts a set of

Figure 63. Longitudinal and transverse Zeeman splitting for spinenergy levels of molecular spin S = 10 affected by uniaxial zero-field splitting DSz2 (D < 0). B stands for magnetic flux density.

Figure 62. Plot of potential energy vs magnetization direction for a single Mn12 molecule in zero-applied magnetic field with a S = 10 ground state. There is an axial zero-field splitting, characterized by H = DSz2, where D < 0.

energy levels for a giant spin, e.g. S = 10, affected by uniaxial zero-field splitting DSz2 (D < 0), which forms a parabolic profile of energy barrier prohibiting spin reversal. As a result, molecular magnetizations hardly overcome the anisotropy energy barrier at low temperatures and show a typical magnetic relaxation behavior with single relaxation time. This monodispersivity of relaxation is the unique point of SMMs distinguished from other nonequilibrium magnets such as spin glasses with temperature-dependent energy barriers or conventional superparamagnets with barrier-height distributions. Another notable feature of SMMs is resonant tunneling of molecular magnetization characteristic of mesoscopic magnets. When a longitudinal or transverse magnetic field is applied on a uniaxial molecular spin whose level scheme is provided by Figure 62, the Zeeman effect lifts the 2-fold degeneracy of upspin and down-spin states as shown in Figure 63. In the case of longitudinal field, the spin levels make crossings each other at even intervals of field. The level crossing greatly enhances magnetization reversal via quantum tunneling through the anisotropy barrier, satisfying energy conservation. This tunneling process is called resonance tunneling, and the crossing fields are thus referred to as resonance fields Breson. As a result, a mesoscopic permanent magnet, SMM, has a strange-shaped hysteresis loop with multiple step structures at each Breson (Figure 64).248 It is essentially a single-molecule property and should be discriminated from well-known cross relaxations in a weakly coupled spin pair. Calorimetry can provide important information characterizing SMMs, e.g., the blocking temperature (TB), level-crossing fields (Breson) where quantum tunneling of magnetization occurs, and the magnitude of hyperfine interactions between electron and nuclear spins, which are supposed to be a dominant factor of magnetic relaxation in SMMs. As stated in chapter 1, heat-capacity calorimetry has an aspect of spectroscopy resolving energy level structures, due to the fact that the heat capacity is related with the canonical partition function,

Figure 64. Magnetization hysteresis loop of [Mn12O12(OAc)16(H2O)4]. Resonant tunneling extremely promotes the reversal of molecular magnetizations, resulting in the stepped loop structure. Reprinted with permission from ref 248. Copyright 2003 Wiley-VCH.

which is exactly a Laplace transform of the density of states. Heat capacity contributed from a finite number of discrete energy levels is known as a Schottky anomaly, rising as exp(−ΔE/kT)/T2 and decaying as T−2. In contrast, a continuous density of states, characteristic of elementary excitations such as phonons and magnons, contributes Tn terms according to the dispersion relation and the spatial (lattice) dimension. Blocking temperature TB is a kind of glass transition temperature, where the characteristic time for activated reversal of molecular magnetization overtakes the observation time scale (typically ∼100 s) and the molecule behaves as a molecularsized permanent magnet below this temperature. Because the spin-up and spin-down states of a molecule are energetically degenerate without any intermolecular interactions, calorimetric detections of TB are usually conducted under a magnetic field giving a Zeeman splitting in the spin doublet. On lowering temperatures, an HT tail ∼T−2 of the Schottky anomaly corresponding to the thermal excitation between the ground doublet Sz = +S and −S is observed, which is quenched below TB where the spin states become nonergodic. Thus, a step indicating a glass transition occurs in the heat-capacity curve at TB. It is noted that such a crossover from ergodic to nonergodic situation in calorimetry can be interpreted as a switching in the selection rule for thermal excitation from allowed to forbidden, similar to other spectroscopic methods using photoexcitation. AJ


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ples251−255 and for single crystals.256−260 In several works, clear heat-capacity drops at TB were observed by application of magnetic field.251,252,255 Because the dipolar fields exerted from neighboring molecules are also effective to remove the degeneracy of the ground doublet, the heat capacity under zero external field was closely examined to obtain the information on the freezing process.254 The results showed lowering of the TB and the blocking process over a certain temperature range. Those were not consistent with the other works performed under finite field, revealing sharp blockings. The effect of transverse magnetic field, which promotes tunneling processes in magnetization reversal, was also examined.258−260 Obvious depression of TB was observed owing to phonon-assisted incoherent quantum tunneling (Orbach process) at relatively lower magnetic fields, while at higher applied fields the spin−lattice relaxation is dominated by coherent tunneling and the blocking phenomena are completely lost. Because the nuclide 55Mn with nuclear spin I = 5/2 has the natural abundance of 100%, a Schottky anomaly arising from the hyperfine splitting of 55Mn nuclear spin levels has a certain contribution below 0.5 K.252,254,255,258−260 Although the hyperfine interaction constants from calorimetric studies were almost consistent with internal field provided by other methods including 55Mn NMR and DFT (density functional theory) calculation,254,255 it was pointed out that the magnetic field dependence of the nuclear heat capacity requires taking into account the slow nuclear spin−lattice relaxation.260 The total heat capacity of Mn12 is well-reproduced by accumulating the contributions stated above. For example, the total heat capacity of [Mn12O12(O2CEt) 16(H2O)3] was analyzed as a sum of four contributions, Clat, Cspin(D1), Cspin(D2), and Chf, shown in Figure 66.255 Clat is the heat capacity of the phonon system and is well-approximated by the Debye function with Debye temperature θD = 30 K. Cspin is a multilevel Schottky heat capacity due to electron spin under uniaxial zero-field splitting (D); however, it is required to adopt two components corresponding to Jahn−Teller isomers. The

Below TB, nuclear spins contained in an SMM molecule feel a rather stationary hyperfine field and the nuclear spin-energy levels contribute another Schottky heat capacity. When the intermolecular interactions are not negligible, SMMs start deviating from the isolated molecule behavior and heat-capacity anomalies accompanying the growth of shortrange order are observed at low temperatures. In such cases, there is a competition between the kinetic freezing of magnetization reversal and the spin correlation leading to a magnetic long-range order. If the supposed phase transition temperature Tc is much lower than TB, the SMM behavior should be sustained despite the intermolecular interactions and the superparamagnetic blocking is detectable by calorimetric methods without any external bias fields. On the other hand, the long-range order is established at the lowest temperatures and the magnetic entropy of R ln 2, corresponding to an SMMlike Ising spin, should be detected by calorimetry, when the Tc is higher than the TB estimated by extrapolation of the Arrhenius plot. There is an interesting discussion whether the long-range order is achieved below TB via quantum tunneling processes, and this question will be revisited later. 6.1. Dodecanuclear Manganese Clusters (Mn12)

The dodecanuclear mixed-valence manganese cluster [Mn12O12(OAc)16(H2O)4] (hereafter, Mn12) consisting of four Mn(IV) and eight Mn(III) ions (see Figure 65)249 has a

Figure 65. Skeletal structure of Mn12O12 core. Four ferromagnetically coupled Mn(IV) ions carrying S = 3/2 are surrounded by eight peripheral Mn(III) ions carrying S = 2. Reprinted with permission from ref 249. Copyright 1997 American Physical Society.

large resultant spin of S = 10 and is known as the first discovered SMM with a high activation barrier of magnetization reversal U eff/k ∼ 60 K. Chemical modifications, e.g., substitution of acetate ligands with other carboxylates and one- or two-electron reduction of the Mn12O12 core, afforded many derivatives also behaving as SMMs. The large magnetic anisotropy of the molecule is attributed to the nearly parallel arrangement of easy axes of eight Jahn−Teller distorted Mn(III) ions. Because of a partial fluxionality of Mn12 molecules, there are some “Jahn−Teller isomers” with one of eight Jahn−Teller elongation axes tending to a wrong direction out of the molecular axis.250 In some Mn12 derivatives, Jahn− Teller isomers are unable to be purified chemically and only a mixture sample is available. The temperature dependence of heat capacity of several Mn12 compounds was reported for polycrystalline sam-

Figure 66. Temperature dependence of heat capacity of [Mn12O12(O2CEt)16(H2O)3]. Clat, Cspin, Chf, and total Cp indicate the Debye heat capacity, the multilevel Schottky heat capacities from two Jahn−Teller isomers, the contribution from the hyperfine interaction, and the net heat capacity, respectively. Reprinted with permission from ref 255. Copyright 2001 American Chemical Society. AK


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processes, breaking the kinetic selection rule of calorimetry. However, these peaks are suppressed below TB. A series of Mn12 isomers, Mn12 wheels, were also examined by calorimetry. 2 6 3 They have a general formula of [Mn12L8(RCOO)14] and showed no evidence of magnetic long-range order in heat capacities, aside from the broad peaks of heat capacity at around 10 K attributable to the Schottky heat capacity of zero-field splitting.

last term Chf is the contribution from hyperfine interactions of 55 Mn nuclei and is well-approximated by the first term of HT expansion, ⎛ hνn ⎞2 nNAk C hf ≈ I(I + 1)⎜ ⎟ ⎝ kT ⎠ 3

(6)

where n is the total number of nuclei having the nuclear spin quantum number I, NA is the Avogadro constant, h is the Planck constant, and νn is the resonance frequencies in 55Mn NMR. a.c. calorimetry utilizing heater power modulation with frequency ω is especially advantageous in research of timedependent phenomena. Time-dependent heat capacities C(ω) of Mn12 were reported with respect to the effect of longitudinal magnetic field.257,261,262 If the spin−lattice relaxation time τ is short enough to achieve thermal equilibrium (ωτ ≪ 1), spin levels at both sides of the energy barrier can participate in population exchange and contribute to the Schottky heat capacity due to Zeeman splitting (bilateral heat capacity, Cbil, in Figure 67).257 When the relaxation time exceeds the measure-

6.2. Octanuclear Iron Clusters (Fe8)

The octanuclear Fe(III) cluster [(tacn)6Fe8O2(OH)12](Br)8 (hereafter, Fe8) is another SMM with a spin ground state of S = 10, thoroughly investigated next to the Mn12. A distinctive feature of the Fe8 is the rhombicity of its magnetic anisotropy, in clear contrast to the Mn12, which has complete tetragonal symmetry. Because of the deviation from axial symmetry, it is the first compound in which the quantum interference effect (Berry’s phase) of magnetization tunneling was demonstrated.248 The temperature dependence of heat capacity of the Fe8 was also reported under zero external field254 and under transverse magnetic fields.258−260,264 The effect of longitudinal field was also investigated by a.c. calorimetry.265a The results for Fe8 resemble ones for Mn12, except Fe8 has a lower anisotropy barrier and some rhombicity. Quite peculiar heat-capacity anomalies were reported at around 2.6 K for an Fe8 single-crystal sample examined by a.c. calorimetry.265b These anomalies were not affected by changing measurement frequency, and the peak temperature was found to be depressed under applied magnetic field up to 4.0 T, showing a close resemblance to a Néel temperature of antiferromagnetic long-range ordering. Recent measurement using relaxation technique266 got rid of the anomalies around 2 K, and the low-temperature heat capacity was fully interpreted with the Schottky contribution due to spin levels and local-field fluctuation. 6.3. Other Single-Molecule Magnets

Among other SMMs, some are known to have magnetic phase transitions yielding long-range order and calorimetric researches on them were carried out. λ-Shaped heat-capacity anomalies were reported at Tc = 1.33 K for [Mn 4 (hmp) 6 Br 2 (H 2 O) 2 ] Br 2 , 2 6 7 T c = 1. 19 K fo r [Fe19(metheidi)10(OH)14(O)6(H2O)12]NO3·24H2O,268 and Tc = 0.99 K for (enH2)2[Fe6μ3-O]2(μ2OH)6(ida)6]·6H2O·2EtOH.268b The intermolecular interactions causing long-range orders in these compounds were supposed to be superexchange rather than magnetic dipolar interactions in origin, which are claimed to be dominant in the case of [Mn6O4Br4(Et2dbm)6] possessing Tc = 0.161 K,264,269 and also in Hpyr[Fe17O16(OH)12(py)12Br4]Br4 with Tc ∼ 0.85 K.266b,270 Although the long-range order in [Fe4(OCH3)6(dpm)6] was suggested,271 no heat-capacity anomalies were found by calorimetry in the measurement temperature range (T > 2 K).272 Heat capacities of a series of manganese tetramers with S = 9/2, [Mn 4 O 3 L(dbm) 3 ] (L = Cl(OAc) 3 , (OAc) 4 , and (O2CC6H4Me)4, abbreviated to Mn4Cl, Mn4Ac, and Mn4Me, respectively), were investigated by thermal relaxation method.264,266b,273,274 The equilibration time (or instrumental relaxation time, in other words) dependence of heat capacities Cp(τe) revealed a remarkable difference among them. The Mn4Cl having a relatively slow spin−lattice relaxation shows a large τe dependence below 1 K, i.e., a longer τe is required to

Figure 67. (a) Magnetic heat capacity of the Mn12 single crystal vs the applied magnetic field, measured at ω = 25 Hz. The zero-fieldcalculated magnetic heat capacity Cmag(0) is 56.0, 47.7, 40.2, and 21.3 nJ K−1 for, respectively, T = 4.75, 4.14, 3.67, and 2.65 K. (b) a.c. heat capacity C(ω) of an Mn12 single-crystal sample under magnetic field applied parallel to the molecular axis. Remarkable enhancement is recognized at the level-crossing fields Breson, which disappears below the temperature where the relaxation time of spin reversal coincides with the time scale of measurement frequency ω. Reprinted with permission from ref 257. Copyright 1999 American Physical Society.

ment time scale (ωτ ≫ 1) by cooling, the population flow between the up-spin side and the down-spin side is kinetically forbidden and a smaller heat capacity (unilateral heat capacity, Cuni) is observed. The remarkable peaks observed at Breson in Figure 67 arise from phonon-assisted incoherent tunneling AL


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S = 0 does not fall into the SMM category, it should be discussed here since it is one of exchange-coupled finite spin systems of great interest in the quantum effect involved. Calorimetric studies266b reported cover [LiFe6(OCH3)12(dbm)6]PF6,284 [NaFe6(OCH3)12(pmdbm)6]PF6,284,285 [LiFe6(OCH3)12(dbm)6]BPh4,286a [Fe10(OCH3)20(C2H2O2Cl)10],285 [Fe12(OCH3)12(dbm)12],286b and [Cr8F8(piv)16].286c,d As shown in Figure 68,286a the six-membered antiferromagnetic iron wheel shows avoided crossing of spin levels, arising

achieve thermal equilibrium. On the other hand, the Mn4Me having a fast electron spin−lattice relaxation shows equilibrium heat capacity with a λ-peak anomaly of magnetic long-range order at Tc = 0.21 K. Despite such a very low Tc, the dominant intermolecular interaction is not attributed to dipole−dipole but superexchange in origin, being |J|/k ≈ 0.14 K.274b It is inferred from these observations that in this temperature range thermal equilibration of the spin systems is promoted via incoherent magnetization tunneling in the lowest Kramers doublet. Thus, the answer to the question, whether the longrange order is achieved below TB via quantum tunneling processes, is presumably yes, since the spin system under incoherent tunneling couples with phonon bath through nuclear spin-mediated (I+S− etc.) spin−lattice interaction. Two heat-capacity results were reported for the Ni4 cubane tetramers with ferromagnetically coupled S = 4 ground states. [Ni4(OH)(OMe)3(Hphpz)4(MeOH)3](MeOH)275 has no magnetic long-range orders down to 300 mK, and the lowtemperature heat capacity was made use of determination of molecular zero-field splitting parameters. In marked contrast, [Ni(hmp)(dbm)Cl]4276 showed a typical λ-shaped singularity of heat capacity at Tc ∼ 300 mK, confirming the ferromagnetic 3D long-range ordering. 6.4. Single-Chain Magnets (SCMs)

Figure 68. Zeeman splitting of lowest-lying spin levels of [LiFe6(OCH3)12(dbm)6]BPh4. Reprinted with permission from ref 286a. Copyright 2002 American Physical Society.

In 1963, Glauber277a found a slowing-down of magnetic relaxation in the 1D Ising model at low temperatures by using Monte Carlo simulation. SCMs are 1D systems consisting of Ising spins, which exhibit blocking phenomena similar to SMMs, as predicted by Glauber, and lack any magnetic longrange orders. Typical examples are an alternating Co(II)-radical chain with highly anisotropic exchange interactions277b and a concatenated-SMM chain possessing remarkable single molecule anisotropy.277c An example of calorimetric studies on alternating radical-metal SCMs was conducted on [MnIII(5TMAMsaltmen)(TCNQ·)](ClO4)2, and it revealed absence of magnetic long-range order down to 0.6 K.278 Heat capacities of concatenated-SMM chains [Mn4(hmp)6R2](ClO4)2 (R = OAc or Cl)279 and a rare-earth transition-metal mixed chain [Ln(bpy)(H2O)4M(CN)4] (Ln = Gd(III), Y(III); M = Fe(III), Co(III))280 were also reported. The former shows a characteristic monodispersive magnetic relaxation,281 and the latter shows similar relaxations under external fields. Except [Gd(bpy)(H2O)4Fe(CN)4], no indications of magnetic long-range order were observed. For the [Mn4(hmp)6R2](ClO4)2 systems, magnetic entropies were well-accounted for by considering the contribution from the lowest zero-field splitting spin levels, R ln(2S + 1) = R ln 19. Concatenated-SMM systems consisting of Mn4-SMM units were later extended from 1D to 2D systems where magnetic long-range orderings are commonly found in heat capacity measurements despite the critical temperatures are rather scattered over a few decades: TN = 4.35 K in [Mn4(hmp)6{N(CN)2}2](ClO4)2, TN = 2.03 K in [Mn4(hmp)4Br2(OMe)2{N(CN)2}2]·2THF·0.5H2O, Tc = 1.96 K in [Mn5(hmp)4(OH)2{N(CN)2}6]·2MeCN·2THF, and Tc = 380 mK in [Mn4(hmp)4(pdm)2{N(CN)2}2](ClO4)2·1.75H2O·2MeCN.282

from anisotropic energy terms responsible for S-mixing, when an external magnetic field is applied in 25° apart from the normal of the ring plane. Because the Schottky heat-capacity anomaly expected from thermal excitation between the lowest two levels has the maximum at the energy separation Δ(B) ≈ 2.5 kT, an isothermal field sweep provides heat-capacity maxima at the fields satisfying this resonance condition and provides zero heat capacity at strict level crossings. Bearing it in mind, the heat-capacity curve in Figure 69 is consistent with the energy level scheme in Figure 68. The nonvanishing heat capacity at the level-crossing field Bc1 and Bc2 clearly demonstrates the presence of finite energy separation between the lowest two spin levels and this minimum gap corresponds to the off-diagonal matrix elements causing quantum coherence. Recently, similar avoided crossing was detected by

Figure 69. Field dependence of the isothermal heat capacity of [LiFe6(OCH3)12(dbm)6]BPh4 measured at 0.78 K. The solid curve is a two-level Schottky anomaly calculated using the energy level scheme in Figure 68. The broken curve is the best fit assuming sample inhomogeneity and no avoided crossings. Reprinted with permission from ref 286a. Copyright 2002 American Physical Society.

6.5. Antiferromagnetic Spin Rings

Antiferromagnetic rings, especially ferric wheels, were extensively investigated aiming to elucidate the quantum coherence found in the avoided crossing in Zeeman-splitting spin levels.283 Although the antiferromagnetic ring with a spin ground state of AM


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calorimetry also in a nonring ferrimagnetic manganese cluster [Mn9(2POAP-2H)6](ClO4)6.286e Recently, not fully antiferromagnetic but a canted-antiferromagnetic spin ring [Dy3(μ3-OH)2(o-van)3Cl(H2O)5]Cl3 was reported and the absence of any long-range order was demonstrated by calorimetry.287 6.6. Other 0D Metal Complex Systems

Paramagnetic polynuclear/mononuclear metal complexes lacking any magnetic long-range orders belong to the zerodimensional (0D) magnet class, similar to SMMs. Their spin Hamiltonians are easily solvable to give spin energy levels, so that they serve nice simple exercises for comparison of theoretical energy spectra with deconvoluted ones from the Schottky heat capacity. An instructive example is [NiII(en)2(H2O)2]SO4,288 where the Schottky heat capacity due to an S = 1 spin was analyzed to determine zero-field splitting. [CuII7(μ3-Cl)2(μ3-OH)6(D-pen-disulfide)3] is a double-cubane complex,289 which has no significant magnetic anisotropy and the Schottky heat capacity under magnetic field was utilized for characterization of the spin ground state. Another example is a basic acetate-type heterotrimer [CrIII2CuIIO(O2CCBr3)6(C4H8O)3],290 where anisotropic exchange couplings between metal ions were resolved based on the heat capacity. All these compounds show no effective energy barriers for spin reversal in contrast to SMMs.

Figure 70. Molar heat capacity of [Fe(NCS)2(phen)2]. Broken curves indicate the normal heat capacities. Reprinted with permission from ref 298. Copyright 1974 Elsevier Ltd.

5 = 13.38 J K−1 mol−1 expected for a conversion from the 1A1g LS state to the 5T2g HS state. The entropy gain due to the change in the spin multiplicity is only 27% of the total entropy of transition. It should be remarked here that the change in the orbital degeneracies between the LS and the HS state does not generally contribute to the entropy gain at the spin crossover transition because the orbital degeneracy, if any, has been lifted by the low-symmetry ligand field to give a nondegenerate orbital. As the spin crossover occurring, for example, in an octahedral symmetry involves a transfer of electrons between the bonding t2g and antibonding eg orbitals, the metal-to-ligand bond distances remarkably change, being about 20 pm shorter in the LS state than in the HS state. This brings about a drastic change in the density of vibrational states, mainly the metal−ligand skeletal vibrational modes.299 Thus, the transition entropy involves a large contribution from the nonelectronic vibrations and the lattice heat capacity exhibits a discontinuity at the transition temperature. As seen in Figure 70 for the complex [Fe(NCS)2(phen)2], a jump of ΔCp (normal) = 18.7 J K−1 mol−1 is observed. On the basis of variable-temperature IR absorption spectroscopy, they accounted for the excess entropy beyond the contribution from the spin multiplicity and the heat-capacity jump at the transition temperature in terms of the phonon contribution. Thermal spin crossover is possible only for the complexes whose ground electronic level is a LS state at low temperatures and the first excited HS level is located within a thermally accessible range. The energy difference is overwhelmed by the entropy term TΔS mainly due to the drastic change in lattice vibrations, and consequently, the Gibbs energy of the HS state becomes lower than that of the LS state above a temperature at which the spin crossover takes place. Sorai and Seki298 concluded that the thermal spin crossover is an entropy-driven phenomenon and coupling between the electronic states and the phonon system plays a fundamental role in the spin crossover phenomena occurring in the solid state. This concept has long since been accepted.

7. SPIN CROSSOVER PHENOMENA Complexes showing spin crossover phenomena are typical molecule-based magnets, because the magnetic properties dramatically change through the low spin (LS) to high spin (HS) transition.291−293 However, the calorimetric studies have already been summarized by one of the authors (M.S.) in review articles,11,12 and thus only a few topics will be picked up here. Thermodynamic data for trinuclear and polynuclear iron(II) spin crossover complexes are compared with those for mononuclear complexes by Berezovskii and Lavrenova.13 7.1. Entropy-Driven Phenomena: Coupling with Phonon

Although the possibility of the spin-state transition was theoretically predicted in 1954 on the basis of the ligand field theory by Tanabe and Sugano,294 it was in 1967 that the first experimental evidence was reported for iron(II) complexes [Fe(NCX)2(phen)2] (X = S or Se; phen =1,10-phenanthroline) by König and Madeja,295 in which the electronic state at room temperature is 5T2g with the spin quantum number of S = 2 while at low temperature it becomes 1A1g with S = 0. They gave a reasonable interpretation (as due to a spin crossover mechanism) for the unusual magnetic behaviors of these complexes reported first by Baker and Bobonich296 in 1964. Because the spin crossover phenomenon is principally based on a change in the quantum state of the electron spin, many researchers believed that the dominant driving force leading to thermal spin crossover transition would be a change in the spin multiplicity. However, in 1972/1974, Sorai and Seki297,298 reported calorimetric study on the spin crossover iron(II) complexes, [Fe(NCS)2(phen)2] and [Fe(NCSe)2(phen)2], for the first time, and their results showed that this interpretation was not correct. As reproduced in Figure 70, a large phase transition was observed for [Fe(NCS)2(phen)2] at 176.29 K with an entropy change of ΔS = (48.8 ± 0.7) J K−1 mol−1. This entropy value is extremely large in comparison with the entropy gain ΔS = R ln AN


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Table 6. Crystallographic Data for Mononuclear Spin Crossover Complexes compoundsa

type of SCOb

[Fe(2-pic)3]Cl2·CH3OH [Fe(2-pic)3]Cl2·C2H5OHd [Fe3(Ettrz)6(H2O)6](CF3SO3)6 [Fe(2-pic)3]Br2·CH3OH [Fe(NCS)2(phen)2] [Fe(NCS)2(py)2(bpym)]·0.25py [FeIII(acpa)2]BPh4 [FeIII(acpa)2]PF6 [Fe(NCS)2(btz)2] [Fe(NCS)2(tap)2]·CH3CN [Fe{5-NO2-sal-N(1,4,7,10)}]e,f

G, 1-step A, 2-step A, 1-step A, 1-step A, 1-step A, 1-step G, 1-step G, 1-step G, 1-step G, 1-step A, 2-step

[Fe(NCS)2(PM-BiA)2] [Fe(NCS)2(PM-PEA)2]e

A, 1-step A, 1-step

[Fe(NCS)2(PM-TeA)2] [Fe(NCS)2(PM-AzA)2] [Fe(btr)3](ClO4)2f [Fe{HC(3,5-Me2pz)3}2](BF4)2e,f

G, 1-step G, 1-step A, 2-step A, 1-step

[Fe(DPEA)(bim)](ClO4)2·0.5H2Of [FeIII(ISQ)2Br]f,g [FeIII(ISQ)2I]g [FeIII(isoxazole)6](ClO4)2f [FeIII(isoxazole)6](BF4)2e,f

G, 2-step G, 1-step G, 1-step A, 1-step G, 2-step

[Fe(DAPP)(abpt)](ClO4)2h [Fe(Hpy-DAPP)](BF4)2h {Fe(pmd)[Ag(CN)2][Ag2(CN)3]} {Fe(py)2[Ag(CN)2]2} {[Fe(mph)2](ClO4)(MeOH)0.5(H2O)0.5}2 [Fe(H4L)2](ClO4)2·H2O·2(CH3)2CO

A, 1-step A and G, 2-step A, 2-steps G, 2-steps A, 2-steps A, 1-step

T (K)c

crystal system

space group

Z

refs

115 (LS), 227 (HS) 90 (LS), 298 (HS) 105 (LS), 300 (HS) 110 (LS), 298 (HS) 130 (LS), 293 (HS) 298 (HS) 120 (LS), 311 (HS) 120 (LS), 290 (HS) 130 (LS), 293 (HS) 135 (LS), 290 (HS) 292 (HS) 153 (IP) 103 (LS) 140 (LS), 298 (HS) 298 (HS) 140 (LS) 11 (LS), 298 (HS) 110 (LS), 298 (HS) 150 (LS), 190 (IP), 260 (HS) 220 (HS) 173 (IP) 123 (LS), 293 (HS) 100 (LS), 295 (HS) 100 (LS), 295 (HS) 150 (LS), 225 (HS), 295 (HS) 260 (HS) 130 (IP) 123 (LS), 293 (HS) 90 (LS), 130 (IP), 298 (HS) 30 (LS), 170 (IP), 290 (HS) 77 (LS), 125 (IP), 200 (HS) 110 (LS), 180 (IP), 296 (HS) 100 (LS), 200 (HS)

orthorhombic monoclinic orthorhombic monoclinic orthorhombic tetragonal triclinic monoclinic orthorhombic triclinic monoclinic monoclinic triclinic orthorhombic monoclinic orthorhombic orthorhombic monoclinic trigonal monoclinic triclinic monoclinic triclinic monoclinic trigonal trigonal triclinic monoclinic monoclinic monoclinic monoclinic triclinic triclinic

Pbna P21/c P3̅1c P21/n Pbcn I41/a P1̅ P2/a Pbcn P1̅ P2/c P2 P1 Pccn P21/c Pccn Pccn P21/c R3̅ C2/c P-1 P21/c P-1 P21/c P3̅ P-3̅ P-1 P21/n P21/c P21/c C2/m P1̅ P1̅

8 4 2 4 4 16 2 2 4 2 2 2 2 4 4 4 4 4 6 4 2 4 4 4 3 3 3 4 4 16 2 2

324,325 325,326,327,328,329 330 331 323 332 333 333 334 335 312 312 312 336 337 337 337 337 313 338 338 339 340 340 341 341 341 342 343 343a 343b 343c 343d

a

2-pic, 2-picolylamine (2-aminomethy-pyridine); Ettrz, 4-ethyl-1,2,4-triazole; phen, 1,10-phenanthroline; bpym, 2,2′-bipyrimidine; Hacpa, N-(1acetyl-2-propylidene)-2-pyridylmethylamine; btz, 2,2′-bi-4,5-dihydrothiazine; tap, 1,4,5,8-tetraazaphenanthrene; 5-NO2-sal-N(1,4,7,10), the Schiff base condensation of 5-nitrosalicylaldehyde with 1,4,7,10-tetraazadecane in a 2:1 ratio; PM-BiA, N-(2′-pyridylmethylene)-4-aminobiphenyl; PMPEA, N-(2′-pyridylmethylene)-4-(phenylethynyl)aniline; PM-TeA, N-(2′-pyridylmethylene)-4-aminoterphenyl; PM-AzA, N-(2′-pyridylmethylene)4-(phenylazo)aniline; btr, 4,4′-bis-1,2,4-triazole; pz, pyrazolyl ring; DPEA, 2-aminoethyl-bis(2-pyridylmethyl)amine; bim, 2,2-bisimidazole; ISQ, Nphenyl-o-imino(4,6-di-tert-butyl)benzosemiquinonate; DAPP, bis(3-aminopropyl)(2-pyridylmethyl)amine; abpt, 4-amino-3,5-bis(pyridin-2-yl)-1,2,4triazole; pmd, pyrimidine; py, pyridine; Hmph, 2-methoxy-6-(pyridine-2-ylhydrazonomethyl)phenol; H4L, 2,6-bis[5-(2-hydroxyphenyl)pyrazol-3yl]pyridine. bSCO, spin crossover; G, gradual type of SCO; A, abrupt type of SCO; 1-step, single-step SCO; 2-step, double-step SCO. cLS, low-spin state; IP, intermediate phase; HS, high-spin state. dOrder−disorder of the solvate molecule. eThe space group is different between the LT phase and the HT phase. fThere exist two inequivalent iron sites or two sublattices. gThe ligand “ISQ” is a free radical with spin S = 1/2. hOrder−disorder of the ligand and the anions.

To verify this conclusion, Bousseksou et al.300,301 performed Raman spectroscopy of this benchmark complex [Fe(NCS)2(phen)2] at 300 and 100 K to obtain the frequency shifts associated with the HS-to-LS transition. Their overall result was consistent with the conclusion originally proposed by Sorai and Seki298 that the vibrational entropy change is primarily attributable to the intramolecular vibrations. Recently, Brehm, Reiher, and Schneider302 calculated the normal-mode frequencies of this complex in both the HS and the LS states on the basis of density functional theory (DFT/BP86) and compared the result with the IR and Raman spectra recorded at 298 and 100 K. The vibration-related entropy change due to the LS to HS transition estimated by their quantum chemical calculation was ΔS = 19.5 J K−1 mol−1. This value corresponds to 40% of the experimental value.298 Because the share of the spin entropy is 27%, the remaining 33% of the entropy is attributed to the change in the lattice vibrations perturbed by

intermolecular interactions. At any rate, a large part of the entropy gain due to the spin-state conversion beyond the spin contribution is caused by the change in the phonon. This situation is always encountered in any spin crossover complexes, irrespective of abrupt and gradual types of spin crossover phenomena.302−319 7.2. Phase Transition without Change in Crystal Symmetry

A very drastic change around the metal environment is involved when the spin-state conversion takes place, as easily recognized by the fact that the interatomic distances between the central metal atom and the ligands are drastically shortened on going from a HS to a LS state.320−322 One may, therefore, anticipate a big change in the crystal symmetry for the spin crossover phenomena. Quite surprisingly, however, this is not the case. For example, Gallois et al.323 reported that the most-studied spin crossover complex [Fe(NCS)2(phen)2] is crystallized in the orthorhombic space group Pbcn with Z = 4 both at 293 K AO


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structure in the monoclinic C2/c space group with Z = 4 and there is one unique iron(II) site. Below 204 K, the compound converts to a 50:50 mixture of HS and LS. There is a radical change in the coordination sphere for half of the iron(II) sites, most notably a shortening of the Fe−N bond distances by ca. 20 pm. The crystal system changes to triclinic P space group with Z = 2.

(HS phase) and at 130 K (LS phase). In disagreement with most of the predictions, the LS-to-HS transition is accompanied neither by a change in the crystal symmetry nor by an order− disorder transition involving NCS− groups. Only a large reorganization of the iron(II) environment is detected. The main structural modifications, when passing from the HS to the LS form, consist of a remarkable shortening of the Fe−N distances (by about 20 to 10 pm) and a noticeable variation of the N−Fe−N angles, leading to a more regular shape of the [Fe−N6] octahedron. The absence of a change in crystal symmetry at the spin-state conversion has been encountered in most spin crossover complexes. Table 6 lists space groups of mononuclear spin crossover complexes in both HS and LS states. The phase transition without change in the space group, which may be designated as “isomorphous” or “isostructural” phase transition, seems to be characteristic of spin crossover phenomena. In the case of usual order−disorder types of phase transitions, their mechanisms involve conversion of molecular orientations from ordered states at low temperature to disordered states at high temperature, which would give rise to a big change in the crystal symmetry. On the other hand, in the case of spin crossover transitions, molecular structures are retained as being approximately similar between LS and HS states, whereas big changes occur in the bond lengths. The isomorphous character of the spin-state conversion has also been supported by spectroscopy. On the basis of ESR study of a precipitated sample of an iron complex doped with manganese(II), [Fe0.99Mn0.01(NCS)2(phen)2], Rao et al.344 found that the zero-field splitting parameters D and E appearing in the spin Hamiltonian for Mn2+ ion decrease steadily with increasing temperature and are not affected by the transition. Consequently, they described this as “It appears that any structural changes that occur during the spin transition are short-range ones inside the iron(II) complex and not longrange ones involving the total lattice.” Although a genuine phase transition is impossible unless a long-range order exists, this description is suggestive to understand the transition mechanisms of spin crossover in the solid state and the spinstate equilibrium in solution. An interesting example of iron(III) spin crossover complex, which exhibits a single-step transition without change in the crystallographic symmetry, is a five-coordinate iron(III) complex, [FeIII(ISQ)2Br], reported by Chun et al.,340 where ISQ is N-phenyl-o-imino(4,6-di-tert-butyl)benzosemiquinonate π radical anion with spin S = 1/2. This complex contains a 1:1 mixture of the total spin St = 3/2 and 1/2 forms in the temperature range of 4.2−150 K. Above 150 K, the latter form undergoes a spin crossover from 1/2 to 3/2. The St = 3/2 ground state is attained via intramolecular antiferromagnetic coupling between a HS iron(II) (SFe = 5/2) and two ligand π radicals, whereas the St = 1/2 form is generated from exchange coupling between an intermediate spin iron(III) (SFe = 3/2) and two ligand π radicals. A few exceptional complexes listed in Table 6 are [Fe{5NO2-sal-N(1,4,7,10)}],312 [Fe(NCS)2(PM-PEA)2],337 [Fe{HC(3,5-Me2pz)3}2]·(BF4)2,338 and [FeIII(isoxazole)6]·(BF4)2,341 in which the space group changes between the LS LT phase, the mixed intermediate-temperature phase, and the HS HT phase. In the case of [Fe{HC(3,5Me2pz)3}2]·(BF4)2, where pz = pyrazolyl ring, an abrupt singlestep spin crossover at 204 K is exhibited. Above this temperature, the compound is completely HS with the

7.3. Cooperativeness of Spin Crossover Phenomena

As widely recognized, spin crossover transformations can phenomenologically assume two limiting cases: One is the so-called abrupt type in which the spin-state transformation takes place abruptly within a narrow temperature range of a few Kelvin, and the other is the gradual type in which the spin crossover occurs gradually over a wide temperature range, typically greater than 100 K. Calorimetric measurements of the gradual type have been made for the spin crossover complexes [FeIII(acpa)2]PF6 [Hacpa = N-(1-acetyl-propylidene)-2-pyridylmethylamine],304 [FeIII(3EtO-salAPA)2]ClO4·C6H5Br (3EtOsalAPAH = the Schiff base condensed from 1 mol of 3ethoxysalicylaldehyde with 1 mol of N-aminopropyl-aziridine),11,308 and [FeII(2-pic)3]Cl2·CH3OH (2-pic =2-picolylamine or 2-aminomethylpyridine).315 These complexes exhibit unusually broad heat-capacity peaks over a wide temperature region. As an example, the molar heat capacity of [Fe(2pic)3]Cl2·CH3OH is shown in Figure 71.

Figure 71. Molar heat capacity of the spin crossover complex [Fe(2pic)3]Cl2·CH3OH. The dotted curve indicates the normal heat capacity. Reprinted with permission from ref 315. Copyright 2001 American Chemical Society.

To elucidate the reason for the gradual spin-state conversion and the degree of cooperativeness inherent in the spin-state conversion, Sorai and Seki298 proposed in 1974 a simple domain model based on the Frenkel theory345 of heterophase fluctuations in liquids. As schematically shown in Figure 72, this model assumes that a crystal lattice consists of N noninteracting domains with uniform size containing n complexes and that the spin-state conversion in each domain takes place simultaneously, where N × n is equated to the Avogadro constant NA. When a system consisting of the HS fraction f HS and the LS fraction (1 − f HS) is in a thermal equilibrium, the Gibbs energy of the system G at a temperature T is given by AP


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Figure 72. Schematic drawing of the domain model. The crystal lattice is considered to consist of domains of uniform size containing equal numbers of spin crossover complexes. T1/2 is the transition temperature at which the number of LS domains becomes equal to that of HS domains.298 Figure 73. Excess heat capacity beyond the normal heat capacity arising from the spin-state conversion in [Fe(2-pic)3]Cl2·CH3OH. Open circles are experimental values, and the solid curve corresponds to the domain model with n = 1.5. Reprinted with permission from ref 315. Copyright 2001 American Chemical Society.

G = fHS G H + (1 − fHS )G L + NkT {fHS ln fHS + (1 − fHS )ln(1 − fHS )}

(7)

where k is the Boltzmann constant and GH and GL are the Gibbs energies of the HT and LT phases at T, respectively. The HS fraction f HS in thermal equilibrium is determined by the equilibrium condition (∂G/∂ f HS)T = 0 as follows:

characterized by n = 1 simply corresponds to a chemical equilibrium between two energy states described by the van’t Hoff scheme, which is often encountered in denaturation of small globular proteins from a native state to a denatured one.346,347 The fact that the spin crossover can proceed even without cooperativeness rationalizes the experimental facts that the iron(II) complex highly diluted with the zinc(II) analogue [FexZn1−x(2-pic)3]Cl2·C2H5OH exhibits a spin-state conversion even for the specimen with x = 0.0009 as seen by Mössbauer spectroscopy reported by Gütlich et al.348 and that the spinstate equilibrium is realized in solution.349−353 Among various models proposed for interpretation of cooperative interactions inherent in the spin crossover phenomena, Wajnflasz and Pick354 introduced a statistical model in 1970 in which the microscopic interaction and the entropy effect are considered. Slichter and Drickamer355 reported an essential model based on the concept of regular solution in 1972. Bolvin and Kahn356 published in 1995 an Ising model within and beyond the mean-field approximation to establish a bridge between the regular solution model of Slichter and Drickamer355 and the domain model proposed by Sorai and Seki298 in 1974. Two main results emerged as follows: One is that the occurrence of a thermal hysteresis is less probable in the nonrandom distribution model and the other is that the like-spin molecules tend to assemble in likespin clusters. The former is favorable to the regular solution model while the latter is favorable to the domain model. It should be remarked that evidence for the existence of domains has been obtained for [Fe(2-pic)3]Cl2·C2H5OH based on EPR measurements in 1990 by Doan and McGarvey.357 It should be remarked that Nishino et al.358 derived in 2003 the relationship between the phenomenological macroscopic parameter n in the Sorai and Seki domain model298 and the microscopic parameters in the Wajnflasz and Pick statistical model.354

fHS = 1/{1 + exp(ΔG /NkT )} = 1/{1 + exp(nΔG /RT )}

(8)

where ΔG is equal to (GH − GL) and R implies the gas constant. The molar heat capacity at constant pressure Cp is obtained by the following relation: Cp

=

∂ {f HH + (1 − fHS )HL} ∂T HS

= {fHS Cp ,H + (1 − fHS )Cp ,L} n(HH − HL)2 exp(nΔG /RT ) + RT 2{1 + exp(nΔG /RT )}2 = Cp(lattice) + ΔCp

(9)

where HH and HL mean the molar enthalpy of the HT and LT phases, respectively. A convenient way to estimate the average number n of complexes per domain is to fit the ΔCp term given by eq with the observed heat-capacity anomaly ΔCp(obsd) at the peak temperature T1/2, at which the HS fraction becomes equal to the LS fraction and ΔG becomes zero: ΔCp(obsd, T1/2) = n{HH(T1/2) − HL(T1/2)}2 /4RT1/2 2 (10)

The excess heat capacity of [Fe(2-pic)3]Cl2·CH3OH beyond the normal heat capacity (see Figure 73)315 is well-reproduced by the domain model (solid curve) when the number of complexes per domain is only n = 1.5. Small values of n have also been encountered in [Fe(acpa)2]PF6 (n = 5),304 [Fe(3EtO-salAPA)2]ClO4·C6H5Br (n = 1),11,308 and [Fe(Hpt)3](BF4)2·2H2O (n = 1.5),318 where Hpt =3-(pyrid-2yl)-1,2,4-triazole. These small numbers of complex per domain make a sharp contrast to n = 95 for [Fe(NCS)2(phen)2],298 77 for [Fe(NCSe)2(phen)2],298 and 2000 for [CrI2(depe)2] [depe = trans-bis{1,2-bis-(diethylphosphino)ethane}],307 which belong to the abrupt type complex. As the value of n is decreased, the cooperativeness of the spin crossover becomes weaker and the heat-capacity peak due to the spin-state conversion becomes broader. The system

7.4. Two-Step Spin-State Conversion

Next to [Fe(NCS)2(phen)2], extensively studied spin crossover complexes are [Fe(2-pic)3]Cl2·(solv).359−361 Gütlich and his collaborators362 reinvestigated the spin transition in [Fe(2pic)3]Cl2·C2H5OH in 1982 by Mössbauer spectroscopy and magnetic susceptibility measurements. They found, for the first time, unusual “two-step” spin conversion in the crossover region with transition temperatures at 120.7 and 114.0 K. In the first step, the HS fraction decreases from 100% at room AQ


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temperature to 50% near 120 K. In the second step below ∼115 K, the HS fraction is further diminished from ∼40% to the pure LS state at 4.2 K. Between ∼115 and ∼120 K, the spin transition is much more gradual than below and above this temperature range. Kaji and Sorai303 reported a molar heat capacity of [Fe(2-pic)3]Cl2·C2H5OH and observed two peaks at 114.0 and 122.2 K corresponding to the two-step spin conversion (see Figure 74). The broken curves drawn in this

IP reveals that, contrary to earlier assertions, the intermediate plateau in the two-step transition of this compound is the expression of a structure with two different iron sites showing long-range order, very much like that found for other compounds with two-step behavior. The absence of discernible diffuse scattering suggests that short-range correlations are very weak. The combination of the isostructural HS and LS phases with the ordered IP implies re-entrant phase transition behavior, a relatively rare phenomenon. In the case of [Fe(btr)3](ClO4)2, where btr = 4,4′-bis-1,2,4triazole, Garcia et al.313 reported that the two-step spin conversion in this complex is associated with the presence of two slightly different iron(II) sites, whereas the space group (trigonal R3̅) remains identical independently of the spin state. Matouzenko et al.339 found a similar situation in the two-step spin-state conversion in [Fe(DPEA)(bim)](ClO4)2·0.5H2O, where DPEA = 2-aminoethyl-bis(2-pyridylmethyl)amine and bim = 2,2-bisimidazole, while the space group is monoclinic P21/c. Matouzenko et al.343 reported that [Fe(Hpy-DAPP)](BF4)2, where DAPP = bis(3-aminopropyl)(2-pyridylmethyl)amine, gives the first example of two-step spin transition in a mononuclear complex presenting a single iron(II) crystallographic site in the whole temperature range. The two-step spin transition behavior of the system is induced by two different geometries of the [FeN6] coordination core generated in the lattice by the disorder in the ligand. The HS-to-LS transition is conjugated with the conformational change in the chelate cycle. In contrast, the iron(II) spin crossover complex [Fe{5-NO2sal-N(1,4,7,10)}], where 5-NO2-sal-N(1,4,7,10) = the Schiff base condensation of 5-nitrosalicylaldehyde with 1,4,7,10tetraazadecane in a 2:1 ratio, changes its space group depending on the spin state. Petrouleas and Tuchagues363 reported a wellseparated two-step spin-state conversion in this complex on the basis of Mössbauer spectroscopy and magnetic susceptibility measurement. As shown in Figure 75, magnetic susceptibility

Figure 74. Molar heat capacity of [Fe(2-pic)3]Cl2·C2H5OH. Broken curves indicate the estimated normal heat capacities. Reprinted with permission from ref 303. Copyright 1985 Elsevier Ltd.

figure represent the normal heat capacities estimated by an effective frequency distribution method.20 Integration of the excess heat capacity beyond the normal heat-capacity curves with respect to ln T gives the entropy gain ΔS due to the phase transition. The observed entropy gain (ΔS = 50.6 J K−1 mol−1) is well accounted for in terms of the contributions from (i) the spin entropy due to the singlet-to-quintet conversion [R ln 5 = 13.4 J K−1 mol−1], (ii) orientational disordering of the ethanol solvate molecule over three sites with occupancies in the ratios 3:2:2 in the HS phase327,328 [−R (3/7) ln(3/7) − 2R (2/7) ln(2/7) = 9.0 J K−1 mol−1], and (iii) the phonon contribution (about 28 J K−1 mol−1). According to single-crystal X-ray studies at temperatures well above and well below the critical transition region by Mikami et al.,327,328 the space group of [Fe(2-pic)3]Cl2·C2H5OH is monoclinic P21/c for both the HS phase and the LS phase. Thus, the presence of a structural phase transition is not necessarily essential for both a single-step and a two-step spinstate conversion. On the basis of a single-crystal XRD study, Chernyshov et al.329 showed quite recently that the structure of [Fe(2-pic)3]Cl2·C2H5OH undergoes two first-order phase transitions on cooling from a HS phase via an intermediate phase (IP) to a LS phase. Comparisons of structural data over large ranges of temperature clearly show that the structural changes from the HS to the LS phase, both with a single iron site, are discontinuous. The unit cell doubles in size in the IP phase. The relationship between the phases is isostructural. The

Figure 75. Temperature dependence of χmT for [Fe{5-NO2-salN(1,4,7,10)}]. Both cooling (○) and heating (●) experimental data are shown. Reprinted with permission from ref 363. Copyright 1987 Elsevier Ltd.

(χm) times T against temperature plot for this complex clearly shows a two-step spin-state conversion at 173 and 136 K on cooling while it shows a two-step spin-state conversion at 146 and 180 K on heating. Boinnard et al.312 studied in detail the two-step spin-state conversion in this complex with special emphasis on the structural changes by use of XRD analysis, IR, AR


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Table 7. Genuine and Apparent Thermal Hysteresis Reported compoundsa [Fe(NCS)2(4,7-(CH3)2-phen)2] [Fe(2-pic)3]Cl2·H2O [Fe(NCS)2(bt)2] [Fe(phy)2](ClO4)2 [Fe(bi)3](ClO4)2 [Fe(phy)2](BF4)2 [Fe(NCS)2{4,7-(CH3)2-phen}2]·α-picoline [Fe(Htrz)2.85(NH2trz)0.15](ClO4)2·nH2O [Fe(Htrz)2(trz)]BF4 [Fe(Htrz)3](BF4)2·H2O (α-form) [Fe(Htrz)3](BF4)2·H2O (β-form) [Fe(NH2trz)3](tos)2·2H2Od [Fe(NH2trz)3](tos)2 [Fe(hyetrz)3](nps)2·3H2Od [Fe(hyetrz)3](nps)2 [Fe(NCS)2(PM-PEA)2] [Fe(NCS)2(btr)2]·H2O [Fe(NCS)2(PM-BiA)2] [Fe(hyetrz)3]Cl2·3H2Od [Fe(hyetrz)3]Cl2 [Fe(hyetrz)3](NO3)2·2H2Od [Fe(hyetrz)3](NO3)2 [Fe(hyetrz)3](PF6)2·2H2Od [Fe(hyetrz)3](PF6)2 [Fe(bzimpy)2](ClO4)2·0.25H2O [Fe(H4L)2](ClO4)2·H2O·2(CH3)2CO [Fe(NCS)2(btr)2]·H2O {Fe(pz)[PtII(CN)4]} {Fe(pz)[PtII/IV(CN)4I]}

Tc↓ (K)

Tc↑ (K)

ΔTc (K)b

experimental method

refs

118.6 204 220 172.3 176.9 239.3 108.3 277 146 288 345 323 276 279 279 100 100 194 124 168 301 301 298 298 195 195 397 133 117 284 383

121.7 295 280.8 181.9 184.3 246.8 114.8 286 202 304 395 345 282 361 296 370 115 231 144 173 364 314 355 315 322 205 409 173 147 304 398

3.1 91c 60.8c 9.6 7.4 7.5 6.5 9 56 16 50 22 6 82c 17 270c 15 37 20 5 63c 13 57c 17 127c 10 12 40 30 20 15

Mössbauer Mössbauer heat capacity Mössbauer magnetic susceptibility Mössbauer Mössbauer Mössbauer Mössbauer magnetic susceptibility magnetic susceptibility magnetic susceptibility magnetic susceptibility optical detection optical detection optical detection optical detection magnetic susceptibility magnetic susceptibility magnetic susceptibility optical detection optical detection optical detection optical detection optical detection optical detection magnetic susceptibility heat capacity DSC magnetic susceptibility magnetic susceptibility

372 361 319 373,374 375 376,377 378 379 380 381,382 311,382 311,382 311,382 383 383 384 384 385 375 336 386 386 386 386 386 386 314,316 343d 389a 389b 389b

a

4,7-(CH3)2-phen, 4,7-dimethyl-1,10-phenanthroline; 2-pic, 2-picolylamine or 2-aminomethylpyridine; bt, 2,2′-bi-2-thiazoline; phy, 1,10phenanthroline-2-carbaldehyde phenylhydrazone; bi, 2,2′-bi-imidazoline; Htrz, 1,2,4−4H-triazole; trz, 1,2,4-triazolato; NH2trz, 4-amino-1,2,4triazole; tos, tosyl (= p-toluenesulfonyl); hyetrz, 4-(2′-hydroxyethyl)-1,2,4-triazole; Hnps, 3-nitrophenylsulfonic acid; PM-PEA, N-(2′pyridylmethylene)-4-(phenylethynyl)aniline; btr, 4,4′-bis-1,2,4-triazole; PM-BiA, N-(2′-pyridylmethylene)-4-aminobiphenyl; bzimpy, 2,6-bis(benzimidazole-2-yl)pyridine; H4L, 2,6-bis[5-(2-hydroxyphenyl)pyrazol-3-yl]pyridine; btr, 4,4′-bis-1,2,4-triazole; pz, pyrazine. bΔTc (K) = Tc↑ (K) − Tc↓ (K). cNot genuine but apparent thermal hysteresis. dSynergy between the spin-state conversion and the removal of noncoordinated solvate water molecules.

molecular distortion model, in which two nonequivalent sites occupied by equivalent iron complexes are assumed in a unit cell of the lattice. Molecular distortions of a pair of the two complexes occupying the two nonequivalent sites couple with each other symmetrically and antisymmetrically. The merit of this model is that various patterns of the two-step spin conversion are obtained by changing the coupling strength of the iron ion with molecular distortion and with lattice strain and also the strength of interpair interactions. Afterward, other theoretical models were reported as follows: a model accounting for two-step spin transitions in binuclear compounds by Real et al.,306 two-sublattice model by Bousseksou et al.,365 Ising-like model taking into account the nonequivalence of the two sublattices by Boinnard et al.,312 the Monte Carlo method treating both the nearest and the next nearest-neighbor interactions by Kohlhaas et al.,366 the Bethe approximation to treat the nearest-neighbor interaction, which is present in addition to the long-range elastic interaction between the spinchanging molecules, by Romstedt et al.,367 the modified Bragg− Williams approximation by Koudriavtsev,368 an Ising-like model consisting of two equivalent sublattices with a “ferro” intrasublattice interaction, and an “antiferro” intersublattice

magnetic susceptibility, Mössbauer spectroscopy, DSC, and theoretical studies. The space group of [Fe{5-NO2-salN(1,4,7,10)}] is altered from triclinic P1 in the LS phase to the monoclinic P2/c in the HS phase via the monoclinic P2 in the intermediate phase (IP), in which a spin-state equilibrium at ∼50% of HS and LS molecules is realized over the ∼35 K temperature range. On the basis of these experimental and theoretical studies, they concluded that the origin of the twostep spin-state conversion is related to the structural phase transitions. An interesting example of iron(III) spin crossover complex showing two-step spin-state transition is [FeIII(isoxazole)6](BF4)2 reported by Hibbs et al.341 This complex undergoes two reversible spin crossover transitions at 91 and 192 K. As listed in Table 6, the crystal symmetry at 260 K is trigonal P3,̅ while it is triclinic P1̅ at 130 K. There exist two different iron sites, Fe1 and Fe2, in a 1:2 ratio. The LS-to-HS transitions at Fe1 and Fe2 sites take place at 91 and 192 K, respectively. Theoretical treatment for two-step spin transition was first made by Sasaki and Kambara364 in 1989. Couplings of the spin states of the iron ion with a molecular distortion and a lattice strain have been taken into account in this cooperative AS


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interaction by Nishino et al.,369 and so on. We may conclude now that the two-step spin-state conversion is accompanied either by structural phase transition involving two sublattices or by order−disorder phase transition of ligand and/or solvent molecules. On the basis of variable-temperature X-ray structural analyses, magnetic susceptibility, and heat capacity measurements, Oshio et al.370 confirmed that the multicomponent system of [Fe(dpp)2][Ni(mnt)2]2·MeNO2 (dpp, 2,6-bis(pyrazol-1-yl)pyridine; mnt, maleonitriledithiolate) undergoes multiple spin-state conversions in both the cationic and anionic components. The asymmetric unit in the crystal contains one [Fe(dpp)2]2+ cation, two [Ni(mnt)2]− anions ([Ni1]− and [Ni2]−), and one solvent molecule. Magnetic susceptibility measurements revealed that a paramagnetic state in the HT phase is abruptly converted to a diamagnetic LT phase below 180 K as the temperature is lowered from 270 K. As the temperature is raised from 125 to 270 K, successive phase transitions occur to the HT phase via intermediate phases (IM1, IM2, and IM3) at 175.5, 186.5, 194.0, and 244.0 K, respectively. In the HT phase [Fe(dpp)2]2+ is in the HS state, and both [Ni1]− and [Ni2]− moieties are arranged in monomeric form with an S = 1/2 spin ground state. In the LT phase [Fe(dpp)2]2+ is in the LS state and the nickel moieties are dimerized and diamagnetic. In the IM1 and IM2 phases the FeII sites are partially in the HS state and both [Ni]− moieties are dimeric, as suggested by 57Fe Mö ssbauer measurements. In the IM3 phase, [Fe(dpp)2]2+ is in the HS state and the anions exist in both their monomeric ([Ni1]−) and dimeric ([Ni2]−) forms. Rapid thermal quenching from 300 to 5 K yields a metastable HS phase, which relaxes to the LT phase via the IM1 phase as the temperature is raised to 150 K.

The most appealing spin crossover complexes are those for which room temperature falls in the middle of a thermal hysteresis loop. They have succeeded in finding such candidates. As seen in Table 7, many complexes have been found until now to exhibit thermal hysteresis. Among them, there are peculiar complexes showing extremely large hysteresis, such as [Fe(NH2trz)3](tos)2·2H2O (ΔTc = 82 K) [NH2trz = 4-amino1,2,4-triazole and tos = tosyl (= p-toluenesulfonyl)],383 [Fe(hyetrz)3](nps)2·3H2O (ΔTc = 270 K) [hyetrz = 4-(2′hydroxyethyl)-1,2,4-triazole and Hnps = 3-nitrophenylsulfonic acid],384 [Fe(hyetrz)3]Cl2·3H2O (ΔTc = 63 K),386 [Fe(hyetrz)3](NO3)2·2H2O (ΔTc = 57 K),386 and [Fe(hyetrz)3](PF6)2·2H2O (ΔTc = 127 K).386 However, as the large thermal hysteresis originates in the synergy between the spin-state conversion and the removal of noncoordinated solvate water molecules, such a hysteresis is realized only for the first thermal cycling. After dehydration, the hysteresis loop is remarkably reduced. For example, the large hysteresis, ΔTc = 270 K, of [Fe(hyetrz)3](nps)2·3H2O is diminished to ΔTc = 15 K for the dehydrated [Fe(hyetrz)3](nps)2.384 Therefore, the large hysteresis observed in these complexes is not genuine but only apparent. As mentioned above, 57Fe Mössbauer spectroscopy for [Fe(2-pic)3]Cl2·H2O gave a wide thermal hysteresis of ΔTc = 91 K [Tc(↓) = 204 K and Tc(↑) = 295 K].361 To confirm this fact by other experimental methods, Nakamoto et al.319 examined the thermal property of this complex by DTA (differential thermal analysis) and adiabatic calorimetry. As the result, it turned out that this thermal hysteresis is caused by the existence of a metastable phase. The cooling DTA run for the as-grown sample always gave rise to an exothermic peak at 199 K and the immediately succeeding heating run showed an endothermic peak at 211 K, while no anomaly was observed around 295 K. This thermal behavior is quite different from that previously derived from Mössbauer spectroscopy.361 To solve this mystery, they measured the heat capacity of this complex with adiabatic calorimetry. The sample was first cooled from room temperature to 10 K. In the course of cooling, an exothermic peak due to a transition from the HT HS phase to the LS phase was detected at 200 K. The heat-capacity measurement was started for the specimen thus cooled. Although a heat-capacity peak originating in the phase transition from the LS phase to the HS phase was actually observed around 220 K, the magnitude of this peak depended on the heat treatment applied to the specimen. When the sample was annealed around 200 K, the more or less metastable LS phase was stabilized to the stable LS phase with evolution of heat and, as the result, the phase transition at 220 K disappeared. Instead, a large heat-capacity peak arising from the transition from the stable LS phase to the stable HS phase was observed at 280 K. The relationships between these phases are schematically shown in Figure 76 in terms of Gibbs energy. The phase transition at 280 K (temperature marked B in Figure 76) was always supercooled to about 200 K (path, A → B → C → D), and the supercooled HS phase was transformed to the metastable LS phase around 200 K (path, D → E). When the heating rate is high, this phase was transformed to the supercooled HS phase at 210 K (temperature marked C) through the path E → F → C → B → A. However, when the specimen was treated slowly around this temperature region, the metastable LS phase was stabilized to the stable LS phase

7.5. Thermal Hysteresis: Bistable States

The thermal hysteresis usually occurs when the phase transition is, at least to some extent, first-order and it involves a discontinuous change of volume.371 Because the spin-state conversions always involve a remarkable change in the volume between the LS and the HS states, spin crossover complexes are favorable for thermal hysteresis, although the most spin-state conversions occurring in the solid state are characterized by the isomorphous phase transition without accompanying a change in the space group. Table 7 chronologically lists the spin crossover complexes showing thermal hysteresis. König and Ritter372 first remarked in 1976 the hysteresis occurring in [Fe(NCS)2(4,7-(CH3)2phen)2], where 4,7-(CH3)2-phen = 4,7-dimethyl-1,10-phenanthroline, and compared it with the Slichter and Drickamer model based on the theory of regular solution.387 On the basis of 57Fe Mössbauer spectroscopy, Sorai et al.361 reported in 1977 unusually large thermal hysteresis as wide as ΔTc = 91 K in [Fe(2-pic)3]Cl2·H2O. Quite recently, however, it turned out319 that this large hysteresis is not genuine but apparent in the sense that an extra metastable LS phase is involved in the mechanism of hysteresis. Details will be described later. To cope with the demand for storing ever more information, Kahn and his collaborators311,336,381−382,388 were turning their attentions to memory storage and display systems based on the bistability realized by thermal hysteresis in spin crossover complexes. Because the magnetic and optical properties are quite different between the HS and LS states, measurement of such a physical property may distinguish clearly the two states. AT


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existence of a metastable LT phase is encountered in ferrocene crystal.390 7.6. Mechanochemical Effect in Spin Crossover Complex

Spin crossover phenomena are often influenced by mechanical treatment for specimen, the so-called mechanochemical effect. Sorai et al.391a reported heat capacity measurements for unperturbed and differently perturbed samples of the FeIII spin crossover complex [Fe(3MeO-salenEt)2]PF6, where 3MeO-salenEt is a monoanion of the Schiff base condensation product from 3-methoxysalicylaldehyde and N-ethylethylenediamine. The unperturbed compound shows a cooperative spin crossover transition at 162.31 K, presenting a hysteresis of 2.8 K. The enthalpy and entropy changes at the phase transition are ΔtrsH = 5.94 kJ mol−1 and ΔtrsS = 36.7 J K−1 mol−1, respectively. By mechanochemical treatments, (i) the phase transition temperature is lowered by 1.14 K, and moreover the enthalpy and entropy gains at the phase transition are remarkably diminished to ΔtrsH = 4.94 kJ mol−1 and ΔtrsS = 31.1 J K−1 mol−1. In spite of different mechanical perturbations (grinding with a mortar and pestle and grinding in a ball-mill), two sets of heat capacity measurements provide basically the same results. The mechanochemical perturbation exerts its effect more strongly on the LS state than on the HS state. On the basis of heat capacity measurements, Miyazaki et al.391b found unusual spin crossover behavior for the FeII complex [Fe(DAPP)(abpt)](ClO 4 ) 2 [DAPP, bis(3aminopropyl)(2-pyridylmethyl)amine; abpt, 4-amino-3,5-bis(pyridin-2-yl)- 1,2,4-triazole]. This complex exhibits a spin crossover phase transition accompanying an order−disorder of the ligands at Ttrs = 185.61 K. The enthalpy and entropy of transition for the as-prepared crystals amount to ΔtrsH = 15.44 kJ mol−1 and ΔtrsS = 83.74 J K−1 mol−1, respectively. As shown in Figure 77A, the transition temperature shifts to hightemperature side and the heat capacity peak due to the phase transition becomes broader whenever the specimen experiences

Figure 76. Gibbs energy relationship between various LS and HS phases realized in [Fe(2-pic)3]Cl2·H2O. Reprinted with permission from ref 319. Copyright 2004 The Chemical Society of Japan.

around 200 K (path, F → G). The heat-capacity measurement for the specimen thus treated reflects the abrupt spin transition at 280 K (path, I → G → B → A). This new finding of the existence of the metastable LS phase clarifies the cause of the apparent thermal hysteresis loop spanning 204 and 295 K found in the earlier Mössbauer experiment.361 As the time for recording a Mössbauer spectrum (usually several hours) is much longer than the annealing time, the sample relaxed near 200 K from the metastable to the stable LS phase (path, F → G). Thus, the cooling branch of the HS fraction derived from the Mössbauer spectra as a function of temperature followed the path A → B → C → D → E → F → G → I. When the sample was heated, the Mössbauer measurements recorded always the stable LS phase following the path I → G → B. At 280 K (B), the transition from the stable LS to the stable HS phase occurred. Thus, the path I → G → B → A in the heating mode is followed in both the Mössbauer and the heat-capacity experiments. A very similar phenomenon caused by the

Figure 77. (A) Molar heat capacities of the [Fe(DAPP)(abpt)](ClO4)2 crystal in the vicinity of the phase transition obtained by 10 series of measurements. (B) Images of the [Fe(DAPP)(abpt)](ClO4)2 crystal by optical microscope. From left to right of upper, middle, and lower lines: (a) original crystal and crystals experiencing phase transition (b) 1, (c) 2, (d) 3, (e) 4, (f) 5, (g) 7, (h) 10, and (i) 20 times, respectively. Reprinted with permission from ref 391b. Copyright 2007 American Chemical Society. AU


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the phase transition. However, the transition enthalpy and entropy remain almost constant: ΔtrsH = 15.36 kJ mol−1 and ΔtrsS = 82.33 J K−1 mol−1 even after 20 times thermal cycling across the phase transition. The excess heat capacities of this compound are well reproduced by the domain model proposed by Sorai and Seki.298 The number of molecules per domain n indicating the cooperativity of the spin crossover transition is lowered from n = 14 for the initial peak to n = 7.5 after 20 thermal cycles. Figure 77B represents the microscopic images of the crystals taken at ambient temperature after thermal cycling between liquid nitrogen temperature and room temperature. Quite interestingly, the crystals disintegrate and become smaller crystallites whenever they experience the phase transition. Typical dimension of the original crystal is 0.1 × 0.3 × 1 mm3 (Figure 77B (a)), but it is reduced to about 50 μm after 20 cycles across the phase transition (Figure 77B (i)). This observation is regarded as a successive self-grinding effect.

Figure 78. Molecular structure of the mixed-valence oxo-centered trinuclear basic metal acetate having a formula [M III 2M II O(O2CCH3)6(py)3] with D3h symmetry in the valence-detrapped state.398

8. INTRAMOLECULAR ELECTRON TRANSFER IN MIXED-VALENCE COMPOUNDS Mixed-valence chemistry is now a very attractive field of science.231,392−395 In the mixed-valence MMX type and N(nC3H7)4[FeIIFeIII(dto)3] complexes shown above, the charge transfer was involved in the relevant phase transitions and thereby the magnetic properties were altered. There is another category of mixed-valence complexes, in which the electronic state is strongly coupled with molecular vibrations and, as a result, labile molecular motions with large amplitude are excited. In these complexes, the magnetic properties remain unchanged between the valence-trapped state at LT and the valence-detrapped state at HT. Well-known examples of this category are the oxo-centered trinuclear basic metal acetate complexes with the formula [MIII2MIIO(O2CCH3)6L3]·(solv) (L, monodentate ligand; solv, solvent molecule) and the 1′,1‴disubstituted biferrocenium salts. Because the phase transitions arising from the intramolecular electron transfer in these mixedvalence complexes have already been reviewed by Sorai and Hendrickson,11,396,397 a brief summary will be given here.

residing on the low-oxidation metal center is coupled with the eg symmetry M3O stretching vibration via the vibronic interactions. In effect, the vibronic interaction includes a pseudo-Jahn−Teller distortion of the molecule and the 3-fold degeneracy may be lifted.404 The adiabatic potential energy surface for the ground state of the mixed-valence M3O complex provides three or four potential minima depending on the value of the electron-transfer integral between two metal ions, the force constant for the eg stretching vibration, and the vibronic coupling constant of the d-electron to the eg vibrational mode. The three potential minima correspond to three different isosceles triangles of a complex formed by the pseudo-Jahn− Teller distortion. In each of these states, the extra d-electron is located mainly on one of the three metal ions. The fourth potential minimum corresponds to a M3O complex that has an equilateral triangular form, and the extra d-electron is coherently delocalized over all three metal ions. Entropy gain due to the valence-detrapping phenomenon is straightforwardly related to a change in the number of microscopic states that are thermally accessible for a given mixed-valence complex. At low temperatures, the mixedvalence complex is valence-trapped and has statically one of the isosceles triangular configurations. However, depending on the magnitude of various parameters mentioned above, the dynamically interconverting mixed-valence complexes in the HT phase will gain either three or four microscopic states. In the former case, the valence-detrapping gives an entropy gain of R ln 3, whereas in the latter case the entropy gain is R ln 4. The rate of intramolecular electron transfer is seriously affected by environmental effects in the solid state. In particular, solvate molecules that are not explicitly coordinated to the central metal ions but are stoichiometrically amalgamated in the crystal lattice play an important role for the intramolecular electron-transfer event. The mixed-valence trinuclear complexes discussed here are [MIII2MIIO(L2)6(L1)3]·(solv) (L1, monodentate ligand; L2, bidentate ligand; solv, solvate molecule). Sorai and his collaborators measured heat capacities of [Fe 3 O(O2CCH3)6(py)3]·(py) (1),405,406 [Mn3O(O2CCH3)6(py)3]·(py) (2),407,408 [Fe3O(O2CCH3)6(py)3]·(CHCl3) (3),409,410 [Fe3O(O2CCH3)6(4Me-py)3]·(CHCl3) (4),411 [Fe3O(O2CCH3)6(3-Me-py)3]·(3-

8.1. Trinuclear Mixed-Valence Compounds

Figure 78 shows the iron complex [Fe3O(O2CCH3)6(py)3] (py, pyridine) at the valence-detrapped state at room temperature. 398 Each of three transition-metal ions is octahedrally coordinated by the central oxygen ion, a neutral pyridine, and four acetate ligands. Many of these complexes have been shown to convert from being valence-trapped at low temperatures to valence-detrapped at high temperatures.396,397,399,400 One of the remarkable discoveries at the initial stage of the investigations was that the solvate molecules that are not explicitly coordinated to the central metal ions have a dramatic effect on the electron-transfer rate, while the nonsolvated analogue [Fe3O(O2CCH3)6(py)3] remains in a valence-trapped state even at 315 K.399,400 Formal states of oxidation in the trinuclear mixed-valence M3O complex are two M(III) and one M(II). The M(II) ion is characterized by having an “extra” d-electron in comparison to the M(III). The electronic state of a free-molecule would be triply degenerated in the sense that the extra d-electron can reside on any one of three metal centers. Intramolecular electron-transfer between the metal centers in a trinuclear mixed-valence complex MIII2MIIO is usually treated by a theory incorporating vibronic interactions.401−403 The extra d-electron AV


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Me-Py) (5),412 and [Fe3O(O2CCH3)6(3-Me-py)3]·(toluene) (6),413 together with a solid solution [FeIII2FeII0.5CoII0.5O(O2CCH3)6(py)3]·(py) (7)414 and a nonmixed-valence analogue [FeIII2CoIIO(O2CCH3)6(py)3]·(py) (8).414 The complexes 1−4 commonly crystallize into an identical rhombohedral crystal system with the space group R32 in the valencedetrapped HT phases. The variable-temperature 57Fe Mössbauer spectra of the complex 1 reported by Hendrickson and his collaborators399,400 showed dramatic changes with temperature. At temperatures below about 100 K, two quadrupole-split doublets are seen as follows: one characteristic of HS iron(II) and the other of HS iron(III). As the temperature is increased, an extra doublet appears abruptly around 110 K at the expense of the initial two doublets. This means that the new doublet arises from the electron-delocalized state. The spectrum eventually becomes a single doublet above about 190 K. This fact indicates that the rate of intramolecular electron-transfer exceeds the rate of 107 s−1, which the Mössbauer technique can sense. Figure 79 shows

altered only by substitution of the central metal ions. Figure 80 represents molar heat capacity of [Mn3O-

Figure 80. Molar heat capacity of the mixed-valence complex [Mn3O(O2CCH3)6(py)3]·(py). Reprinted with permission from ref 407. Copyright 1989 American Chemical Society.

(O2CCH3)6(py)3]·(py).407,408 This complex brings about a sharp first-order phase transition at 184.7 K, and the transition entropy amounts to ΔS = 35.8 J K−1 mol−1, which is much greater than that of the homologous iron complex. Thermodynamic data concerning the phase transition are summarized in Table 8. Because the entropy due to the valence-detrapping is either R ln 3 (= 9.1 J K−1 mol−1) or R ln 4 (= 11.5 J K−1 mol−1) at most, the observed transition entropies obviously contain extra contributions other than the intramolecular electron transfer. A clue for the extra candidates lies behind its crystal structure. A schematic drawing of the molecular packing in the valence-detrapped HT phase of [Fe3O(O2CCH3)6(py)3]·(py) is shown in Figure 81.398 For simplicity, the acetato ligands are not drawn. As described above, the space group of this crystal is R32. Along the crystallographic C3 axis, the metal complexes and pyridine solvate molecules occupy alternating sites of the 32 symmetry. That is, each pyridine solvate molecule is sandwiched between two metal complexes. The plane of the pyridine solvate molecule is perpendicular to the metal-complex plane. As required by the presence of the C3 axes along which the metal complexes are stacked, the pyridine solvate molecules are disordered, with at least three orientational positions. On the other hand, solid-state 2H NMR for the iron complex with deuterated pyridine solvate molecule studied by Hendrickson and his collaborators398 revealed detailed molecular dynamics. As illustrated in Figure 82, the molecular plane of the pyridine jumps among three positions about the crystallographic C3 axis. In each planar position, the pyridine solvate molecule librates among four positions, where in each of these positions two carbon atoms are on the C3 axis. As the result, the pyridine solvate molecule converts from being static to dynamically interconverting among 12 different orientations.

Figure 79. Molar heat capacity of the mixed-valence complex [Fe3O(O2CCH3)6(py)3]·(py). The broken curve indicates the normal heat capacity. Reprinted with permission from ref 406. Copyright 1986 American Chemical Society.

the molar heat capacity of this complex.405,406 This complex exhibits essentially two kinds of phase transitions: One is a firstorder phase transition at about 112 K and the other is a higherorder phase transition around 190 K. Interestingly, the LT phase transition takes place at the temperature at which the first change in the Mössbauer spectrum occurs and the peak temperature of the HT phase transition is identical with the temperature where the Mössbauer spectrum becomes a single average doublet. Moreover, the large temperature range involved in the HT phase transition fully overlaps with the temperature region where the drastic change in the Mössbauer spectrum is occurring. Therefore, the observed phase transitions are concluded to arise from the intramolecular electron-transfer event in the mixed-valence complex. The entropy gain associated with the phase transitions is (30.6 ± 0.8) J K−1 mol−1. The phase transition behavior is drastically AW


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Table 8. Entropy Gain at the Phase Transitions Observed in the Mixed-Valence Complexes with the Formula [MIII2MIIO(O2CCH3)6L3]·(solv) (in the Unit of J K−1 mol−1)a complex 1 8 2 3 4 5 6

entropy gain

M(III)

M(II)

L

solv

Ttrs (K)

ΔS(metal)

Fe Fe Mn Fe Fe Fe Fe

Fe Co Mn Fe Fe Fe Fe

py py py py 4-Me-py 3-Me-py 3-Me-py

py py py CHCl3 CHCl3 3-Me-py toluene

112, 191 150 185 208 94 282 293

R ln 3 R R R R R

ln ln ln ln ln

4 4 4 3 3

ΔS(solv) R R R R R R R

ln ln ln ln ln ln ln

12 3 18 8 2 2 2

ΔS(calcd) 29.8 9.1 35.6 28.8 17.3 14.9 14.9

ΔS(obsd) 30.6 10.3 35.8 28.1 17.2 13.7 15.1

± 0.8

± 0.4 ± 1.4 ± 0.7

refs 405,406 414 407,408 409,410 411 412 413

a ΔS(metal), entropy gain due to conformational change of the M3O complex; ΔS(solv), entropy gain due to orientational disordering of the solvate molecule; and ΔS(calcd), ΔS(metal) + ΔS(solv).

As evidenced by many instances (see Table 8), the intramolecular electron transfer occurs in these trinuclear mixed-valence complexes by coupling with the onset of orientational disordering of the solvate molecules. The intermolecular interactions responsible for such a coupling may be electric dipole−dipole interactions between the complex and the solvate molecules. Actually, when the solvate molecule is benzene having no electric dipole moment, the mixed-valence complex does not bring about a phase transition arising from the electron transfer. A simple but fundamental question arises here as to why only the [Fe3O(O2CCH3)6(py)3]·(py) complex exhibits two phase transitions (a first-order phase transition at low temperature and a higher-order transition at high temperature) in contrast to a single first-order phase transitions in other complexes. The first statistical theoretical model to describe the phase transitions observed for mixed-valence Fe3O complexes was proposed by Kambara et al.404 Stratt and Adachi415 took an imaginative and insightful approach. Although their theory ignores the effects of the solvate molecules, qualitative aspects inherent in the phase transitions occurring in the trinuclear mixed-valence complexes are well-interpreted. It should be remarked here that the onset of conformational change of solvate molecule is not necessarily a condition for intramolecular electron transfer. For example, although a crystal of mixed-valence complex [Fe3O(O2CCH2CN)6(H2O)3] does not contain any solvate molecules, a phase transition due to valence-detrapping really occurs at 128 K.416 Nakamoto et al.417 reported that the mixed-valence trinuclear iron monoiodoacetate complex previously reported as [Fe3O(O2CCH2I)6(H2O)3]418 is an unusual compound. On the

Figure 81. Schematic drawing of the crystal structure of [Fe3O(O2CCH3)6(py)3]·(py). The acetato ligands are not shown for clarity. Crystallographic C3 axis runs through the central oxygen atoms of the Fe3O complexes. Reprinted with permission from ref 398. Copyright 1987 American Chemical Society.

Now, the transition entropy can be interpreted as follows. If each metal complex converts from being statically distorted in one state to interconverting dynamically between three vibronic states when the complex is heated from low temperatures, this contributes R ln 3 to the entropy gain. The sum of R ln 3 for the metal complex and R ln 12 for the pyridine solvate molecules is R ln 36 (= 29.8 J K−1 mol−1). This value agrees well with the observed entropy gain (30.6 ± 0.8) J K−1 mol−1. This fact provides definite evidence that the intramolecular electron transfer does proceed in cooperation with orientational disordering of the solvate molecule.

Figure 82. Orientational disordering of the solvate molecules in the crystal lattice of the trinuclear complexes [M3O(O2CCH3)6(L1)3]·(solvate), designating [M, L1, solvate] for short. (a) Pyridine in [Fe, py, py],398 (b) pyridine in [Fe2CoO(O2CCH3)6(py)3](py),414 (c) pyridine in [Mn, py, py],408 (d) deuterated chloroform in [Fe, py, CDCl3],409 and (e) chloroform in [Fe, 4-Me-py, CDCl3].411 AX


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vibrational mode.397,421 If these two vibronic states interact, one can expect an adiabatic potential shown in Figure 84a.421 The

basis of XRD analysis, they found that its correct formula should be [Fe I I I 2 Fe I I O(O 2 CCH 2 I) 6 (H 2 O) 3 ][Fe I I I 3 O(O2CCH2I)6(H2O)3]I. The two kinds of Fe3O molecules (FeIII2FeIIO and FeIII3O) are crystallographically indistinguishable. All of the iron atoms are crystallographically equivalent because of a crystallographic 3-fold symmetry. The heat capacity of this complex seems to exhibit no thermal anomaly in the temperature range of 5.5−309 K, although the valencedetrapping phenomenon has been observed in this temperature range. This fact indicates that the valence-detrapping phenomenon in this complex occurs without any phase transition, leading this complex to a glassy state, probably because the crystal of this complex is just like a solid solution of distorted mixed-valence FeIII2FeIIO molecules and permanently undistorted FeIII3O molecules, which may act as an inhibitor for a cooperative valence-trapping. Stadler et al.419 reported electron transfer in a trinuclear oxocentered mixed-valence iron complex [FeIII2FeIIOL3], in which L2− is a pentadentate ligand designed to coordinate all three metal atoms in the central cluster. The crystal structure of this complex shows 3-fold symmetry, attributed to rotational disorder. The magnetization data indicate strong superexchange between basis oxidation states Fe(3+, 3+, 2+). The Mössbauer spectra indicate significant valence delocalization even at a low temperature (4.2 K) with estimated valences Fe(2.9+, 2.9+, 2.2+) in the solid state. At higher temperatures, no lifetime broadening is observed but additional Mössbauer absorptions are consistent with increasing proportions of trinuclear complexes with greater delocalization, Fe(2.75+, 2.75+, 2.5+).

Figure 84. Potential energy surface E(q) drawn as a function of the out-of-phase combination (Q = QA − QB) of the two symmetric metal−ligand breathing vibrational modes (QA and QB) of the two halves of a binuclear mixed-valence species (a) for a symmetric mixedvalence complex in the absence of environmental effects and (b) for results if the environment about the binuclear mixed-valence complex is asymmetric. Reprinted with permission from ref 421. Copyright 1987 American Institute of Physics.

ground-state surface has two minima, both at the same energy. In general, there will be several vibrational energy levels below the barrier height ΔEB. In Figure 84a, only the zeroth vibrational level and one fundamental at hν are shown, where h is the Planck constant and ν is the vibrational frequency. For a binuclear FeIIFeIII complex in an asymmetric environment, just as in the solid state, the potential energy diagram is as shown in Figure 84b. The two vibronic states are not at the same energy. There is an energy difference ΔE between the two states. This difference reflects the different environments near the two iron ions in the binuclear complex. The FeaIIFebIII vibronic state is stabilized relative to the FeaIIIFebII vibronic state. When the thermal energy kT is less than ΔE, the complex is found in the FeaIIFebIII state. If the complex is to convert from FeaIIFebIII to FeaIIIFebII, it needs to be thermally excited by ΔE or more. Because there will be many vibrational levels on both sides of the asymmetric double well, it can then tunnel from a “left” vibrational level to a vibrational level on the “right” side. Vibronic coupling works efficiently to make a small polaron migrate over the metal centers within a complex molecule. Those compounds possess a potential of charge ordering in their crystalline phases at low temperatures.422,423 Transitions between such a valence-trapped and a valence-detrapped state have been observed by Mössbauer spectroscopy in some binuclear complexes. Nonsubstituted biferrocenium triiodide424 and many dialkylbiferrocenium triiodides425,426 are known to bring about a transition from a valence-trapped to a valencedetrapped state above 200 K. 1′,1‴-Diiodobiferrocenium hexafluoroantimonate427 really gave rise to a phase transition arising from the valencedetrapping at 134 K. The entropy gain (6.0 ± 0.5) J K−1 mol−1 is very close to R ln 2 (= 5.76 J K−1 mol−1), the value expected for two energetically equal vibronic states as shown in Figure 84a. In contrast to this, biferrocenium triiodide428 and 1′,1‴diethylbiferrocenium triiodide429 bring about very small heatcapacity anomalies at 328 and 250 K with the entropy gains of only (1.77 ± 0.06) and 1.0 J K−1 mol−1, respectively. Interestingly, when the substituent X in Figure 83 is Br or I and A− is I3−, the 1′,1‴-disubstituted-biferrocenium triiodides remain in a valence-detrapped state even at 4.2 K.430,431

8.2. Binuclear Mixed-Valence Compounds

Electron transfer in binuclear mixed-valence complexes consisting of 1′,1‴-disubstituted biferrocenium cations (see Figure 83) is also of interest. Electron transfer in mixed-valence

Figure 83. Schematic drawing of molecular structure of mixed-valence 1′,1‴-disubstituted biferrocenium cation (X, substituent, A−, anion such as I3−, PF6−, or SbF6−).

molecular systems is generally described by the PKS model232 based on a vibronic approach. As mixed-valence complexes are electronically labile, the lowest energy electronic states are vibronic, and as a result, the complexes are very sensitive to their environments.420 In the absence of environmental effects, a binuclear FeIIFeIII mixed-valence complex has two electronic states, which would be electronically degenerate. The two potential energy curves would be superimposed and strong vibronic coupling would result. Certain vibrational coordinates, such as the out-of-phase combination of the totally symmetric ring−metal−ring breathing modes on each metallocene moiety in Figure 83 would couple to the two electronic states. Two energetically degenerate vibronic states result, for which the two parabolic potential-energy curves are displaced relative to each other in the nuclear coordinate space Q0 of the coupled AY


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corresponds to a two-level Schottky heat capacity with an energy difference of δ/hc = 1 cm−1, where c is the velocity of light. The presence of the Schottky heat-capacity anomaly demonstrates the existence of a double-well potential: There is a tunnel splitting between the vibronic ground states and hence a charge-ordering phase transition in I2Fe2+I3− is absent. This type of nuclear tunneling has been expected to exist in the mixed-valence systems both from experiment428 and from a theoretical treatment by Kambara et al.421,434 However, it is indeed in this complex that the nuclear tunneling was actually and directly observed for the first time. In the sense that the quantum effect prevents the charge ordering with ferroelectric long-range order, the I2Fe2+I3− complex is regarded as a molecular quantum paraelectric. In fact, a Curie-like behavior characteristic of a paraelectric was observed in a preliminary dielectric measurement with the 0.4−20 kHz frequency range between 4 and 300 K.433

Structural analysis for 1′,1‴-diiodobiferrocenium triiodide (hereafter, I2Fe2+I3− for short)432 revealed that two iron atoms in a cation are related to each other by a symmetry center and thus are equivalent at room temperature. This fact means that I2Fe2+I3− is in the valence-detrapped state at room temperature. Even at 4.2 K, the two iron atoms in the cation remain equivalent, because only one quadrupole-split doublet was observed by 57Fe Mössbauer spectroscopy and EPR measurements at 4.2 K gave a relatively isotropic g tensor characteristic of a valence-detrapped state.430 It is noted, however, that the valence-detrapped phase should be interpreted not as a statically electron-delocalized state (i.e., single-minimum potential) but as a dynamically disordered state, because IR spectroscopy shows the existence of a potential barrier separating two valence-trapped configurations of the cation.432 These two configurations are, therefore, degenerate at high temperatures and contribute R ln 2 to the entropy. To elucidate how this entropy is removed at low temperatures, say below 4.2 K, following the requirement of the third law of thermodynamics, Nakano et al.433 measured the heat capacity of I2Fe2+I3− complex by adiabatic calorimetry between 80 mK and 25 K. One of two ways might be anticipated as follows: either a phase transition or none. If a phase transition occurs, this complex would have a charge-ordered ground state similar to those in other biferrocenium salts.427−429 On the other hand, if no phase transitions take place, a quantummechanical tunneling would be incorporated into the doubleminimum adiabatic potential so as to lift the degeneracy without long-range order. The observed heat capacity of the I2Fe2+I3− complex is shown in Figure 85 on a logarithmic scale.

9. THERMOCHROMIC COMPLEXES Many transition metal complexes change color in the solid state depending on the temperature. This phenomenon, the so-called thermochromism,435,436 involves a change in electronic state of a molecule and thus a change in the magnetic properties. Because a review for calorimetric studies on thermochromic complexes has already been given elsewhere,11,14 a brief summary will be presented here. The electronic energy of transition-metal complexes may be perturbed by changes in (i) electron configuration, (ii) coordination geometry, (iii) coordination number, (iv) molecular motion of ligands, and so on. A typical example due to cause (i) is encountered in the spin crossover complexes described in chapter 7. Thermochromism due to (ii) is observed in structural isomers of coordination compounds. Some nickel complexes are known to exhibit two structural isomers: One is a diamagnetic green form with square-planar coordination geometry, and the other is a tetrahedral paramagnetic brown form. A calorimetric study has been done for bis[N-(3-methoxysalicylidene)isopropylaminato]nickel(II).437a Recently Wang et al.437b succeeded in determining the standard thermodynamic quantities for both the paramagnetic brown form (B) and the diamagnetic green isomer (G) of bis(N-isopropyl-5,6-benzosalicylideneiminato)nickel(II):437b S°(B) − S°(G) = 32.8 J K−1 mol−1, H°(B) − H°(G) = 16.0 kJ mol−1, and G°(B) − G°(G) = 6.25 kJ mol−1, respectively. 9.1. Thermochromism Due to a Change in the Coordination Number

Figure 85. Molar heat capacity of the mixed-valence complex 1′,1‴diiodobiferrocenium triiodide. The solid curve shows the two-energy level Schottky anomaly with δ/hc = 1 cm−1. Reprinted with permission from ref 433. Copyright 1992 Elsevier Ltd.

Thermochromism due to (iii) is seen in (i-PrNH3)[CuCl3], where the color of the solid abruptly changes from brown to orange when the crystal is heated above room temperature. On the basis of XRD, NMR, EPR, DSC, and magnetic susceptibility measurements, Roberts et al.438 revealed that the LT phase of (i-PrNH3)[CuCl3] belongs to a triclinic crystal system, while the HT phase is orthorhombic and that the coordination geometry around the central copper atom changes grossly through the thermochromic transition at 324 K: The coordination number of a copper atom changes from five in the LT phase to six in the HT phase. The isopropylammoniumu cations are ordered and hydrogen-bonded to one of the terminal chloride ions in the LT phase. In contrast, in the HT phase, the isopropylammoniumu cations are disordered and thereby the hydrogen bonds are weakened. As a result, the

Open circles plotted below 2 K correspond to a pellet sample, while solid circles between 1 and 25 K are for a powder sample. A discrepancy detected between the two sets of data is based on a result in the pellet sample.433 Two heat-capacity anomalies were found as follows: One takes place below ca. 0.2 K, and the other is a broad peak centered around 0.6 K. The former arises from a magnetic interaction between the paramagnetic cations. The anomaly occurring around 0.6 K is obviously a Schottky type heat capacity corresponding to a thermal excitation between two energy levels. The solid curve in Figure 85 AZ


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terminal chloride ion that is otherwise hydrogen-bonded to isopropylammonium cation bridges two adjacent copper ions. The HT phase contains tribridged linear chains of [(CuCl3) − ]∞, while the LT phase contains bibridged linear chains of [(Cu2Cl6)2−]∞. The heat capacity determined by Nishimori and Sorai439 revealed a phase transition at 335.6 K. The enthalpy and entropy gains at the thermochromic phase transition were ΔH = 5.54 kJ mol−1 and ΔS = 16.5 J K−1 mol−1, respectively. Of the observed entropy gain, ΔS = 16.5 J K−1 mol−1, the 2-fold disordering of the cations contributes R ln 2 (= 5.8 J K−1 mol−1) to the entropy, while there is substantially no contribution from the acoustic lattice vibrations because of no volume change at the transition. The remaining entropy, 10.7 J K−1 mol−1, may be attributed to the excess contribution from the optical metal−ligand skeletal vibrations due to the change in the coordination number. 9.2. Thermochromism Due to Molecular Motion of the Ligands

A typical example of the thermochromism due to (iv) is that occurring in square-planar copper(II) complex [Cu(daco)2](NO3)2 (daco, 1,5-diazacyclooctane, an eight-membered ring molecule shown in Figure 86).440 The color of the crystal is

Figure 87. Molar heat capacity of the thermochromic compound [Cu(daco)2](NO3)2. The broken curve shows the normal heat capacity. Reprinted with permission from ref 442. Copyright 1995 Elsevier Ltd.

Figure 86. Molecular structure of eight-membered ring molecule 1,5diazacyclooctane (daco for short).

brilliant orange but changes to violet discontinuously when heated above 360 K and reverts to orange on cooling. On the basis of X-ray study for a single crystal, Hoshino et al.441 concluded that the thermochromic mechanism for [Cu(daco)2](NO3)2 might involve a ligand−anion interaction rather than metal−anion bonding. Figure 87 illustrates the molar heat capacity of [Cu(daco)2](NO3)2 determined by adiabatic calorimetry by Hara and Sorai.442 A large heatcapacity anomaly associated with the thermochromic phenomenon was found at 359 K. This phase transition exhibits both superheating and supercooling effects, indicating a first-order phase transition. The thermodynamic quantities due to the phase transition were ΔH = (8.40 ± 0.22) kJ mol−1 and ΔS = (23.4 ± 0.8) J K−1 mol−1. The large entropy gain implies that softening of molecular or lattice vibrations and/or onset of orientational disordering of the daco ligand or the nitrate anion would be involved in the phase transition. For the present thermochromic compound, however, the possibility of softening of optical vibrational modes is precluded, because the IR spectra showed no detectable shift of absorption bands on going from one phase to the other. The authors442 speculated that dynamic interconversion of a daco-ligand between a chair and a boat form might take place in the HT phase. If this is the case, each daco-ring will gain four conformations as shown in Figure 88 (i.e., chair−chair, chair−boat, boat−chair, and boat− boat), and the total number of conformations per [Cu(daco)2]2+ cation amounts to 16 (= 4 × 4), because a cation contains two daco ligands. According to the Boltzmann principle, the conformational entropy is estimated to be R ln 16 (= 23.05 J K−1 mol−1). This entropy gain accounts

Figure 88. Schematic drawing of four conformations of the eightmembered daco ring having almost equal energy. Reprinted with permission from ref 442. Copyright 1995 Elsevier Ltd.

surprisingly well for the observed transition entropy of ΔS = (23.4 ± 0.8) J K−1 mol−1. The relationship between the color change and the change in the electronic state of the [Cu(daco)2]2+ ion is interpreted as follows. A single-crystal X-ray study441 revealed that the structure is centrosymmetric with the center of symmetry on the site of Cu and the CuN4 chelate plane is planar, but it is not precisely square since the two Cu−N coordination bonds of a daco ligand differ in length significantly. On the other hand, in the HT phase, eight-membered daco ligands undergo dynamic conformational changes between the boat and the chair forms, and thereby, the CuN4 chelate plane is dynamically distorted from a planar form probably to a distorted tetrahedral coordination geometry. When such a planar-to-tetrahedral geometrical change occurs, a red-shift in the d−d transition is expected, as indicated by a theoretical angular overlap method.443 The visible spectrum of [Cu(daco)2](NO3)2 indeed exhibits such a red shift.440,441 BA


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Thermodynamic gains were ΔH = 17.4 kJ mol−1 and ΔS = 55.2 J K−1 mol−1. XRD analysis444,445 provides a useful clue to the molecular freedom responsible for the entropy gain. The most remarkable change in the structure occurring through the phase transition is the motion of the chelate rings, which is easily seen from anisotropic thermal ellipsoids of the constituent atoms. The chelate rings are puckering up and down from the CuN4 plane in the HT phase. Provided that this chelate ring puckering is responsible for an essential part of the transition mechanism, the enthalpy gain at the phase transition is estimated as follows. One assumes that the chelate rings are static at very low temperatures, whereas they pucker in the HT phase. There are two chelate rings in a cation, consisting of a five-membered CuN2C2 ring. When the plane formed by the CuN2 is fixed, the five-membered ring has four different conformations: two conformations in which two carbon atoms are tipped off to the same side of the plane and two conformations in which two carbon atoms are tipped off to the opposite sides of the plane. It should be remarked that the number of degrees of freedom for the puckering motion of a five-membered ring is two. Moreover, because there exist two chelate rings in a cation, so the total number of the puckering modes amounts to four. These four modes are roughly assumed to be degenerate and can be simply approximated by the Einstein harmonic oscillator. If one assumes that this puckering motion is responsible for the total entropy of transition, the Einstein characteristic frequency is estimated to be 115 cm−1 for [Cu(dieten)2](ClO4)2. On the basis of this frequency, the heatcapacity jump at the transition temperature is estimated as ΔC(ligand) = 33 J K−1 mol−1 and ΔH(ligand) = 8.0 kJ mol−1. The heat-capacity jump is comparable with the observed value of 46 J K−1 mol−1. The contribution of the chelate-ring puckering ΔH(ligand) to the observed ΔH(obsd) is about 46%. According to the X-ray structural analysis of [Cu(dieten)2](ClO4)2,444,445 there is no significant change in the Cu−N distances, whereas the trans bond angles of N−Cu−N change slightly through the phase transition. In the LT phase, the angles are 180.0° while in the HT phase they are 178.0 and 174.7°. The [Cu(dieten)2]2+ cation is characterized by a squareplanar coordination geometry in the LT phase. This geometry is slightly distorted toward a tetrahedral coordination in the HT phase. Despite such a small geometrical change, the color of complexes changes dramatically when the phase transition occurs. The color of the complexes depends obviously on the absorption of visible light by a molecule. The absorption spectra in the visible region depend on the energy-level splitting of the d-orbitals. Variable-temperature d−d transition of [M(dieten) 2 ]X 2 has already been studied. 446,447 These complexes show a red shift in the d−d transition when they are heated. To investigate the relationship between the absorption spectra and the d-orbital energy levels, the angular overlap model (AOM)443 was adopted. The relative energies of the five d-orbitals can be calculated as a function of the polar angles (θ, φ) of the ligand position vectors. The calculated energy levels are illustrated in Figure 90 as a function of θ.450 The visible light absorption maximum (νmax) is shifted from 20700 cm−1 in the LT phase to 19305 cm−1 in the HT phase for [Cu(dieten)2](ClO4)2.446,447 The absorption maximum in the electronic spectrum corresponds to the dyz, dxz → dx2−y2 transition. Namely, the transition energy is the energy difference between these two orbitals, ΔE(θ) in Figure 90. In

9.3. Enthalpy Change Due to Thermochromism

A quite similar situation has been encountered in a homologous series of thermochromic complexes with formula [M(dieten)2]X2, where M is Cu(II) or Ni(II), dieten is N,N-diethylethylenediamine, and X is BF4− or ClO4−. The copper complexes show a color change from red to blue−violet, and the color of the nickel complexes changes from orange to red when they undergo phase transition in the solid state. It was initially thought that the thermochromism was caused by axial approach of the anions to the copper or nickel ion. However, when the crystal structures of both the LT and the HT phases were determined, that idea turned out to be wrong.444,445 There exists no axial coordination of the counteranions in either of the phases because the bulky alkyl groups bonded to the nitrogen atoms prevent the counteranions from approaching the central metal atom. A new mechanism of thermochromism was proposed as follows: In the HT phase, the chelate rings pucker up and down while they remain static in the LT phase. Such ring motion affects the ligand field strength, leading to the color change. This mechanism seems to be consistent with various experimental results so far available. However, the relationship between the microscopic aspects444−448 hitherto reported and the macroscopic energetic and entropic aspects was still unclear. Nishimori et al. measured heat capacities of [Cu(dieten)2]X2 by adiabatic calorimetry.449,450 Figure 89 represents the molar heat capacity of [Cu(dieten)2](ClO4)2.450 A large single peak due to the thermochromic phase transition was found at 317.64 K. This phase transition is characterized by a long heat-capacity tail extending down to about 200 K. Moreover, the normal heat-capacity curve has a jump of 46 J K−1 mol−1 at the transition temperature between the LT and the HT phases.

Figure 89. Molar heat capacity of the thermochromic compound [Cu(dieten)2](ClO4)2. Dotted curves show the normal heat capacities of the LT and HT phases, resulting in a heat capacity jump at the transition temperature. Reprinted with permission from ref 450. Copyright 1996 Taylor & Francis Group. BB


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10. SPIN-PEIERLS TRANSITIONS Owing to strong quantum fluctuations, interesting phenomena occur in low-dimensional quantum spin systems.158 SpinPeierls (SP) transition451−456 takes place in a system consisting of quantum spin (S = 1/2) antiferromagnetic (AF) Heisenberg chains through the interchain spin-phonon coupling. In 1974, Pytte451 revealed that at the SP transition temperature (TSP) lattice instability takes place and a system of uniform AF Heisenberg chains in the high-temperature phase is transformed to a system of dimerized or alternating AF chains in the low-temperature phase. The essential concept of the SP transition is the instability of 1D spin systems, leading to dimerization in the chains and the formation of a singlet ground state with a magnetic energy gap, which are caused by the coupling between spin and lattice. The SP transition is thus the magnetic analogue of the electronic Peierls transition.457,458 Because the SP transition is related to a spin-gap system, as in the case of the Haldane gap,157,459 magnetic susceptibility is rapidly reduced below TSP and also magnetic heat capacity manifests characteristic temperature dependence at low temperature.

Figure 90. Energy level diagram correlating tetrahedral and planar coordination geometries calculated on the basis of the angular overlap model. Reprinted with permission from ref 450. Copyright 1996 Taylor & Francis Group.

10.1. Organic Spin-Peierls Compounds

The first experimental evidence for the SP transition was reported in 1975 by Bray et al.452,453 for organic π-donor− acceptor compounds TTF-CuBDT and TTF-AuBDT, where TTF is tetrathiafulvalene and BDT means S4C4(CF3)4. Magnetic susceptibility of each compound has a broad maximum near 50 K, while the second-order phase transitions occur at 12.4 and 2.1 K for TTF-CuBDT and TTF-AuBDT, respectively. This type of behavior clearly shows the characteristics of the SP transition in reasonably good agreement with a mean-field theory. The SP transition is characterized by a progressive spin−lattice dimerization occurring below the transition temperature in a system of 1D AF Heisenberg chains. Direct evidence for the dimerization was obtained by Xray diffraction.460,461 Heat capacity measurements by Wei et al.462 revealed a mean-field like transition at 12.4 K for TTFCuBDT and a broadened peak by 1D fluctuation effect at 2.06 K for TTF-AuBDT. More examples of organic SP compounds have been found thereafter. In 1979 Huizinga et al.463 measured magnetic susceptibility and heat capacity on MEM-(TCNQ)2, where MEM+ stands for N-methyl-N-ethylmorpholinium, and found a phase transition at 18 K. The data are described reasonably well by a mean-field SP transition theory. The transition is accompanied by a dimerization of the spin sites, which means that the TCNQ chain is tetramerized. de Jonge and Kopinga464 measured heat capacity of similar compound DEM(TCNQ)2,where DEM+ is N,N- diethylmorpholinium, and observed a SP-like anomaly at around 30 K. At low temperatures, the heat capacity contains a term linear in temperature, indicating that one-half of the TCNQ stacks is still in its unperturbed state. As described in section 2.3, Mukai et al.82 reported in 1996 the evidence of SP transition at 15 K for a verdazyl radical crystal p-CyDOV (s, Figure 9) on the basis of magnetic susceptibility and high-field magnetization data up to 35 T. In advance of the discovery of the SP transition, Vegter et al.465−467 had reported that radical-ion salts M-TCNQ (M, alkali metal ion or NH4+) belonging to low conducting 1D AF systems exhibit structural phase transitions accompanied by dimerization of TCNQ radical ions in the temperature range of

the LT phase, because the coordination geometry is D4h, ΔE(90°) is equal to the absorption maximum observed at lower temperature, while in the HT phase ΔE(θ) is equal to the absorption maximum recorded at higher temperature. Therefore, the following relation is obtained hν at low temperature ΔE(90°) = max hνmax at high temperature ΔE(θ )

(11)

The enthalpy (actually the internal energy) of the d-orbitals is the sum of each electron’s energy. The d-electrons are accommodated in the energy level diagram shown in Figure 90, one by one from bottom to top. There are nine d-electrons in the copper complexes. The energy of the d-orbitals in the LT phase is calculated from the energy scheme at θ = 90° whereas that in the HT phase is calculated from the scheme at the θ estimated from the equation given above. Consequently, the enthalpy change of the d-orbitals arising from the change in the coordination geometry is straightforwardly determined as the difference between the electron energies of the LT and the HT phases. The contribution of the d-orbital energy to the transition enthalpy amounted to 8.4 kJ mol−1 for [Cu(dieten)2](ClO4)2. The observed transition enthalpy may be regarded as consisting of two contributions: One is the enthalpy gain due to the onset of the puckering motion of the metal−ligand chelate rings and the other is the electronic energy gained by a change in the coordination geometry across the phase transition. These two contributions were analyzed in terms of the harmonic oscillator model and the angular overlap model, respectively. As estimated above, the former is ΔH(ligand) = 8.0 kJ mol−1 and the latter amounts to ΔH(AOM) = 8.4 kJ mol−1. The sum of these contributions corresponds to ΔH(total) = 16.4 kJ mol−1. Surprisingly, this theoretical value well accounts for the observed value, ΔH (obsd) = 17.4 kJ mol−1. Even better agreement has been attained for [Cu(dieten)2](BF4)2, in which ΔH(obsd) and ΔH(total) are 16.6 and 16.8 kJ mol−1, respectively. BC


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(200−400) K. Although the transition temperatures are much higher than those of traditional SP compounds, their magnetic susceptibilities465−468 and X-ray diffraction469−472 show similar behavior to usual SP transition. Takaoka and Motizuki473 theoretically treated these phase transitions occurring in MTCNQ salts as the SP transition. On the basis of X-ray diffraction analysis, Kobayashi472 found in 1978 two crystalmodifications in NH4-TCNQ: The form I shows a dimerization transition at (301 ± 5) K, while the form II at around 215 K. Both phase transitions are related to a SP transition. Kotani et al.474 measured heat capacities of the form I of NH4-TCNQ crystal at temperatures in the (12−350) K range by adiabatic calorimetry and found a λ-type heat capacity anomaly arising from a SP transition at 301.3 K. The enthalpy and entropy of transition are ΔtrsH = (667 ± 7) J mol−1 and ΔtrsS = (2.19 ± 0.02) J K−1 mol−1, respectively. Fundamental concept of the SP transition is to describe the quantum spin system of AF linear chain in terms of the pseudofermion and to take into account its interaction with phonon. For such a system, a lattice distortion arising from dimerization is expected to appear at low-temperature side. Thus, one should take into account both the spin and the lattice systems to interpret the nature of the SP transition. By assuming a uniform 1D AF Heisenberg chain consisting of quantum spin (S = 1/2) in the high-temperature phase and an alternating AF nonuniform chain in the low-temperature phase, one can schematically draw temperature dependence of the magnetic heat capacity (ΔCmag) and the magnetic entropy (Smag) as depicted in Figure 91. T0 is a temperature, below which the progressive dimerization substantially ceases. ΔSm and ΔSd are the magnetic entropy of the uniform (m: monomer) and nonuniform (d: dimer) AF Heisenberg chain models at Ttrs and T0, respectively. In the actual situation, the value ΔSd(T0) is buried in the estimated normal heat capacity. ΔSsro is the entropy due to short-range-order effect in the monomer chain above Ttrs. The observed entropy of transition ΔtrsS may consist of the magnetic and the lattice contributions:

Figure 91. Schematic drawing of the magnetic heat capacity (top) and the entropy gain due to the magnetic origin in NH4-TCNQ crystal (down). T0 is the temperature, below which the progressive dimerization substantially ceases. ΔSm and ΔSd are the magnetic entropy of the uniform (monomer) and nonuniform (dimer) AF Heisenberg chain models at Ttrs and T0, respectively. In the actual situation, the value ΔSd(T0) is buried in the estimated normal heat capacity. ΔSsro is the entropy due to the short-range-order effect in the monomer chain above Ttrs. Reprinted with permission from ref 474. Copyright 2009 Elsevier Ltd.

As to the magnetic system, the monomer−dimer transition occurs at Ttrs in the 1D S = 1/2 AF Heisenberg chains. As schematically shown in Figure 91, the total magnetic entropy, Smag(T = ∞), consists of the following components:

from the lattice system is determined as ΔtrsSlat = 1.58 J K−1 mol−1 from eq 12. The contribution from the magnetic origin to the entropy of transition is as small as 28%, and the remaining 72% arises from the phonon system. Moreover, the entropy due to the short-range-order effect in the uniform (monomer) chain above Ttrs is estimated as ΔSsro(Ttrs < T < ∞) = 4.57 J K−1 mol−1 from eq 13, which is 80% of the total magnetic entropy (R ln 2). The fact that the majority of the magnetic entropy remains above Ttrs as the short-range-order effect is characteristic of the low-dimensional magnets. The essential difference between the SP and the Peierls system in the half-filled state is the ratio of the on-site Coulomb interaction U to the electron-transfer energy t, resulting in the different optical, magnetic, and transport properties. The material of the SP system is an insulator and the magnetic susceptibility reduces to zero as a function of temperature with lattice dimerization. On the basis of optical, magnetic, and structural properties of a segregated-stack charge-transfer complex DAP-TCNQ (DAP, 1,6-pyrenediamine), Sekiwaka et al.475 suggested in 1997 that the transition observed around 160 K in this complex belongs to a SP system, even though the

where ΔSd(T0) is the magnetic entropy of the nonuniform (dimer) AF Heisenberg chain at T0 ≈ 240 K, below which the progressive dimerization substantially ceases, and ΔSsro(Ttrs < T < ∞) is the entropy due to the short-range-order effect in the uniform (monomer) chain above Ttrs. In the actual situation, the value ΔSd(T0) is buried in the estimated normal heat capacity. On the other hand, the magnetic entropy of the uniform chain (monomer) at Ttrs is the sum of two contributions:

On the basis of the 1D AF Heisenberg model, one can theoretically calculate the values as ΔSd(T0) = 0.58 J K−1 mol−1 and ΔSm(Ttrs) = 1.19 J K−1 mol−1, and hence ΔtrsSmag = 0.61 J K−1 mol−1. Because the observed entropy gain at the phase transition is ΔtrsS = (2.19 ± 0.02) J K−1 mol−1, contribution BD


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band filling deviates from the half-filled regime. The temperature dependence of the electric conductivity along the stacking axis does not show metal-to-insulator transition and is always semiconducting. Thus, DAP-TCNQ is considered a Mott insulator. Detailed thermodynamic investigation of the quasi-1D sulfurbased organic salts (TMTTF)2PF6 and (TMTTF)2Br was performed by Lasjaunias et al.476 in 2002 in the temperature range from 30 mK to 7 K (TMTTF, tetramethyltetrathiafulvalene). They considered the general aspects of the lowtemperature heat capacity of these materials in relation to their ground states and compared them with those previously measured for selenium members of the same family. All these compounds exhibit very similar thermodynamic behavior in spite of various electronic ground states: A SP transition at TSP = 15 K for (TMTTF)2PF6, commensurate AF spin modulation for (TMTTF)2Br, incommensurate spin-density wave for (TMTSF)2PF6 (TMTSF, tetramethyltetraselenafulvalene). In 2009, de Souza et al.477 reported results of high-resolution thermal expansion measurements on the charge-ordering (TCO) and SP (TSP) transition in (TMTTF)2X for X = AsF6 and PF6: TCO = 65 K (X = PF6) and 105 K (AsF6) while TSP = 16.4 K (PF6) and 11.4 K (AsF6). On the basis of distinct lattice effects observed at TCO in the c*-axis expansivity, they proposed a scheme which involves a charge modulation along the TMTTF stacks and its coupling to displacements of the anions. The c*axis expansivity is most strongly affected by the SP transition. By combining the result of expansivity and that of heat capacity for the X = AsF6 salt they separated the contribution associated with the SP transition from the ordinary lattice effects. Organic charge transfer salts consisting of a strong acceptor molecule of 2,5-R1,R2-dicyanoquinodiimine (R1,R2-DCNQI; R1 and R2 being alkyl groups; see Figure 103) and a metal cation (M) have attracted wide interest both experimentally and theoretically because they provide various electronic phases such as metallic, charge-density-wave (CDW), charge-ordered (CO), SP, AF ordered state, and so on. By use of a.c. and thermal relaxation calorimetry techniques, Nakazawa et al.478 reported in 2003 thermodynamic investigation of the SP systems (DMe-DCNQI)2Ag and (DMeDCNQI)2Li, where DMe-DCNQI is 2,5-dimethyl-N,N′dicyanoquinonediimine. The fully gapped nature characteristic of the SP ground state was confirmed by the absence of T-linear γ term in the low-temperature heat capacity for a single crystal. As shown in Figure 92, two distinct peaks were observed at 71 and 86 K in the heat capacity of (DMe-DCNQI)2Ag, while only a single peak was found at 52 K for (DMe-DCNQI)2Li. The two-step structure in the Ag salt suggests that the SP transition of this material involves an intermediate state, which is probably attributable to the degrees of freedom inside the dimers. For the specimen consisting of a compact pellet of numerous pieces of tiny crystals, however, the thermal anomaly around the SP transition disappeared, but instead a γ term probably associated with the 1D AF spin excitations appeared. This implies that the present SP state is quite fragile against disorder or stress. These findings display novel features of the SP transition in the quarter-filled band systems. ̈ character manifests itself also in the heat capacity Such naive at low temperature. Okuma et al.479 measured low-temperature heat capacity on pellet samples of (DMe-DCNQI)2M (M = Li, Ag) and found a T-linear term for both samples. It was attributable to the spin-wave excitation induced by inhomogeneous pressure effect produced in the course of pellet

Figure 92. CpT−1 vs T plot of the a.c. calorimetry data of (a) (DMeDCNQI)2Ag and (b) (DMe-DCNQI)2Li. The absolute values have been calibrated by use of the thermal relaxation calorimetry data. The solid curves are baselines represented by polynomial functions of the temperature. The hatched areas are used for calculation of the entropy gain due to the transitions. Reprinted with permission from ref 478. Copyright 2003 American Physical Society.

formation. Although temperature dependence of magnetic susceptibility is almost the same in both materials, the coefficient of T-linear term of the Ag salt is three times larger than that of the Li salt. They interpreted this peculiar electronic state in terms of competition of the SP mechanism and the Coulomb repulsion. 10.2. Inorganic Spin-Peierls Compounds

In addition to those organic compounds, the SP transition occurs even in pure inorganic compounds. Hase et al.480 found in 1993 that the magnetic susceptibilities for single-crystal of CuGeO3, a linear Cu2+ (spin 1/2) chain compound rapidly drops to small constant values in all directions with decreasing temperature below a phase transition temperature near 14 K. Because the magnetic-field dependence of the transition temperature quantitatively agrees with both theoretical predictions and experimental results for organic SP systems, they claimed that their finding provides an unambiguous evidence for the existence of the SP transition in an inorganic compound. Using an a.c. calorimetry, Kuroe et al.481 measured heat capacity of CuGeO3 and found a clear peak at 12.5 K. On the other hand, Sahling et al.482 determined its heat capacities in the (40 mK to 35 K) temperature range by use of a usual transient heat-pulse technique and found a sharp anomaly at TSP = 14.2 K corresponding to a second-order phase transition with strong critical fluctuations. Below 6 K, the heat capacity BE


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Figure 93. Molecular structures of donors and acceptors which yield conducting organic charge transfer salts.

conclusion is that the phase transition in α′-NaV2O5 at 35 K cannot be explained by a SP transition alone.

showed an exponential decay, corresponding to an energy gap inherent in the SP systems. As the second possible inorganic SP system, Isobe and Ueda483 reported a candidate α′-NaV2O5, which belongs to a typical quantum spin (S = 1/2) 1D AF Heisenberg linear chain system. In this crystal, there exist two kinds of pyramidal chains: One is magnetic V4+O5 (S = 1/2) chain and the other is nonmagnetic V5+O5 (S = 0) chain. The magnetic susceptibility shows a good fit to the equation for an S = 1/2 AF Heisenberg linear chain above 34 K, while below 34 K it rapidly decreases with decreasing temperature to a constant value which is reasonable for spin-singlet V4+−V4+ pairs. This rapid reduction of the spin susceptibility below 34 K suggests the existence of a SP transition. However, the SP transition found in this compound is somewhat unusual. Köppen et al.484 presented the results of high-resolution thermal expansion and heat capacity measurements on single crystal of α′-NaV2O5 and found a clear evidence for two almost degenerate phase transitions associated with the formation of the dimerized state with a nonmagnetic ground state around 33 K. They concluded that the secondorder transition at T2 = (32.7 ± 0.1) K is the signature of a SP transition as to the occurrence of spontaneous strains and the sharp first-order transition at T1 = (33 ± 0.1) K is assigned to a structural transformation induced by the incipient SP order parameter above T > T2. Heat capacity of this complex is also reported by Powell et al.485 A large sharp peak was observed at TSP = 33 K, but its size was more than an order of magnitude greater than the value estimated from mean-field theory. This fact is consistent with large fluctuations expected from its 1D magnetic structure. However, shape of the peak was somewhat sample dependent and does not have a symmetric form about TSP expected for either Gaussian or critical fluctuations, suggesting either that the samples are near a multicritical point or that there is some heterogeneous broadening. Hemberger et al.486 reported that the temperature dependence of the heat capacity can be explained assuming a sum of a linear (magnetic) and a Debye (lattice) contribution, but the heat capacity jump at TSP = 35 K cannot be described by the opening of a gap in the mean-field approximation. Their

11. ORGANIC CONDUCTORS Organic conducting systems are attracting wide interests from people working in both fundamental researches and application because they are chemically synthesized metals and semiconductors which do not exist in nature. In general, organic molecules have self-contained closed-shell structures, where two electrons with alternative spins occupy their HOMOs (highest occupied molecular orbitals). Therefore, charge carriers should be produced by introducing electrons into LUMOs (lowest unoccupied molecular orbitals) or removing electrons from HOMOs. It is well-known that charge transfer salts of donors and acceptors (Figure 93) with their counterions make various kind of electronic phases, such as superconductive, spin density wave (SDW), charge density wave (CDW), and antiferromagnetic (AF) ones.487−489 Thermodynamic features and phase transitions detected by heat capacity measurements on donor−acceptor type charge transfer systems, BEDT-TTF (bisethylenedithiotetrathiafuluvalene) based donor-anion systems, and π−d interacting systems containing 3d-transition metal ions are summarized. Heat capacity of quantum spin-liquid systems, which have 2D triangle structure, is also mentioned. 11.1. Donor−Acceptor Type Charge Transfer Salts

TTF-TCNQ is the most classical salt with long history of research since the first observation of metallic conductivity by Coleman et al.490 in 1973. TTF and TCNQ stack in a segregated manner in this crystal and form a double column structure. The charge transfer rate from TTF to TCNQ is evaluated as 0.59, a quite reasonable value for conductive carriers in both TTF and TCNQ columns. The metal to insulator transition takes place around 50 K due to the Peierls instability of electronic structure of quasi-1D bands. In spite of various experiments performed until now, the transition mechanisms and detailed electronic structure are not fully understood yet. Heat capacity measurements of TTF-TCNQ have been performed by several groups using a.c. calorimetry and relaxation techniques. Craven et al.491 observed a clear peak BF


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Figure 94. (a) Temperature dependence of heat capacity of TTF-TCNQ obtained by adiabatic calorimetry for powdered sample. Heat capacities of neutral TTF, TCNQ samples and their summation are also shown. (b) Enlarged plot around the metal−insulator transition and those obtained by a.c. calorimetry with varying frequencies for a single crystal sample are shown in the right figure. Reprinted with permission from ref 494. Copyright 1999 The Physical Society of Japan.

step transition at 28 and 32 K, each of which is associated with the transition in TSF and TTF columns. They speculated that relatively larger peak at 54 K in TTF-TCNQ as compared with two peaks in TSF-TCNQ and absence of thermal anomaly around 45−49 K in TTF-TCNQ mean that CDW formation in each column occurs simultaneously in TTF-TCNQ, while it occurs independently in each column in the TSF-TCNQ. The electronic state of charge transfer salts consisting of BEDT-TTF (bisethylendithia tetrathiafuluvalene) and TCNQ molecules is also studied by means of transport and magnetic properties.496 In this compound, BEDT-TTF molecules tend to form 2D network structure separated by TCNQ layers. The donor arrangement in the 2D plane is classified as β′- and β″type, which are famous structures in the BEDT-TTF based compound discussed in section 11.2. Although the charge transfer rate is considered as ∼0.5, which is close to typical 2:1 salts consisting of donors and counteranions, the electronic properties are different between β′ and β″ structures. The former salt shows Mott insulating state due to strong dimerization in the donor arrangement, while the latter shows metallic properties with γ = 21.5 mJ K−2 mol−1. Although the thermodynamic details are not published yet, absence of thermal anomalies in β′-salt is indicated in the paper by Yamashita et al.497 by thermal relaxation method. In the case of mixed stack compound such as TTF-CA (CA, p-chloranil), the electronic state is insulating or semiconductive because of the alternating arrangement of donor and acceptor molecules. The neutral state, which is realized when the charge transfer ρ is smaller than 0.5, changes to the ionic state with ρ > 0.5 by applying pressure or by decreasing temperatures. Electronic state of TTF-CA shows a drastic phase transition from neutral state to ionic state around 80 K. Adiabatic heat capacity measurement performed by Kawamura et al.498 detected a large peak due to the neutral to ionic transition at 82.5 K. The transition entropy and enthalpy are (540 ± 5) J

structure in heat capacity and estimated that the transition temperature is Tc = (54.8 ± 0.4) K. The subsequent measurement due to relaxation calorimetry by Djurek et al.492 showed three anomalies at 49, 46, and 38 K. It was suggested that the 2kF CDW occurs in TCNQ column around 52 K, and 4kF CDW is formed at 48 K in the TTF column, followed by the third transition related to the phase locking of 2kF CDW and 4kF CDW at lower temperatures.493 In 1999, Saito et al.494 performed adiabatic measurements of TTF-TCNQ in a wide temperature range of 10−300 K for powdered sample together with high resolution a.c. calorimetry for a single crystal sample. As shown in Figure 94, the thermal anomalies with peaks at 52 and 37 K were clearly observed, while the transition around 46−49 K was not detected. They also studied frequency dependence of the a.c. calorimetry and found a downward shift of the low temperature peak around 37 K with increasing frequencies. They claimed that this transition is related to the formation of the lock-in structure of two CDWs and has a firstorder character as predicted previously.493 A possibility of glasslike freezing around 33 K is also reported.494 The powdered sample in Figure 94b has a broader peak than that of the single crystal data obtained by a.c. calorimetry. This fact means that sample quality is important for the discussion of thermodynamic properties. Viswanathan and Johnston495 studied low temperature heat capacity and observed that both γ (the electronic heat capacity coefficient) and β (the coefficient of T3 term in heat capacity reflecting on the low temperature approximation of Debye model) depend on sample quality. The existence of impurities suppresses the metal insulator transition and recovers metallic nature at low temperature, which is commonly observed in organic conductors with relatively soft lattices. Thermodynamic properties of TSF-TCNQ, which has isostructure as TTF-TCNQ, were also reported by Saito et al.494 They observed that the Peierls transition occurs as a twoBG


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mol−1 and (6.12 ± 0.06) J K−1 mol−1, respectively. TMBTCNQ (TMB, 3,3′,5,5′-tetramethylbenzidine) and DTTF-CA (DTTF, 4,4′-dimethylteterathiafuluvalene) also show similar neutral to ionic transition. In structurally interesting charge transfer compounds such as STB-TCNQ and STB- F4-TCNQ, molecular dynamics related to the crankshaft motion around the central double bond of CC bring about a glasslike freezing.499,500 The charge transfer rate is considered as 0.1−0.2 in both compounds. Heat capacity shows the glass transition at 240 K for STB-F4TCNQ and 250 K for STB-TCNQ. The charge transfer salt with [Pd(H2−xedag)(Hedag)]TCNQ (x ≈ 1/3) (H2edag, ethylenediaminoglyoxime) was studied by adiabatic technique.501 When x = 0, the average valence of TCNQ is −1, and the system is insulating. However, the increase of x enhances metallic conductivity through the charge transfer mechanism. The heat capacity shows a broad peak around 170 K with the transition enthalpy of 100 J mol−1 and the transition entropy of 0.7 J K−1 mol−1. This result is consistent with gap formation at the Fermi level by the formation of CDW structure by the Peierls mechanism. Tanaka et al.502 performed adiabatic calorimetry for tetrakis(alkylthio)tetrathiafulvalenes (TTCnTTF) with n = 3, 8, where the electric conductivity changes with increasing the length of alkyl chain. For n = 1, the electric resistivity is 1010 Ω m, while it decreases down to 105 Ω cm for n = 10 compound due to the change of molecular shape from boat-like structure (n ≤ 3) to chair-like one (n ≥ 4). The alkyl chains work as molecular fasteners to increase intermolecular interactions.

Figure 95. Spatial arrangement of donor molecules in 2D layer in organic charge transfer salts with (BEDT-TTF)2X composition.

respectively, while M = NH4 salt does not show this transition and gives rise to superconductivity around 1 K. The density wave transitions in the former three are characterized by a steplike decrease of resistivity with decreasing temperatures, as is observed by Sasaki et al.504,505 in M = K salt for the first time. The anomalous steps in resistivity also correspond to the onset temperatures of large magnetoresistance, and an abrupt increase in the temperature dependence of the Hall coefficient. Magnetic susceptibility shows a sharp decrease below this temperature with anisotropic character as in the case of typical AF magnetic orderings. To clarify the nature of the transition, thermodynamic measurement was performed by Nakazawa et al.506,507 by semiadiabatic and thermal relaxation techniques. From the CpT −1 vs T 2 plot (Figure 96), the γ and β values were determined as 6.4 mJ K−2 mol−1 and 11.6 mJ K−4 mol−1 for M = K salt and 7.1 mJ K−2 mol−1 and 11.1 mJ K−4 mol−1 for M = Rb salt. This is nearly 1/3 of the γ value for M = NH4 salt (γ = 25−26 mJ K−2 mol−1). The small γ in the former two salts means that a

11.2. (BEDT-TTF)2X Salts

Charge transfer salts consisting of a donor molecule of BEDTTTF and counterions (X) usually have a composition of (BEDT-TTF)2X. In addition to the face-to-face overlap of the orbitals in stacking direction, the S−S contact between neighboring BEDT-TTF molecules makes a fairly large transverse overlap of the molecular orbitals. These overlaps of molecular orbitals resulting in large transfer energies in various directions lead to a variety of 2D networked arrangements in the donor layers. The donor arrangements in 2D plane are sorted by using Greek characters. Typical arrangements of BEDT-TTF molecules in 2D plane503 are shown in Figure 95. Among these salts, those possessing the α-type, β-type, and κtype structures are studied extensively by calorimetric measurements. The α-type structure is a typical example of the nondimeric arrangement in (BEDT-TTF)2X systems. The BEDT-TTF donors form a herringbone type arrangement which tends to make a metallic state with 1/4 filling. In this structure, dimensionality and correlation parameters are controlled by the dihedral angle determined by external pressure and the anion sizes. The α-(BEDT-TTF)2MHg(SCN)4 (M = K, Rb, Tl and NH4) are widely studied organic conductors, showing metallic conductivity down to milli-Kelvin region. According to the band calculation, these salts have two different Fermi surfaces of which detailed structures have been experimentally observed by quantum oscillation measurements such as SdvH and angleresolved magnetoresistance. One is an open Fermi surface with quasi-1D character, and the other is a large 2D Fermi surface with cylindrical structure. In the case of M = K, Rb, and Tl salt, a nesting effect mainly appearing in quasi-1D Fermi surface makes a kind of density wave state at 8, 12, and 10 K,

Figure 96. Low temperature heat capacity of α-(BEDT-TTF)2MHg(SCN)4 shown in CpT −1 vs T 2 plot. The M = K and Rb salts have smaller γ values than that of NH4 salt. The peak structure of CpT −1 is associated with the superconductive transition. Reprinted with permission from ref 506. Copyright 1995 American Physical Society. BH


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relatively high transition temperature in this structure, the γ values are comparable with those of nonsuperconductors and a low Tc superconductor such as α-(BEDT-TTF)2NH4Hg(SCN)4 and other organic conductors, for example, (DMeDCNQI)2Cu. This situation is similar to the case of high Tc oxides in that AF fluctuations are detected but the electronic heat capacity coefficient is not so large as compared with enhanced metallic systems consisting of d- or f-electrons. Thermodynamic studies on κ-type compounds were performed by Andraka and Stewart and their group. The γ values of 10 K class superconductive salt of κ-(BEDT-TTF)2Cu(NCS)2 is (25 ± 3) mJ K−2 mol−1 by Andraka et al.518 and that of κ-(BEDTTTF)2Cu[N(CN)2]Br is 22 mJ K−2 mol−1 in the initial stage of investigation.519 Following this, several groups have reported heat capacity data up to now. The list of thermodynamic parameters of κ-(BEDT-TTF)2X system is given in Table 9. To establish the 2D dimer−Mott picture reported by Kanoda et al.,514,515 characterization of the insulating phase by low temperature heat capacity measurements were performed by his group520,521 for the Mott insulating sample of κ-(BEDTTTF)2Cu[N(CN)2]Cl and deutrated κ-(d8:BEDT-TTF)2Cu[N(CN)2]Br. Although the pristine salt of κ-(BEDT-TTF)2Cu[N(CN)2]Br is a superconductor with Tc = 11.4 K, the deuterated salt shifts to the antiferromagnetic region due to a chemical pressure effect. The low temperature heat capacity of κ-(BEDT-TTF)2Cu[N(CN)2]Cl and κ-(d8:BEDT-TTF)2Cu[N(CN)2]Br clearly show vanishing γ term in CpT −1 vs T 2 plot. The magnetic fields up to 8 T do not affect on the heat capacity in the insulating sample. The absence of electron density of states at the Fermi energy, even though the electronic structure is effectively half filling, supports the picture of the dimer−Mott system. Another important point reported by the heat capacity works is the absence of thermal anomaly in the Néel temperatures in both compounds. The effective J = 2t2/Udimer in this system is evaluates as 200−250 K and a large amount of magnetic entropy by short-range correlations exist in the higher temperatures. Furthermore, quantum fluctuations in the Mott insulating state obscure the long-range nature of phase transitions. The absence of thermal anomaly around the Néel temperatures in such a Mott−Hubbard system with a layered structure has later been investigated by Yamashita et al.497 in κ-(BEDT-TTF) 2 Cu[N(CN) 2]Cl and a similar dimerized salt of β′-(BEDT-TTF)2ICl2 system. The latter compound is also a Mott insulating system, which does not show any γ value and relatively hard lattice expressed by smaller β = 7.46 mJ K−4 mol−1. This compound was converted to 14.6 K superconductor under external pressure of about 80 GPa.522 Thermodynamic research on 10 K class superconductors in the κ-type salt have been conducted over 20 years because this subject has common aspects with d- and f-electron systems. The first thermodynamic measurements of superconductive transition of κ-(BEDT-TTF)2Cu(NCS)2 system was performed by Katsumoto et al.523 in 1988 by a.c. technique. Heat capacity measurement of this compound by relaxation calorimetry was performed by Andraka et al.518 in 1989. From the CpT −1 vs T2 plot under a magnetic field up to 12.5 T, they estimated that the normal state γ value is (25 ± 3) mJ K−2 mol−1 and claimed that the superconductivity is in a strong coupling region with ΔCp/γTc > 2. Graebner et al.524 developed a new highresolution a.c. technique and observed a clear anomaly related to superconductive phase transition. They also reported the data obtained under in-plane and out-of-plane magnetic fields. The heat capacity peak reported by them means that this salt is

part of the Fermi surface disappears as a nesting which results in a decrease of the electronic density of states. As a matter of fact, the isostructural salt of α-(BEDT-TTF)2TlHg(SeCN)4, which does not show any phase transition and keeps metallic down to low temperatures, shows γ value of 20 mJ K−2 mol−1 similar to the normal state of M = NH4 salt.508 Numerical values for the characteristic parameters such as γ and β derived from calorimetric measurements are summarized in Table 9. The existence of thermal anomaly around the transition temperature in the K and Rb salts was confirmed by highresolution a.c. technique by Henning et al.509 They estimated fairly large ΔCp values of 968 mJ K−1 mol−1 for Rb and 247 mJ K−1 mol−1 for M = K salt. Kovalev et al.510,511 reported a heat capacity jump in M = K system with smaller ΔCp ≈ 100 mJ K−1 mol−1, which seems to be reasonable as a nesting type transition. Thermodynamic data at low temperatures and around the transitions temperatures demonstrate that density wave type phase transitions related to the disappearance of density of states occur in this system. Heat capacity measurement on α-(BEDT-TTF)2I3 was performed by Fortune et al.512 by a.c. calorimetry. This salt has weak dimerization in stacking direction and has semimetallic electronic structure with two small Fermi pockets having electron and hole characters. At the metal−insulator transition temperature of 135 K, this salt exhibits a sharp peak with large transition entropy of about 94% of R ln 2. Hysteretic behavior of the peak position observed on cooling and heating with a rate of 6 K h−1 demonstrates that the transition is firstorder. This compound is recently found to show zero-gapped states produced by crossing of dispersion curvatures by Tajima et al.513 through systematic analysis of transport properties under pressures and with magnetic fields. The importance of intersite Coulomb and mechanism to make charge disproportionation is discussed in this compound. The heat capacity measurements under pressure are important for understanding the peculiar electronic states of this salt. It is well-known that BEDT-TTF based charge transfer salts with κ-type structure shows interesting physical characteristics produced by electron correlations in 2D bands. These systems involve the Mott insulating salts with antiferromagnetic (AF) transitions around 10−30 K, nonordered magnetic salts such as spin liquid systems, and metallic/superconductive salts with relatively high transition temperatures. To understand the origins and relations between these states are important subjects of organic conductors. Important parameters dominating electronic properties are on-site Coulomb interaction U and bandwidth W because the κ-type structure is known as a dimerized structure. Donor dimers with face-to-face contact are arranged in nearly orthogonal position, and consequently a zigzag networked structure (Figure 95) is constructed. The intradimer transfer energy is usually larger than other transfers, and thus each dimer is considered as a structural unit in the 2D plane. Rigid dimerization in the κ-type structure leads to unique electronic structure characterized by a simple Mott−Hubbard physics resulting from competition of the on-site (on-dimer) U and bandwidth W. When U is larger than W, the system becomes a Mott insulating state with AF ground state. On the other hand, in the region where W exceeds U, the ground state is a correlated metallic state. The superconductivity with relatively high Tc (≈ 10 K) appears in this region. The P−T phase diagram, in which the AF phase is neighboring to the superconducting phase, is established by Kanoda et al. (Figure 97)514−517 Although there are many superconductors with BI


BJ

∼4 (under 0.37 GPa)

0.1 (under 17 T) 0.1 1.1 4.8

3−4 3.4 5 1.5(βL), 8(βH) 4.88 14.6 (under 80 GPa) 4.5

2.4

10.4 12.8 (under 0.03 GPa) 11.6

0.9

Tc (K)

14.4 25 10 12.6 19.9

6.4 7.1 25−26 20 25 0 22 0 50 52 55 18.9 30 24 20 0 18.7 ∼0 (slow cooling) ∼100 (rapid cooling) 40

γ (mJ K−2 mol−1)

21 24

14.5

7.46 12.2

10.3

12 18

10.9

11.6 11.1

β (mJ K−4 mol−1)

197

221

197

218

219 210 212 203

223 226 221

θD (K)

1.37

2.1

0.54 1.6 1.1

2

>2, 1.5

1.2−1.3

ΔCp/γTc

spin liquid spin liquid

AF transition at 8.3 K AF transition at 0.45 K AF transition at 2.4 K

the fitting includes a δT5 term

the fitting includes a δT5 term

remarks

refs 506 506 506 508 518 514 519 518 528 529 529 525,526 527 549 553 508 555 556 556 563−565 560 558 566 574 571 585 586

a The γ and β are coefficients of the T-linear term and the T3 terms, respectively. The Debye temperature (θD) was determined from the value of β by low temperature approximation of the Debye model. The Tc denotes the critical temperature of the superconductive transition and ΔCp is magnitude of heat capacity discontinuity appears at Tc.

α-(BEDT-TTF)2KHg(SCN)4 α-(BEDT-TTF)2RbHg(SCN)4 α-(BEDT-TTF)2NH4Hg(SCN)4 α-(BEDT-TTF)2TlHg(SeCN)4 κ-(BEDT-TTF)2Cu(NCS)2 κ-(BEDT-TTF)2Cu[N(CN)2]Cl κ-(BEDT-TTF)2Cu[N(CN)2]Br κ-(d8:BEDT-TTF)2Cu[N(CN)2]Br κL-(BEDT-TTF)2Ag(CF3)4TCE κ-(BEDT-TTF)4Hg2.78Cl8 κ-(BEDT-TTF)4Hg2.89Br8 κ-(BEDT-TTF)2I3 κ-(BEDT-TTF)2Ag(CN)2H2O β-(BEDT-TTF)2I3 β-(BEDT-TTF)2AuI2 β′-(BEDT-TTF)2ICl2 β″-(BEDT-TTF)2SF5CH3CF2SO3 θ-(BEDT-TTF)2RbZn(SCN)4 θ-(BEDT-TTF)2CsZn(SCN)4 λ-(BETS)2FeCl4 κ-(BETS)2FeCl4 κ-(BETS)2FeBr4 λ-(BETS)2GaCl4 DMe-DCNQI)2Cu DMeO-DCNQI)2Cu κ-(BEDT-TTF)2Cu2(CN)3 EtMe3Sb[Pd(dmit)2]2

compound

Table 9. Thermodynamic Parameters Obtained by Low Temperature Heat Capacity Measurements of Organic Conductors and Superconductorsa

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relaxation calorimetry under magnetic field up to 14 T and obtained γ = (22 ± 3) mJ K−2 mol−1 and Debye temperature of 210 K. They also estimated ΔCp/γTc = 2 ± 0.5, which is comparable with the data of κ-(BEDT-TTF)2Cu(NCS)2. Thermodynamic data of other superconductive salts are listed in Table 9 for κ-(BEDT-TTF)2I3,525,526 κ-(BEDT-TTF)2Ag(CN)2H2O,527 and κL-(BEDT-TTF)2Ag(CF3)4TCE,528 where TCE is 1,1,2-trichloroethane, and κ-(BEDTTTF)4Hg2.89Br8.529−531 To discuss low energy excitations over the superconductive gap, it is necessary to derive information on the gap structure around the Fermi surface by observing quasiparticle excitations. Nakazawa and Kanoda532 performed heat capacity measurements of κ-(BEDT-TTF)2Cu[N(CN)2]Br in dilution temperature region for this purpose. By comparing the data of superconductive κ-(BEDT-TTF)2Cu[N(CN)2]Br and insulating κ-(d8:BEDT-TTF)2Cu[N(CN)2]Br at low temperatures, they found that the electronic heat capacity below about 3 K has a T2 term in electronic heat capacity with a coefficient of 2.2 mJ K−3 mol−1. They also claimed that even in the superconductive state, the T-linear term expressed as γresT with γres = 1.2 mJ K−2 mol−1 exists even in good quality samples. The existence of this residual γres means that nearly 5% of electrons remain as normal electrons. The temperature dependence of electronic heat capacity shown in Figure 98 deviates from the BCS curve calculated by assuming a full gap with 2Δ/kBTc = 3.52. They also studied how the γ recovers in the vortex state by measuring heat capacity under magnetic fields and observed that γ obeys γ = kγn(H/Hc2)1/2 where γn is a normal state γ value and k is a coefficient of order of unity. These results demonstrate that the superconductivity is d-wave with line nodes around the Fermi surface. Almost the same tendency was observed in κ-(BEDT-TTF)2Cu(NCS)2, which also has larger quadratic terms and almost the similar γ values.533 The heat capacity results at extremely low temperature were consistent

Figure 97. Electronic phase diagram of κ-(BEDT-TTF)2X system. The 10 K class superconductive phase is neighboring to the AF insulating phase. The horizontal axis can be tuned by external pressure or chemical pressure. Reprinted with permission from ref 517. Copyright 2006 The Physical Society of Japan.

in the region of strong coupling of BCS theory. The data under magnetic field gives −dHc2/dT = 16 T K−1 in the H|| plane and −dHc2/dT = 0.75 T K−1 in the H⊥ plane direction. Twodimensional nature of organic superconductor was suggested by these data. In the case of κ-(BEDT-TTF)2Cu[N(CN)2]Br salt, Andraka et al.519 also performed heat capacity measurements by

Figure 98. Temperature dependence of the electronic heat capacity of κ-(BEDT-TTF)2Cu[N(CN)2]Br obtained by subtracting a curve determined based on the heat capacity of insulating d8 salt as a lattice background. The solid line demonstrates existence of T2 term in heat capacity. The dashed curve is the isotropic BCS curvature (left). The magnetic field dependence of low temperature electronic heat capacity coefficient, γ in the vortex states. (right) The inset shows the result of similar analysis taking into account the residual γres term. Reprinted with permission from ref 532. Copyright 1997 American Physical Society. BK


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with the assertion of d-wave paring proposed by T3 dependence of T1−1 in the superconductive state V-shape spectrum of I−V curves in tunneling spectroscopy by STM534 and 13C NMR works.535−537 Elsinger et al.538 has claimed that the lattice estimation using deuterated salt in ref makes ambiguity for accurate discussion on the electronic state. They measured heat capacity of κ(BEDT-TTF)2Cu[N(CN)2]Br between 1.7 and 21 K under magnetic fields of 0 and 14 T. They subtracted the data under magnetic fields from 0 T data and discussed overall peak shape in this temperature range. Assuming a full gap model with strong-coupling region, they succeeded to fit the electronic heat capacity with a model with α = 2.7. After the work of Elsinger et al., Müller et al.539 performed similar analysis on κ-(BEDTTTF)2Cu(NCS)2 between 2 and 30 K and with magnetic fields 0 and 8 T. They also claimed that the strong coupling model with full gap structure holds for organic superconductors. However, these works were performed in the higher temperature region, where lattice contribution gives relatively large contribution. The relatively larger peak than that expected from the BCS weak coupling theory is also confirmed by Yamashita et al.540 for κ-(BEDT-TTF)2Cu(NCS)2, and these data are also consistent with earlier works performed by Andraka et al.518 The inconsistency between low-temperature analysis below 2 K532 and the peak analysis by Elsinger et al.538 and Müller et al.539 was explained by Taylor et al.541 in 2010. They measured heat capacity of two high quality compounds with high resolution technique at 0 and 14 T and analyzed the peak by αmodel which takes into account the strength of coupling. They analyzed in s-wave and d-wave cases and compared the results. Their high resolution data even at low temperature region clearly indicated that the low temperature behaviors below 4−6 K can be well fitted by d-wave model with strong coupling region of α = 1.73, γn = 26.6 mJ K−2 mol−1. The quadratic temperature dependence existing at low temperatures is consistent with ref 532, although the deviation from T2 above 3 K is probably attributable to the underestimation of lattice terms. Malone et al.542 studied in-plane magnetic fields dependence in heat capacity at extremely low temperature region below 1 K for κ-(BEDT-TTF)2Cu[N(CN)2]Br and κ-(BEDT-TTF)2Cu(NCS)2. They observed clear 4-fold symmetric oscillation in γ which arises from crystal. The 4-fold oscillation is arisen from the existence of nodes in the superconductive states. Their results strongly suggestive of the dxy order parameter of the superconductivity in these materials. As a novel aspect of 10 K class superconductors, thermodynamic properties at extremely high magnetic field region studied by Lortz et al.543 is mentioned here. The 2D layered superconductor has a cylindrical Fermi surface and possesses a very large Horb. When the in-plane external fields exceeds the Pauli limit value of HP = 21 T, the superconductive state changes to an unusual one where part of the Cooper pairs are sacrificed to be normal. Spatial modulation in the superconductive order parameter occurs in the real space, and this state is called as Fulde−Ferrell−Larkin−Ovchinnikov (FFLO) state. As shown in Figure 99, they performed heat capacity measurements with strong magnetic fields up to 28 T applied parallel to the plane and found that heat capacity peak changes from a second-order structure to drastic first-order structure peak around 23 T and Hc2 becomes larger at low temperature regions as is predicted by FFLO theories.

Figure 99. Temperature dependence of the electronic heat capacities, CeT −1 of κ-(BEDT-TTF)2Cu(NCS)2 under strong magnetic fields up to 28 T. The magnetic fields are applied parallel to the conducting planes. The first-order transition means that the superconductivity is converted to FFLO state. Reprinted with permission from ref 543. Copyright 2007 American Physical Society.

The nature of superconductivity of κ-(BEDT-TTF)2Cu[N(CN)2]Br and its deuterated salt is affected by disorders produced by the rapid cooling process. Because the deuterated sample is located just close to the boundary, the drastic change in charge-transport properties for example insulator to superconductive transition occurs by controlling the cooling rate. This phenomenon is considered as electronic origin specific for the Mott−Hubbard system where metallic and AF insulating phases are neighboring by a first-order phase boundary. Akutsu et al.544 performed high-resolution a.c. heat capacity measurements of κ-(BEDT-TTF)2Cu[N(CN)2]Br and observed a glasslike anomaly around 90 K with a frequency of 0.24 Hz. The heat capacity anomaly is shifted to about 105 K at 1 Hz and 110 K at 4 Hz (Figure 100). They concluded that the ethylene groups at the edge of BEDT-TTF molecules have two stable conformations and glasslike freezing occurs during the cooling process. The glass transition temperature in the static limit coincides with the temperature at which resistivity and thermal expansion show anomalies (Tg = 77 K). Similar glasslike freezing is also observed in κ-(BEDT-TTF)2Cu[N(CN)2] Cl and also in (DMET)2BF4, (DMET)2ClO4, and (DIMET)2BF4 as discussed in ref 545. To see the variation of superconductive characters in the phase diagram of κ-type salts, a.c. heat capacity measurements under controlled pressure are performed by Tokoro et al.546,547 in κ-(BEDT-TTF)2Ag(CN)2H2O, κ-(BEDT-TTF)2Cu(NCS)2, and κ-(BEDT-TTF1−xBEDSe-TTFx)2Cu[N(CN)2]Br system. The substitution of BEDT-TTF by BEDSe-TTF molecules less than 20% produces a chemical pressure effect and decreases the transition temperature. Tokoro et al. studied by a.c. heat capacity under pressure and pursued the shift of a thermal anomaly associated with superconductive transition. However, the background value is so large and it is still in the developing process to attain better resolution because the lattice heat capacity above 5 K is very large in the organic systems. A new technique using a microchip calorimeter was also performed for a tiny sample of κ-(BEDT-TTF)2Cu[N(CN)2]Br under magnetic fields using tiny crystals of a few mg by Muraoka et al.548 The β-type structure is known in BEDT-TTF salts with linear-type anions such as I3−, IBr2−, I2Br−, and AuI2−, etc. They BL


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disappears by partial nesting of Fermi surface by a kind of SDW, CDW formation. They discussed that the reduction of Fermi surface is reflected on the density of states around the Fermi surface, resulting a difference of the electronic heat capacity between βH phase and βL phase below Tc = 1.5 K. Fortune et al.552 also indicated possibility of reconstruction in the Fermi surface structure from closed orbit to open orbit due to the formation of a superlattice, giving an incommensurate potential to conduction electrons. They observed that a well annealed sample shows a thermal anomaly related to superconductive transition at 6−7 K for the first time in this paper. Heat capacity of β-(BEDT-TTF)2AuI2 was reported by Stewart et al.553 They found a thermal anomaly around Tc = (4.88 ± 0.05) K. The γ value was evaluated as 20 mJ K−2 mol−1. Andres et al.554 also performed heat capacity measurement and observed an anomaly around 4.0 K. The difference of the heat capacity data obtained under 0 and 3 T applied perpendicular to the plane is discussed using a model based on the Gaussian distribution of transition temperature. The discussion on low energy excitations and gap structure were not performed yet in β-type organic superconductors. 11.3. π−d Interacting Systems

Studying the interplay of conductivity and long-range magnetic ordering is an important subject for organic conducting systems. The magnetic interactions between π-electrons and d-electrons and the possibility of π−d hybridization in electronic bands have been discussed for about 20 years. In 2001, the first observation of magnetic field induced superconductivity was reported by Uji et al.557 in the π-d system in (BETS)2X (BETS, bisethylenedithiotetraselenafulubalene). This molecule was developed as an analogous molecule of BEDT-TTF, and inner surfer atoms of BEDT-TTF are replaced by selenium. The charge transfer compounds consisting of BETS donors and FeBr4− and FeCl4− anions are known to crystallize into κ- and λ-type structures in which donor dimers are constructed. In the case of λ-(BETS)2FeCl4, it is known that antiferromagnetic transition occurs around 8.3 K. This transition appears simultaneously with the metal to insulator transition. However, by applying extremely large magnetic fields above 17 T just parallel to the 2D plane, metallic conductivity recovers and field-induced superconductivity, which is explained by the Jaccarino−Peter compensation effect, appears.557 On the other hand, coexistence of superconductivity and magnetic long-range ordering was found by Fujiwara and Kobayashi et al. in refs 558−561 as for κ-(BETS)2FeBr4 and κ(BETS)2FeCl4. Thermodynamic study on κ-(BETS)2FeBr4 was performed in ref for the first time, where a single crystal of about 0.19 mg was measured down to 0.9 K by relaxation technique in a 3He cryostat. Zero-field heat capacity shown in Figure 101 shows a sharp λ-type anomaly at 2.4 K, just as the same temperature where a step-like decrease of resistivity occurs with decreasing temperature. The magnetic entropy reaches to about 15 J K−1 mol−1, corresponding to the spin entropy of R ln 6 for Fe3+. The bulk ordering of 3d electron spins was confirmed by this discussion. Application of magnetic field of 1.0 and 2.0 T perpendicular to the plane reduced the peak temperature down to 2.36 and 2.05 K, respectively, but the peak shape remained sharp. Otsuka et al.560 performed heat capacity measurement by adiabatic technique using 5.80 mg of a polycrystalline sample of κ-(BETS)2FeCl4 and obtained similar sharp heat capacity peak

Figure 100. Temperature dependence of heat capacity obtained by high-resolution a.c. calorimetry technique. The frequency dependence of hump structure is related to the dynamics of ethylene motion in the BEDT-TTF molecules. Reprinted with permission from ref 544. Copyright 2000 American Physical Society.

all are known to show superconductivity. Thermodynamic studies based on heat capacity measurements are performed for β-(BEDT-TTF)2I3 and β-(BEDT-TTF)2AuI2. The electronic state of former salt shows complicated characters coupled with disorders in ethylene group as in κ-type salts. When the sample is cooled down in ambient-pressure, disordered freezing of ethylene group of BEDT-TTF molecules leads to an incommensurate structure concerning molecular packing around T = 175 K. The low temperature phase obtained by the ambient-pressure cooling is called as βL-phase and exhibits superconductive phase transition at Tc = 1.5 K. If this sample is annealed at 100 K for several hours, the long-range superlattice structure of ethylene groups disappears and changes to an ordered phase called as βH-phase. This βH-phase shows different transport and magnetic properties from those of βLphase and its Tc is enhanced up to about 8 K. The βH-phase can be obtained also by cooling under pressures higher than 0.04 GPa. The first thermodynamic measurement for β-(BEDTTTF)2I3 was performed between 0.7 and 18 K by Stewart et al.549 in 1986. They reported that electronic heat capacity coefficient γ is (24 ± 3) mJ K−2 mol−1 and the Debye temperature is θD = (197 ± 5) K, but they could not detect thermal anomaly. Fortune et al.550,551 studied thermodynamic properties of this salt systematically by a.c. calorimetry and found a distinct second-order phase transition at 175 K, which is related to the formation of long-range superlattice structure in ethylene conformation. They also found a step-like anomaly at 22.25 K. Using the absolute value of Cp given by Stewart et al.,549 they evaluated that ΔCp corresponds to about 4% of Cp and entropy release by this transition is ΔS = 0.2 J K−1 mol−1. This entropy is nearly 40% of electronic entropy evaluated from γ and transition temperature of 22.25 K. The entropy release at this temperature means that part of the Fermi surface BM


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occurs around 8.3 K. Akiba et al.564,565 examined heat capacity of λ-(BETS)2FeCl4 precisely and indicated that the low temperature hump can be fitted by a Schottky splitting of Fe3+ produced by an effective internal field of 4 T. This field corresponds to the Hπ−d field determined by the ordered π electrons in the BETS layers, which means that the effective field on the 3d electron in the FeCl4− layer just originates in the π−d interactions. They also suggested that the 3d spins in the FeCl4− anions are not ordered and behave as paramagnetic as shown schematically in Figure 102. The charge transfer salts consisting of BEDT-TTF molecules and magnetic anions form π−d systems. The well-known salt is (BEDT-TTF)6Cu2Br6, which was studied by Suzuki et al.567 Two crystallographically independent donor molecules exist in the donor sheet of which valence is considered as +3/4 and +0. The planar Cu(II)Br42− and linear Cu(I)Br2− anions are located randomly in the anion layer with the ratio of 1:1. Therefore the chemical formula is considered as (BEDT-TTF+3/4)4(BEDTTTF0)2CuIIBr42−CuIBr2−. Suzuki et al.567 observed a sharp thermal anomaly with first-order character at 59 K. The structural phase transition originated from the Jahn−Teller distortion of CuBr42− anions is suggested. An AF transition due to the d-electrons in the Cu(II) was found at 7.5 K by magnetization measurements, but the thermodynamic study to discuss π−d interaction is not performed at present. A charge transfer complex consisting of BEDT-TTF and dioxalatocuprate with a chemical formula of (BEDT-TTF)4Cu(C2O4) was studied by Wang et al.568 by a.c. heat capacity. They observed a sharp peak at 262 K and broad peak around 170 K, both attributed to the structural changes. The electronic states of charge transfer salts of acceptor molecules of R1,R2-DCNQI (2,5-R1,R2-dicyanoquinodiimine) (Figure 103) and metal cations give an interesting characters

Figure 101. Temperature dependence of heat capacity of κ(BETS)2FeBr4 around the Néel temperature obtained under 0 T, 1 T, and 2 T. Reprinted with permission from ref 559. Copyright 2005 The Chemical Society of Japan.

at 0.45 K, which is also consistent with the resistivity measurement. In this case, the entropy gain at 1.5 TN was 81% of R ln 6, suggesting the presence of low-dimensional character. Attempts to detect thermal anomalies around the superconductive transition temperatures and quasiparticle excitations at low temperatures have been unsuccessful because the heat capacity of π-electrons was masked by the huge magnetic contribution. The λ-type salts of (BETS)2FeCl4 are recently unveiled to show curious thermodynamic behaviors. This compound shows metal−insulator transition around 8.3 K as we mentioned. According to the magnetic properties measurements by Tokumoto et al.,562 AF ordering Fe3+ takes place simultaneously with this transition. The paramagnetic metal state above this temperature shows curious dielectric properties by microwave conductivity measurements and the low temperature AF states changes to field induced superconductive phase above 17 T. The dimer−Mott physics in the π-electrons in a conducting plane is considered as affecting the spin ordering in Fe3+ of d-electrons. Heat capacity measurement was performed by Negishi et al.,563 and they reported that λ-type anomaly

Figure 103. Molecular structures and abbreviation of various kind of R1, R2-DCNQI.

Figure 102. Temperature dependence of heat capacity of λ-(BETS)2FeCl4. The low temperature shoulder of heat capacity demonstrates a Schottky type contribution of Fe3+ in the anion layers. Reprinted with permission from ref 564. Copyright 2009 The Physical Society of Japan. BN


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electron mass enhancement in the re-entrant region of group III. The realization of Kondo effect by π−d hybridization is proposed for the first time in an organic system. On the other hand, (DMeO-DCNQI)2Cu, which has bulky group as R1,R2 and is located in good metallic region in group I, shows normal γ value of 10 mJ K−2 mol−1 as an organic conducting system. Kagoshima et al.572 performed a systematic work for the (DMe-DCNQI)2Cu system in the reentrant region to confirm whether the heat capacity enhancement is realized especially in the low temperature metallic region in group III compounds. To avoid disorder effects due to the mixed crystal of different R1,R2 molecules like that of DMe-DCNQI and MeBr-DCNQI, they used a mixed crystal sample consisting of solid solution of pristine h8(DMe-DCNQI)2Cu and deuterated d8(DMeDCNQI)2Cu molecules. The 29% deuterated compound [(h 8 :DMeDCNQI) 0.71 (d 8 :DMeDCNQI) 0.29 ] 2 Cu shows metal−insulator−metal transitions. The heat capacity of this mixed compound, however, gives a small γ term of 22 mJ K−2 mol−1, which is comparable with organic conducting systems. They also reported similar behaviors in 40% substituted sample,573 and the γ values were evaluated as (20−30) mJ K−2 mol−1 from the estimation of the extremely low temperature data below 0.4 K. In this system, Kato et al.569 made it possible to study physical properties in the phase diagram systematically by preparing high-quality single crystals using partial deuterated or 13 C substituted molecules so as to tune material in the phase diagram by chemical pressure precisely. The isotope-substituted molecules can make high-quality single crystals free from disorders inevitably produced by a solid solution of molecules with slightly different size. The sensitive electronic structure against pressure in (DMe-DCNQI)2Cu can be studied thoroughly using these crystals. The heat capacity measurements of h8:DMeDCNQI (denoted as d0[0,0,:0] according to the notification in Figure 105) sample (in group I) and

originating from coupling of magnetic and conductive properties. When R1 and R2 are methyl groups, it is called DMeDCNQI (dimethyl-dicyanoquinodiimine). The compound (DMe-DCNQI)2Cu has been studied by various methods to investigate an π−d hybridization of conducting π-electrons and 3d electrons in Cu2+ which also possesses strong electron correlations. According to the phase diagram by Kato et al.,569,570 the electronic state is sorted into three different regions as is shown in Figure 104. The salts in the group I

Figure 104. Peculiar phase diagram of R1, R2-DCNQI system. The group I region is metallic, while the group II is AF insulator. The group III region shows reentrant metallic behavior at low temperatures. Reprinted with permission from ref 569. Copyright 2000 The Chemical Society of Japan.

region have a metallic character, and those in the group III region show curious reentrant metal−insulator−metal transition. The group II region is an insulating region, and the salts belonging to this region possess AF ground states. Because of the existence of charge transfer between organic molecules and metal, the average valence of Cu is known as +4/3 and that of organic molecule is −2/3. In the insulating region, Cu+ and Cu2+ are arranged in an ordered structure like Cu+Cu+Cu2+ in the cation chain assisted by the cooperative Jahn−Teller distortion of CN coordination. DCNQI molecules undergo a Peierls transition to make an insulating state simultaneously with this ordering in the Cu site. The reentrant metallic region in group III was speculated to have strong hybridization effect of 2π-electrons with the localized 3d-electrons to make anomalous metallic states. The thermodynamic researches on DCNQI system started with intention to clarify electronic state of this point. In 1991, Nishio et al.571 performed heat capacity measurements of (DMe-DCNQI)2Cu, (DMeO-DCNQI)2Cu, and (DMe1−xMeBrx-DCNQI)2Cu system. They suggested that electronic heat capacity coefficient of (DMe-DCNQI)2Cu may exceed 40 mJ K−2 mol−1 and enhanced up to 64 mJ K−2 mol−1 in (DMe0.92MeBr0.08)2Cu. The latter salt was a solid solution system of DMe-DCNQI and MeBr-DCNQI molecules. They reported that π−d hybridization may induce an

Figure 105. The molecular structure of isotope labeled DMe-DCNQI molecules.

d2:DMeDCNQI samples (denoted as d2[2,0,:0] and d2[1,1,:0]) and (in group III) were performed. Both samples belonging to group I and group III show similar γ values: γ = (25 ± 3) mJ K−2 mol−1 for d0[0,0,:0], (22 ± 3) mJ K−2 mol−1 for d2[2,0,:0], and 25 ± 3 mJ K−2 mol−1 for d2[1,1,:0], indicating that the enhancement is not so large as compared with BEDT-TTF systems. In the d8 sample located in the insulating region (group II), the γ value is nearly 0 mJ K−2 mol−1. The electronic state in the reentrant region has a similar character as a high temperature metallic region. The appearance of the reentrant region was explained using the Gibbs energy including the electronic state and lattice state by Nishio et al.574,575 The systematic heat capacity and thermal analysis works were performed for well characterized samples. BO


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Nishio et al.576 also detected a magnetic transitions by heat capacity in d8[3,3;2] sample, which shows AF transition around 6.8 K (Figure 106). The sample of d3[3,0:0], which is also

of quantum behavior in a bulk feature and studied mainly on the theoretical side. To unveil magnetic behaviors in this spinliquid has been an important subject of thermodynamic investigations. This is because the thermodynamic measurements have prominent points for proving novel excitations produced by many-body effects at the low energy region. The well-known theory is the long-range resonating valence bond predicted by Anderson580 in 1973 as a possible ground state of S = 1/2 triangular lattices. Recently, there has been great progress in material developments and experimental techniques. Fascinating candidate compounds, which give spin-liquid ground states, have been synthesized. Eight compounds with triangular and Kagomé lattices with 2D structure; 3D Hyper Kagomé lattices are known as suitable candidates to study spinliquid science as was reviewed by Balents581 in 2010. Two organic triangular salts of κ-(BEDT-TTF)2Cu2(CN)3 and EtMe3Sb[Pd(dmit)2]2 are included in them. As mentioned in section 11.2, organic charge transfer compounds have a low-dimensional arrangement of donor or acceptor molecules. The κ-type salts of (BEDT-TTF)2X are known as 10 K class organic superconductors, and interesting physics related to electron correlations is widely discussed. The superconductive salts such as κ-(BEDT-TTF)2Cu(NCS)2 and κ-(BEDT-TTF)2Cu[N(CN)2]Br have square lattices due to the smaller transverse transfer integral t′ than that of nearest neighbor t. If t′ increases and becomes comparable with t, the magnetic fluctuations due to frustration become dominant. In the case of κ-(BEDT-TTF)2Cu2(CN)3, the t′/t value is 1.06 and a nearly ideal triangular lattice is established. The anion radical salt of EtMe3Sb[Pd(dmit)2]2 also has a similar 2D triangular lattice of Pd(dmit)2 dimers. In this case, the ratio of transfer integral expressed using (tr =) t′/t (= ts ∼ tB) is 0.92. By studying low temperature heat capacity of these salts, we can get unique insights into frustration physics. Both compounds are molecular-dimer based Mott insulators with a two-dimensional structure. The electronic properties are dominated by strong magnetic interactions |J|/kB larger than 200 K . In these salts, the spin-liquid nature has been unveiled by 13C NMR,582−584 heat capacity,585,586 and thermal transport587,588 measurements. A possible relation between the superconductivity of conductive π-electrons and coupling with charge degrees of freedom in the dimer unit is stimulating extensive interests in frustration problems and electron correlations. Measurements of heat capacity of the two salts are reported by Yamashita et al. They reported that both salts do not show any evidence of long-range ordering down to about 75 mK for κ-(BEDT-TTF)2Cu2(CN)3585 and 100 mK for EtMe3Sb[Pd(dmit)2]2.586 The data are shown in Figures 107 and 108. The heat capacity divided by temperature obeys a linear relation with T2 at a low temperature region below 2.5 K, except for the low temperature upturn originated from nuclear Schottky or other Schottky contributions due to methyl rotation. The β values are evaluated as 21 mJ K−4 mol−1 for κ-(BEDTTTF)2Cu2(CN)3 and 24.1 mJ K−4 mol−1 for EtMe3Sb[Pd(dmit)2]2, which are larger than usual organic conducting and superconducting systems. These plots show distinct finite γ values in spite of the insulating ground state with S = 1/2 electrons/holes being localized on each dimer. The γ values are evaluated as 12.6 mJ K−2 mol−1 for κ-(BEDT-TTF)2Cu2(CN)3 and 19.9 mJ K−2 mol−1 for EtMe3Sb[Pd(dmit)2]2. In both figures, the Mott insulating salts with AF ordered ground state (κ-(BEDT-TTF)2Cu[N(CN)2]Cl, κ-(d8;BEDT-TTF)2Cu[N-

Figure 106. Heat capacity around the AF transition temperature of the group II region of isotope labeled (DMe-DCNQI)2Cu system. Reprinted with permission from ref 574. Copyright 2000 The Physical Society of Japan.

located in the AF region, shows exactly the same peak at 6.8 K, demonstrating that the magnetic transition is a bulk transition originating from localized Cu2+ ions. The transition entropy is 1.8 J K−1 mol−1, which is 1/3 of the spin entropy of Cu2+ ion (R ln 2). Although the metallic state of (DMe-DCNQI)2Cu does not show large mass enhancement expected from π−d hybridization, this behavior is realized in (DI-DCNQI)2Cu. In this salt, the bandwidth is suppressed in comparison to that of (DMe-DCNQI)2Cu owing to the decrease of intrachain transfer energy of t||. The interchain interaction increases to make the band structure more 3D-like, and consequently the metallic state is stabilized down to low temperatures. The enhanced Pauli susceptibility and increase of γ term up to about 50 mJ K−2 mol−1 in heat capacity was reported by Tamura et al.577 Nakazawa et al.578 and Okuma et al.579 also confirmed that this salt shows a large γ term of 41.7 mJ K−2 mol−1 after subtracting the Schottky contribution due to iodine atoms in the molecule. This value is twice as large as that of (DMeDCNQI)2Cu and nearly four times larger than that of the (DMeO-DCNQI) 2Cu system. The existence of narrow hybridized band with 3D character suggested by Kanoda et al. by site selective NMR experiments is consistent with this thermodynamic feature. Okuma et al. further reported in the same paper that 10% of Ag doping in Cu site enhances the γ value up to 62.6 mJ K−2 mol−1 without changing the β term. It is speculated that the electron correlation which produces CO ground state in pure (DI-DCNQI)2Ag may be responsible for enhancing the electronic heat capacity coefficient in the 10% doped system. 11.4. Organic Spin Liquid Systems

The spin-liquid state appears when the frustrated spin systems lose their energies by keeping strong magnetic interactions at low temperatures. It is considered as an unusual manifestation BP


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worse quality samples, existence of Cu2+ ions or localized spins produced by disorders behaves as paramagnetic impurities which vary with magnetic fields. The finite γ and β of both salts are not affected by rather strong magnetic fields up to 8 T in good quality samples. The question whether the gapless liquidlike state is stable as the ground-state- like Fermi liquid systems is still an open question. Another important subject observed by thermodynamic measurements for organic spin-liquid systems is an existence of a broad hump structures. In κ-(BEDT-TTF)2Cu2(CN)3, the hump structure was observed around 6 K as an extra contribution to the natural trend of the lattice heat capacity (Figure 109).585 At this temperature, Mana et al. observed a

Figure 107. Heat capacity of the spin liquid system of κ-(BEDTTTF)2Cu2(CN)3, which shows temperature linear term down to low temperature. Reprinted with permission from ref 585. Copyright 2008 Nature Publication Group.

Figure 109. The heat capacity of κ-(BEDT-TTF)2Cu2(CN)3 system around the hump temperature of 5. 7K. Reprinted with permission from ref 585. Copyright 2008 Nature Publication Group.

distinct anomaly in thermal expansion measurements as well as heat capacity measurements.590 Similar broad hump was observed around 3.7 K in the temperature dependence of the heat capacity of EtMe3Sb[Pd(dmit)2]2.586 The peak temperature of the hump structure corresponds to the dip-like structure of temperature dependence of T1−1 in refs and 583, and they are probably related to the change of spin degrees of freedom. From this fact, Yamashita et al. claimed that these anomalies mean a kind of condensation of the spin-liquid from the Heisenberg state where the nearest neighbor interaction is dominant.585,586 The higher-order exchange interaction is thermally disturbed at high temperatures. However, in the low-temperature region under discussion, the system gradually obtains a quantum character which cannot be explained by the Heisenberg model with the nearest neighbor interaction alone. Such a crossover would not have a long-range nature like typical phase transitions. The entropy related to the hump structure is evaluated as 700−1000 mJ K−1 mol−1 for κ-(BEDTTTF)2Cu2(CN)3 and that of EtMe3Sb[Pd(dmit)2]2 is 70−100 mJ K−1 mol−1 in refs and 586. The larger entropy in κ-(BEDTTTF)2Cu2(CN)3 may probably contain some kind of lattice entropy because the thermal expansion measurements observed a clear anomaly and dielectric fluctuations dominated by interdimer interactions exist above this temperature. The thermodynamic study using microchip devices for a tiny single

Figure 108. Temperature dependences of the heat capacity of the spin-liquid system of EtMe3Sb[Pd(dmit)2]2 under magnetic field up to 10 T, which show appearance of gapless spin liquid state. Reprinted with permission from ref 586. Copyright 2011 Nature Publication Group.

(CN)2]Br, β′-(BEDT-TTF)2ICl2, and EtMe3As[Pd(dmit)2]2) of charge ordered ground state EtMe3P[Pd(dmit)2]2 are compared, and the existence of γ was found to be specific for these two salts which behave as spin liquids. The existence of a finite γ term in both compounds demonstrates that the thermal excitations have a gapless character dominated by a kind of continuous density of states in spin excitations. The magnitude of γ scales to the magnetic susceptibility extrapolated down to T = 0 with a Wilson ratio values are nearly 1 for κ-(BEDTTTF)2Cu2(CN)3 and EtMe3Sb[Pd(dmit)2]2. This fact demonstrates that the spin-liquid characters and origin of spin excitations are not so different between two salts. Because the spin-liquid state is delicate against disorders or impurities, Yamashita et al.589 reported a sample dependence of the low temperature heat capacity of κ-(BEDT-TTF)2Cu2(CN)3. They observed that the worse quality samples have slightly larger heat capacity and have extra magnetic field dependent terms explained by a paramagnetic impurities, while a good quality sample does not show any magnetic fields up to 8 T. In the BQ


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crystal of κ-(BEDT-TTF)2Cu2(CN)3 and its deuterated salt reproduced the existence of hump structures.591 The origin of the heat capacity hump seems to be essential for discussing the mechanism of the formation of organic spinliquids because this hump is a common feature of two systems consisting of different molecules. Theoretical researches are performed for example from the standpoint of instability of the Fermi surface of spinons.592 The spin-liquid research in organic systems contains various factors related to low-dimensional quantum characters and electron correlation problems. Furthermore relation with the superconductivity as is predicted by Anderson et al.593 is the next important subject to be solved. Heat capacity measurements play important roles in the research of quantum spin-liquid systems.

simultaneously concerned, unexpected ambiguity is involved when one partitions the observed physical quantity into individual contributions from different degrees of freedom. Serious endeavor to minimize such ambiguity is necessary for quantitative discussion. Combination of calorimetric probe with other experiments is required to pursue such complicated and delicate problems. To confirm the mechanisms of phase transition based on calorimetric measurements, comparison with quantum chemistry calculation and/or theoretical models is highly desirable. It should be remarked that a theory, which fails to explain the observed thermodynamic quantities, is regarded as being incomplete because thermodynamic quantities basically reflect every degree of freedom. In this sense, the degree of coincidence with thermodynamic data can serve as a litmus test for goodness of a theoretical model. Development of heat capacity measurements under extreme conditions such as low temperatures below 100 mK, strong magnetic fields above 20 T, high pressure, and light emitted conditions, etc., can further cultivate novel aspects of moleculebased magnets in the future. It is also plausible to develop advanced techniques such as microchip calorimetry for tiny single crystals and thermal analyses in partial area of a sample using fabricated nanoprobe. These techniques can also work to excavate novel thermodynamic phenomena and characters of phase transitions in molecule-based magnets. Complementary roles played by macroscopic thermodynamic studies and microscopic spectroscopic studies are crucially important for deep understanding of materials and phenomena. In that sense, molecular thermodynamics related to statistical mechanics and quantum chemistry will continue to provide us with important information concerning energetic and entropic aspects.

12. CONCLUDING REMARKS Important and unique roles played by heat-capacity calorimetry in the fields of molecule-based magnetism have been reviewed by focusing on phase transitions occurring not only under ambient pressure but also under applied pressure and magnetic fields. The subjects cover genuine organic free radicals, hybrid compounds between organic and inorganic moieties, assembled-metal complexes, single-molecule magnets, spin crossover complexes, mixed-valence complexes, thermochromic complexes, spin-Peierls compounds, and organic conductors. The molecule-based magnets exhibit low dimensionality concerning the magnetic lattices formed by magnetic interactions owing to structural anisotropy inherent in the constituent molecule moieties. This aspect is clearly reflected in the heat-capacity calorimetry. By revealing the mechanisms of magnetic interactions inherent in such low-dimensional systems, one can gain fundamental insights into fascinating molecule-based magnets. The most precise and accurate heat-capacity calorimetry has so far been accomplished by traditional adiabatic calorimetry. This will also be the case in the future. However, because this calorimetry requires special skills of researchers to construct calorimeters and to operate them, the number of papers, in which the adiabatic calorimetry is used, has been not so many. Thanks to recent development of relaxation calorimetry, in particular commercially available apparatuses, calorimetric studies based on this calorimetry have become popular. It is a great merit that the amount of sample required for this method is only ∼0.5 to ∼10 mg. Easy measurement under an applied magnetic field is also a big merit for the studies of moleculebased magnets. Research in the field of molecule-based magnets were mainly focused on genuine magnetic systems at the dawn, and then the research interest has extended to composite magnetic systems, in which multiple degrees of freedom are cooperatively coupled with spin. Spin crossover and spin-Peierls transitions, for instance, belong to such a category. Magnetic features coupled with spin and lattice dimensionalities, softness of lattice, and electron correlations, etc., have clearly been unveiled from entropic discussion by calorimetric experiments up to now. Recent examples of more complicated systems are magnetoelectronics known as “spintronics” and “multiferroics,” defined as materials that exhibit more than one ferroic order parameter simultaneously. Because nonexistence of the selection rule to excitation is one of the most characteristic aspects inherent in equilibrium thermodynamics, those systems are likewise suited to calorimetric investigations. However, in the case of complicated systems in which many degrees of freedom are

AUTHOR INFORMATION Corresponding Author

*Phone: +81 72 753 0922. Fax: +81 72 753 0922. E-mail: [email protected]. Notes

The authors declare no competing financial interest. Biographies

Michio Sorai started chemical thermodynamic studies under the supervision of Professor Syûzô Seki at Osaka University. After receiving his Ph.D. degree in 1968, he was appointed to Research Associate at Osaka University, Associate Professor in 1981, and then full Professor of physical chemistry in 1987. As a postdoctoral fellow, he joined the research group of Professor Philipp Gütlich at BR


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Technische Hochschule Darmstadt (1974−1976) to study microscopic mechanisms inherent in spin crossover phenomena by use of 57 Fe Mö ssbauer spectroscopy. He served as Director of the Microcalorimetry Research Center (1993−1999) and the Research Center for Molecular Thermodynamics (1999−2003). He retired from Osaka University in 2003. His research has focused on molecular thermodynamics of phase transitions occurring in functional materials. He was a committee person of the IUPAC I.2 Commission on Thermodynamics (1998−2001). He served as President of the Japan Society of Calorimetry and Thermal Analysis (1999−2001). He is the Editor-in-Chief of the book “Comprehensive Handbook of Calorimetry and Thermal Analysis” (John Wiley & Sons, 2004) and the author of the book “Molecular Thermodynamics for Phase Transitions” (Asakura Publishing Co., 2007). He received the Hugh M. Huffman Memorial Award at the Calorimetry Conference in 2001. The distinguished title “Honorary Professor” was awarded to him in 2006 at Henryk Niewodniczański Institute of Nuclear Physics in Kraków, Polish Academy of Sciences.

Motohiro Nakano was born in 1963 in Matsumoto, Japan. He received his B.Sc. in Chemistry from Osaka University in 1985 and a Ph.D. in Inorganic and Physical Chemistry from Osaka University in 1990, working on heat-capacity calorimetry of paramagnetic compounds at very low temperatures under the supervision by Professor Michio Sorai. Then he moved to Professor Gen-etsu Matsubayashi’s group at Osaka University, where chemical and physical properties of sulfur-rich molecular conductors and transition metal complexes were investigated. In 1998−1999, he worked with Professor David N. Hendrickson at University of California at San Diego on the magnetism of single-molecule magnets. He was promoted to Assistant Professor in 2003 and then Associate Professor since 2007 at the Graduate School of Engineering, Osaka University. His research interests are mainly in energy level structures and dynamics of the transition metal complexes showing molecular bistability, including magnetization reversal processes of single-molecule magnets, dielectric responses of mixed-valence complexes, and field-induced spin crossover phenomena.

Yasuhiro Nakazawa was born in 1962 in Tokyo, Japan. He received his Ph.D. from the University of Tokyo in 1991, supervised by Professor Masayasu Ishikawa. After working as a JSPS Post-Doctor in the same laboratory, he moved to Professor Kazushi Kanoda’s group at the Institute for Molecular Science, Okazaki, Japan, as a Research Associate and started thermodynamic research on organic conducting compounds. In 2000, he joined with Professor Michio Sorai’s group in the Research Center for Molecular Thermodynamics, Osaka University, as a Research Associate. Then he moved as an Associate Professor of Department of Chemistry, Graduate School of Science and Engineering, at the Tokyo Institute of Technology. In 2005, he was appointed to a Professor of Department Chemistry, Osaka University. He is interested in thermodynamic research in organic conductors, organic superconductors, organic spin-liquid systems, and metal complexes by the thermal relaxation technique.

Yuji Miyazaki was born in Nagasaki in Japan in 1964. After graduating from Osaka University in 1987, he received his M.Sc. and his Ph.D. in Inorganic and Physical Chemistry from Osaka University in 1989 and 1993, respectively. Under the supervision of Professors Hiroshi Suga and Takasuke Matsuo, his thesis work dealt with the development of an adiabatic heat-capacity microcalorimeter and the LT thermal behavior of some crystalline proteins. He then joined the group of Professor Michio Sorai at the Microcalorimetry Research Center (now the Research Center for Structural Thermodynamics) at the same university as a Research Associate and now Associate Professor. His current research interests are mainly dynamics of biomolecules and synthetic macromolecules, magnetic properties of molecule-based magnets, and thermal properties of organic charge-transfer complexes.

ACKNOWLEDGMENTS M.S. expresses his sincere thanks to Prof. Emer. Syûzô Seki (Osaka University), who was the supervisor when he started chemical thermodynamic study, and Prof. Emer. Hiizu Iwamura (The University of Tokyo) for recommending him as a contributor to this journal. Although this is a review article covering papers of various research groups, many papers of the present authors and their co-workers are involved. On this occasion, we express our sincere thanks to all of the colleagues, students, and collaborators over the world. It is a great pleasure to acknowledge Prof. Emer. Philipp Gütlich (University of Mainz), Prof. David N. Hendrickson (University of California at San Diego), the late Prof. Olivier Kahn (ICMC Bordeaux), and Prof. Emer. Wolfgang Haase (Technical University of Darmstadt) with whom the Cooperative Research Projects were concerned. BS


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