This content has been downloaded from IOPscience. Please scroll down to see the full text. Download details: IP Address: 186.217.236.152 This content was downloaded on 14/10/2015 at 20:52 Please note that terms and conditions apply. Probing the Mott physics in -(BEDT-TTF)2X salts via thermal expansion View the table of contents for this issue, or go to the journal homepage for more 2015 J. Phys.: Condens. Matter 27 053203 (http://iopscience.iop.org/0953-8984/27/5/053203) Home Search Collections Journals About Contact us My IOPscience iopscience.iop.org/page/terms http://iopscience.iop.org/0953-8984/27/5 http://iopscience.iop.org/0953-8984 http://iopscience.iop.org/ http://iopscience.iop.org/search http://iopscience.iop.org/collections http://iopscience.iop.org/journals http://iopscience.iop.org/page/aboutioppublishing http://iopscience.iop.org/contact http://iopscience.iop.org/myiopscience Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 27 (2015) 053203 (27pp) doi:10.1088/0953-8984/27/5/053203 Probing the Mott physics in κ-(BEDT-TTF)2X salts via thermal expansion Mariano de Souza1,2,4 and Lorenz Bartosch3 1 Departamento de Fı́sica, Instituto de Geociências e Ciências Exatas—IGCE, Unesp—Universidade Estadual Paulista, Cx. Postal 178, 13506-900 Rio Claro (SP), Brazil 2 Physikalisches Institut, Goethe-Universität, Max-von-Laue-Str. 1, 60438 Frankfurt am Main, Germany 3 Institut für Theoretische Physik, Goethe-Universität, Max-von-Laue-Str. 1, 60438 Frankfurt am Main, Germany E-mail: mariano@rc.unesp.br and lb@itp.uni-frankfurt.de Received 2 October 2014, revised 25 November 2014 Accepted for publication 5 December 2014 Published 21 January 2015 Abstract In the field of interacting electron systems the Mott metal-to-insulator (MI) transition represents one of the pivotal issues. The role played by lattice degrees of freedom for the Mott MI transition and the Mott criticality in a variety of materials are current topics under debate. In this context, molecular conductors of the κ-(BEDT-TTF)2X type constitute a class of materials for unraveling several aspects of the Mott physics. In this review, we present a synopsis of literature results with focus on recent expansivity measurements probing the Mott MI transition in this class of materials. Progress in the description of the Mott critical behavior is also addressed. Keywords: Mott insulators, Mott criticality, lattice effects (Some figures may appear in colour only in the online journal) 1. Introduction A tiny change in physical parameters can sometimes have dramatic consequences. This is particularly true in correlated condensed matter systems in which a small alteration in pressure, chemical composition, or temperature can tune a system across a phase transition. In this context, the Mott metal-to-insulator (MI) transition represents one of the most remarkable examples of electronic correlation phenomena. In contrast to conventional band insulators in which all bands are either filled or empty, the Mott insulating state is driven by interactions and emerges once the ratio of the electron– electron interaction U to the hopping matrix element (kinetic energy) t exceeds a critical value. It is essentially this ratio which is tuned in many experiments. Molecular conductors of the κ-phase of (BEDT-TTF)2X (where BEDT-TTF refers to bisethylenedithio-tetrathiafulvalene, i.e. C10S8H8 and X to a monovalent counter anion), usually called charge-transfer salts, have been recognized as model systems to investigate 4 Current address: Institute of Semiconductor and Solid State Physics, Johannes Kepler University Linz, 4040 Linz, Austria. electronic correlations in two dimensions. In particular, single crystals of wholly deuterated charge-transfer salts of the κ phase with X = Cu[N(CN)2]Br have been attracting broad interest due to fact that they lie in the vicinity of a first- order phase transition line in the phase diagram [1], which separates the metallic from the insulating state, thus giving the possibility of investigating the Mott MI transition using temperature as a tuning parameter. One result to be discussed in detail in the frame of this paper refers to the first experimental observation of the role played by lattice degrees of freedom for the Mott MI transition in the above-mentioned materials [2]. Both the discontinuity and the anisotropy of the lattice parameters observed via high-resolution thermal expansion experiments indicate a complex role of the lattice effects at the Mott MI transition for the κ-(BEDT-TTF)2X charge- transfer salts, which cannot be interpreted simply by taking into account a purely 2D electronic model. Furthermore, in the frame of the present work, a model based on the rigid-unit modes scenario is proposed to describe the negative thermal expansion above the so-called glass-like transition temperature Tg � 77 K in deuterated charge-transfer salts of κ-(BEDT- TTF)2Cu[N(CN)2]Br and parent compounds. 0953-8984/15/053203+27$33.00 1 © 2015 IOP Publishing Ltd Printed in the UK http://dx.doi.org/10.1088/0953-8984/27/5/053203 mailto: mariano@rc.unesp.br mailto: lb@itp.uni-frankfurt.de http://crossmark.crossref.org/dialog/?doi=10.1088/0953-8984/27/5/053203&domain=pdf&date_stamp=2015-01-21 J. Phys.: Condens. Matter 27 (2015) 053203 M de Souza and L Bartosch From a theoretical point of view, perturbative approaches and their generalizations are restricted to the regimes U � t and t � U and have difficulties in addressing phase transitions. In fact, they entirely fail in addressing the critical behavior in the immediate vicinity of a putative critical point. Instead, it is advantageous to start from the critical point itself and use concepts of scaling to describe its vicinity. We will review here how some aspects of the lattice response observed in [2] and [3] can be understood in terms of such a scaling theory [4, 5]. The compound κ-(BEDT-TTF)2Cu2(CN)3 is another material which attracted great interest in the last few years, see, for instance [6] and references cited therein. This system has an almost perfect quasi-2D frustrated triangular lattice with a ratio of hopping integrals close to t ′/t � 1. Interestingly enough, while the ratio U /t shows a monotonic temperature dependence, the degree of frustration t ′/t as a function of temperature presents a non-monotonic behavior and becomes more pronounced at low temperatures [7]. Up to now, no clear experimental evidence of magnetic ordering in κ-(BEDT-TTF)2Cu2(CN)3 has been reported in the literature [8]. Thus, this system has been recognized as a candidate for the realization of a spin-liquid [9]. At T = 6 K, a crossover from a paramagnetic Mott insulator to a genuine quantum spin-liquid has been proposed [6]. Nevertheless, the physical nature of such a crossover, sometimes called hidden ordering, is still controversial. Expansivity measurements performed on this material revealed a pronounced phase- transition-like-anomaly at T = 6 K, which coincides nicely with specific heat results reported in the literature [6]. These results provide strong evidence that this lattice instability at T = 6 K is directly linked to the proposed spin-liquid phase. The electronic properties of κ-(BEDT-TTF)2X charge- transfer salts have been reported in various review articles, see, for instance [10–24]. Here we focus on topics linked to dilatometric studies of the κ-(BEDT-TTF)2X salts. Some parts of this review are based on the Ph.D. thesis of one of us [3]. Before giving a detailed outline of the main body of this review article at the end of this introductory chapter, let us first briefly review some general properties of the κ-(BEDT- TTF)2X salts and discuss their crystallographic structure in section 1.1, their phase diagram in section 1.2 and the most relevant experimental results from the literature related to this work in section 1.3. 1.1. The (BEDT-TTF) molecule and crystallographic structure of the κ-ET salts The basic entity which furnishes the structure of the present class of organic conductors is the (BEDT-TTF) molecule, usually abbreviated simply to ET. The structure of the ET molecule is shown in figure 1. As a result of the two distinct possible configurations of the ethylene end groups in their extremes (known as staggered and eclipsed configurations), the ET molecule is not completely planar. It has been proposed that the degrees of freedom of the ethylene end groups can influence the electronic properties of the salts of the κ-family, see [25] and references cited therein. Employing infra-red imaging spectroscopy, the authors of [26] reported on the Figure 1. (BEDT-TTF), or simply ET, molecule, which is the basic entity that furnishes the quasi-2D (BEDT-TTF)-based charge-transfer salts. Figure reproduced with permission from [15]. sensitivity of the vibration mode of the central C=C bonds, the so-called ag(ν3) mode, upon going from the metallic to the insulating state. In the following, let us briefly discuss the various phases in which ET charge-transfer salts can crystallize. During the crystallization process of the (ET)2X salts, the ET molecules can assume different spatial arrangements, giving rise to different phases. Several thermodynamically stable phases, usually referred to by the Greek letters α, β, θ , λ and κ , are known, see figure 2 for a schematic representation of the main known phases. This review is focussed on salts of the κ-phase family. Interestingly enough, different crystalline arrangements imply different band fillings and consequently different physical properties. A detailed discussion about each phase is presented in [21, 22]. The packing pattern of the κ-phase differs distinctly from the others in the sense that it consists of two face-to-face ET molecules, see figure 2. The two molecules interact with each other via overlap between the highest occupied molecular orbitals (HOMOS), as the HOMOS of each molecule are partially occupied [21]. Due to the intrinsic strong dimerization in the κ-(ET)2X family, pairs of ET molecules can be treated as a dimer unity. As a result of such an assumption, an anisotropic triangular lattice is built by such dimer units as shown in figure 3 and the system can be described by the Hubbard model. The counter anion X governs the ratio between the inter-dimer transfer integrals t and t ′. For example, for the κ-(ET)2Cu2(CN)3 salt t ′/t = 0.83 [28, 29], suggesting a strongly frustrated triangular lattice in this compound, to be discussed in more detail below. It is well to note that, more recently, a new phase, called mixed phase, with alternating α- and κ-type arrangements of the ET molecules, has been synthesized [30]. In addition, a dual-layered superconductor with alternating α′- and κ-type packing has been reported in the literature [31]. The crystallographic structure of the κ-(ET)2Cu[N(CN)2] Br salt [32] is displayed in figure 4. The structure of the latter substance is orthorhombic with the space group Pnma with 2 J. Phys.: Condens. Matter 27 (2015) 053203 M de Souza and L Bartosch Figure 2. Schematic top view, i.e. view along the ET molecule long axis of the α-, β-, κ- and θ -phase of the ET-based organic conductors. The ET molecules and the unit cells are, respectively, illustrated by solid blue and open red rectangles. The dashed lines indicate the π -orbital overlaps. Figure reproduced with permission from [27]. Figure 3. Crystal structure of the κ-(ET)2X salts (top view). The largest hopping terms between the dimers are indicated by t and t ′. Note that the hopping terms t , t ′ and t form a triangular lattice. It should be noted that the axes labeling refers to the monoclinic lattices, e.g. the salts with X = Cu(NCS)2 or Cu2(CN)3. Figure taken from [21]. four dimers and four anions in a unit cell. An interesting aspect is that although the compounds κ-(ET)2Cu(NCS)2 and κ-(ET)2Cu2(CN)3 [33] also belong to the κ-phase family, their crystallographic structure is monoclinic. Since the counter anion X− = Cu[N(CN)2]Br− assumes a closed-shell configuration, metallic properties are observed only within the ET layers, which lie in the crystallographic ac-plane and correspond to the large face of the crystal. In the perpendicular direction to the referred layers, i.e. the b-direction5, the insulating anion layer partially blocks charge transfer, so that the electrical conductivity along this direction (⊥ to the layers) 5 For the κ-(ET)2Cu2(CN)3 salt, the a-axis is the out-of-plane direction. Figure 4. The crystallographic structure of κ-(ET)2Cu[N(CN)2]Br. While a and c are the in-plane axes, b is along the out-of-plane direction. ‘ET’ and ‘polymeric anion’ indicate layers of ET molecules and anions, respectively. The various atoms are represented by different colors, as indicated in the label. For clarity, protons at the end of the ET molecules are omitted. Figure taken from [3]. 3 J. Phys.: Condens. Matter 27 (2015) 053203 M de Souza and L Bartosch Figure 5. Layered structure formed by the Cu[N(CN)2]X− polymeric anion in κ-(ET)2Cu[N(CN)2]X, view along the b-axis. Here, X refers to either Br or Cl. Dashed lines highlight the X...N weak contacts between the anion chains. The rectangle indicates the contour of the unit cell. Figure reprinted from [34]. Copyright 1991, with permission from Elsevier. is reduced by a factor of 100–1000, depending on the counter anion. Due to this anisotropy, these materials are called quasi- two-dimensional (quasi-2D) conductors. As displayed in figure 5, the Cu[N(CN)2]Br− (or Cu[N(CN)2]Cl−) polymeric anion structure is composed of planar triply-coordinated Cu(I) atoms with two bridging dicyanamide [(NC)N(CN)]− ligands, forming a zig-zag pattern chain along the a-axis. The terminal Br− ions, which overlap slightly with the central N atoms of neighboring polymeric chains, make the third bond at each Cu(I) atom [10]. As shown in figure 6, the counter anion Cu2(CN)−3 differs distinctly from the Cu[N(CN)2]Br− anion in that it does not form chains, but a planar reticule of Cu(I) ions coordinated in a triangular fashion and bridging cyanide groups. Notably, one of the cyanide groups (labeled N/C11 in figure 6) is located on an inversion center and therefore must be crystallographically disordered [33]. As will be discussed in section 5, this particular arrangement is likely to be responsible for the absence of the so-called glass-like transition in κ-(ET)2Cu2(CN)3. Directional-dependent thermal expansion experiments, to be discussed in section 3, will show that the more pronounced lattice effects, associated with the Mott MI transition for κ-D8- Br, occur along the in-plane direction which is parallel to the anion chains direction (a-axis). In addition, we will propose a model based on the collective vibration modes of the CN groups of the polymeric anion chains to explain the negative thermal expansion along the a-axis above Tg � 77 K. For completeness, the lattice parameters of the κ-(ET)2X charge-transfer salts discussed in this work are listed in table 1. 1.2. Phase diagram of κ-(ET)2X Charge-transfer salts of the κ-(ET)2X family have been recognized as strongly correlated electron systems. Among other features, the latter statement is based on the unusual metallic behavior inherent to these systems, as will be Figure 6. View along the out-of-plane direction (a-axis) of the polymeric anion layer of Cu2(CN)−3 in κ-(ET)2Cu2(CN)3. The rectangle indicates the contour of the unit cell. The N/C11 cyanide ion is located on an inversion center, being thus crystallographically disordered. The ellipsoids therefore represent either a carbon or a nitrogen atom with a 50% probability. Figure reprinted from [33]. Copyright 1991 American Chemical Society. discussed below. Band structure calculations suggest that these materials should be metals down to low temperatures [21]. However, due to correlation effects, the phase diagram embodies various phases, including the paramagnetic Mott insulator (PI), the antiferromagnetic Mott insulator (AF), the superconductor and the anomalous metal, see the phase diagram depicted in figure 7. A remarkable feature is the competition between antiferromagnetic ordering and superconductivity. The similarities between the present charge-transfer salts and the high-Tc cuprate superconductors have been discussed, see e.g. [14, 36, 37]. The major difference, however, between these two classes of materials is that while for the cuprate superconductors one deals with a doping-controlled Mott transition, for the charge-transfer salts to be discussed here one has a bandwidth-controlled Mott transition. In addition, no spin-glass phase is observed in organic conductors between the antiferromagnetic and superconducting phases. The symmetry of the superconducting order parameter in these materials is still a topic of debate, see e.g. [38] and references therein. From the experimental point of view, a remarkable feature is the tunability of such systems. In contrast to the cuprate superconductors, where the ground states are tuned by doping, i.e. band filling and therefore disorder is unavoidable, the ground states of organic conductors can be tuned by chemical substitution, namely counter anion substitution and/or by applying external pressure. Pressure mainly induces strain which in turn leads to modified hopping matrix elements t and t ′ and thereby modifies the bandwidth W . As a consequence, applying pressure leads to a change of the ratio W/U . A pressure of a few hundred bar, which is easily attainable by employing the 4He-gas pressure technique, is enough to make a wide sweep over relevant regions of the pressure versus temperature (P -T ) phase diagram6. 6 For the Mott insulator V2O3, for example, a pressure of a few kbar is required to cross the first-order line in the P –T phase diagram [127]. 4 J. Phys.: Condens. Matter 27 (2015) 053203 M de Souza and L Bartosch Table 1. Lattice parameters at room temperature and structure of the investigated κ-(ET)2X salts [33, 34]. Anion a (Å) b(Å) c(Å) V (Å3) α β γ Struc. D8-Br — — — — ⊥ ⊥ ⊥ Orth. H8-Br 12.949 30.016 8.539 3317 ⊥ ⊥ ⊥ Orth. H8-Cl 12.977 29.977 8.480 3299 ⊥ ⊥ ⊥ Orth. CuCN 16.117 8.5858 13.397 1701.2 ⊥ 113.42◦ ⊥ Monoc. Note: CuCN refers to Cu2(CN)3. Data for κ-D8-Br are not available in the literature. ⊥ corresponds to 90◦. Note that in contrast to the other compounds, for the Cu2(CN)3 salt the a-direction is the out-of-plane direction. ) K( er utare p meT 100 10 1 X = Cu[N(CN)2]Cl Cu[N(CN)2]Br Cu(NCS)2 antiferro- magnetic insulator superconductor metal paramagnetic insulator Pressure (P0,T0) TMI (1st order) T* “crossover” W/U Figure 7. Conceptual pressure versus temperature (P -T ) phase diagram for the κ-(ET)2X charge-transfer salts, after [35]. Arrows pinpoint the positions of the various salts with their counter anion X at ambient pressure. The ratio of the on-site Coulomb repulsion relative to the bandwidth, i.e. U /W , is reduced by application of hydrostatic pressure. The phase diagram shown in figure 8 has been mapped by NMR and ac susceptibility [39], transport [41] and ultra- sonic [42] measurements under helium gas pressure on the salt with X = Cu[N(CN)2]Cl. In this generic phase diagram, the antiferromagnetic transition line, not observed in ultra-sound experiments [42], was obtained from the NMR relaxation rate [39]. From the splitting of NMR lines, the estimated magnetic moment per dimer is (0.4– 1.0) µB. The superconducting transition line was determined from the ac susceptibility [39] and its fluctuations in the immediate vicinity of the Mott transition by means of Nernst effect measurements [40], whereas the S-shaped first-order MI transition line via ultra-sound [42], transport [41] and ac susceptibility [39] measurements. This first-order Mott MI transition line ends in a critical point (P0, T0), which has been studied by several groups [39, 41, 42, 44, 45]. Among possible scenarios, this critical end-point has been discussed in analogy with the liquid-gas transition, see e.g. [46]. In this scenario, above (P0, T0) the metallic state cannot be distinguished from the paramagnetic insulating state. More recent studies suggest that lattice effects are relevant close to the Mott critical end-point [2, 4, 5]. By means of resistivity measurements under pressure, the critical behavior in the proximity of the point (P0, T0) was investigated by Kagawa et al [45]. These authors found critical exponents which do not fit in the most common universality Figure 8. Pressure versus temperature phase diagram of κ-(ET)2Cu[N(CN)2]Z, with Z = Br or Cl. Lines indicate the respective phase transitions. D8-Br and H8-Br indicate the positions (estimated according to [43]) of the κ-D8-Br and κ-H8-Br salts, respectively. The orange broken line indicates the crossover from the metallic to the insulating regime above the critical temperature. Thin solid lines (red), demarcating the hatched area, refer to the positions of the anomalies observed via ultrasonic measurements (see figure 10). The vertical (black) dotted line indicates T sweeps of the κ-D8-Br salt, depicting a crossing of the S-shaped first-order Mott MI phase transition line. PI, AFI and SC denote the paramagnetic insulator, the antiferromagnetic insulator and the superconducting phase, respectively. The open circle refers to the Mott critical end point. At the point marked by a closed circle in the phase diagram, TMI matches TN, see the discussion in section 3.2. As can be seen in the figure, the slope of dTMI/dP can be both positive and negative, indicating that upon crossing the MI line the entropy can either increase or decrease. While the entropy increases for dTMI/dP > 0 when crossing the MI line from the metallic to the PI state, for dTMI/dP < 0, when going from the metallic to the AFI state, the entropy associated with the metallic state is decreased due to magnetic ordering. Figure taken from [2]. classes and they assigned this to the quasi-2D structure of the present substances. Such a scenario shall be examined in more detail in section 4. The possibility of tuning the system from the insulating to the metallic side of the P versus T phase diagram, i.e. crossing the first-order Mott MI line by application of pressure, is indicative of a bandwidth-controlled Mott MI transition. Owing to the position of the salts with different counter anions X, as shown in figure 8, the fully hydrogenated salt with X = Cu[N(CN)2]Br superconducts below 12 K, the highest Tc under ambient pressure among all organic conductors till the present date, whereas the salt with counter anion X = Cu[N(CN)2]Cl is a Mott insulator with TN � 24 K. 5 J. Phys.: Condens. Matter 27 (2015) 053203 M de Souza and L Bartosch It has very recently been shown that the antiferromagnetic Néel state is accompanied by charge ordering, i.e. the salt κ-(ET)2Cu[N(CN)2]Cl is in fact a multiferroic [47]. Interestingly enough, fully deuteration of the ethylene end groups of the ET molecules in the salt with X = Cu[N(CN)2]Br, as described in detail above, results in a shift of the latter towards the boundary of the S-shaped first-order Mott MI transition line. It is due to their close position to this line [1] that the deuterated salt with X = Cu[N(CN)2]Br, hereafter named κ-D8-Br, has attracted particular interest. As a matter of fact, the C–H and C–D bond lengths are slightly different. It is well known that the deuterium nucleus has one proton and one neutron, while the hydrogen nucleus is formed simply by one proton. Thus deuterium has a higher mass than hydrogen and, as consequence C–D chemical bonds have a smaller eigen- frequency than the C–H bonds. Hence, the displacement of the deuterium atoms from their equilibrium positions during the vibration process is smaller than that estimated for the hydrogen atoms. Thus the C–D chemical bonds are slightly smaller than the C–H bonds. Furthermore, the contact between the C–D2 end groups and the anions is weaker than that between the C–H2 end groups and the anions. As a consequence, the lattice of the fully deuterated salt is softer than the lattice of the fully hydrogenated salt. This process of exchanging atoms is usually called applying ‘chemical pressure’. In fact, application of pressure (chemical or external) changes the distance between the ET molecules so that the overlap between π -orbitals’ is increased and, as a consequence, the bandwidth (W) is changed. The ratio W/U is considered the pivotal parameter which defines the various phases of these substances [35]. Interestingly enough, for the salt with the anion X = Cu2(CN)3, magnetic ordering is absent in the whole insulating region. The P versus T phase diagram of the κ- (ET)2Cu2(CN)3 salt is depicted in figure 9. 1.3. Key experimental results As mentioned above, the metallic state of κ-(ET)2X presents some distinct properties as compared to those of conventional metals. For instance, in the κ-H8-Br salt, the out-of-plane resistance as a function of temperature has a non-monotonic behavior [49] with a maximum around 90 K, which was observed to be sample-dependent. Decreasing temperature, this maximum is followed by a sudden drop and the resistivity sharply changes its slope around T ∗ � 40 K. Below Tc � 11.5 K superconductivity is observed. On applying pressure, T ∗ shifts to higher temperatures and disappears at roughly 2 kbar. From about T ∗ � 40 K down to Tc � 11.5 K the resistance obeys a T 2 behavior which is frequently assigned to a coherent Fermi liquid state. NMR studies revealed a peak in the spin-relaxation rate divided by temperature (1/T T1) around T ∗ for the present compound [50]. The anomalous metallic behavior around T ∗ was also observed by thermal, magnetic, elastic and optical measurements, see [49] and references therein. The origin of the T ∗ anomaly is still controversial. Ultrasonic velocity measurements under pressure on the compound κ-(ET)2Cu[N(CN)2]Cl performed Figure 9. P -T phase diagram of the κ-(ET)2Cu2(CN)3 salt, obtained from resistance and NMR measurements under pressure using an oil pressure cell. The Mott MI transition and crossover lines are associated with the temperature at which the quantities spin-lattice relaxation rate divided by temperature (1/T1T ) and dR/dT show a maximum, see labels indicated in the figure. The Fermi-liquid regime (yellow area) was fixed by the temperature regime where the resistance R follows the R0 + AT 2 relation and (1/T1T ) is constant. The superconducting transition line was obtained via in-plane resistance measurements. Reprinted with permission from [48]. Copyright 2005 by the American Physical Society. by Fournier et al [42] revealed the existence of two new lines in the phase diagram (not indicated in figure 8) which merge at the critical end point. These lines indicate the softening of the lattice response and should therefore coincide with anomalies expected in future dilatometric studies under pressure. It is important to keep in mind that the two lines observed in ultrasound experiments are different from the metal-to-insulator crossover line above Tc. The experimental findings obtained by Fournier et al are shown in figure 10. From this data set, pronounced anomalies, depending on the pressure, are observed below around 40 K. The maximum amplitude of the softening in the sound velocity (at T ≈ 34 K, P ≈ 210 bar) marks the critical end point (P0, T0) and corresponds to a softening of roughly 20% of the sound velocity. On increasing the pressure, the peak position shifts its position to higher temperatures. Below P0 the anomaly becomes gradually less pronounced and saturates at around 32 K. The peak position was thus used by the authors to draw the two crossover lines in the P -T phase diagram (see figure 11). The large anomalies in the sound velocity observed by Fournier et al were discussed in terms of a diverging electronic compressibility. As a matter of fact, acoustic and lattice anomalies are, according to dynamical mean-field theory (DMFT) calculations, expected at the Mott MI transition as a reaction to the softening of the electronic channel, see [51, 52]. The explanation for this is that in the metallic state the conduction electrons contribute more to the cohesion of the solid than in the insulating one. Based on this argument, according to [51], as an effect of the localization, abrupt 6 J. Phys.: Condens. Matter 27 (2015) 053203 M de Souza and L Bartosch T (K) 10 20 30 40 50 60 70 80 90 ∆ V / V -0.20 -0.15 -0.10 -0.05 0.00 210 bars 220 275 352 460 26 28 30 32 34 36 38 40 150 bars 200 215 226 230 Figure 10. Main panel: Relative sound velocity V/V versus T under pressure for the κ-(ET)2Cu[N(CN)2]Cl salt. The various pressure values are indicated in the label of the figure. Inset: data for pressures below 230 bar. Reprinted with permission from [42]. Copyright 2003 by the American Physical Society. P (bar) 0 100 200 300 400 500 600 T ( K ) 0 10 20 30 40 50 60 SC-I SC-II SC-III M I Figure 11. Phase diagram obtained from ultrasound experiments under pressure for the κ-(ET)2Cu[N(CN)2]Cl salt. Different symbols refer to the various anomalies observed on three different samples. The critical point (P0, T0) is indicated by the gray circle. SC-I and -II indicate metastable superconductivity, while SC-III indicates bulk superconductivity. Dotted hexagon indicates the pressure point, obtained from microwave resistivity measurement at ambient pressure. Reprinted with permission from [42]. Copyright 2003 by the American Physical Society. changes of the lattice parameters are expected at the Mott MI transition. We shall discuss the lattice effects for the present material in section 3. Regarding the Fermi surface, the weak interlayer transfer integrals were nicely estimated from magnetore- sistance measurements for deuterated κ-(ET)2Cu(NCS)2 (figure 12). From a more fundamental point of view, the determination of the interlayer transfer integral in a quasi-2D system is quite helpful, for instance, to understand the validity of the criterion usually adopted to classify whether the interlayer charge transport in quasi-2D systems is incoherent or coherent. For a detailed discussion, see e.g. [53] and references cited therein. Thus charge-transfer salts of the κ-(ET)2X phase serve as a good platform to investigate the fermiology of quasi-2D systems. Furthermore, knowing precisely the relevant parameters that govern the Mott Physics, theoretical approaches can be employed to understand the fundamental aspects of the Mott transition, see e.g. [7]. Owing to the magnetic properties of the κ-(ET)2X family, several studies employing different experimental methods have been carried out on the salt with X = Cu[N(CN)2]Cl [54–56]. By means of resistance measurements, a magnetic-field- induced Mott MI transition was observed by Kagawa et al [57] by varying temperature, pressure and the magnetic field. From an analysis of their NMR line shape, relaxation rate and magnetization data, Miyagawa et al [56] were able to describe the spin structure of this state. Below T = 26– 27 K, they found a commensurate antiferromagnetic ordering with a moment of (0.4–1.0) µB/dimer, as already mentioned above. The observation of an abrupt jump in the magnetization curves for fields applied perpendicular to the conducting layers, i.e. along the b-axis, was discussed in terms of a spin-flop (SF) transition. Furthermore, a detailed discussion about the SF transition, taking into account the Dzyaloshinskii– Moriya exchange interaction, was presented by Smith et al [58, 59]. Similarly to that for the pressurized chlorine salt, a magnetic-field-induced MI transition was also observed in partially deuterated κ-(ET)2Cu[N(CN)2]Br [60]. In addition, a discussion on the phase separation and SF transition in κ- D8-Br was reported in the literature [61]. As already mentioned above, the organic charge-transfer salt κ-(ET)2Cu2(CN)3 has the peculiarity that the ratio of its hopping matrix elements t ′ to t is close to unity, more precisely 0.83 [28, 29], leading to a strongly frustrated isotropic S = 1/2 triangular lattice with the coupling constant J = 250 K, where this coupling constant is obtained by fitting the magnetic susceptibility using the triangular-lattice Heisenberg model. Magnetic susceptibility and NMR measurements revealed no traces of long-range magnetic ordering down to 32 mK [8]. Based on these results, this system has been proposed to be a candidate for the realization of a spin-liquid state [62]. Interestingly, upon applying pressure (see figure 9), the system becomes a superconductor [48], i.e. superconductivity appears in the vicinity of a spin-liquid state. As shown in figure 13, specific heat experiments revealed the existence of a hump at 6 K, insensitive to magnetic fields up to 8 T. This feature was assigned to a crossover from the paramagnetic Mott insulating to the quantum spin-liquid state [6]. Such a ‘crossover’ has been frequently referred to as ‘hidden ordering’. Below 6 K, the specific heat presents a distinct T -dependence, including a T -linear dependence in the temperature window 0.3–1.5 K, as predicted theoretically for a spin-liquid [63]. The spin entropy in this T -range is roughly 2.5% of R ln 2 [64], indicating that below 6 K only 2.5% of the total spins contribute to the supposed spin-liquid state. Extrapolating the low-T specific heat data, the authors of [6] found a linear specific heat coefficient γ = (20 ± 5) mJ mol−1 K−2 (see figure 14). The latter is sizable given the insulating behavior of the material and contrasts with a vanishing γ value for the related compounds κ-(ET)2Cu[N(CN)2]Cl and fully deuterated κ-(ET)2Cu[N(CN)2]Br. A finite γ could indicate a spinon Fermi surface. 7 J. Phys.: Condens. Matter 27 (2015) 053203 M de Souza and L Bartosch Figure 12. (a) Fermi surface cross section and Brillouin zone for κ-(ET)2Cu(NCS)2. (b) Interlayer magnetoresistance Rzz for deuterated κ-(ET)2Cu(NCS)2 as a function of the tilt angle (θ ≡ angle formed between the applied magnetic field B and the normal to the quasi-2D planes formed by the BEDT-TTF layers) taken under a static magnetic field B = 45 T at T = 520 mK. The central peak (around θ = 90◦) is associated with the closed orbits shown in (a), while the features in the edges are due to angle-dependent magnetoresistance oscillations (AMRO). Reprinted with permission from [53]. Copyright 2002 by the American Physical Society. Figure 13. Main panel: specific heat divided by T , i.e. Cp/T , data for the κ-(ET)2Cu2(CN)3 salt plotted against T 2 until 10 K. A broad hump anomaly around T = 6 K can be clearly observed. Inset: specific heat data for κ-(ET)2Cu2(CN)3, assuming that the data of the superconductor κ-(ET)2Cu(NCS)2 describe the phonon background. Reprinted by permission from Macmillan Publishers Ltd: [6], copyright 2008. A gauge theory with an attractive interaction between spinons mediated via a non-compact U(1) gauge field was proposed by Lee et al [65] to describe a possible spin- liquid state in κ-(ET)2Cu2(CN)3. Within this model, a pairing of spinons on the same side of the Fermi surface should occur at low temperatures. The attractive interaction between spinons with parallel momenta is quite analogous to Figure 14. Low-temperature specific heat data for the κ-(ET)2Cu2(CN)3 salt as well as κ-D8-Br, κ-(ET)2Cu[N(CN)2]Cl and β ′-(ET)2I Cl2, plotted is Cp/T versus T 2. The extrapolation of the red solid line towards zero temperature indicates the existence of a T -linear contribution for the spin-liquid candidate κ-(ET)2Cu2(CN)3. Data under magnetic fields for κ-(ET)2Cu2(CN)3 are also shown. Reprinted by permission from Macmillan Publishers Ltd: [6], copyright 2008. Ampère’s force law which states that two current carrying wires with parallel currents attract. The authors deduce that the pairing of spinons is accompanied by a spontaneous breaking of the lattice symmetry, which in turn should couple to a lattice distortion (analogously to the Spin-Peierls transition) and should be detectable experimentally via x-ray scattering. While Lee et al [65] derive a specific heat which 8 J. Phys.: Condens. Matter 27 (2015) 053203 M de Souza and L Bartosch scales as T 2/3 and therefore contributes even stronger than linear at low T , it should be kept in mind that the very- low temperature behavior of κ-(ET)2Cu2(CN)3 is dominated by a diverging nuclear contribution to the specific heat. Indeed, the data set reported by Yamashita et al [6] is also reasonably well described by a T 2/3 behavior at intermediate temperatures [64]. Even though specific heat measurements by Yamashita et al [6] are compatible with a spinon Fermi surface with gapless excitations, thermal conductivity measurements reported by Yamashita et al in [66] show a vanishing low- temperature limit of the thermal conductivity divided by temperature, indicating the absence of gapless excitations. The excitation gap was estimated to be κ � 0.46 K. Applying a theory based on Z2 spin liquids, Qi et al argued that thermal transport properties should be dominated by topological vison excitations [67]. The gap found by Yamashita et al [66] should therefore be interpreted as a vison gap. It was pointed out by Ramirez [64] that before a final declaration of κ- (ET)2Cu2(CN)3 as a spin-liquid is made some points should be clarified, among them the anomaly observed in the specific heat at 6 K. We will come back to the system κ-(ET)2Cu2(CN)3 in section 5 where we will discuss its thermal expansion properties. Regarding the fascinating properties of this salt, recently the existence of two magnetic field-induced quantum critical points was reported in the literature [68]. Yet, based on a careful analysis of their muon spin rotation results, performed under extreme conditions, the authors of [68] discuss a possible interpretation for the features observed in several physical quantities around 6 K in terms of the formation of bosonic pairs emerging from a portion of the fermionic spins in κ- (ET)2Cu2(CN)3. The remainder of this article is organized as follows: In section 2, we discuss experimental aspects like sample preparation and high-resolution thermal measurements. Section 3 is dedicated to an exhaustive discussion on the thermal expansivity of fully deuterated salts of κ- (ET)2Cu[N(CN)2]Br, while in section 4 the Mott criticality is comprehensively reviewed. Section 5 is devoted to the discussion of the thermal expansivity of the spin- liquid candidate κ-(ET)2Cu2(CN)3. Finally, conclusions and perspectives are presented in section 6. 2. Experimental setup 2.1. Sample preparation As thermal expansion measurements under external hydro- static pressure are extremely challenging, using compounds of different chemical composition currently represents the most practical way to probe the phase diagram of the κ-(ET)2X fam- ily via dilatometry. Deuterated (98%) bis(ethylenedithiolo)- tetrathiofulvalene (D8-ET) was grown according to Hartke et al [69] and Mizumo et al [70], making the reduction of car- bon disulfide with potassium in dimethylformamide and subse- quent reaction with deuterated (98%) 1.2-dibromoethane. The intermediate thione C5D4S5 was mixed with triethylphosphite in an inert atmosphere kept at 1200 ◦C and recrystallized sev- eral times from chlorobenzene. By investigating the CH2 and Table 2. Samples of the organic conductor κ-(ET)2X on which thermal expansion measurements were performed. Crystal #4 was used to perform preliminary studies of Raman spectroscopy. In [2] crystal #2 is referred to as crystal #3. Crystal Anion X Number Direction Batch D8-Br 1 a, b, c A2907a D8-Br 2 a, b A2995a D8-Br 3 a, b, c A2907a D8-Br 4 — A2907a H8-Br 7 b A2719a H8-Cu2(CN)3 1 a SKY 1050b H8-Cu2(CN)3 1 c KAF 5078b H8-Cu2(CN)3 2 b KAF 5078b a Refers to samples provided by Schweitzer (University of Stuttgart). b To samples provided by Schlueter (Argonne National Laboratory). CD2 stretch vibration modes, no signs of any CH2 or CDH vibrations could be detected, see [71]. Hence, the grade of deuteration of κ-(D8-ET)2Cu[N(CN)2]Br is at least 98% [72]. As discussed in detail in [49], by employing the preparation technique described above, hydrogenated single crystals of the same substance resulted in samples whose resistivity re- vealed reduced inelastic scattering and enhanced residual re- sistivity ratios. In general, the crystals have bright surfaces in shapes of distorted hexagons with typical dimensions of ap- proximately 1×1×0.4 mm3. Due to the small size and fragility of the crystals, the experimental challenge lies in assem- bling them in the dilatometer. The samples of the deuterated and hydrogenated variants of κ-(ET)2Cu[N(CN)2]Br and κ- (ET)2Cu2(CN)3 salts used for measurements discussed in this work are listed in table 2. The fully deuterated (hydrogenated) salts of κ-(ET)2Cu[N(CN)2]Br will be subsequently referred to as κ-D8-Br (κ-H8-Br). Samples of κ-(ET)2Cu2(CN)3 studied here were prepared according to [33]. 2.2. Thermal expansion measurements As already mentioned in the introduction, thermal expansion coefficient measurements are a very useful and powerful tool to explore phase transitions in solid state physics research [3, 4, 47, 73–85]. For instance, combining specific heat and thermal expansion data one can employ the Ehrenfest relation to determine the pressure dependence of a second- order phase transition temperature. Such an analysis may serve also as a check of its direct measurement as a function of hydrostatic pressure. Furthermore, contrasting with specific heat which is an isotropic property, anisotropic effects may be studied via thermal expansion measurements. Regarding the charge-transfer salts of the κ-phase, to our knowledge, the first high-resolution thermal expansion experiments along the three crystallographic directions were carried out about twenty years ago by Kund and collaborators [86]. The measurements presented in the present work were carried out by making use of an ultra-high-resolution capacitance dilatometric cell, built after [87], with a maximum resolution of l/l = 10−10, in the temperature range 1.3 9 J. Phys.: Condens. Matter 27 (2015) 053203 M de Souza and L Bartosch C Sa m pl e Figure 15. Schematic representation of the cell used for the thermal expansion measurements. Details are discussed in the main text. Figure adapted from [3]. (pumping of Helium bath) to 200 K under magnetic fields up to 10 T. In order to avoid external vibrations, the cryostat is equipped with shock absorbers. A detailed description of the dilatometer used in this work has been presented in several works [90–92]. The principle of measurement is described in the following. As sketched in figure 15, the cell, which is entirely made of high-purity copper to ensure good thermal conductivity and covered with gold is constituted basically of a frame (brown line) and two parallel pistons (orange), the upper one being movable. As a matter of fact, the gold layer should work like a protection, avoiding oxidation of the cell. Nevertheless, after some years in use, two striking anomalies have been observed at T � 212 K and 230 K. These anomalies are assigned to the formation of copper-oxide (CuO) [93] in the body of the cell. Due to this, experimental data taken in this T window are not reliable. The sample is placed between these two pistons and by moving the upper piston carefully, the starting capacitance is fixed. The lower piston is mechanically linked to a parallel plate capacitor, as schematically represented by springs in figure 15. The variation of the sample length, i.e. contraction or expansion, as the temperature is lowered or increased, together with cell effects corresponds exactly to the change of the distance between the plates of the capacitor and consequently to a change of the capacitance, so that very tiny length changes can be detected. In this construction, the distance between the plates of the capacitor is about 100 µm. For the sake of avoiding stray electric fields, the parallel plate capacitor is surrounded by guard rings [87]. The most remarkable characteristic in this capacitance dilatometer, however, is its high resolution ( l/l = 10−10), corresponding to absolute length changes of l = 0.01 Å for a specimen of length 10 mm. This resolution is roughly five orders of magnitude higher than that of conventional methods like neutrons or x-ray diffraction [73] and is mainly due to the high resolution of the capacitance bridge and the high quality of the plate capacitor in the dilatometer cell, which allows the detection of very small changes, i.e.changes of about 10−7 pF can be detected in the capacitance of the system. Two different high precision capacitance bridges have been used in these experiments: (a) Andeen Hagerling—Model 2500A 1 kHz and (b) General Radio—Model 1616. However, the above-mentioned resolution holds only until T � 40 K, where a precise PID temperature control becomes difficult due to the large time constant involved in the experiment. As deduced in [87], the sensitivity of the measurements is proportional to the square of the starting capacitance (C2). This means that the higher the starting capacitance, the higher the sensitivity. However, specimens of the molecular conductors investigated here are quite sensitive to the pressure applied by the dilatometric cell so that one cannot set a starting capacitance too high since this would consequently lead to a break in the sample. Hence, the starting capacitance for measurements under normal pressure (see Subsection 2.3) was limited to ∼18 pF. Just for completeness, it is worth mentioning that the empty capacitance of the system reads 16.7 pF. For measurements under magnetic fields, a magnet power supply (model PS120-3) supplied by Oxford Instruments was used. In all performed measurements under magnetic fields reported here the field was applied parallel to the measured direction. For the direction-dependent thermal expansion measurements, the alignment of the crystal orientation was made using an optical microscope and guaranteed with an error margin of ±5◦. To obtain the intrinsic thermal expansivity of the investigated specimen the thermal expansivity of copper of the dilatometric cell body was subtracted from the raw data. Apart from that, no further treatments like splines or any other kind of mathematical fittings were done. The linear thermal expansion coefficient αi in the direction i at constant P is defined as αi = 1 l (∂l(T ) ∂T ) P , (1) where l refers to the sample length. The physical quantity described by equation (1) will be frequently used in this work. From the length changes of the sample l(T ) = l(T ) − l(T0) (T0 is a fixed temperature), which is the physical quantity measured, the linear thermal expansion coefficient [equation (1)] was approximated by the differential quotient as α(T ) ≈ l(T2) − l(T1) l(300 K) × (T2 − T1) , (2) where T = (T1 + T2)/2 and T1 and T2 are two subsequent temperatures of data collection. Unless otherwise stated, in order to guarantee thermal equilibrium in all thermal expansion measurements reported here, a very low temperature sweep rate (±1.5 K h−1) was employed. 2.3. Thermal expansion measurements under quasi-uniaxial pressure One of the challenges in the context of thermodynamic measurements is the realization of high-resolution thermal 10 J. Phys.: Condens. Matter 27 (2015) 053203 M de Souza and L Bartosch expansion measurements under hydrostatic pressures, see e.g. [88]. However, by changing the starting capacitance and taking into account the contact area between the dilatometer and the sample, one can perform quasi-uniaxial thermal expansion experiments under pressures up to ∼100 bar employing the above-described setup, see e.g. [89]. The term ‘quasi-uniaxial’ is used here in the sense that one may have a pressure-gradient on the sample volume. The dependence between the force (F ) exerted by the springs and the capacitance change ( C) can be easily measured and is given by F C = 0.124 · N pF (3) Making use of equation (3), the pressure applied by the dilatometric cell on the sample can be estimated. For example, taking a contact area of 0.2×10−6 m2 and a capacitance change of 2 pF and using the definition of pressure P = F/A results in a quasi-uniaxial pressure P of roughly 12 bar. Hence, a slight change of the contact area and/or starting capacitance results in a significant change of the pressure exerted by the dilatometer on the sample. The crucial point of such experiments lies in the assembling and disassembling of the sample holder from the cryostat without damaging the sample, as such procedure is necessary to set the new starting capacitance (new pressure). As depicted in figure 11, for the (BEDT-TTF) based charge-transfer salts a hydrostatic pressure of a few hundred bar is enough to make a significant sweep across a wide range of the pressure-temperature phase diagram. Measurements under quasi-uniaxial pressure, to be discussed in section 3.4, will show that a quasi-uniaxial pressure of a few tenth of bar changes the shape of the thermal expansion coefficient curves dramatically, indicating therefore that the properties of this material are altered. 3. Thermal expansion measurements on fully deuterated salts of κ-(ET)2Cu[N(CN)2]Br (‘κ-D8-Br’) As otherwise stated, in order to reduce cooling-speed dependent effects, the temperature was decreased by a rate of ∼ − 3 K h−1 through the so-called glass-like transition around 80 K7 in all experiments reported here. figure 16 depicts the linear thermal expansivity (upper part) along the in-plane a-axis for crystals #1 and #3 together with the out- of-plane (b-axis) electrical resistivity8 ρ⊥ (lower part) for crystal #1. As depicted in figure 16, upon cooling, ρ⊥ achieves a broad maximum centered at ∼45 K, then abruptly drops and levels out at ∼30 K. The electrical resistivity remains metallic, i.e. dR/dT > 0, down to ∼20 K, below which ρ⊥ increases again (see upper inset in figure 16), indicating a phase transition to an insulating state. A comparable ρ⊥ behavior was observed for the single crystal #3 as well as for the crystal investigated in [43], except for small differences around the maximum and some details in the insulating 7 This cooling speed (∼−3 K h−1) was employed in the temperature range ∼85–65 K. From room temperature down to ∼85 K as well as from 65 K down to 4.2 K, an average cooling speed of −40 K h−1 was employed. 8 Resistance measurements shown in figure 20 were carried out by Strack. Figure 16. Left scale: thermal expansion coefficient along the in-plane a-axis, αa(T ), for crystals #1 (red) and #2 (blue); right scale: out-of-plane electrical resistivity, ρ⊥, for crystal #1 of the κ-D8-Br salt. The upper inset depicts a zoom of the low temperature ρ⊥(T ) data employing the temperature scale depicted in the main panel. Figure taken from [2]. regime. Interestingly enough, for all three crystals studied, ρ⊥ vanishes below ∼11.5 K. A zero electrical resistivity followed by a very small feature in the thermal expansivity αi(T ) suggests percolative superconductivity coexisting with an antiferromagnetic insulating phase for κ-D8-Br [61], see the phase diagram presented in figure 8. In [1], Kawamoto et al performed a.c. magnetic susceptibility measurements on κ-D8-Br under two distinct conditions: (a) cooling the system slowly, namely employing a cooling-rate of 0.2 K min−1; and (b) using a high cooling-rate of 100 K min−1. From the measurements carried out after slow cooling, below the onset of superconductivity at Tc = 11.5 K, they did not observe a jump in the real part of the a.c. magnetic susceptibility, as expected for bulk superconductivity, but a gradual transition towards low temperatures. From the latter behavior they estimated a percolative superconducting volume fraction of 20% . In the case of measurements after rapid cooling, according to the authors, the superconducting volume fraction is reduced to 1–2%. As can be seen in figure 16, some features observed in the electrical resistivity manifest themselves in the coefficient of thermal expansion as well. Indeed, the rather abrupt drop of ρ⊥ is accompanied by a pronounced maximum in αa(T ) centered at a temperature Tp � 30 K. As will be discussed in more detail below, this behavior can be assigned to a crossover phenomenon in the vicinity of a nearby critical point. Upon cooling the system further, α(T ) reveals a huge negative peak, indicating a phase transition, see the phase diagram depicted in figure 8. The concomitant change in ρ⊥ from metallic to 11 J. Phys.: Condens. Matter 27 (2015) 053203 M de Souza and L Bartosch Figure 17. Main panel: relative length changes data li(T )/ li , with i = a, b, c for crystal #1 of the κ-D8-Br salt along the in-plane a- and c- and out-of-plane b-axis. For clarity, the data set have been offset. Lower inset: hysteresis in la(T )/ la associated with the Mott MI transition, measured under a very low temperature sweep-rate of ±1.5 K h−1. Upper inset: 2D triangular-lattice dimer model with hopping terms t and t ′, the dots represent dimers of ET molecules. Figure taken from [2]. insulating behavior indicates that the peak in the expansivity data, αa(T ), is due to the Mott MI transition9. Note that a similar behavior of αa(T ) is observed for crystal #2, albeit with somewhat reduced (∼20%) peak anomalies at Tp and TMI. This reduction might be related to small misalignments of the crystal as well as to different pressures exerted by the dilatometer cell on the crystals, approximately 4 bar for crystal #1 and 6 bar for crystal #2. More information on the nature of such transitions can be obtained via investigation of possible anisotropic lattice effects, namely by measuring and analysing the relative length changes along the three crystallographic directions, i.e. li(T )/ li = (li(T ) − li(0 K))/li(300 K) (with i = a, b, c), as depicted in figure 17. As can be seen in figure 17, the most pronounced effects appear along the a-axis (in-plane), i.e. parallel to the polymeric (Cu[N(CN)2]Br) anion chains, see figure 5. Of special interest is the ‘S-shaped’ anomaly near Tp which corresponds to the peak at Tp in αa(T ) (see figure 16) and does not show any trace of hysteresis upon cooling or warming, at least within the resolution of the present experiments. This is a generic feature of strong fluctuations in the vicinity of a critical end point. Upon further cooling through TMI, the length of the crystal along the a-axis shows a distinct increase of roughly la/la = 3.5·10−4 within a very limitedT -window, consistent 9 In employing α = −80 ·10−6 K−1, obtained from the height of the anomaly in the thermal expansion coefficient for crystal #1 (figure 16), using the entropy change S = −0.074 J mol−1 K−1 at the Mott MI transition, whose detailed deduction is presented in section 3.2, results in a jump of the specific heat of about 0.2 J mol−1 K−1. Details of this estimate are provided in appendix 2. This value corresponds roughly to 0.4% of the background at TMI in the specific heat (specific heat measurements on κ-D8-Br crystal #1 were performed by Köhler) employing ac calorimetric method, so that no signatures in the specific heat at TMI could be detected. with a broadened first-order phase transition. The assumption of a first-order phase transition is also supported by the observation of a hysteresis around TMI of ∼0.4 K, as depicted in the lower inset of figure 17, which nicely agrees with the hysteresis observed in ρ⊥(T ) (see upper inset of figure 16). The associated anomalies observed along the b-axis (out-of- plane) are clearly less pronounced. Notably, for the second in-plane direction (c-axis), anomalous behavior in lc/ lc can neither be detected at TMI nor at Tp. The reliability of the previous results is supported by the same anisotropy observed for crystal #3, investigated in [43]. Based on the quasi- 2D electronic structure of the present material, the observed anisotropic lattice effects at the Mott MI transition are very intriguing and deserve to be analyzed in more detail. The pronounced effects observed in the expansivity data along the in-plane a-axis, along which dimer–dimer overlap is absent, agree with the hypothesis that the diagonal hopping transfer integrals, t , play a crucial role in this process. Considering that these transfer integrals carry a net component along the in-plane c-axis, which is expected to be softer than the out-of- plane b-axis and second in-plane a-axis, a noticeable effect along the c-axis would then be naturally expected at TMI. According to the authors of [94], the uniaxial isothermal compressibilities ki = l−1 i (dli/dP) for the superconducting compound κ-(ET)2Cu(NCS)2 are robustly anisotropic with k1:k2:k3 = 1:0.53:0.17 (ratio estimated under 1 bar), where k1 refers to the isothermal compressibility along the in-plane c direction, k2 along the second in-plane b-axis and k3 along the direction parallel to a projection of the long axis of the ET donor molecules, i.e. along the out-of-plane a-axis for this salt, since its crystallographic structure is monoclinic [95]. Note that along the c-axis a direct dimer–dimer interaction t ′, as indicated in the upper inset of figure 17, exists, making a zero-effect along this axis even more surprising. A possible explanation for this feature would be a cancellation of counteracting effects related to the transfer integrals t and t ′, which seems to be improbable. Furthermore, it is not clear how these in-plane interactions may produce comparatively strong effects in the out-of-plane b-axis. These findings provide strong evidence that other degrees of freedom, namely the lattice, should be somehow coupled to the π electrons, thus indicating that the Mott MI transition in the present material cannot be completely understood solely by taking into account a purely 2D electronic scenario [2]. 3.1. Anisotropic lattice effects in κ-D8-Br and rigid-unit modes In figure 1810 the relative length changes in the T -range 4.2 K < T < 200 K for the a-, b- and c-axes are shown. In the following, a possible explanation for the unusual anisotropic lattice effects observed in κ-D8-Br is presented. Upon cooling, strongly anisotropic effects are observed in the whole temperature range. The more pronounced effects are observed along the in-plane a-axis, which is marked by a minimum around 150 K and an abrupt change in slope at Tg . This feature will be discussed in more detail below. 10 In [72] Grießhaber refers to a cooling rate of∼−1 K min−1 as a slow cooling, while the rapid cooling rate is not mentioned. 12 J. Phys.: Condens. Matter 27 (2015) 053203 M de Souza and L Bartosch 0 40 80 120 160 200 0 40 80 120 0 20 40 60 80 100 0 200 400 600 Tg TP ∆V / V (1 0-5 ) T(K) TMI a-axis b- c- ∆l i/ i ( 10 -4 ) T(K) T g NTE Tp T MI Figure 18. Main panel: relative length changes along the a- and c-axis (in-plane) and b-axis (out-of-plane) for κ-D8-Br crystal #1 up to 200 K. Data along the b- and c-axes are shifted for clarity. Position of Tg estimated according to [25]. NTE indicates negative thermal expansion along the a-axis (in-plane) in the T -range Tg < T � 150 K. Inset: relative volume change V/V = la/ la + lb/ lb + lc/ lc versus T . Low temperature data will be used for estimating the entropy change associated with the Mott MI transition (section 3.2). Figure after [3]. An interpretation in terms of disorder due to the freezing- out of the ethylene groups of the ET molecules either in the staggered or in the eclipsed configuration was proposed by Müller et al [25]. By substituting the terminal (CH2)2 groups by (CD2)2, one observes a shift of Tg to higher temperatures. A feature that is expected because of the higher mass of the (CD2)2 in comparison to the mass of the (CH2)2 groups, which in turn implies a higher relaxation time at a given temperature. Hence, the model proposed by Müller et al is supported by the so-called isotope effect. However, it does not explain the negative expansivity along the a-axis (in-plane) observed above Tg , namely in the interval Tg < T � 150 K. More recently, high-resolution synchrotron x- ray diffraction experiments [97] have provided indications that such an anomaly is not solely linked to the ethylene groups freezing-out. Essentially, the claims of the authors are based on the comparison between the estimated number of the disordered ethylene groups (staggered configuration) at 100 K (∼11%) with those at extrapolated zero temperature (3 ± 3%). They conclude that such a transition is most likely related to structural changes, probably involving the insulating polymeric anion chains. Upon further cooling, the anomalies at Tp and TMI, already discussed in detail in the previous section, show up. From figure 18 it becomes clear that the lattice effects at Tg , Tp and TMI are more pronounced along the polymeric anion chain a-axis, with negative thermal expansion (NTE), i.e. a decrease of length with increasing temperature, observed below TMI (due to charge carrier localization) and also in the temperature window Tg < T � 150 K. It should be noted that the in-plane anisotropy at both TMI and Tg corresponds to an orthorhombic distortion. Recently, NTE was also observed by Goodwin et al in other materials containing cyanide bridges in their structures, see [98, 99] and references therein. These authors proposed a simple model [99] to explain the origin of this unusual behavior. In this model, the transverse vibrational (a) (b) (c) Metal C N Metal Figure 19. Schematic representation of the M-CN-M linkage. (a) Linear configuration. (b) Shift of the carbon (C) and nitrogen (N) atoms aside from the metal–metal axis in the same direction. (c) Shift in the opposite direction. Note that the total length along the metal–metal axis is reduced due to the displacement of the C-N atoms. This is a mechanism giving rise to negative thermal expansion. Figure reproduced after [99]. displacement of the CN units, away from the metal–metal axes (figures 19(b) and (c)), is assigned to be the mechanism responsible for NTE in these materials. As a matter of fact, NTE is also observable in other materials. For example, the NTE of liquid water below 4 ◦C is connected with a breaking of the tetrahedral H bonding. Below 4 ◦C, NTE is required to overcompensate the increase of entropy due to such a structural transition [101]. Another example can be found in ZrW2O8. This material exhibits NTE over a wide temperature window of 50–400 K [102]. The origin of this anomalous behavior is assigned to its structural arrangement, which is built up by WO4 tetrahedra and rigid ZrO6 octahedra. Rigid-unit modes in this material show up due to the fact that each WO4 unit has one vertex not shared by another unit. The displacement of the C and N atoms aside from the metal–metal axis has the effect of decreasing the distance between the metal atoms as the temperature is increased, see figures 19(b) and (c). According to the authors of [99], in polymeric crystalline materials, these vibrational modes will not occur separately, but rather they will affect the vibration modes of other metal-CN-metal linkages, giving rise to the NTE phenomenon. Nevertheless, the deformation of the metal-cyanide bridge will carry a high-energy penalty. Hence, assuming that the metal coordination geometries are preserved, the vibration modes illustrated in figures 19(b) and (c) can be observed only if they are coupled with the crystal lattice in the form of phonon modes. Yet, according to Goodwin et al such modes are referred to as rigid-unit modes (RUM) and can be experimentally observed in the low-energy window 0–2 THz, i.e. in the range of typical phonon frequencies [103]. Since the Cu[N(CN)2]Br− (figure 5) counter anion has a polymeric nature, the above-described model is a good candidate to explain why NTE is observed only along the in-plane anion chain a-axis. The latter might have their origin in the RUM of dicyanamide [(NC)N(CN)]− ligands along the anion 13 J. Phys.: Condens. Matter 27 (2015) 053203 M de Souza and L Bartosch polymeric chain. Note that the contraction of the lattice with growing T above Tg along the a-axis is accompanied by an expansion along the b- and c-axes. In addition, preliminary Raman studies (not shown here), carried out on κ-D8-Br (crystal #4), revealed the appearance of a double peak structure in the frequency window 50–200 cm−1 at 20 K (below Tg), not observed at 130 K (above Tg , but below the onset of NTE) [100]. Further systematic studies are required to verify if such splitting shows up above or below Tp. Note that this frequency window fits roughly into the above-mentioned energy window predicted for RUM. Interestingly enough, the double peak structure is not observed at 5 K (below TMI), indicating that vibration modes in this energy window are no longer active below TMI. Hence, these results support the idea of strong coupling between lattice and electronic degrees of freedom at the Mott transition. Still owing to the anomalous lattice effects above Tg , based on the above discussion, it seems that the freezing-out of the degrees of freedom of the ethylene end-groups of the ET molecules in the staggered/eclipsed configuration alone cannot be responsible for a NTE along the a-axis. Indeed, as pointed out in [104], the glass-like anomaly is not observed in the organic superconductors β ′′-(ET)2SF5CH2CF2SO3 (Tc = 5 K, large discrete anion) and in κ-(ET)2I3 (Tc = 3.5 K, linear anion) as well as in the nonsuperconducting α-(ET)2KHg(SCN)4 (polymeric anion). For such materials, a smooth Debye-like behavior along the three crystallographic directions up to 200 K is observed, i.e. no glass-like transition is observed. It is appropriate to mention here that the glass-like anomaly is not observed in the κ-(ET)2Cu2(CN)3 salt, to be discussed in section 5. Thus, a possible scenario to explain this feature for the κ-(ET)2Cu[N(CN)2]Z salts would be the existence of a complicated entwinement between the ethylene end groups and the vibration modes of the polymeric anion chain, not knowingly reported as yet in the literature. According to this idea, the ethylene end groups delimit cavities, where the anions are trapped [34], so that below Tg an anion ordering transition occurs in a similar way to the anion ordering transition observed in the quasi-1D (TMTCF)2X charge-transfer salts [105]. Further systematic Raman and infra-red studies are necessary to provide more information about the nature of the glass-like transition. 3.2. Entropy change associated with the Mott MI transition More information about the Mott physics in the present materials can be obtained by estimating the amount of entropy linked to the Mott MI transition. Such entropy change is directly associated with the slope of the first-order phase transition line in the universal P -T phase diagram figure 8 and the volume change associated with the Mott MI transition by the Clausius–Clapeyron equation, which reads dTc dP = V S . (4) Here, V = (VI − VM) ( S = (SI − SM)) refer to the difference in volume (entropy) between the insulating (I) and metallic (M) states. Employing dTMI/dP = (−2.7 ± 0.1) K MPa−1, as obtained from the slope of the S-shaped line at TMI in figure 8 and V/V = (4.2 ± 0.5) × 10−4, as estimated from the inset of figure 18, one obtains S = −0.074 J mol−1 K−1. This small entropy change at TMI represents a subtle fraction of the full entropy related to the metallic state of the κ-H8-Br salt at T � 14 K of S = γ · T � 0.375 J mol−1 K−1, using the Sommerfeld coefficient γ = 0.025 J mol−1 K−2 [106]. Electronic specific heat data by Nakazawa et al [107] revealed that by means of gradual deuteration it is possible to tune the system from the Mott insulating state to the metallic regime of the phase diagram. As observed in these experiments, γ decreases towards zero in the insulating phase. Based on these literature results, γ of the fully deuterated salt discussed in this work should actually be somewhat smaller than that of the fully hydrogenated salt. It follows that the small entropy change of −0.074 J mol−1 K−1, estimated as above-described, demonstrates that the spin entropy of the paramagnetic insulating state at elevated T must be almost completely removed. In addition, this result is consistent with the Néel temperature TN coinciding with TMI at this point in the phase diagram [108]. 3.3. Magnetic field effects on κ-D8-Br Having detected unusual features in the thermal expansivity of κ-D8-Br, additional information about the role played by lattice degrees of freedom for the Mott MI transition in κ- D8-Br can be obtained by carrying out thermal expansion measurements under magnetic fields. In what follows we discuss the effects of magnetic fields in the expansivity data at the immediate vicinity of the Mott transition. However, before doing so, let us consider first the low-temperature case at zero magnetic fields. The thermal expansivity along the b- axis (out-of-plane) for B = 0 is shown in figure 20 together with the electrical resistance11 data normalized to its value at room temperature. Upon cooling, a negative anomaly in the expansivity data at around T = 13.6 K is observed. The latter is linked to the Mott MI transition and mimics the lattice response upon crossing the first-order Mott MI transition line in the phase diagram, see the discussion in the previous sections. Note that the anomaly in αb(T ) is directly connected to an enhancement of the electrical resistance around the same temperature, which corroborates the assumption that the anomaly in the expansivity αb(T ) is triggered by the Mott MI transition. The drop of the electrical resistance at Tc = 11.6 K, accompanied by a very small kink (indicated by an arrow) in αb(T ), are fingerprints of percolative SC in small portions of the sample volume coexisting with the AF ordered insulating state, see e.g. [109]. In the temperature window 12 K � T � 16 K, jumps in the electrical resistance are noticeable. In fact, such jumps in R(T ) are frequently observed in organic charge-transfer salts. The huge expansivity of the crystal, which in turn is due to the softness of this class of materials, induces stress on the electrical contacts, giving rise to such jumps. In general, the appearance of jumps in resistivity data are attributed to cracks in the crystal. For the quasi one- dimensional (TMTTF)2X and (TMTSF)2X salts, in particular, 11 See footnote 8. 14 J. Phys.: Condens. Matter 27 (2015) 053203 M de Souza and L Bartosch Figure 20. Left scale: thermal expansion coefficient along the out-of-plane b-axis. Right scale: electrical resistance data normalized to its value at room temperature, being both data set for κ-D8-Br crystal #3. TMI denotes to the Mott MI transition temperature and Tc refers to the percolative SC critical temperature. Jumps in the resistance around 13 K and 15 K are most likely related to the large expansivity of the material in this temperature range, as discussed in the text. Figure taken from [3]. such jumps in the resistivity data are assigned to stress caused by localized defects and/or to mechanical twinning [12]. The thermal expansivity data along the b-axis (out-of- plane) at low-T under external magnetic fields of 0, 0.5, 1, 2, 4, 6 and 10 T [85] is depicted in figure 21. From this data one can directly see that upon reducing T under a magnetic field up to 10 T, the Mott MI transition temperature TMI stands essentially unaffected. Nevertheless, upon reducing T further under a low magnetic field of 0.5 T, a second field-induced anomaly at around TFI = 9.5 K is observed. Increasing the magnetic field, this anomaly becomes more and more pronounced, until it saturates at around∼4 T. A careful analysis of the data shown in figure 21 reveals the presence of a double- peak structure for magnetic fields higher than 1 T. However, further experiments are necessary to gain more insights into this feature. No such magnetic field-induced effects were observed along the in-plane a- and c-axis (not shown). As known from the literature, see e.g. [110], orientational field- dependent magnetic phenomena may indicate a spin-flop (SF) transition. Such a transition takes place when the system is in an AFI state and a magnetic field, exceeding a certain critical value (critical field), is applied parallel to the so-called easy-axis. In the present case, this field dependence of the thermal expansivity αb(T ) is suggestive of a SF transition with strong magneto-elastic coupling. As can be directly seen from figure 21, the magnetic critical field Hc is smaller than 0.5 T for κ-D8-Br, which is consistent with Hc ∼ 0.4 T for the κ-(ET)2Cu[N(CN)2]Cl salt, as obtained by Miyagawa Figure 21. Thermal expansion coefficient perpendicular to the conducting layers under selected fields for the κ-D8-Br crystal #3. TMI refers to the Mott MI transition temperature and TFI to the field-induced phase transition temperature, as discussed in the main text. Figure reproduced after [85]. et al. in [56]. The physical explanation why the anomaly in the expansivity αb(T ) at T = TFI becomes more evident for magnetic fields higher than Hc, however, cannot be based solely in terms of a SF transition, so that a possibly different mechanism might be related to this feature. It was pointed out by Ramirez et al in [111] that the emergence of pronounced anomalies under finite magnetic fields in thermodynamic quantities, like the specific heat or thermal expansion, have only been observed in a few systems and can be considered as unusual physical phenomena. Let us discuss two possible distinct scenarios to explain this unusual feature: (i) First of all, electrical resistance measurements carried out on partially deuterated κ-(ET)2Cu[N(CN)2]Br salts under magnetic field sweeps, being the magnetic field applied along the out-of-plane b-axis and the temperature fixed at T = 5.50 K and 4.15 K for 50% and 75% salts, respectively, unveiled a sudden change from SC to high resistive-states for magnetic fields of about 1 T [85, 112]. Such a state was found to transform into a high-resistive state via jump-like increases in the electrical resistance upon further increasing the applied magnetic field to 10 T [112]. The authors of [112] interpreted this behavior as a magnetic field-induced first-order SC-to-Insulator transition and related it to the theoretical description 15 J. Phys.: Condens. Matter 27 (2015) 053203 M de Souza and L Bartosch based on the SO(5) symmetry for superconductivity and antiferromagnetism, as proposed for the high-Tc cuprates by Zhang [113]. Moreover, upon tuning the κ- (ET)2Cu[N(CN)2]Cl salt close to the Mott MI transition by hydrostatic pressure [44] a pronounced B-induced increase in the electrical resistance was observed also for this salt. In the latter case, it was proposed that minor metallic/superconducting phases near the Mott MI transition undergo a magnetic field-induced localization transition see theoretical predictions, see, e.g. [114] and references cited therein. Thus, based on these findings, it is natural to expect that also for the fully deuterated κ- (ET)2Cu[N(CN)2]Br salt, located on the verge of the Mott MI transition line in the phase diagram, the electronic states may undergo drastic changes upon increasing the strength of the applied magnetic field. Based on the above discussion, the growth of the anomaly in αb(T) with increasing fields H > Hc may be related with the sensitivity of the electronic channel to applied magnetic fields in this particular region of the phase diagram. In fact, the initial rapid growth and the tendency to saturation for field above about 4 T is quite similar to the evolution of the electrical resistance, i.e. the increase of R(B, T = const.) with field observed in the above- mentioned transport studies [44, 112]. It is worth mentioning that no such magnetic field-induced effects were observed for fields applied parallel to the in-plane a- and c-axis (not shown), supporting thus the hypothesis of a SF transition. Indeed, it was pointed out in [115] that the type of the inter-layer magnetic ordering strongly depends on the direction of the applied magnetic field. In particular, based on a detailed analysis of NMR and magnetization data, taking into account the Dzyaloshinskii–Moriya interaction, the atuhors of [44, 112] found that inter- plane antiferromagnetic ordering can be observed only for magnetic fields exceeding Hc applied along the out- of-plane b-axis. We stress that the term ‘inter-plane afm ordering’ is used here as defined in figures 4 and 5 of [115]. Thus, given the absence of lattice effects at TFI for magnetic fields applied along the in-plane a- and c-axis, the present results suggest a close relation between inter- plane antiferromagnetic ordering and the lattice response observed at TFI. Employing the same notation used by the authors of [115], the magnetization of the +(−) sublattice at the layer l is M+(−)l ., being the staggered and ferromagnetic moments given by M† l = (M+l − M−l)/2 and MF l = (M+l + M−l)/2, respectively. For fields above Hc applied along the out-of-plane b-axis, MF l is along the b-axis and M† l lies in the a-c plane. M† A and M† B are antiparallel, giving rise to an inter-plane antiferromagnetic ordering. The pronounced negative peak anomaly in αb(T ) observed at T = TFI suggests that in order to obtain this particular spin configuration, the increase in exchange energy forces the (ET)+ 2 layers to move apart from each other. (ii) A possible second interpretation for the sharp peaks in αb(T ) induced by a magnetic field is that upon exceeding the critical field Hc, percolative superconductivity is destroyed, the remaining electron spins do not couple with the applied magnetic field due to the correlated motion among them. These field-decoupled spins are strongly coupled to the lattice and give rise to the minimum at TFI. The term field-decoupled spins is used in [111] to refer to a similar effect (double peak structure) observed in specific heat measurements under magnetic field for the ‘spin ice’ compound Dy2Ti2O7. According to the authors of [111], for magnetic fields applied parallel to the [1 0 0] crystallographic direction, due to correlated motion among the spins, half the spins have their Ising- axis orientated perpendicular to the magnetic field. The latter are called decoupled-field spins. It was observed that magnetic fields exceeding a certain critical value lead to the ordering of these field-decoupled spins. Monte Carlo calculations, also reported by the authors in [111], support the proposed model. The term ‘spin ice’ is used by the authors to refer to the spin orientations in analogy with the degeneracy of ground states observed in ice (H2O in solid phase), where hydrogen atoms are highly disordered and give rise to a finite entropy as T → 0. This is because the oxygen atoms in water form a well defined structure, while the hydrogen atoms remain disordered as a result of the two inequivalent O-H bond lengths, as first pointed by Pauling [116]. The present results do not enable us to determine the exact orientation of these field-decoupled spins. It is worth mentioning that the above-discussed scenarios should be seen as ‘possible scenarios’ and that the concrete physical origin of the features observed at TFI remain elusive. Another particularity in the data presented in figure 21 is that for magnetic fields H > Hc, in a temperature range between TMI and TFI, an intermediary phase appears, probably paramagnetic, as depicted in the schematic phase diagram in figure 22. It is tempting to speculate that this feature can be seen as a separation of the TN and TMI lines in the phase diagram. A possible physical description to this would be related to the energetic competition between the AFI ordered phase and the paramagnetic phases after crossing the MI first- order line, resulting thus in a decrease of TN with increasing magnetic field. Similar experiments were performed in two other κ-D8-Br crystals (crystals #1 and #2). However, for both crystals, in contrast to the pronounced anomaly observed at TFI ≈ 9.5 K under magnetic fields (figure 21), the application of magnetic fields results in a smooth change of the out-of- plane expansivity αb(T ) around the same temperature. In this regard, two factors should be considered as a possible explanation for the absence of the above-described effects: (i) As described in e.g. [117], the SF transitions are very sensitive to the alignment between the applied magnetic field and the easy-axis. A subtle misalignment between the magnetic field and the easy-axis can therefore give rise to a suppression of the transition; (ii) The absence of a sharp transition can also be due to sample inhomogeneities and defects. Sample inhomogeneities would imply, for instance, that portions of percolative SC may vary from sample to sample, reflecting therefore differences in their magnetic properties [1]. 16 J. Phys.: Condens. Matter 27 (2015) 053203 M de Souza and L Bartosch 0 5 10 15 20 0 2 4 6 8 10 SF+Perc. SC Spin-reoriented phase AFI+Perc. SC PI PM H / / b- ax is (T ) T (K) Figure 22. Schematic H -T diagram for κ-D8-Br for a magnetic field applied along the out-of-plane b-axis. Symbols indicate the peak anomaly observed in the expansivity αb(T ), while lines refer to the various phase transitions, see the discussion in the main text. PI and PM stand, respectively, for the paramagnetic insulator and the paramagnetic metal. The hatched area marks the temperature interval in which a double-peak structure in αb(T ) is observed. Spin-reoriented phase denotes the region where M† A and M† B are antiparallel, giving way to an inter-plane antiferromagnetic ordering, see the discussion in the main text. Figure reproduced after [3, 85]. The present findings are compiled in the schematic T versus H diagram shown in figure 22. The dashed line around ∼0.5 T defines the separation boundary between the AFI and the SF phases. The thick line indicates the suppression of percolative SC giving place to field-decoupled and/or flopped spins, here referred to as mixed states. Concluding this section, the thermal expansion results on κ-D8-Br along the out-of-plane b-axis under a magnetic field show that the Mott MI transition in this salt is largely field- independent under magnetic fields up to at least 10 T. Such findings are consistent with the picture of a Mott insulator formed by a set of hole localized in dimers formed by two ET molecules [22]. At TFI = 9.5 K a phase transition induced by a magnetic field is observed, indicative of a SF transition with strong magneto-elastic coupling, accompanied by an enhancement of αb(T ) due to the suppression of the percolative SC under magnetic fields above 1 T. Further experiments, like magnetostriction measurements above and below TFI are highly desired to shed more light on the above-discussed magnetic field-induced lattice effects. 3.4. Thermal expansion under quasi-uniaxial pressure Given the high anisotropy of the κ-(ET)2X charge-transfer salts, thermal expansion measurements under quasi-uniaxial pressure could provide more insights to better understand the transitions at TMI, Tp and Tg . The effect of quasi-uniaxial pressure on the sample was studied for crystal #2 as shown in figure 23. In these experiments, a pressure of (65 ± 5) bar was applied along the out-of-plane b-axis. Roughly speaking, pressure application along the b-axis means a change of the contact between the polymeric-anion chains and the ET molecules and/or enhancement of the tilt of the ET molecules. Figure 23. Main panel: thermal expansion coefficient along the out-of-plane b-axis for κ-D8-Br crystal #2 under ambient (10 bar) and quasi-uniaxial pressure of 65 bar. Dashed lines are guide for the eyes. For clarity, data are shifted. Inset: blowup of the low-T αb(T ) data together with measurement under pressure (65 bar) and magnetic field of 10 T. ZF refers to zero magnetic field. The black dashed arrow indicates the peak position under quasi-uniaxial pressure of 65 bar, while the pink dashed arrow highlights a shift of the peak position to lower T when a magnetic field of 10 T is applied. Figure after [3]. As shown in the main panel of figure 23, the shape of the expansivity curves at TMI and Tp is dramatically changed by applying a quasi-uniaxial pressure of 65 bar. These findings reveal that the anomalies observed in the out-of-plane expansivity αb(T ) at TMI and Tp are strongly affected upon applying pressure, while Tg remains practically unaffected. Since κ-D8-Br is located on the boundary of the Mott MI transition (figure 8), this feature might be associated with a slight shift of its position from the insulating to the metallic side of the phase diagram. Assuming the peak position TMI = 13.6 K as the transition temperature under ambient pressure (actually under a pressure of roughly 10 bar) and TMI = 11.8 K as the transition temperature under quasi-uniaxial pressure (65 bar), one obtains dTMI/dPb � −33 K kbar−1. Such a value is roughly one order of magnitude smaller than the hydrostatic pressure dependence of TMI (dTMI/dP � −380 K kbar−1), estimated from the slope of the Mott MI transition line in figure 8 at T = 11.8 K. Such discrepancy might reflect the anisotropy in the uniaxial-pressure effects. However, the shift of Tp (TMI) to high (low) temperatures is in perfect agreement with a positive (negative) pressure dependence of Tp (TMI) in the phase diagram (figure 8). Applying a magnetic field of 10 T (pink curve in the inset of figure 23), the peak position of the transition, indicated by black and pink dashed arrows, shifts to lower temperatures. As TMI is insensitive to magnetic fields up to 10 T (section 3.3), this observation suggests that the anomaly in αb(T ) under quasi-uniaxial pressure should be triggered by superconductivity. The present thermal expansion results under quasi-uniaxial pressure can be seen as a starting point for experiments under hydrostatic pressure, which will 17 J. Phys.: Condens. Matter 27 (2015) 053203 M de Souza and L Bartosch Figure 24. Thermal expansion coefficient as a function of temperature for κ-D8-Br crystal #3 along the in-plane a-axis. Measurements taken on warming after employing different cooling speeds (−3 K h−1 and −100 K h−1). Data are shifted for clarity. Arrows indicate the respective transition temperatures for the slowly cooled crystal. Dashed lines are used to show the shift of TMI (towards higher temperatures), Tp (towards lower temperatures) and Tg (towards higher temperatures) under fast cooling across Tg . Figure after [3]. provide important information about the physics in the vicinity of the Mott MI transition region in the phase diagram. 3.5. Influence of the cooling speed on Tg in κ-D8-Br In this section, the influence of the cooling speed below the glass-like transition at Tg ≈ 77 K on TMI and Tp is discussed. In fact, the role of the cooling speed on the physical properties of fully and partially deuterated salts of κ-(ET)2Cu[N(CN)2]Br has been intensively discussed in the literature, see e.g. [109, 118]. As shown in figure 24, the expansivity along the in-plane a-axis for a crystal of κ-D8-Br was measured after cooling the sample through the so-called glass-like phase transition by employing two distinct cooling speeds (−3 K h−1 and −100 K h−1). A broadening of the transition, accompanied by a reduction in the size of the peak anomaly at Tp � 30 K, which is related to the critical end-point of the first-order Mott MI transition line in the phase diagram and the Mott MI temperature TMI � 13.6 K, is observed for fast cooling (−100 K h−1). Another remarkable feature is that the peak position of the Tp = 30 K anomaly shifts toward lower temperatures, while the anomaly at TMI = 13.6 K shifts slightly to higher temperatures. Based on these observations, one can conclude that the increase of the cooling speed through Tg results in an opposite effect to that observed upon applying quasi-uniaxial pressure (section 3.4). The observed shift of Tg towards higher temperatures under fast cooling is in agreement with literature results [25]. Resistivity measurements on κ- D8-Br salts [72, 119], synthesized by employing the same method as that used for the salts studied in this work, revealed that under a fast cooling speed12 the resistance increases dramatically below Tg � 80 K, superconductivity is suppressed, giving way to a residual resistance ratio of ∼5. This feature together with the observed enhancement of the anomaly in αa on fast cooling (figure 24) can be assigned to an increase of the scattering provoked by the randomly distributed potential of the polymeric Cu[N(CN)2]Br− anion. A similar situation is encountered in the quasi-1D (TMTSF)2ClO4 superconductor salt, where by rapid cooling (>50 K min−1) superconductivity is destroyed, giving way to a first-order anion-ordering transition at T = 24 K, which is accompanied by a spin-density wave transition around 6 K [12]. 4. Mott criticality Before discussing the Mott criticality in detail, let us first present a general discussion about critical behavior and universality classes. 4.1. Critical behavior and universality classes Continuous phase transitions involve fluctuations on all length scales, leading to the remarkable phenomenon called universality. While details of the interaction do not matter, only robust properties of the system like its effective dimensionality, the absence or presence of long-range interactions or the symmetry of relevant low-energy fluctuations determine the low-energy behavior. As a consequence, it is possible to classify continuous phase transitions according to their universality classes. In real materials, upon approaching the critical temperature, fluctuations are no longer negligible. The more the critical temperature is approached, the stronger are the fluctuations. Finally, fluctuations succeed in destroying the original phase and a new phase, often with distinct symmetries, emerges. To distinguish the two different phases, one usually tries to introduce an order parameter as the expectation value of some field which vanishes in one phase but not the other. The transition is, in many cases, accompanied by broadening effects due to crystal defects or inhomogeneities. Amazingly, around Tc some physical quantities tend to obey power laws in the reduced temperature t = (T −Tc)/T0, which measures the relative distance to the transition temperature. As a consequence of universality, the divergence of the compressibility of water in the vicinity of its critical point is described by exactly the same power-law dependence as the divergence of the susceptibility of a uniaxial magnet in the vicinity of the Curie temperature. In such a uniaxial magnet, the spins can only align themselves parallel or anti-parallel to a given crystal axis. As the magnetization is finite in the symmetry-broken state below Tc and vanishes above Tc, the expectation value of the magnetization may serve as an order parameter. 12 See footnote 10. 18 J. Phys.: Condens. Matter 27 (2015) 053203 M de Souza and L Bartosch Concerning thermodynamic properties, one usually defines the four critical exponents α̃, β, γ and δ, which in the case of magnets are related to the specific heat (at constant pressure) Cp(T ), the spontaneous magnetization Ms(T ) and the susceptibility χ(T ). More generally, Ms(T ) and χ(T ) describe the order parameter and the order parameter susceptibility. Usually, α is used in the literature to refer to the specific heat critical exponent. In order to avoid any confusion with the linear thermal expansion coefficient α(T ), here, α̃ is used to refer to the specific heat critical exponent. Close to criticality, one expects the following power-law behavior, Cp(T ) ∼ A± |t |−α̃ α̃ , (5) Ms(T ) ∼ B|t |β, t � 0, (6) χ(T ) ∼ C±|t |−γ , (7) Ms(H) ∼ DH 1/δ, t = 0. (8) While the amplitudes A±, B, C± and D are non-universal and the coefficients A+ and C+ governing the behavior for t > 0 are generally different from the corresponding coefficients A− and C− for t < 0, the ratios A+/A− and C+/C− are in fact universal, i.e. they are not material-specific and only depend on the underlying universality class. In fact, not all of the above exponents are independent. Using concepts of scaling or the renormalization group, the following identities can be derived [120–124], α̃ + 2β + γ = 2, (9) γ = β(δ − 1). (10) These two relations are frequently called the Rushbrooke and Widom identities. It is striking to note that in the study of critical behavior near a second-order phase transition, materials displaying completely different crystal structures as well as quite different subsystems obey the same critical behavior near Tc, giving thus rise to the universality classes. The theoretical values for the critical exponents of different universality classes accompanied by an example of a phase transition are listed in table 3. It should be noted that sufficiently far away from the transition, fluctuations can largely be ignored and mean field behavior prepails. More precisely, corrections to mean field theory can only be expected to become important for temperatures satisfying |t | � tG, (11) where tG = C ( kB cvξ d 0 )2/(4−d) . (12) is the Ginzburg scale. Here, d is the effective dimensionality of the system, cv is the jump of the specific heat across the phase transition and ξ0 is the bare coherence length. In three dimensions, the universal dimensionless proportionality constant C introduced here is given by C = 1/(32π2) [125, 126], which is much smaller than 1. Quite generally, a crossover is expected near tG from mean field to non-mean field behavior. If, however, the bare coherence length is sufficiently large, the non-mean field regime can turn out to be unmeasurably small. In conventional superconductors, for example, the size of the bare coherence length can be one thousand the size of the lattice spacing, leading to tG ≈ 10−18. As a consequence, mean field theory is essentially exact. From the experimental point of view, the estimate of critical exponents can be a hard task. For instance, owing to the specific heat critical exponent α̃, a reliable estimate of the phonon background can be one of the crucial points. In addition, for a reliable estimate of the critical behavior of a system, fine measurements close to Tc are necessary. Accurate measurements of e.g. the specific heat are required over several orders of magnitude of t . For real materials, a broadening of the transition due to inhomogeneities (impurities or crystal defects) is frequently observed. Due to this, Tc cannot be measured directly, but is rather obtained indirectly via self consistent fittings. 4.2. Scaling Ansatz for the Mott metal-insulator transition In 2005, in a stimulating article entitled ‘Unconventional Critical Behaviour in a Quasi-2D Organic Conductor’, Kagawa and collaborators reported on the criticality at the pressure-induced Mott transition in the organic κ-(BEDT- TTF)2Cu[N(CN)2]Cl charge-transfer salt [45]. In this study, the authors made use of the isothermal pressure-sweep technique, using helium as a pressure transmitting medium, to explore the critical behavior of this organic salt through conductance measurements. The pressure-sweep technique had been previously applied by Limelette and collaborators [127] to study the Mott critical behavior of Cr-doped V2O3, which is now recognized as a canonical Mott insulator system and behaves mean-field like. Very close to the transition the authors also claim Ising universality. In contrast to κ-(BEDT-TTF)2Cu[N(CN)2]Cl, chromium-doped vanadium sesquioxide is a truly three-dimensional material. In the study of the quasi-two-dimensional charge-transfer salt κ-(BEDT- TTF)2Cu[N(CN)2]Cl, the critical behavior of the conductance data at the critical endpoint was analyzed in the framework of the scaling theory of the liquid-gas transition [134]. For simplicity, in the so-called non-mixing approximation, the rescaled pressure and the rescaled temperature were used as two independent scaling variables to obtain the critical exponents β = 1, γ = 1 and δ = 2, as listed in table 3. Substituting these values in the Rushbrooke relation one obtains α̃ = −1. As pointed out by the authors, the obtained critical exponents do not fit in the well-known universality classes, indicating the experimental discovery of a new universality class. A possible explanation of the experimentally observed exponents of unconventional criticality was given by Imada and collaborators [135, 136]. The appearance of unconventional critical exponents came as a surprise to many who expected Ising universality. This expectation was based on an argument by Castellani et al [137] who argued that, even though there is no symmetry breaking, the double occupancy of lattice sites can serve as an order parameter. While doubly occupied (or empty) sites are 19 J. Phys.: Condens. Matter 27 (2015) 053203 M de Souza and L Bartosch Table 3. Different universality classes with their respective critical exponents, accompanied by a proposed example of the phase transition. The theoretical estimates for the critical exponents are from [130] (3D Ising), [131] (3D XY ) and [132] (3D Heisenberg). Universality class α̃ β γ δ Examples of phase transition Mean-field 0 0.5 1.0 3.0 Superconducting transition in conventional superconductors or Mott MI transition in (V1−xCrx)2O3 [127] 2D Ising 0 0.125 1.75 15 Preroughening transition in GaAs [128] 3D Ising 0.110(1) 0.3265(3) 1.2372(5) 4.789(2) Liquid-gas transition 3D XY −0.0151(3) 0.3486(1) 1.3178(2) 4.780(1) Superfluid transition in 4He [129] 3D Heisenberg −0.1336(15) 0.3689(3) 1.3960(9) 4.783(3) Ferromagnetic transition in a clean and isotropic ferromagnet Unconventional criticality −1 1 1 2 Mott MI transition in κ-(BEDT-TTF)2Cu[N(CN)2]Cl [45] Note: The mean-field values are derived in any textbook on critical phenomena (see e.g. [121–124]), values quoted for the 2D Ising model are the exact values from Onsager’s famous solution [133]. localized on the insulating side of the phase transition, they do proliferate on the metallic site. The order parameter was therefore expected to be of an Ising type. The Mott criticality was also studied in the framework of dynamical mean-field theory of the Hubbard model by Kotliar et al [140]. These authors confirmed the statement that the Mott transition should lie in the Ising universality class. Having a phase transition between a low-density gas of localized doubly occupied or empty lattice sites to a high-density fluid of unbound doubly occupied or empty lattice sites is quite analogous to the well-known liquid-gas transition which lies also in the Ising universality class. An attempt to reconcile the experimentally observed unconventional critical exponents β = 1, γ = 1 and δ = 2 with the well-established two-dimensional Ising universality class was made by Papanikolaou and collaborators [46]. These authors pointed out that the critical exponents derived from transport measurements do not necessarily coincide with the thermodynamic critical