Home Search Collections Journals About Contact us My IOPscience

Probing the Mott physics in -(BEDT-TTF)2X salts via thermal expansion

This content has been downloaded from IOPscience. Please scroll down to see the full text.

2015 J. Phys.: Condens. Matter 27 053203

(http://iopscience.iop.org/0953-8984/27/5/053203)

View the table of contents for this issue, or go to the journal homepage for more

Download details:
This content was downloaded by: marianodesouza
IP Address: 193.171.32.17
This content was downloaded on 22/01/2015 at 17:36

Please note that terms and conditions apply.

 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)2 X 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 electronic correlations in two dimensions. In particular, single

crystals of wholly deuterated charge-transfer salts of the κ
A tiny change in physical parameters can sometimes have phase with X = Cu[N(CN)2 ]Br have been attracting broad
dramatic consequences. This is particularly true in correlated interest due to fact that they lie in the vicinity of a first-
condensed matter systems in which a small alteration in order phase transition line in the phase diagram [1], which
pressure, chemical composition, or temperature can tune a separates the metallic from the insulating state, thus giving
system across a phase transition. In this context, the Mott the possibility of investigating the Mott MI transition using
metal-to-insulator (MI) transition represents one of the most temperature as a tuning parameter. One result to be discussed
remarkable examples of electronic correlation phenomena. In in detail in the frame of this paper refers to the first experimental
contrast to conventional band insulators in which all bands observation of the role played by lattice degrees of freedom for
are either filled or empty, the Mott insulating state is driven the Mott MI transition in the above-mentioned materials [2].
by interactions and emerges once the ratio of the electron– Both the discontinuity and the anisotropy of the lattice
electron interaction U to the hopping matrix element (kinetic parameters observed via high-resolution thermal expansion
energy) t exceeds a critical value. It is essentially this ratio experiments indicate a complex role of the lattice effects
which is tuned in many experiments. Molecular conductors at the Mott MI transition for the κ-(BEDT-TTF)2 X charge-
transfer salts, which cannot be interpreted simply by taking
of the κ-phase of (BEDT-TTF)2 X (where BEDT-TTF refers
into account a purely 2D electronic model. Furthermore, in
to bisethylenedithio-tetrathiafulvalene, i.e. C10 S8 H8 and X to
the frame of the present work, a model based on the rigid-unit
a monovalent counter anion), usually called charge-transfer
modes scenario is proposed to describe the negative thermal
salts, have been recognized as model systems to investigate
expansion above the so-called glass-like transition temperature
4 Current address: Institute of Semiconductor and Solid State Physics, Tg
77 K in deuterated charge-transfer salts of κ-(BEDT-
Johannes Kepler University Linz, 4040 Linz, Austria. TTF)2 Cu[N(CN)2 ]Br and parent compounds.

0953-8984/15/053203+27$33.00 1 © 2015 IOP Publishing Ltd Printed in the UK

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)2 Cu2 (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 Figure 1. (BEDT-TTF), or simply ET, molecule, which is the basic
temperature presents a non-monotonic behavior and becomes entity that furnishes the quasi-2D (BEDT-TTF)-based
charge-transfer salts. Figure reproduced with permission from [15].
more pronounced at low temperatures [7].
Up to now, no clear experimental evidence of magnetic
ordering in κ-(BEDT-TTF)2 Cu2 (CN)3 has been reported in sensitivity of the vibration mode of the central C=C bonds, the
the literature [8]. Thus, this system has been recognized as a so-called ag (ν3 ) mode, upon going from the metallic to the
candidate for the realization of a spin-liquid [9]. At T = 6 K, insulating state.
a crossover from a paramagnetic Mott insulator to a genuine In the following, let us briefly discuss the various phases
quantum spin-liquid has been proposed [6]. Nevertheless, the in which ET charge-transfer salts can crystallize.
physical nature of such a crossover, sometimes called hidden During the crystallization process of the (ET)2 X salts,
ordering, is still controversial. Expansivity measurements the ET molecules can assume different spatial arrangements,
performed on this material revealed a pronounced phase- giving rise to different phases. Several thermodynamically
transition-like-anomaly at T = 6 K, which coincides nicely stable phases, usually referred to by the Greek letters α, β, θ ,
with specific heat results reported in the literature [6]. These λ and κ, are known, see figure 2 for a schematic representation
results provide strong evidence that this lattice instability at of the main known phases. This review is focussed on salts of
T = 6 K is directly linked to the proposed spin-liquid phase. the κ-phase family. Interestingly enough, different crystalline
The electronic properties of κ-(BEDT-TTF)2 X charge- arrangements imply different band fillings and consequently
transfer salts have been reported in various review articles, different physical properties. A detailed discussion about each
see, for instance [10–24]. Here we focus on topics linked phase is presented in [21, 22].
to dilatometric studies of the κ-(BEDT-TTF)2 X salts. Some The packing pattern of the κ-phase differs distinctly from
parts of this review are based on the Ph.D. thesis of one of the others in the sense that it consists of two face-to-face ET
us [3]. Before giving a detailed outline of the main body of molecules, see figure 2. The two molecules interact with
this review article at the end of this introductory chapter, let us each other via overlap between the highest occupied molecular
first briefly review some general properties of the κ-(BEDT- orbitals (HOMOS), as the HOMOS of each molecule are
TTF)2 X salts and discuss their crystallographic structure in partially occupied [21].
section 1.1, their phase diagram in section 1.2 and the most Due to the intrinsic strong dimerization in the κ-(ET)2 X
relevant experimental results from the literature related to this family, pairs of ET molecules can be treated as a dimer unity.
work in section 1.3. 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
1.1. The (BEDT-TTF) molecule and crystallographic structure
anion X governs the ratio between the inter-dimer transfer
of the κ -ET salts
integrals t and t  . For example, for the κ-(ET)2 Cu2 (CN)3 salt
The basic entity which furnishes the structure of the present t  /t = 0.83 [28, 29], suggesting a strongly frustrated triangular
class of organic conductors is the (BEDT-TTF) molecule, lattice in this compound, to be discussed in more detail
usually abbreviated simply to ET. The structure of the ET below.
molecule is shown in figure 1. As a result of the two distinct It is well to note that, more recently, a new phase, called
possible configurations of the ethylene end groups in their mixed phase, with alternating α- and κ-type arrangements of
extremes (known as staggered and eclipsed configurations), the the ET molecules, has been synthesized [30]. In addition, a
ET molecule is not completely planar. It has been proposed dual-layered superconductor with alternating α  - and κ-type
that the degrees of freedom of the ethylene end groups can packing has been reported in the literature [31].
influence the electronic properties of the salts of the κ-family, The crystallographic structure of the κ-(ET)2 Cu[N(CN)2 ]
see [25] and references cited therein. Employing infra-red Br salt [32] is displayed in figure 4. The structure of the latter
imaging spectroscopy, the authors of [26] reported on the 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)2 X 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)2 Cu(NCS)2 and
κ-(ET)2 Cu2 (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
Figure 4. The crystallographic structure of κ-(ET)2 Cu[N(CN)2 ]Br.
the ET layers, which lie in the crystallographic ac-plane and
While a and c are the in-plane axes, b is along the out-of-plane
correspond to the large face of the crystal. In the perpendicular direction. ‘ET’ and ‘polymeric anion’ indicate layers of ET
direction to the referred layers, i.e. the b-direction5 , the molecules and anions, respectively. The various atoms are
insulating anion layer partially blocks charge transfer, so that represented by different colors, as indicated in the label. For clarity,
the electrical conductivity along this direction (⊥ to the layers) protons at the end of the ET molecules are omitted. Figure taken
from [3].
5 For the κ-(ET)2 Cu2 (CN)3 salt, the a-axis is the out-of-plane direction.

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− Figure 6. View along the out-of-plane direction (a-axis) of the
polymeric anion in κ-(ET)2 Cu[N(CN)2 ]X, view along the b-axis. polymeric anion layer of Cu2 (CN)− 3 in κ-(ET)2 Cu2 (CN)3 . The
Here, X refers to either Br or Cl. Dashed lines highlight the X...N rectangle indicates the contour of the unit cell. The N/C11 cyanide
weak contacts between the anion chains. The rectangle indicates the ion is located on an inversion center, being thus crystallographically
contour of the unit cell. Figure reprinted from [34]. Copyright 1991, disordered. The ellipsoids therefore represent either a carbon or a
with permission from Elsevier. nitrogen atom with a 50% probability. Figure reprinted from [33].
Copyright 1991 American Chemical Society.

is reduced by a factor of 100–1000, depending on the counter

discussed below. Band structure calculations suggest that these
anion. Due to this anisotropy, these materials are called quasi-
materials should be metals down to low temperatures [21].
two-dimensional (quasi-2D) conductors.
However, due to correlation effects, the phase diagram
As displayed in figure 5, the Cu[N(CN)2 ]Br− (or
embodies various phases, including the paramagnetic Mott
Cu[N(CN)2 ]Cl− ) polymeric anion structure is composed of
insulator (PI), the antiferromagnetic Mott insulator (AF),
planar triply-coordinated Cu(I) atoms with two bridging
the superconductor and the anomalous metal, see the phase
dicyanamide [(NC)N(CN)]− ligands, forming a zig-zag pattern
diagram depicted in figure 7.
chain along the a-axis. The terminal Br− ions, which overlap
A remarkable feature is the competition between
slightly with the central N atoms of neighboring polymeric
antiferromagnetic ordering and superconductivity. The
chains, make the third bond at each Cu(I) atom [10].
similarities between the present charge-transfer salts and the
As shown in figure 6, the counter anion Cu2 (CN)− 3 high-Tc cuprate superconductors have been discussed, see
differs distinctly from the Cu[N(CN)2 ]Br− anion in that it
e.g. [14, 36, 37]. The major difference, however, between
does not form chains, but a planar reticule of Cu(I) ions
these two classes of materials is that while for the cuprate
coordinated in a triangular fashion and bridging cyanide
superconductors one deals with a doping-controlled Mott
groups. Notably, one of the cyanide groups (labeled N/C11
transition, for the charge-transfer salts to be discussed here
in figure 6) is located on an inversion center and therefore
one has a bandwidth-controlled Mott transition. In addition, no
must be crystallographically disordered [33]. As will be
spin-glass phase is observed in organic conductors between the
discussed in section 5, this particular arrangement is likely
antiferromagnetic and superconducting phases. The symmetry
to be responsible for the absence of the so-called glass-like
of the superconducting order parameter in these materials is
transition in κ-(ET)2 Cu2 (CN)3 .
still a topic of debate, see e.g. [38] and references therein.
Directional-dependent thermal expansion experiments, to
From the experimental point of view, a remarkable feature
be discussed in section 3, will show that the more pronounced
is the tunability of such systems. In contrast to the cuprate
lattice effects, associated with the Mott MI transition for κ-D8-
superconductors, where the ground states are tuned by doping,
Br, occur along the in-plane direction which is parallel to the
i.e. band filling and therefore disorder is unavoidable, the
anion chains direction (a-axis). In addition, we will propose
ground states of organic conductors can be tuned by chemical
a model based on the collective vibration modes of the CN
substitution, namely counter anion substitution and/or by
groups of the polymeric anion chains to explain the negative
applying external pressure. Pressure mainly induces strain
thermal expansion along the a-axis above Tg
77 K.
which in turn leads to modified hopping matrix elements t and
For completeness, the lattice parameters of the κ-(ET)2 X
t  and thereby modifies the bandwidth W . As a consequence,
charge-transfer salts discussed in this work are listed in table 1.
applying pressure leads to a change of the ratio W/U . A
pressure of a few hundred bar, which is easily attainable by
1.2. Phase diagram of κ -(ET)2 X employing the 4 He-gas pressure technique, is enough to make
Charge-transfer salts of the κ-(ET)2 X family have been a wide sweep over relevant regions of the pressure versus
recognized as strongly correlated electron systems. Among temperature (P -T ) phase diagram6 .
other features, the latter statement is based on the unusual 6 For the Mott insulator V2 O3 , for example, a pressure of a few kbar is required
metallic behavior inherent to these systems, as will be 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)2 X 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.

paramagnetic
100 insulator “crossover”
T*
Temperature (K)

(P0,T0)
TMI metal
(1st order)
10

antiferro-
magnetic
insulator superconductor

1
Pressure
X = Cu[N(CN)2]Cl Cu[N(CN)2]Br Cu(NCS)2 W/U
Figure 7. Conceptual pressure versus temperature (P -T ) phase
diagram for the κ-(ET)2 X charge-transfer salts, after [35]. Arrows
pinpoint the positions of the various salts with their counter anion X Figure 8. Pressure versus temperature phase diagram of
at ambient pressure. The ratio of the on-site Coulomb repulsion κ-(ET)2 Cu[N(CN)2 ]Z, with Z = Br or Cl. Lines indicate the
relative to the bandwidth, i.e. U /W , is reduced by application of respective phase transitions. D8-Br and H8-Br indicate the positions
hydrostatic pressure. (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.
The phase diagram shown in figure 8 has been mapped Thin solid lines (red), demarcating the hatched area, refer to the
by NMR and ac susceptibility [39], transport [41] and ultra- positions of the anomalies observed via ultrasonic measurements
sonic [42] measurements under helium gas pressure on (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
the salt with X = Cu[N(CN)2 ]Cl. In this generic phase Mott MI phase transition line. PI, AFI and SC denote the
diagram, the antiferromagnetic transition line, not observed paramagnetic insulator, the antiferromagnetic insulator and the
in ultra-sound experiments [42], was obtained from the superconducting phase, respectively. The open circle refers to the
NMR relaxation rate [39]. From the splitting of NMR 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.
lines, the estimated magnetic moment per dimer is (0.4–
As can be seen in the figure, the slope of dTMI /dP can be both
1.0) µB . The superconducting transition line was determined positive and negative, indicating that upon crossing the MI line the
from the ac susceptibility [39] and its fluctuations in the entropy can either increase or decrease. While the entropy increases
immediate vicinity of the Mott transition by means of Nernst for dTMI /dP > 0 when crossing the MI line from the metallic to the
effect measurements [40], whereas the S-shaped first-order 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
MI transition line via ultra-sound [42], transport [41] and ac to magnetic ordering. Figure taken from [2].
susceptibility [39] measurements.
This first-order Mott MI transition line ends in a critical classes and they assigned this to the quasi-2D structure of the
point (P0 , T0 ), which has been studied by several groups present substances. Such a scenario shall be examined in more
[39, 41, 42, 44, 45]. Among possible scenarios, this critical detail in section 4. The possibility of tuning the system from the
end-point has been discussed in analogy with the liquid-gas insulating to the metallic side of the P versus T phase diagram,
transition, see e.g. [46]. In this scenario, above (P0 , T0 ) the i.e. crossing the first-order Mott MI line by application of
metallic state cannot be distinguished from the paramagnetic pressure, is indicative of a bandwidth-controlled Mott MI
insulating state. More recent studies suggest that lattice effects transition. Owing to the position of the salts with different
are relevant close to the Mott critical end-point [2, 4, 5]. counter anions X, as shown in figure 8, the fully hydrogenated
By means of resistivity measurements under pressure, the salt with X = Cu[N(CN)2 ]Br superconducts below 12 K,
critical behavior in the proximity of the point (P0 , T0 ) was the highest Tc under ambient pressure among all organic
investigated by Kagawa et al [45]. These authors found critical conductors till the present date, whereas the salt with counter
exponents which do not fit in the most common universality 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)2 Cu[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 Figure 9. P -T phase diagram of the κ-(ET)2 Cu2 (CN)3 salt, obtained
smaller than the C–H bonds. Furthermore, the contact between from resistance and NMR measurements under pressure using an oil
pressure cell. The Mott MI transition and crossover lines are
the C–D2 end groups and the anions is weaker than that between
associated with the temperature at which the quantities spin-lattice
the C–H2 end groups and the anions. As a consequence, the relaxation rate divided by temperature (1/T1 T ) and dR/dT show a
lattice of the fully deuterated salt is softer than the lattice maximum, see labels indicated in the figure. The Fermi-liquid
of the fully hydrogenated salt. This process of exchanging regime (yellow area) was fixed by the temperature regime where the
atoms is usually called applying ‘chemical pressure’. In resistance R follows the R0 + AT 2 relation and (1/T1 T ) is constant.
The superconducting transition line was obtained via in-plane
fact, application of pressure (chemical or external) changes
resistance measurements. Reprinted with permission from [48].
the distance between the ET molecules so that the overlap Copyright 2005 by the American Physical Society.
between π -orbitals’ is increased and, as a consequence, the
bandwidth (W) is changed. The ratio W/U is considered the by Fournier et al [42] revealed the existence of two new
pivotal parameter which defines the various phases of these lines in the phase diagram (not indicated in figure 8) which
substances [35]. merge at the critical end point. These lines indicate the
Interestingly enough, for the salt with the anion X = softening of the lattice response and should therefore coincide
Cu2 (CN)3 , magnetic ordering is absent in the whole insulating with anomalies expected in future dilatometric studies under
region. The P versus T phase diagram of the κ- pressure. It is important to keep in mind that the two lines
(ET)2 Cu2 (CN)3 salt is depicted in figure 9. observed in ultrasound experiments are different from the
metal-to-insulator crossover line above Tc .
1.3. Key experimental results The experimental findings obtained by Fournier et al are
shown in figure 10. From this data set, pronounced anomalies,
As mentioned above, the metallic state of κ-(ET)2 X presents depending on the pressure, are observed below around 40 K.
some distinct properties as compared to those of conventional The maximum amplitude of the softening in the sound velocity
metals. For instance, in the κ-H8-Br salt, the out-of-plane (at T ≈ 34 K, P ≈ 210 bar) marks the critical end point
resistance as a function of temperature has a non-monotonic (P0 , T0 ) and corresponds to a softening of roughly 20% of
behavior [49] with a maximum around 90 K, which was the sound velocity. On increasing the pressure, the peak
observed to be sample-dependent. Decreasing temperature, position shifts its position to higher temperatures. Below P0
this maximum is followed by a sudden drop and the resistivity the anomaly becomes gradually less pronounced and saturates
sharply changes its slope around T ∗
40 K. Below Tc
at around 32 K. The peak position was thus used by the authors
11.5 K superconductivity is observed. On applying pressure, to draw the two crossover lines in the P -T phase diagram (see
T ∗ shifts to higher temperatures and disappears at roughly figure 11).
2 kbar. From about T ∗
40 K down to Tc
11.5 K the The large anomalies in the sound velocity observed by
resistance obeys a T 2 behavior which is frequently assigned Fournier et al were discussed in terms of a diverging electronic
to a coherent Fermi liquid state. NMR studies revealed compressibility. As a matter of fact, acoustic and lattice
a peak in the spin-relaxation rate divided by temperature anomalies are, according to dynamical mean-field theory
(1/T T1 ) around T ∗ for the present compound [50]. The (DMFT) calculations, expected at the Mott MI transition as a
anomalous metallic behavior around T ∗ was also observed by reaction to the softening of the electronic channel, see [51, 52].
thermal, magnetic, elastic and optical measurements, see [49] The explanation for this is that in the metallic state the
and references therein. The origin of the T ∗ anomaly is conduction electrons contribute more to the cohesion of the
still controversial. Ultrasonic velocity measurements under solid than in the insulating one. Based on this argument,
pressure on the compound κ-(ET)2 Cu[N(CN)2 ]Cl performed according to [51], as an effect of the localization, abrupt

6
J. Phys.: Condens. Matter 27 (2015) 053203 M de Souza and L Bartosch

of quasi-2D systems. Furthermore, knowing precisely the

0.00 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].
-0.05
Owing to the magnetic properties of the κ-(ET)2 X family,
several studies employing different experimental methods have
∆V/V

-0.10 been carried out on the salt with X = Cu[N(CN)2 ]Cl [54–56].
By means of resistance measurements, a magnetic-field-
210 bars
-0.15 220 150 bars induced Mott MI transition was observed by Kagawa et al [57]
200
275 215 by varying temperature, pressure and the magnetic field.
352 226
460 230 From an analysis of their NMR line shape, relaxation rate
-0.20
and magnetization data, Miyagawa et al [56] were able to
26 28 30 32 34 36 38 40
describe the spin structure of this state. Below T = 26–
10 20 30 40 50 60 70 80 90
27 K, they found a commensurate antiferromagnetic ordering
T (K)
with a moment of (0.4–1.0) µB /dimer, as already mentioned
Figure 10. Main panel: Relative sound velocity V /V versus T above. The observation of an abrupt jump in the magnetization
under pressure for the κ-(ET)2 Cu[N(CN)2 ]Cl salt. The various curves for fields applied perpendicular to the conducting layers,
pressure values are indicated in the label of the figure. Inset: data i.e. along the b-axis, was discussed in terms of a spin-flop
for pressures below 230 bar. Reprinted with permission from [42].
Copyright 2003 by the American Physical Society.
(SF) transition. Furthermore, a detailed discussion about
the SF transition, taking into account the Dzyaloshinskii–
60
Moriya exchange interaction, was presented by Smith et al
[58, 59]. Similarly to that for the pressurized chlorine salt,
50
a magnetic-field-induced MI transition was also observed in
partially deuterated κ-(ET)2 Cu[N(CN)2 ]Br [60]. In addition,
40
a discussion on the phase separation and SF transition in κ-
D8-Br was reported in the literature [61].
T (K)

30 M
As already mentioned above, the organic charge-transfer
I
salt κ-(ET)2 Cu2 (CN)3 has the peculiarity that the ratio of its
20
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
10
SC-I SC-II SC-III triangular lattice with the coupling constant J = 250 K, where
this coupling constant is obtained by fitting the magnetic
0
0 100 200 300 400 500 600 susceptibility using the triangular-lattice Heisenberg model.
P (bar) Magnetic susceptibility and NMR measurements revealed no
traces of long-range magnetic ordering down to 32 mK [8].
Figure 11. Phase diagram obtained from ultrasound experiments Based on these results, this system has been proposed to be a
under pressure for the κ-(ET)2 Cu[N(CN)2 ]Cl salt. Different
symbols refer to the various anomalies observed on three different candidate for the realization of a spin-liquid state [62].
samples. The critical point (P0 , T0 ) is indicated by the gray circle. Interestingly, upon applying pressure (see figure 9), the
SC-I and -II indicate metastable superconductivity, while SC-III system becomes a superconductor [48], i.e. superconductivity
indicates bulk superconductivity. Dotted hexagon indicates the appears in the vicinity of a spin-liquid state. As shown in
pressure point, obtained from microwave resistivity measurement at figure 13, specific heat experiments revealed the existence of
ambient pressure. Reprinted with permission from [42]. Copyright
2003 by the American Physical Society. a hump at 6 K, insensitive to magnetic fields up to 8 T.
This feature was assigned to a crossover from the
changes of the lattice parameters are expected at the Mott MI paramagnetic Mott insulating to the quantum spin-liquid
transition. We shall discuss the lattice effects for the present state [6]. Such a ‘crossover’ has been frequently referred to
material in section 3. as ‘hidden ordering’. Below 6 K, the specific heat presents a
Regarding the Fermi surface, the weak interlayer distinct T -dependence, including a T -linear dependence in the
transfer integrals were nicely estimated from magnetore- temperature window 0.3–1.5 K, as predicted theoretically for
sistance measurements for deuterated κ-(ET)2 Cu(NCS)2 a spin-liquid [63]. The spin entropy in this T -range is roughly
(figure 12). 2.5% of R ln 2 [64], indicating that below 6 K only 2.5% of
From a more fundamental point of view, the determination the total spins contribute to the supposed spin-liquid state.
of the interlayer transfer integral in a quasi-2D system is Extrapolating the low-T specific heat data, the authors of [6]
quite helpful, for instance, to understand the validity of the found a linear specific heat coefficient γ = (20 ± 5) mJ mol−1
criterion usually adopted to classify whether the interlayer K−2 (see figure 14). The latter is sizable given the insulating
charge transport in quasi-2D systems is incoherent or coherent. behavior of the material and contrasts with a vanishing γ value
For a detailed discussion, see e.g. [53] and references for the related compounds κ-(ET)2 Cu[N(CN)2 ]Cl and fully
cited therein. Thus charge-transfer salts of the κ-(ET)2 X deuterated κ-(ET)2 Cu[N(CN)2 ]Br. A finite γ could indicate a
phase serve as a good platform to investigate the fermiology 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)2 Cu(NCS)2 . (b) Interlayer magnetoresistance Rzz for deuterated
κ-(ET)2 Cu(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 Figure 14. Low-temperature specific heat data for the
for the κ-(ET)2 Cu2 (CN)3 salt plotted against T 2 until 10 K. A broad κ-(ET)2 Cu2 (CN)3 salt as well as κ-D8-Br, κ-(ET)2 Cu[N(CN)2 ]Cl
hump anomaly around T = 6 K can be clearly observed. Inset: and β  -(ET)2 I Cl2 , plotted is Cp /T versus T 2 . The extrapolation of
specific heat data for κ-(ET)2 Cu2 (CN)3 , assuming that the data of the red solid line towards zero temperature indicates the existence of
the superconductor κ-(ET)2 Cu(NCS)2 describe the phonon a T -linear contribution for the spin-liquid candidate
background. Reprinted by permission from Macmillan Publishers κ-(ET)2 Cu2 (CN)3 . Data under magnetic fields for κ-(ET)2 Cu2 (CN)3
Ltd: [6], copyright 2008. are also shown. Reprinted by permission from Macmillan
Publishers Ltd: [6], copyright 2008.
A gauge theory with an attractive interaction between Ampère’s force law which states that two current carrying
spinons mediated via a non-compact U (1) gauge field was wires with parallel currents attract. The authors deduce
proposed by Lee et al [65] to describe a possible spin- that the pairing of spinons is accompanied by a spontaneous
liquid state in κ-(ET)2 Cu2 (CN)3 . Within this model, a breaking of the lattice symmetry, which in turn should
pairing of spinons on the same side of the Fermi surface couple to a lattice distortion (analogously to the Spin-Peierls
should occur at low temperatures. The attractive interaction transition) and should be detectable experimentally via x-ray
between spinons with parallel momenta is quite analogous to 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 Table 2. Samples of the organic conductor κ-(ET)2 X on which
linear at low T , it should be kept in mind that the very- thermal expansion measurements were performed. Crystal #4 was
used to perform preliminary studies of Raman spectroscopy. In [2]
low temperature behavior of κ-(ET)2 Cu2 (CN)3 is dominated crystal #2 is referred to as crystal #3.
by a diverging nuclear contribution to the specific heat.
Indeed, the data set reported by Yamashita et al [6] is also Crystal
Anion X Number Direction Batch
reasonably well described by a T 2/3 behavior at intermediate
temperatures [64]. D8-Br 1 a, b, c A2907a
Even though specific heat measurements by Yamashita D8-Br 2 a, b A2995a
et al [6] are compatible with a spinon Fermi surface D8-Br 3 a, b, c A2907a
D8-Br 4 — A2907a
with gapless excitations, thermal conductivity measurements H8-Br 7 b A2719a
reported by Yamashita et al in [66] show a vanishing low- H8-Cu2 (CN)3 1 a SKY 1050b
temperature limit of the thermal conductivity divided by H8-Cu2 (CN)3 1 c KAF 5078b
temperature, indicating the absence of gapless excitations. The H8-Cu2 (CN)3 2 b KAF 5078b
excitation gap was estimated to be κ
0.46 K. Applying a a
Refers to samples provided by Schweitzer
theory based on Z2 spin liquids, Qi et al argued that thermal (University of Stuttgart).
transport properties should be dominated by topological vison b
To samples provided by Schlueter (Argonne National
excitations [67]. The gap found by Yamashita et al [66] Laboratory).
should therefore be interpreted as a vison gap. It was pointed
out by Ramirez [64] that before a final declaration of κ- CD2 stretch vibration modes, no signs of any CH2 or CDH
(ET)2 Cu2 (CN)3 as a spin-liquid is made some points should vibrations could be detected, see [71]. Hence, the grade of
be clarified, among them the anomaly observed in the specific
deuteration of κ-(D8-ET)2 Cu[N(CN)2 ]Br is at least 98% [72].
heat at 6 K. We will come back to the system κ-(ET)2 Cu2 (CN)3
As discussed in detail in [49], by employing the preparation
in section 5 where we will discuss its thermal expansion
technique described above, hydrogenated single crystals of
properties. Regarding the fascinating properties of this salt,
the same substance resulted in samples whose resistivity re-
recently the existence of two magnetic field-induced quantum
vealed reduced inelastic scattering and enhanced residual re-
critical points was reported in the literature [68]. Yet, based on
sistivity ratios. In general, the crystals have bright surfaces
a careful analysis of their muon spin rotation results, performed
in shapes of distorted hexagons with typical dimensions of ap-
under extreme conditions, the authors of [68] discuss a possible
proximately 1×1×0.4 mm3 . Due to the small size and fragility
interpretation for the features observed in several physical
of the crystals, the experimental challenge lies in assem-
quantities around 6 K in terms of the formation of bosonic
bling them in the dilatometer. The samples of the deuterated
pairs emerging from a portion of the fermionic spins in κ-
and hydrogenated variants of κ-(ET)2 Cu[N(CN)2 ]Br and κ-
(ET)2 Cu2 (CN)3 .
(ET)2 Cu2 (CN)3 salts used for measurements discussed in this
The remainder of this article is organized as follows:
In section 2, we discuss experimental aspects like sample work are listed in table 2. The fully deuterated (hydrogenated)
preparation and high-resolution thermal measurements. salts of κ-(ET)2 Cu[N(CN)2 ]Br will be subsequently referred to
Section 3 is dedicated to an exhaustive discussion on as κ-D8-Br (κ-H8-Br). Samples of κ-(ET)2 Cu2 (CN)3 studied
the thermal expansivity of fully deuterated salts of κ- here were prepared according to [33].
(ET)2 Cu[N(CN)2 ]Br, while in section 4 the Mott criticality
is comprehensively reviewed. Section 5 is devoted to 2.2. Thermal expansion measurements
the discussion of the thermal expansivity of the spin-
As already mentioned in the introduction, thermal expansion
liquid candidate κ-(ET)2 Cu2 (CN)3 . Finally, conclusions and
coefficient measurements are a very useful and powerful tool
perspectives are presented in section 6.
to explore phase transitions in solid state physics research
[3, 4, 47, 73–85]. For instance, combining specific heat
2. Experimental setup and thermal expansion data one can employ the Ehrenfest
relation to determine the pressure dependence of a second-
2.1. Sample preparation
order phase transition temperature. Such an analysis may
As thermal expansion measurements under external hydro- serve also as a check of its direct measurement as a function of
static pressure are extremely challenging, using compounds of hydrostatic pressure. Furthermore, contrasting with specific
different chemical composition currently represents the most heat which is an isotropic property, anisotropic effects may
practical way to probe the phase diagram of the κ-(ET)2 X fam- be studied via thermal expansion measurements. Regarding
ily via dilatometry. Deuterated (98%) bis(ethylenedithiolo)- the charge-transfer salts of the κ-phase, to our knowledge, the
tetrathiofulvalene (D8-ET) was grown according to Hartke first high-resolution thermal expansion experiments along the
et al [69] and Mizumo et al [70], making the reduction of car- three crystallographic directions were carried out about twenty
bon disulfide with potassium in dimethylformamide and subse- years ago by Kund and collaborators [86].
quent reaction with deuterated (98%) 1.2-dibromoethane. The The measurements presented in the present work were
intermediate thione C5 D4 S5 was mixed with triethylphosphite carried out by making use of an ultra-high-resolution
in an inert atmosphere kept at 1200 ◦ C and recrystallized sev- capacitance dilatometric cell, built after [87], with a maximum
eral times from chlorobenzene. By investigating the CH2 and 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

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

Sample
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 (C 2 ). 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
C 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
Figure 15. Schematic representation of the cell used for the thermal
expansion measurements. Details are discussed in the main text.
that the empty capacitance of the system reads 16.7 pF.
Figure adapted from [3]. 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
(pumping of Helium bath) to 200 K under magnetic fields up
reported here the field was applied parallel to the measured
to 10 T. In order to avoid external vibrations, the cryostat is
direction. For the direction-dependent thermal expansion
equipped with shock absorbers. A detailed description of the
measurements, the alignment of the crystal orientation was
dilatometer used in this work has been presented in several
made using an optical microscope and guaranteed with an error
works [90–92]. The principle of measurement is described
margin of ±5◦ . To obtain the intrinsic thermal expansivity of
in the following. As sketched in figure 15, the cell, which is
the investigated specimen the thermal expansivity of copper of
entirely made of high-purity copper to ensure good thermal
the dilatometric cell body was subtracted from the raw data.
conductivity and covered with gold is constituted basically of
Apart from that, no further treatments like splines or any other
a frame (brown line) and two parallel pistons (orange), the
kind of mathematical fittings were done.
upper one being movable. As a matter of fact, the gold layer The linear thermal expansion coefficient αi in the direction
should work like a protection, avoiding oxidation of the cell. i at constant P is defined as
Nevertheless, after some years in use, two striking anomalies
have been observed at T
212 K and 230 K. These anomalies 1
∂l(T ) 
αi = , (1)
are assigned to the formation of copper-oxide (CuO) [93] in l ∂T P
the body of the cell. Due to this, experimental data taken in where l refers to the sample length. The physical quantity
this T window are not reliable. described by equation (1) will be frequently used in this
The sample is placed between these two pistons and by work. From the length changes of the sample l(T ) =
moving the upper piston carefully, the starting capacitance is l(T ) − l(T0 ) (T0 is a fixed temperature), which is the physical
fixed. The lower piston is mechanically linked to a parallel quantity measured, the linear thermal expansion coefficient
plate capacitor, as schematically represented by springs in [equation (1)] was approximated by the differential quotient as
figure 15. The variation of the sample length, i.e. contraction
or expansion, as the temperature is lowered or increased, l(T2 ) − l(T1 )
α(T ) ≈ , (2)
together with cell effects corresponds exactly to the change l(300 K) × (T2 − T1 )
of the distance between the plates of the capacitor and where T = (T1 + T2 )/2 and T1 and T2 are two subsequent
consequently to a change of the capacitance, so that very temperatures of data collection. Unless otherwise stated, in
tiny length changes can be detected. In this construction, order to guarantee thermal equilibrium in all thermal expansion
the distance between the plates of the capacitor is about measurements reported here, a very low temperature sweep
100 µm. For the sake of avoiding stray electric fields, the rate (±1.5 K h−1 ) was employed.
parallel plate capacitor is surrounded by guard rings [87]. The
most remarkable characteristic in this capacitance dilatometer,
2.3. Thermal expansion measurements under quasi-uniaxial
however, is its high resolution ( l/ l = 10−10 ), corresponding
pressure
to absolute length changes of l = 0.01 Å for a specimen
of length 10 mm. This resolution is roughly five orders One of the challenges in the context of thermodynamic
of magnitude higher than that of conventional methods like 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 N
= 0.124 · (3)
C pF
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 Figure 16. Left scale: thermal expansion coefficient along the
of the sample holder from the cryostat without damaging the in-plane a-axis, αa (T ), for crystals #1 (red) and #2 (blue); right
sample, as such procedure is necessary to set the new starting scale: out-of-plane electrical resistivity, ρ⊥ , for crystal #1 of the
capacitance (new pressure). As depicted in figure 11, for the κ-D8-Br salt. The upper inset depicts a zoom of the low temperature
(BEDT-TTF) based charge-transfer salts a hydrostatic pressure ρ⊥ (T ) data employing the temperature scale depicted in the main
panel. Figure taken from [2].
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 regime. Interestingly enough, for all three crystals studied,
in section 3.4, will show that a quasi-uniaxial pressure of a ρ⊥ vanishes below ∼11.5 K. A zero electrical resistivity
few tenth of bar changes the shape of the thermal expansion followed by a very small feature in the thermal expansivity
coefficient curves dramatically, indicating therefore that the αi (T ) suggests percolative superconductivity coexisting with
properties of this material are altered. an antiferromagnetic insulating phase for κ-D8-Br [61], see
the phase diagram presented in figure 8. In [1], Kawamoto
3. Thermal expansion measurements on fully et al performed a.c. magnetic susceptibility measurements on
deuterated salts of κ-(ET)2 Cu[N(CN)2 ]Br (‘κ-D8-Br’) κ-D8-Br under two distinct conditions: (a) cooling the system
slowly, namely employing a cooling-rate of 0.2 K min−1 ; and
As otherwise stated, in order to reduce cooling-speed (b) using a high cooling-rate of 100 K min−1 . From the
dependent effects, the temperature was decreased by a rate measurements carried out after slow cooling, below the onset of
of ∼ − 3 K h−1 through the so-called glass-like transition superconductivity at Tc = 11.5 K, they did not observe a jump
around 80 K7 in all experiments reported here. figure 16 in the real part of the a.c. magnetic susceptibility, as expected
depicts the linear thermal expansivity (upper part) along the for bulk superconductivity, but a gradual transition towards
in-plane a-axis for crystals #1 and #3 together with the out- low temperatures. From the latter behavior they estimated
of-plane (b-axis) electrical resistivity8 ρ⊥ (lower part) for a percolative superconducting volume fraction of 20% . In
crystal #1. As depicted in figure 16, upon cooling, ρ⊥ achieves the case of measurements after rapid cooling, according to
a broad maximum centered at ∼45 K, then abruptly drops the authors, the superconducting volume fraction is reduced
and levels out at ∼30 K. The electrical resistivity remains to 1–2%.
metallic, i.e. dR/dT > 0, down to ∼20 K, below which As can be seen in figure 16, some features observed in
ρ⊥ increases again (see upper inset in figure 16), indicating the electrical resistivity manifest themselves in the coefficient
a phase transition to an insulating state. A comparable ρ⊥ of thermal expansion as well. Indeed, the rather abrupt drop
behavior was observed for the single crystal #3 as well as for of ρ⊥ is accompanied by a pronounced maximum in αa (T )
the crystal investigated in [43], except for small differences centered at a temperature Tp
30 K. As will be discussed in
around the maximum and some details in the insulating more detail below, this behavior can be assigned to a crossover
phenomenon in the vicinity of a nearby critical point. Upon
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 cooling the system further, α(T ) reveals a huge negative peak,
to 4.2 K, an average cooling speed of −40 K h−1 was employed. indicating a phase transition, see the phase diagram depicted
8 Resistance measurements shown in figure 20 were carried out by Strack.
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

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
Figure 17. Main panel: relative length changes data li (T )/ li , that these transfer integrals carry a net component along the
with i = a, b, c for crystal #1 of the κ-D8-Br salt along the in-plane in-plane c-axis, which is expected to be softer than the out-of-
a- and c- and out-of-plane b-axis. For clarity, the data set have been plane b-axis and second in-plane a-axis, a noticeable effect
offset. Lower inset: hysteresis in la (T )/ la associated with the along the c-axis would then be naturally expected at TMI .
Mott MI transition, measured under a very low temperature
sweep-rate of ±1.5 K h−1 . Upper inset: 2D triangular-lattice dimer According to the authors of [94], the uniaxial isothermal
model with hopping terms t and t  , the dots represent dimers of ET compressibilities ki = li−1 (dli /dP ) for the superconducting
molecules. Figure taken from [2]. compound κ-(ET)2 Cu(NCS)2 are robustly anisotropic with
k1 :k2 :k3 = 1:0.53:0.17 (ratio estimated under 1 bar), where
insulating behavior indicates that the peak in the expansivity k1 refers to the isothermal compressibility along the in-plane
data, αa (T ), is due to the Mott MI transition9 . Note that a c direction, k2 along the second in-plane b-axis and k3 along
similar behavior of αa (T ) is observed for crystal #2, albeit the direction parallel to a projection of the long axis of the ET
with somewhat reduced (∼20%) peak anomalies at Tp and donor molecules, i.e. along the out-of-plane a-axis for this
TMI . This reduction might be related to small misalignments salt, since its crystallographic structure is monoclinic [95].
of the crystal as well as to different pressures exerted by the Note that along the c-axis a direct dimer–dimer interaction
dilatometer cell on the crystals, approximately 4 bar for crystal t  , as indicated in the upper inset of figure 17, exists,
#1 and 6 bar for crystal #2. More information on the nature of making a zero-effect along this axis even more surprising. A
such transitions can be obtained via investigation of possible possible explanation for this feature would be a cancellation
anisotropic lattice effects, namely by measuring and analysing of counteracting effects related to the transfer integrals t and
the relative length changes along the three crystallographic t  , which seems to be improbable. Furthermore, it is not clear
directions, i.e. li (T )/ li = (li (T ) − li (0 K))/li (300 K) (with how these in-plane interactions may produce comparatively
i = a, b, c), as depicted in figure 17. strong effects in the out-of-plane b-axis.
As can be seen in figure 17, the most pronounced effects These findings provide strong evidence that other degrees
appear along the a-axis (in-plane), i.e. parallel to the polymeric of freedom, namely the lattice, should be somehow coupled to
(Cu[N(CN)2 ]Br) anion chains, see figure 5. Of special interest the π electrons, thus indicating that the Mott MI transition in
is the ‘S-shaped’ anomaly near Tp which corresponds to the the present material cannot be completely understood solely
peak at Tp in αa (T ) (see figure 16) and does not show any by taking into account a purely 2D electronic scenario [2].
trace of hysteresis upon cooling or warming, at least within
the resolution of the present experiments. This is a generic 3.1. Anisotropic lattice effects in κ -D8-Br and rigid-unit modes
feature of strong fluctuations in the vicinity of a critical end
In figure 1810 the relative length changes in the T -range 4.2 K
point. Upon further cooling through TMI , the length of the
< T < 200 K for the a-, b- and c-axes are shown. In the
crystal along the a-axis shows a distinct increase of roughly
following, a possible explanation for the unusual anisotropic
la / la = 3.5·10−4 within a very limited T -window, consistent
lattice effects observed in κ-D8-Br is presented.
9 In employing α = −80·10−6 K−1 , obtained from the height of the anomaly Upon cooling, strongly anisotropic effects are observed
in the thermal expansion coefficient for crystal #1 (figure 16), using the entropy in the whole temperature range. The more pronounced effects
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
are observed along the in-plane a-axis, which is marked by
about 0.2 J mol−1 K−1 . Details of this estimate are provided in appendix 2. a minimum around 150 K and an abrupt change in slope at
This value corresponds roughly to 0.4% of the background at TMI in the specific Tg . This feature will be discussed in more detail below.
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 10In [72] Grießhaber refers to a cooling rate of ∼−1 K min−1 as a slow cooling,
heat at TMI could be detected. while the rapid cooling rate is not mentioned.

12
J. Phys.: Condens. Matter 27 (2015) 053203 M de Souza and L Bartosch

120 600
Tg Metal
400

∆V /V (10 )
-5

200
80 TMI
TP b-
0 C
∆li/i (10 )
-4

0 20 40 60 80 100
T(K) c-
40
N
NTE
a-axis
0 T Tg
MI
Tp Metal
0 40 80 T(K) 120 160 200
(a) (b) (c)
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 Figure 19. Schematic representation of the M-CN-M linkage. (a)
to 200 K. Data along the b- and c-axes are shifted for clarity. Linear configuration. (b) Shift of the carbon (C) and nitrogen (N)
Position of Tg estimated according to [25]. NTE indicates negative atoms aside from the metal–metal axis in the same direction. (c)
thermal expansion along the a-axis (in-plane) in the T -range Shift in the opposite direction. Note that the total length along the
Tg < T
150 K. Inset: relative volume change metal–metal axis is reduced due to the displacement of the C-N
V /V = la / la + lb / lb + lc / lc versus T . Low temperature data atoms. This is a mechanism giving rise to negative thermal
will be used for estimating the entropy change associated with the expansion. Figure reproduced after [99].
Mott MI transition (section 3.2). Figure after [3].

displacement of the CN units, away from the metal–metal

An interpretation in terms of disorder due to the freezing-
axes (figures 19(b) and (c)), is assigned to be the mechanism
out of the ethylene groups of the ET molecules either in the
responsible for NTE in these materials. As a matter of fact,
staggered or in the eclipsed configuration was proposed by
NTE is also observable in other materials. For example, the
Müller et al [25]. By substituting the terminal (CH2 )2 groups
NTE of liquid water below 4 ◦ C is connected with a breaking
by (CD2 )2 , one observes a shift of Tg to higher temperatures.
of the tetrahedral H bonding. Below 4 ◦ C, NTE is required to
A feature that is expected because of the higher mass of
the (CD2 )2 in comparison to the mass of the (CH2 )2 groups, overcompensate the increase of entropy due to such a structural
which in turn implies a higher relaxation time at a given transition [101]. Another example can be found in ZrW2 O8 .
temperature. Hence, the model proposed by Müller et al This material exhibits NTE over a wide temperature window
is supported by the so-called isotope effect. However, it of 50–400 K [102]. The origin of this anomalous behavior
does not explain the negative expansivity along the a-axis is assigned to its structural arrangement, which is built up by
(in-plane) observed above Tg , namely in the interval Tg < WO4 tetrahedra and rigid ZrO6 octahedra. Rigid-unit modes
T
150 K. More recently, high-resolution synchrotron x- in this material show up due to the fact that each WO4 unit has
ray diffraction experiments [97] have provided indications one vertex not shared by another unit.
that such an anomaly is not solely linked to the ethylene The displacement of the C and N atoms aside from the
groups freezing-out. Essentially, the claims of the authors metal–metal axis has the effect of decreasing the distance
are based on the comparison between the estimated number between the metal atoms as the temperature is increased, see
of the disordered ethylene groups (staggered configuration) figures 19(b) and (c). According to the authors of [99], in
at 100 K (∼11%) with those at extrapolated zero temperature polymeric crystalline materials, these vibrational modes will
(3 ± 3%). They conclude that such a transition is most likely not occur separately, but rather they will affect the vibration
related to structural changes, probably involving the insulating modes of other metal-CN-metal linkages, giving rise to the
polymeric anion chains. Upon further cooling, the anomalies at NTE phenomenon. Nevertheless, the deformation of the
Tp and TMI , already discussed in detail in the previous section, metal-cyanide bridge will carry a high-energy penalty. Hence,
show up. From figure 18 it becomes clear that the lattice effects assuming that the metal coordination geometries are preserved,
at Tg , Tp and TMI are more pronounced along the polymeric the vibration modes illustrated in figures 19(b) and (c) can be
anion chain a-axis, with negative thermal expansion (NTE), observed only if they are coupled with the crystal lattice in the
i.e. a decrease of length with increasing temperature, observed form of phonon modes. Yet, according to Goodwin et al such
below TMI (due to charge carrier localization) and also in the modes are referred to as rigid-unit modes (RUM) and can be
temperature window Tg < T
150 K. It should be noted that experimentally observed in the low-energy window 0–2 THz,
the in-plane anisotropy at both TMI and Tg corresponds to an i.e. in the range of typical phonon frequencies [103]. Since
orthorhombic distortion. Recently, NTE was also observed by the Cu[N(CN)2 ]Br− (figure 5) counter anion has a polymeric
Goodwin et al in other materials containing cyanide bridges nature, the above-described model is a good candidate to
in their structures, see [98, 99] and references therein. These explain why NTE is observed only along the in-plane anion
authors proposed a simple model [99] to explain the origin of chain a-axis. The latter might have their origin in the
this unusual behavior. In this model, the transverse vibrational 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 0.1) K MPa−1 , as obtained from the slope of the S-shaped
growing T above Tg along the a-axis is accompanied by an line at TMI in figure 8 and V /V = (4.2 ± 0.5) × 10−4 ,
expansion along the b- and c-axes. In addition, preliminary as estimated from the inset of figure 18, one obtains S =
Raman studies (not shown here), carried out on κ-D8-Br −0.074 J mol−1 K−1 . This small entropy change at TMI
(crystal #4), revealed the appearance of a double peak structure represents a subtle fraction of the full entropy related to the
in the frequency window 50–200 cm−1 at 20 K (below Tg ), metallic state of the κ-H8-Br salt at T
14 K of S =
not observed at 130 K (above Tg , but below the onset of γ · T
0.375 J mol−1 K−1 , using the Sommerfeld coefficient
NTE) [100]. Further systematic studies are required to verify γ = 0.025 J mol−1 K−2 [106]. Electronic specific heat data
if such splitting shows up above or below Tp . Note that by Nakazawa et al [107] revealed that by means of gradual
this frequency window fits roughly into the above-mentioned deuteration it is possible to tune the system from the Mott
energy window predicted for RUM. Interestingly enough, the insulating state to the metallic regime of the phase diagram.
double peak structure is not observed at 5 K (below TMI ), As observed in these experiments, γ decreases towards zero in
indicating that vibration modes in this energy window are the insulating phase. Based on these literature results, γ of the
no longer active below TMI . Hence, these results support fully deuterated salt discussed in this work should actually be
the idea of strong coupling between lattice and electronic somewhat smaller than that of the fully hydrogenated salt. It
degrees of freedom at the Mott transition. Still owing to follows that the small entropy change of −0.074 J mol−1 K−1 ,
the anomalous lattice effects above Tg , based on the above estimated as above-described, demonstrates that the spin
discussion, it seems that the freezing-out of the degrees of entropy of the paramagnetic insulating state at elevated T
freedom of the ethylene end-groups of the ET molecules in the must be almost completely removed. In addition, this result is
staggered/eclipsed configuration alone cannot be responsible consistent with the Néel temperature TN coinciding with TMI
for a NTE along the a-axis. Indeed, as pointed out in at this point in the phase diagram [108].
[104], the glass-like anomaly is not observed in the organic
superconductors β  -(ET)2 SF5 CH2 CF2 SO3 (Tc = 5 K, large 3.3. Magnetic field effects on κ -D8-Br
discrete anion) and in κ-(ET)2 I3 (Tc = 3.5 K, linear anion)
as well as in the nonsuperconducting α-(ET)2 KHg(SCN)4 Having detected unusual features in the thermal expansivity
(polymeric anion). For such materials, a smooth Debye-like of κ-D8-Br, additional information about the role played by
behavior along the three crystallographic directions up to 200 K lattice degrees of freedom for the Mott MI transition in κ-
is observed, i.e. no glass-like transition is observed. It is D8-Br can be obtained by carrying out thermal expansion
appropriate to mention here that the glass-like anomaly is measurements under magnetic fields. In what follows we
not observed in the κ-(ET)2 Cu2 (CN)3 salt, to be discussed discuss the effects of magnetic fields in the expansivity data
in section 5. Thus, a possible scenario to explain this feature at the immediate vicinity of the Mott transition. However,
for the κ-(ET)2 Cu[N(CN)2 ]Z salts would be the existence of before doing so, let us consider first the low-temperature case
a complicated entwinement between the ethylene end groups at zero magnetic fields. The thermal expansivity along the b-
and the vibration modes of the polymeric anion chain, not axis (out-of-plane) for B = 0 is shown in figure 20 together
knowingly reported as yet in the literature. According to with the electrical resistance11 data normalized to its value
this idea, the ethylene end groups delimit cavities, where at room temperature. Upon cooling, a negative anomaly
the anions are trapped [34], so that below Tg an anion in the expansivity data at around T = 13.6 K is observed.
ordering transition occurs in a similar way to the anion The latter is linked to the Mott MI transition and mimics the
ordering transition observed in the quasi-1D (TMTCF)2 X lattice response upon crossing the first-order Mott MI transition
charge-transfer salts [105]. Further systematic Raman and line in the phase diagram, see the discussion in the previous
infra-red studies are necessary to provide more information sections. Note that the anomaly in αb (T ) is directly connected
about the nature of the glass-like transition. 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
3.2. Entropy change associated with the Mott MI transition
transition. The drop of the electrical resistance at Tc = 11.6 K,
More information about the Mott physics in the present accompanied by a very small kink (indicated by an arrow) in
materials can be obtained by estimating the amount of entropy αb (T ), are fingerprints of percolative SC in small portions of
linked to the Mott MI transition. Such entropy change is the sample volume coexisting with the AF ordered insulating
directly associated with the slope of the first-order phase state, see e.g. [109]. In the temperature window 12 K

transition line in the universal P -T phase diagram figure 8 T
16 K, jumps in the electrical resistance are noticeable. In
and the volume change associated with the Mott MI transition fact, such jumps in R(T ) are frequently observed in organic
by the Clausius–Clapeyron equation, which reads charge-transfer salts. The huge expansivity of the crystal,
which in turn is due to the softness of this class of materials,
dTc V induces stress on the electrical contacts, giving rise to such
= . (4)
dP S jumps. In general, the appearance of jumps in resistivity data
are attributed to cracks in the crystal. For the quasi one-
Here, V = (VI − VM ) ( S = (SI − SM )) refer to the
dimensional (TMTTF)2 X and (TMTSF)2 X salts, in particular,
difference in volume (entropy) between the insulating (I)
and metallic (M) states. Employing dTMI /dP = (−2.7 ± 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]. 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
such jumps in the resistivity data are assigned to stress caused field-induced phase transition temperature, as discussed in the main
by localized defects and/or to mechanical twinning [12]. text. Figure reproduced after [85].
The thermal expansivity data along the b-axis (out-of-
plane) at low-T under external magnetic fields of 0, 0.5, 1, et al. in [56]. The physical explanation why the anomaly
2, 4, 6 and 10 T [85] is depicted in figure 21. From this data in the expansivity αb (T ) at T = TFI becomes more evident
one can directly see that upon reducing T under a magnetic for magnetic fields higher than Hc , however, cannot be based
field up to 10 T, the Mott MI transition temperature TMI stands solely in terms of a SF transition, so that a possibly different
essentially unaffected. Nevertheless, upon reducing T further mechanism might be related to this feature. It was pointed out
under a low magnetic field of 0.5 T, a second field-induced by Ramirez et al in [111] that the emergence of pronounced
anomaly at around TFI = 9.5 K is observed. Increasing anomalies under finite magnetic fields in thermodynamic
the magnetic field, this anomaly becomes more and more quantities, like the specific heat or thermal expansion, have
pronounced, until it saturates at around ∼4 T. A careful analysis only been observed in a few systems and can be considered
of the data shown in figure 21 reveals the presence of a double- as unusual physical phenomena. Let us discuss two possible
peak structure for magnetic fields higher than 1 T. However, distinct scenarios to explain this unusual feature:
further experiments are necessary to gain more insights into
this feature. No such magnetic field-induced effects were (i) First of all, electrical resistance measurements carried
observed along the in-plane a- and c-axis (not shown). As out on partially deuterated κ-(ET)2 Cu[N(CN)2 ]Br salts
known from the literature, see e.g. [110], orientational field- under magnetic field sweeps, being the magnetic field
dependent magnetic phenomena may indicate a spin-flop (SF) applied along the out-of-plane b-axis and the temperature
transition. Such a transition takes place when the system fixed at T = 5.50 K and 4.15 K for 50% and 75% salts,
is in an AFI state and a magnetic field, exceeding a certain respectively, unveiled a sudden change from SC to high
critical value (critical field), is applied parallel to the so-called resistive-states for magnetic fields of about 1 T [85, 112].
easy-axis. In the present case, this field dependence of the Such a state was found to transform into a high-resistive
thermal expansivity αb (T ) is suggestive of a SF transition state via jump-like increases in the electrical resistance
with strong magneto-elastic coupling. As can be directly upon further increasing the applied magnetic field to
seen from figure 21, the magnetic critical field Hc is smaller 10 T [112]. The authors of [112] interpreted this behavior
than 0.5 T for κ-D8-Br, which is consistent with Hc ∼ 0.4 T as a magnetic field-induced first-order SC-to-Insulator
for the κ-(ET)2 Cu[N(CN)2 ]Cl salt, as obtained by Miyagawa 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 destroyed, the remaining electron spins do not couple with
and antiferromagnetism, as proposed for the high-Tc the applied magnetic field due to the correlated motion
cuprates by Zhang [113]. Moreover, upon tuning the κ- among them. These field-decoupled spins are strongly
(ET)2 Cu[N(CN)2 ]Cl salt close to the Mott MI transition coupled to the lattice and give rise to the minimum at
by hydrostatic pressure [44] a pronounced B-induced TFI . The term field-decoupled spins is used in [111] to
increase in the electrical resistance was observed also refer to a similar effect (double peak structure) observed
for this salt. In the latter case, it was proposed that in specific heat measurements under magnetic field for
minor metallic/superconducting phases near the Mott MI the ‘spin ice’ compound Dy2 Ti2 O7 . According to the
transition undergo a magnetic field-induced localization authors of [111], for magnetic fields applied parallel to
transition see theoretical predictions, see, e.g. [114] and the [1 0 0] crystallographic direction, due to correlated
references cited therein. Thus, based on these findings, motion among the spins, half the spins have their Ising-
it is natural to expect that also for the fully deuterated κ- axis orientated perpendicular to the magnetic field. The
(ET)2 Cu[N(CN)2 ]Br salt, located on the verge of the Mott latter are called decoupled-field spins. It was observed that
MI transition line in the phase diagram, the electronic magnetic fields exceeding a certain critical value lead to
states may undergo drastic changes upon increasing the the ordering of these field-decoupled spins. Monte Carlo
strength of the applied magnetic field. Based on the calculations, also reported by the authors in [111], support
above discussion, the growth of the anomaly in αb (T) the proposed model. The term ‘spin ice’ is used by the
with increasing fields H > Hc may be related with the authors to refer to the spin orientations in analogy with the
sensitivity of the electronic channel to applied magnetic degeneracy of ground states observed in ice (H2 O in solid
fields in this particular region of the phase diagram. phase), where hydrogen atoms are highly disordered and
In fact, the initial rapid growth and the tendency to give rise to a finite entropy as T → 0. This is because
saturation for field above about 4 T is quite similar to the oxygen atoms in water form a well defined structure,
the evolution of the electrical resistance, i.e. the increase while the hydrogen atoms remain disordered as a result
of R(B, T = const.) with field observed in the above- of the two inequivalent O-H bond lengths, as first pointed
mentioned transport studies [44, 112]. It is worth by Pauling [116]. The present results do not enable us to
mentioning that no such magnetic field-induced effects determine the exact orientation of these field-decoupled
were observed for fields applied parallel to the in-plane a- spins.
and c-axis (not shown), supporting thus the hypothesis of It is worth mentioning that the above-discussed scenarios
a SF transition. Indeed, it was pointed out in [115] that the should be seen as ‘possible scenarios’ and that the concrete
type of the inter-layer magnetic ordering strongly depends physical origin of the features observed at TFI remain elusive.
on the direction of the applied magnetic field. In particular, Another particularity in the data presented in figure 21 is
based on a detailed analysis of NMR and magnetization that for magnetic fields H > Hc , in a temperature range
data, taking into account the Dzyaloshinskii–Moriya between TMI and TFI , an intermediary phase appears, probably
interaction, the atuhors of [44, 112] found that inter- paramagnetic, as depicted in the schematic phase diagram in
plane antiferromagnetic ordering can be observed only figure 22. It is tempting to speculate that this feature can
for magnetic fields exceeding Hc applied along the out- be seen as a separation of the TN and TMI lines in the phase
of-plane b-axis. We stress that the term ‘inter-plane afm diagram. A possible physical description to this would be
ordering’ is used here as defined in figures 4 and 5 of related to the energetic competition between the AFI ordered
[115]. Thus, given the absence of lattice effects at TFI for phase and the paramagnetic phases after crossing the MI first-
magnetic fields applied along the in-plane a- and c-axis, order line, resulting thus in a decrease of TN with increasing
the present results suggest a close relation between inter- magnetic field. Similar experiments were performed in two
plane antiferromagnetic ordering and the lattice response other κ-D8-Br crystals (crystals #1 and #2). However, for both
observed at TFI . Employing the same notation used by crystals, in contrast to the pronounced anomaly observed at
the authors of [115], the magnetization of the +(−) TFI ≈ 9.5 K under magnetic fields (figure 21), the application
sublattice at the layer l is M+(−)l ., being the staggered and of magnetic fields results in a smooth change of the out-of-
ferromagnetic moments given by Ml† = (M+l − M−l )/2 plane expansivity αb (T ) around the same temperature. In
and MlF = (M+l + M−l )/2, respectively. For fields above this regard, two factors should be considered as a possible
Hc applied along the out-of-plane b-axis, MlF is along explanation for the absence of the above-described effects:
the b-axis and Ml† lies in the a-c plane. MA †
and MB† are (i) As described in e.g. [117], the SF transitions are very
antiparallel, giving rise to an inter-plane antiferromagnetic sensitive to the alignment between the applied magnetic
ordering. The pronounced negative peak anomaly in field and the easy-axis. A subtle misalignment between
αb (T ) observed at T = TFI suggests that in order to obtain the magnetic field and the easy-axis can therefore give rise
this particular spin configuration, the increase in exchange to a suppression of the transition;
energy forces the (ET)+2 layers to move apart from each (ii) The absence of a sharp transition can also be due to sample
other. inhomogeneities and defects. Sample inhomogeneities
(ii) A possible second interpretation for the sharp peaks in would imply, for instance, that portions of percolative
αb (T ) induced by a magnetic field is that upon exceeding SC may vary from sample to sample, reflecting therefore
the critical field Hc , percolative superconductivity is differences in their magnetic properties [1].

16
J. Phys.: Condens. Matter 27 (2015) 053203 M de Souza and L Bartosch

10

8
H // b-axis (T)

Spin-reoriented phase
6
PI PM

4
SF+Perc. SC
2 AFI+Perc. SC

0
0 5 10 15 20
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 Figure 23. Main panel: thermal expansion coefficient along the
paramagnetic metal. The hatched area marks the temperature out-of-plane b-axis for κ-D8-Br crystal #2 under ambient (10 bar)
interval in which a double-peak structure in αb (T ) is observed. and quasi-uniaxial pressure of 65 bar. Dashed lines are guide for the
Spin-reoriented phase denotes the region where MA† and MB† are eyes. For clarity, data are shifted. Inset: blowup of the low-T αb (T )
antiparallel, giving way to an inter-plane antiferromagnetic ordering, data together with measurement under pressure (65 bar) and
see the discussion in the main text. Figure reproduced after [3, 85]. 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 present findings are compiled in the schematic T the peak position to lower T when a magnetic field of 10 T is
versus H diagram shown in figure 22. The dashed line around applied. Figure after [3].
∼0.5 T defines the separation boundary between the AFI and
the SF phases. The thick line indicates the suppression of As shown in the main panel of figure 23, the shape
percolative SC giving place to field-decoupled and/or flopped of the expansivity curves at TMI and Tp is dramatically
spins, here referred to as mixed states. changed by applying a quasi-uniaxial pressure of 65 bar. These
Concluding this section, the thermal expansion results on findings reveal that the anomalies observed in the out-of-plane
κ-D8-Br along the out-of-plane b-axis under a magnetic field expansivity αb (T ) at TMI and Tp are strongly affected upon
show that the Mott MI transition in this salt is largely field- applying pressure, while Tg remains practically unaffected.
independent under magnetic fields up to at least 10 T. Such Since κ-D8-Br is located on the boundary of the Mott MI
findings are consistent with the picture of a Mott insulator transition (figure 8), this feature might be associated with a
formed by a set of hole localized in dimers formed by two ET slight shift of its position from the insulating to the metallic
molecules [22]. At TFI = 9.5 K a phase transition induced side of the phase diagram. Assuming the peak position TMI =
by a magnetic field is observed, indicative of a SF transition 13.6 K as the transition temperature under ambient pressure
with strong magneto-elastic coupling, accompanied by an (actually under a pressure of roughly 10 bar) and TMI = 11.8 K
enhancement of αb (T ) due to the suppression of the percolative as the transition temperature under quasi-uniaxial pressure
SC under magnetic fields above 1 T. Further experiments, (65 bar), one obtains dTMI /dPb
−33 K kbar−1 . Such a value
like magnetostriction measurements above and below TFI are is roughly one order of magnitude smaller than the hydrostatic
highly desired to shed more light on the above-discussed pressure dependence of TMI (dTMI /dP
−380 K kbar−1 ),
magnetic field-induced lattice effects. 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
3.4. Thermal expansion under quasi-uniaxial pressure
of Tp (TMI ) to high (low) temperatures is in perfect agreement
Given the high anisotropy of the κ-(ET)2 X charge-transfer with a positive (negative) pressure dependence of Tp (TMI ) in
salts, thermal expansion measurements under quasi-uniaxial the phase diagram (figure 8). Applying a magnetic field of
pressure could provide more insights to better understand the 10 T (pink curve in the inset of figure 23), the peak position
transitions at TMI , Tp and Tg . The effect of quasi-uniaxial of the transition, indicated by black and pink dashed arrows,
pressure on the sample was studied for crystal #2 as shown shifts to lower temperatures. As TMI is insensitive to magnetic
in figure 23. In these experiments, a pressure of (65 ± 5) bar fields up to 10 T (section 3.3), this observation suggests that
was applied along the out-of-plane b-axis. Roughly speaking, the anomaly in αb (T ) under quasi-uniaxial pressure should be
pressure application along the b-axis means a change of triggered by superconductivity. The present thermal expansion
the contact between the polymeric-anion chains and the ET results under quasi-uniaxial pressure can be seen as a starting
molecules and/or enhancement of the tilt of the ET molecules. point for experiments under hydrostatic pressure, which will

17
J. Phys.: Condens. Matter 27 (2015) 053203 M de Souza and L Bartosch

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)2 ClO4
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.
Figure 24. Thermal expansion coefficient as a function of
temperature for κ-D8-Br crystal #3 along the in-plane a-axis. 4.1. Critical behavior and universality classes
Measurements taken on warming after employing different cooling
speeds (−3 K h−1 and −100 K h−1 ). Data are shifted for clarity. Continuous phase transitions involve fluctuations on all
Arrows indicate the respective transition temperatures for the slowly length scales, leading to the remarkable phenomenon called
cooled crystal. Dashed lines are used to show the shift of TMI universality. While details of the interaction do not matter, only
(towards higher temperatures), Tp (towards lower temperatures) and
Tg (towards higher temperatures) under fast cooling across Tg . robust properties of the system like its effective dimensionality,
Figure after [3]. 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
provide important information about the physics in the vicinity
to classify continuous phase transitions according to their
of the Mott MI transition region in the phase diagram.
universality classes.
In real materials, upon approaching the critical
3.5. Influence of the cooling speed on Tg in κ -D8-Br temperature, fluctuations are no longer negligible. The more
In this section, the influence of the cooling speed below the the critical temperature is approached, the stronger are the
glass-like transition at Tg ≈ 77 K on TMI and Tp is discussed. fluctuations. Finally, fluctuations succeed in destroying the
In fact, the role of the cooling speed on the physical properties original phase and a new phase, often with distinct symmetries,
of fully and partially deuterated salts of κ-(ET)2 Cu[N(CN)2 ]Br emerges. To distinguish the two different phases, one usually
has been intensively discussed in the literature, see e.g. tries to introduce an order parameter as the expectation value
[109, 118]. As shown in figure 24, the expansivity along of some field which vanishes in one phase but not the other.
the in-plane a-axis for a crystal of κ-D8-Br was measured The transition is, in many cases, accompanied by
after cooling the sample through the so-called glass-like phase broadening effects due to crystal defects or inhomogeneities.
transition by employing two distinct cooling speeds (−3 K h−1 Amazingly, around Tc some physical quantities tend to obey
and −100 K h−1 ). power laws in the reduced temperature t = (T −Tc )/T0 , which
A broadening of the transition, accompanied by a measures the relative distance to the transition temperature.
reduction in the size of the peak anomaly at Tp
30 K, As a consequence of universality, the divergence of the
which is related to the critical end-point of the first-order compressibility of water in the vicinity of its critical point is
described by exactly the same power-law dependence as the
Mott MI transition line in the phase diagram and the Mott
divergence of the susceptibility of a uniaxial magnet in the
MI temperature TMI
13.6 K, is observed for fast cooling
vicinity of the Curie temperature. In such a uniaxial magnet,
(−100 K h−1 ). Another remarkable feature is that the peak
the spins can only align themselves parallel or anti-parallel
position of the Tp = 30 K anomaly shifts toward lower
to a given crystal axis. As the magnetization is finite in the
temperatures, while the anomaly at TMI = 13.6 K shifts
symmetry-broken state below Tc and vanishes above Tc , the
slightly to higher temperatures. Based on these observations,
expectation value of the magnetization may serve as an order
one can conclude that the increase of the cooling speed through
parameter.
Tg results in an opposite effect to that observed upon applying
quasi-uniaxial pressure (section 3.4). The observed shift of Tg 12 See footnote 10.

18
J. Phys.: Condens. Matter 27 (2015) 053203 M de Souza and L Bartosch

Concerning thermodynamic properties, one usually bare coherence length is sufficiently large, the non-mean field
defines the four critical exponents α̃, β, γ and δ, which regime can turn out to be unmeasurably small. In conventional
in the case of magnets are related to the specific heat (at superconductors, for example, the size of the bare coherence
constant pressure) Cp (T ), the spontaneous magnetization length can be one thousand the size of the lattice spacing,
Ms (T ) and the susceptibility χ (T ). More generally, Ms (T ) leading to tG ≈ 10−18 . As a consequence, mean field theory
and χ (T ) describe the order parameter and the order parameter is essentially exact.
susceptibility. Usually, α is used in the literature to refer to the From the experimental point of view, the estimate of
specific heat critical exponent. In order to avoid any confusion critical exponents can be a hard task. For instance, owing
with the linear thermal expansion coefficient α(T ), here, α̃ is to the specific heat critical exponent α̃, a reliable estimate of
used to refer to the specific heat critical exponent. the phonon background can be one of the crucial points. In
Close to criticality, one expects the following power-law addition, for a reliable estimate of the critical behavior of a
behavior, system, fine measurements close to Tc are necessary. Accurate
|t|−α̃ measurements of e.g. the specific heat are required over several
Cp (T ) ∼ A± , (5)
α̃ orders of magnitude of t. For real materials, a broadening of
Ms (T ) ∼ B|t|β , t  0, (6) the transition due to inhomogeneities (impurities or crystal
± −γ
defects) is frequently observed. Due to this, Tc cannot be
χ (T ) ∼ C |t| , (7) measured directly, but is rather obtained indirectly via self
Ms (H ) ∼ DH 1/δ , t = 0. (8) consistent fittings.
± ±
While the amplitudes A , B, C and D are non-universal and
the coefficients A+ and C + governing the behavior for t > 0 4.2. Scaling Ansatz for the Mott metal-insulator transition
are generally different from the corresponding coefficients A−
In 2005, in a stimulating article entitled ‘Unconventional
and C − for t < 0, the ratios A+ /A− and C + /C − are in fact
Critical Behaviour in a Quasi-2D Organic Conductor’,
universal, i.e. they are not material-specific and only depend
Kagawa and collaborators reported on the criticality at the
on the underlying universality class. In fact, not all of the
pressure-induced Mott transition in the organic κ-(BEDT-
above exponents are independent. Using concepts of scaling
TTF)2 Cu[N(CN)2 ]Cl charge-transfer salt [45]. In this study,
or the renormalization group, the following identities can be
the authors made use of the isothermal pressure-sweep
derived [120–124],
technique, using helium as a pressure transmitting medium,
α̃ + 2β + γ = 2, (9) to explore the critical behavior of this organic salt through
conductance measurements. The pressure-sweep technique
γ = β(δ − 1). (10) had been previously applied by Limelette and collaborators
These two relations are frequently called the Rushbrooke [127] to study the Mott critical behavior of Cr-doped V2 O3 ,
and Widom identities. It is striking to note that in the which is now recognized as a canonical Mott insulator system
study of critical behavior near a second-order phase transition, and behaves mean-field like. Very close to the transition
materials displaying completely different crystal structures the authors also claim Ising universality. In contrast to
as well as quite different subsystems obey the same critical κ-(BEDT-TTF)2 Cu[N(CN)2 ]Cl, chromium-doped vanadium
behavior near Tc , giving thus rise to the universality classes. sesquioxide is a truly three-dimensional material. In the study
The theoretical values for the critical exponents of different of the quasi-two-dimensional charge-transfer salt κ-(BEDT-
universality classes accompanied by an example of a phase TTF)2 Cu[N(CN)2 ]Cl, the critical behavior of the conductance
transition are listed in table 3. It should be noted that data at the critical endpoint was analyzed in the framework
sufficiently far away from the transition, fluctuations can of the scaling theory of the liquid-gas transition [134]. For
largely be ignored and mean field behavior prepails. More simplicity, in the so-called non-mixing approximation, the
precisely, corrections to mean field theory can only be expected rescaled pressure and the rescaled temperature were used
to become important for temperatures satisfying as two independent scaling variables to obtain the critical
exponents β = 1, γ = 1 and δ = 2, as listed in
|t|
tG , (11) table 3. Substituting these values in the Rushbrooke relation
one obtains α̃ = −1. As pointed out by the authors,
where  2/(4−d) the obtained critical exponents do not fit in the well-known
kB
tG = C . (12) universality classes, indicating the experimental discovery
cv ξ0d of a new universality class. A possible explanation of
is the Ginzburg scale. Here, d is the effective dimensionality the experimentally observed exponents of unconventional
of the system, cv is the jump of the specific heat across criticality was given by Imada and collaborators [135, 136].
the phase transition and ξ0 is the bare coherence length. In The appearance of unconventional critical exponents came
three dimensions, the universal dimensionless proportionality as a surprise to many who expected Ising universality. This
constant C introduced here is given by C = 1/(32π 2 ) expectation was based on an argument by Castellani et al [137]
[125, 126], which is much smaller than 1. who argued that, even though there is no symmetry breaking,
Quite generally, a crossover is expected near tG from the double occupancy of lattice sites can serve as an order
mean field to non-mean field behavior. If, however, the 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−x Crx )2 O3 [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 4 He [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)2 Cu[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 is tuned to its critical value. Instead, the usage of a scaling
do proliferate on the metallic site. The order parameter was theory was proposed in [4]. While such an approach is
therefore expected to be of an Ising type. The Mott criticality extremely general and can be applied to any universality class,
was also studied in the framework of dynamical mean-field it was shown in [4] that the expansivity data of [2] is in fact
theory of the Hubbard model by Kotliar et al [140]. These consistent with 2D Ising criticality: Within the scenario of
authors confirmed the statement that the Mott transition should two-parameter scaling, the Gibbs free energy can be written as
lie in the Ising universality class. Having a phase transition  
between a low-density gas of localized doubly occupied or f (t, h) = |h|d/yh  t/|h|yt /yh . (15)
empty lattice sites to a high-density fluid of unbound doubly
occupied or empty lattice sites is quite analogous to the Here, (x) is a universal scaling function which only
well-known liquid-gas transition which lies also in the Ising depends on the universality class, t and h are the two
universality class. (suitably normalized) scaling variables and yt and yh are their
An attempt to reconcile the experimentally observed corresponding RG eigenvalues. These are the two relevant
unconventional critical exponents β = 1, γ = 1 and δ = 2 eigenvalues of the underlying universality class and determine
with the well-established two-dimensional Ising universality all critical exponents. For example, α̃ = 2 − d/yt or β = (d −
class was made by Papanikolaou and collaborators [46]. These yh )/yt (see e.g. [121–124]). Conventionally, t = (T − Tc )/T0
authors pointed out that the critical exponents derived from and h = (P − Pc )/P0 can be thought of as temperature-
transport measurements do not necessarily coincide with the and pressure-like quantities, but more generally there are also
thermodynamic critical exponents which are defined in terms linear mixing terms such that t = (T − Tc − ζ (P − Pc ))/T0
of an order parameter. According to Papanikolaou et al [46], and h = (P − Pc − λ(T − Tc ))/P0 . In fact, very close to
the singular part of the conductivity  does not only depend the transition Hooke’s law of elasticity breaks down and one
on the singular part of the order parameter, but also depends on should expect compressive strain to enter linearly the scaling
the related energy density. As a consequence,  is given by variables instead of pressure, see below and [5].
The appearance of the mixing terms is a consequence
 = Am + sgn(m)B|m|θ . (13) of the experimental fact that the first-order transition line
ending in the critical point (Pc , Tc ) is not parallel to the T
Here, A and B are non-universal coefficients, m is the order axis, see figure 7. While the linear mixing terms turn out to
parameter and θ = (1 − α̃)/β. In an extended regime not too be important for a quantitative description of the expansivity
close to the critical point the first term on the r.h.s. can be much data [4], let us for simplicity focus here on the much simpler
smaller than the second one such that case of no mixing. For the 2D Ising universality class, yt = 1
and yh = 15/8. Having one integer-valued RG eigenvalue,
 ∝ sgn(m)|m|θ . (14)
logarithmic corrections to scaling should be expected [122].
It then follows that the critical exponents as obtained from the As it turns out, the scaling ansatz for the 2D Ising universality
conductivity can be expressed in terms of the thermodynamic class requires a correction term and reads [138]
critical exponents as follows [46]: βσ = θβ, δσ = δ/θ and t2  
γσ = γ + β(1 − θ ). In particular, for the 2D Ising universality f (t, h) = ln t 2 + |h|d/yh  t/|h|yt /yh . (16)
8π
class one obtains with θ = 8 the exponents βσ = 1, δσ = 7/4
Differentiating twice with respect to t and setting h equal to
and γσ = 7/8 which within the experimental resolution are
zero one then obtains the well-known logarithmic divergence
consistent with the exponents obtained by Kagawa et al [45].
of the specific heat αsing . Assuming no mixing terms, the
To explain the lattice response in the vicinity of the
singular part of the thermal expansivity is given by [46]
MI finite temperature critical end point and explain the
 ∂ 2 f 
corresponding peak in the thermal expansivity a description in 
αsing  ∝  ∝ sgn(h) (−t)−1+β , t < 0. (17)
terms of critical exponents is not sufficient unless the pressure h=±0 ∂h ∂t h=±0

20
J. Phys.: Condens. Matter 27 (2015) 053203 M de Souza and L Bartosch

As a consequence, the Grüneisen ratio sing = αsing /csing

diverges for t < 0 as [4]

αsing 
sing = ∝ sgn (h) (−t)−1+α̃+β . (18)
csing h=±0

Let to emphasize here that the true divergence of the

Grüneisen parameter is a consequence of neglecting linear
mixing terms. Including such mixing terms effectively leads
to a saturation of the Grüneisen parameter at a very large
value. It was estimated in [4] that the crossover scale for the
D8-Br crystals is of order 0.1 bar and thus for most practical
purposes beyond experimental resolution. The smallness of
this crossover scale can be related via the Clausius–Clapeyron
equation to comparatively small entropy differences between Figure 25. Plot of the scaling function α (x) of the thermal
the metallic and insulating phases. expansivity for the 2D Ising universality class. Figure reproduced
The Grüneisen ratio is also known to diverge at a quantum from [4].
critical point [139].
In the case α̃ < 0 there is no divergence in the specific heat,
implying that the specific heat is bounded and can safely be
replaced by a constant. This has interesting consequences for
the universality class of unconventional criticality for which
α̃ = −1 and β = 1. In this case both the expansivity and
the specific heat stay finite such that there is no divergence
of the Grüneisen ratio. Still, there is no reason to expect
the expansivity to be proportional to the specific heat, i.e.
Grüneisen scaling breaks down [4].
To describe the expansivity of the κ-ET salts away but
close to criticality, i.e. for h = 0, we can express the singular
part of the expansivity in terms of derivatives of the scaling
function (x). Differentiating equation (16) with respect to t
and h, one obtains [4]
αsing (t, h) ∝ sgn(h) |h|−1+(d−yt )/yh α (t/|h|yt /yh ), (19)
with the scaling function α (x) given by

d − yt  yt
α (x) =  (x) − x (x). (20)
yh yh
Figure 26. Plot of the singular part of the thermal expansivity αsing
Even though the 2D Ising model can be solved exactly, there as a function of the scaling variables t and h. The first-order
is no known exact expression for the scaling function (x) transition line along the negative t-axis manifests itself by a jump in
and hence no exact expression for α (x). Nevertheless (x) the expansivity which for t → 0 diverges as
[and hence α (x)] can be calculated with very high accuracy αsing ∝ (−t)−1+β = (−t)−7/8 , see equation (17). It should be noted
numerically [4, 138], for a plot see figure 25. With this scaling that every cut at constant h is just given by the (rescaled) scaling
function α (x), as graphically depicted in figure 25. Figure
function at hand, the singular part of the thermal expansivity reproduced from [4].
can now be plotted as a function of t and h, see figure 26.
This singular part of the thermal expansivity corresponds to
the critical region in the pressure-temperature phase diagram. form of the thermal expansivity based on equation (19) on top
The first-order transition line is simply represented by the of a linear background including linear mixing terms gives
jump in αsing (t, h) for h = 0 and t < 0 which describes indeed an excellent fit for the Ising universality class [4], see
the change in volume V ∝ (−t)β . It should be noted figure 27.
that every cut of the graph for constant h is just a rescaled Even though the Ising universality class leads to an
version of the scaling function α (x), as shown graphically excellent fit of the dilatometric data, we would like to stress
in figure 25. Also, the change of sign of αsing with a change here that this is by far no proof of Ising universality. NMR
of sign of h can clearly be seen. While a constant pressure mearsurements of 1/T T1 obey roughly (1/T T1 ) ∝ (P −
corresponds to a constant h in the non-mixing approximation, Pc )1/δ , with δ ≈ 2, as also obtained from conductance
in the case of linear mixing terms a constant P corresponds to measurements [45]. As far as we know, the NMR
cutting the graph in figure 26 at an angle. Fitting the thermal measurements have not found a natural explanation based on
expansivity of the D8-Br crystals [2] # 1 and # 3 by a scaling Ising criticality yet.

21
J. Phys.: Condens. Matter 27 (2015) 053203 M de Souza and L Bartosch

Figure 27. Fit of the expansivity data of the two D8-Br crystals # 1
(diamonds ♦) and # 3 (triangles ∇) from [2] to the scaling theory
outlined in the text. In total, there are six fitting parameters
involved. Two fitting parameters determine the height and width of
the peak, two fitting parameters determine the orientation of the
temperature and pressure axes (and thereby describes linear mixing
terms) and finally two fitting parameters are necessary to describe
the analytic background contribution, as described by the dashed
line. Figure reproduced from [4].
Figure 28. Expansivity along the in-plane c-axis for a single crystal
Notwithstanding the analogy between the Mott MI and the of κ-(ET)2 Cu2 (CN)3 . T anom indicates the temperature at which,
liquid-gas transition, differences do exist and the compressible according to [6], the crossover (hidden ordering) to a spin-liquid
state occurs. The dashed line is used to indicate a hypothetical linear
MI with its long-range shear forces should rather be classified background. Broad hump anomalies at T
150 K and
as a solid–solid transition [5]. Hopping matrix integrals do Tmax,χ
70 K are indicated by the arrows, see the discussion in the
depend on the distance between neighboring atoms, such that text. Figure taken from [3].
the Mott MI transition is sensitive to elastic strain and couples
only indirectly to the applied stress, e.g. compressive pressure. Recall that this is the spin-liquid candidate. Upon cooling, αc
As described in detail in [5] and similarly also in the context decreases monotonously down to Tmin
30 K. Around T =
of the compressible Hubbard model in [51], the homogeneous 150 K, indications of a small and broad hump are observed.
component E of the strain can be obtained by minimizing the In [8], Shimizu and collaborators observed an enhancement of
effective potential density the spin relaxation rate above 150 K, which was attributed to
K0 2 the freezing of the thermally activated vibration of the ethylene
V (E) = E − (P − P0 )E + fsing (t (T , E), h(T , E)),
2 end groups. Note that no traces of a glass-like anomaly around
(21) T = 77–80 K are observed. This behavior is quite distinct
where both t (T , E) and h(T , E) are linear functions of T and from that observed in κ-(ET)2 Cu[N(CN)2 ]Cl (see [25]) and
E, K0 is the bare bulk modulus and P0 is an offset pressure. κ-D8-Br (discussed in section 3) and κ-H8-Br [25], where
For simplicity, we have replaced all tensorial quantities by clear signatures in the thermal expansion coefficient show
scalars. It was shown in [5] that while not too close to the up at Tg
77 K. This discrepancy indicates that the lattice
critical end point Hooke’s law holds and our above treatment dynamic for the κ-(ET)2 Cu2 (CN)3 salt is different from that
was legitimate, the feedback of the electronic subsystem back of the above-mentioned compounds. In fact, according to the
on the lattice drives the effective bulk modulus to zero even RUM model, introduced in section 3.1, the absence of a glass-
above the phase transition described by fsing alone, thereby like anomaly in the κ-(ET)2 Cu2 (CN)3 salt can be understood
leading to a violation of Hooke’s law and a preempted phase in the following way: the anion Cu2 (CN)− 3 (figure 6) consists
transition. This phase transition was explicitly shown to be of a 2D network of Cu(I) and bridging cyanide groups [33].
governed by Landau criticality with mean-field exponents. As the Cu2 (CN)− 3 anion is arranged in a network fashion,
While there is no final answer on the universality class which is quite different from the polymeric arrangement of
underlying the Mott transition in the κ-ET charge-transfer salts Cu[N(CN)2 ]Cl− and Cu[N(CN)2 ]Br− (figure 5), the vibration
yet, we hope that future experiments will shed more light on modes of the CN groups are confined between nearest Cu(I)
this issue. atoms, so that they cannot propagate along the structure.
Hence, the formation of RUM in κ-(ET)2 Cu2 (CN)3 is very
5. Thermal expansion measurements on unlikely and as a consequence no signatures of the glass-
κ-(ET)2 Cu2 (CN)3 like transition can be observed. Cooling the system further,
another broad hump is observed at Tmax,χ
70 K. The
The thermal expansion coefficient along the in-plane c-axis for latter coincides roughly with a broad maximum observed in
crystal #1 of the salt κ-(ET)2 Cu2 (CN)3 is shown in figure 28. magnetic susceptibility measurements [8]. Below T
50 K,

22
J. Phys.: Condens. Matter 27 (2015) 053203 M de Souza and L Bartosch

10

6
c-axis
αi (10-6 K-1)

4
Tanom

2
Thump
0
b-
-2

-4 Figure 30. Main panel: relative length changes (in arbitrary units)
for κ-(ET)2 Cu2 (CN)3 along the in-plane c-axis measured at very
0 2 4 6 8 10 12 14 low sweep-rate of ±1.5 K h−1 , showing the absence of hysteretic
behavior. Inset: thermal expansion data under zero magnetic field
T (K) and 8 T. Such measurements were taken on warming, with the
Figure 29. Blow-up of the low-temperature expansivity data along
magnetic field applied at 1.3 K. Figure taken from [3].
the in-plane c- (crystal #1) and b-axis (crystal #2) of
κ-(ET)2 Cu2 (CN)3 on expanded scales, showing the sharpness of the reproducibility. The latter feature has not been observed in
transition at T anom
6 K and a hump in αc at Thump
2.8 K along specific heat measurements [6], most likely due to the lack of
the c-axis. Figure taken from [3], see also [76].
resolution of such experiments.
αc assumes negative values. Cooling the system further, a The signatures observed in αc have their direct
broad minimum is observed at Tmin
30 K. Below T
14 K, correspondence in the spin-lattice relaxation rate (T1−1 ) and
αc starts to assume positive values. A possible scenario for magnetic susceptibility (χ ) [8]. Below 50 K, both quantities
explaining the negative thermal expansion in the temperature decrease monotonously with temperature down to 4 K, where
range 14 K
T
50 K is discussed in the following. In T1−1 starts to increase and shows a broad maximum at
this temperature range, lattice vibration modes (most likely 1 K, while χ varies smoothly [8]. Thermal expansion
from the anion) become soft, the Grüneisen parameter in measurements under magnetic fields (inset of figure 30)
turn assumes negative values and the lattice expands upon revealed that under 8 T, the peak position around 6 K as well
cooling. Interestingly enough, negative thermal expansion as the hump at T
2.8 K remain unaltered, in agreement
has been also observed in amorphous Y100−x Fex (x = 92.5 with specific heat measurements under a magnetic field [6],
and 84) alloys over the temperature range in which these indicating that the anomaly at 6 K is unlikely to be due to
systems go from the paramagnetic to the spin-glass state. The long-range magnetic ordering, at least for magnetic fields
latter feature is assigned to the thermal dependence of the applied parallel to the c-axis. At this point, it is worthwhile
spin fluctuations [141]. Thus, for the spin-liquid candidate mentioning that for the ‘spin ice’ system Dy2 Ti2 O7 [111],
the hypothesis of negative thermal expansion driven by spin already mentioned previously, the spins’ disorder is suppressed
degrees of freedom cannot be ruled out. In fact, the connection when a magnetic field is applied perpendicular to their Ising
between negative thermal expansion and frustration has been axis. In addition, measurements on cooling and warming (main
reported in the literature, but a theory able to describe these panel of figure 30) revealed no traces of hysteretic behavior, at
phenomena is still lacking [142]. Upon further cooling, a huge least in the resolution of the present experiments, so that the
anomaly is observed at T anom
6 K. The latter coincides with hypothesis of a first-order transition can be ruled out.
the hump anomaly observed in specific heat measurements [6]. Hence, the actual origin of the anomaly in αc at 6 K
This finding constitutes the first observation of lattice effects remains unclear. However, at first glance, the present thermal
associated with the transition (or crossover/hidden ordering) at expansion findings appear to fit in the model proposed by
6 K. A further analysis of the present data is difficult because Lee et al [65], which we described briefly at the end of
the actual phonon background cannot be estimated. The the introduction. According to this model, spin pairing on
low-temperature data will be discussed in more detail in the the Fermi surface generates a spontaneous breaking of the
following. figure 29 shows the thermal expansion coefficient lattice symmetry, giving rise to a phase transition at finite
below 14 K on expanded scales. temperature, which in turn is coupled to a lattice distortion.
Upon cooling below 6 K, a hump at T
2.8 K is In other words, upon cooling, the spin entropy has to be
observed. Several runs were performed in order to check for (partially) frozen. The only way for this, without long-range

23
J. Phys.: Condens. Matter 27 (2015) 053203 M de Souza and L Bartosch

magnetic ordering, is to introduce a lattice distortion. This in-plane c-axis, along which the polymeric anion chains are
process presents some similarities with a classical Spin-Peierls linked via Br-N weak contacts. Still more amazing is the
transition, where the formation of a singlet state requires that observation of pronounced lattice effects along the out-of-
the spin entropy of the triplet state goes to zero by introducing plane b-axis. These findings provide strong evidence that
a lattice distortion. In the present case, the amount of entropy the Mott transition for the present class of materials cannot
in the temperature range 0.3 K < T < 1.5 K corresponds to be fully understood in the frame of a purely 2D electronic
only a few percent of R ln 2 [6], indicating therefore that only triangular dimer model. Even though there is still an ongoing
a minor part of the total number of spins contribute to the spin- debate about the universality class and possible crossover
liquid phase. Interestingly enough, the shape of the transition scales of the Mott transition, the scaling theory reviewed here
at 6 K (figure 29) resembles the shape of the anomaly in the can be used with any universality class. Candidates suggested
thermal expansion coefficient associated with the Spin-Peierls for the Mott transition in the κ-ET salts are unconventional
transition for the quasi-1D organic conductors (TMTTF)2 X criticality [45, 135, 136], 2D Ising [46] and a crossover to
(X = PF6 and AsF6 ) [74]. Hence, the sharp lattice distortion Landau criticality [5]. It was explicitly shown that the thermal
at T anom
6 K seems to be associated with a real phase expansivity data for the κ-D8-Br salt is consistent with 2D
transition and not with a crossover, as proposed in [6]. In fact, Ising criticality [4], which of course does not provide a
thermal expansion measurements along the second in-plane b- proof of this universality class. Finally, the experimentally
axis revealed striking anisotropic in-plane lattice effects [76]. observed critical exponent δ = 2 was also argued to be due to
Interestingly enough, along the in-plane b-axis (see figure 29) subleading quantum effects preceding an asymptotic critical
a negative anomaly is observed at T anom
6 K, indicating regime [143].
that below T anom a lattice expansion (shrinkage) along the c- In order to achieve a better understanding of the
axis (b-axis) occurs. The observed distinct in-plane anisotropy Mott transition in the present material, thermal expansion
implies that the hopping integral terms t and t  are strongly measurements were taken under magnetic field, quasi-uniaxial
affected. pressure as well as by employing different cooling speeds
It remains to be seen whether κ-(ET)2 Cu2 (CN)3 is in across the glass-like transition. For example, measurements
fact in a spin-liquid state for T → 0. In this sense, under magnetic field reveal that the Mott MI insulator
thermal expansion measurements at very low temperatures transition temperature is insensitive to magnetic fields up
along the a-, b- and c-axes are required in order to achieve to 10 T, which is in agreement with the picture of a Mott
a better understanding of the physics of this exciting material. insulator with a hole localized on a dimer formed by an
Such experiments will reveal, for example, whether below ET molecule. A magnetic field-induced first-order phase
1.3 K α/T behaves linearly and remains finite as T → 0, transition at TSF
9.5 K for a field applied along the out-of-
as observed in specific heat measurements [6] or if α/T plane b-axis was observed, indicative of a spin-flop transition
decreases exponentially. As already mentioned at the end of with strong magneto-elastic coupling. Measurements under
the introduction, it was more recently reported [66] that the quasi-uniaxial pressure applied along the out-of-plane b-axes
thermal conductivity of κ-(ET)2 Cu2 (CN)3 is described by an revealed effects in contrast to those observed upon increasing
activated behavior with a gap κ
0.46 K. The latter results the cooling speed across the glass-like transition. While quasi-
are at odds with the specific heat measurements performed by uniaxial pressure shifts TMI towards lower temperatures and Tp
Yamashita et al [6]. Based on such results, one can see that a to higher temperatures, the opposite situation is observed upon
full picture of the fundamental aspects of the spin-liquid phase fast cooling through the glass-like transition. Interestingly
in κ-(ET)2 Cu2 (CN)3 is still incomplete and requires further enough, the so-called glass-like transition temperature Tg
investigations. remains unaffected under quasi-uniaxial pressure. Although
by means of thermal expansion measurements one has access
6. Conclusions and perspectives to the macroscopic behavior of the sample studied, the
strong anisotropic lattice effects observed, marked by a
In this review, we have addressed one of the central questions negative thermal expansion above Tg , enable us to draw
in the field of strongly correlated electron systems, whether some conclusions at the microscopic level. Hence, a model
lattice degrees of freedom play a role for several physical within the rigid-unit-mode scenario, taking into account the
phenomena, among them the Mott MI transition. Molecular low-energy vibration modes of the cyanide ligands of the
conductors of the κ-(BEDT-TTF)2 X family have been revealed polymeric anions Cu[N(CN)2 ]Br− , was proposed to describe
as model systems for the study of the latter phenomena. the negative thermal expansion above the Tg
77 K. In this
The main findings discussed in this review are summarized model, low-energy vibration modes of the cyanide groups of
below. the polymeric anion chains become frozen upon cooling down
High-resolution thermal expansion measurements on fully to Tg , shrinking thus the a lattice parameter upon warming the
deuterated salts of κ-(BEDT-TTF)2 Cu[N(CN)2 ]Br revealed system above Tg .
pronounced lattice effects at the Mott metal-to-insulator Additional experiments, including magnetostriction
transition temperature TMI = 13.6 K, accompanied by a measurements both above and below the magnetic field-
striking anisotropy. While huge effects are observed along induced phase transition temperature TFI as well as magnetic
the in-plane a-axis, along which the polymeric anion chains measurements with B thoroughly aligned along the out-
are aligned, an almost zero effect is observed along the second of-plane b-axis, will certainly help to achieve a better

24
J. Phys.: Condens. Matter 27 (2015) 053203 M de Souza and L Bartosch

understanding of the behavior of almost localized correlated References

electrons in this interesting region of the phase diagram
of the κ-phase of (BEDT-TTF)2 X charge-transfer salts. A [1] Kawamoto A, Miyagawa K and Kanoda K 1997 Phys. Rev. B
theoretical study of the stability of the Mott insulating 55 14140
[2] de Souza M, Brühl A, Strack C, Wolf B, Schweitzer D and
state and the neighboring superconducting phase, taking Lang M 2007 Phys. Rev. Lett. 99 037003
into account magnetic field-induced lattice effects, is highly [3] de Souza M 2008 PhD Thesis University of Frankfurt,
desired as well. We have here presented a phenomenological Germany
description of the Mott metal-insulator transition within a (http://publikationen.ub.uni-frankfurt.de/volltexte/
scaling theory. While transport measurements revealed 2009/6240/)
[4] Bartosch L, de Souza M and Lang M 2010 Phys. Rev. Lett.
unconventional critical exponents, it would be desirable to 104 245701
have direct access to thermodynamic quantities. In particular, [5] Zacharias M, Bartosch L and Garst M 2012 Phys. Rev. Lett.
we hope that future expansivity measurements under pressure 109 176401
might give rise to a better understanding. [6] Yamashita S, Nakazawa Y, Oguni M, Oshima Y, Nojiri H,
We have also examined in this review thermal expansion Shimizu Y, Miyagawa K and Kanoda K 2008 Nat. Phys.
4 459
measurements on the κ-(BEDT-TTF)2 Cu2 (CN)3 salt, a system [7] Jeschke H O, de Souza M, Valentı́ R, Manna R S, Lang M
which is proposed as a candidate for the realization of a and Schlueter J A 2012 Phys. Rev. B 85 035125
spin-liquid ground state. Measurements taken along the in- [8] Shimizu Y, Miyagawa K, Kanoda K, Maesato M and Saito G
plane c-axis reveal some interesting aspects, which can be of 2003 Phys. Rev. Lett. 91 107001
extreme importance for a wider understanding of the physical [9] Balents L 2010 Nature 464 199
[10] Lang M 1996 Supercond. Rev. 2 1
properties of this material. For example, the absence of the [11] Lang M and Müller J 2004 Organic superconductors The
so-called glass-like transition in κ-(BEDT-TTF)2 Cu2 (CN)3 is Physics of Superconductors vol 2, ed K H Bennemann and
in line with the above-mentioned model within the rigid- J B Ketterson (Berlin: Springer) p 435–554
unit modes scenario. While the anion Cu[N(CN)2 ]Br− is [12] Ishiguro T and Yamaji K 1990 Organic Superconductors
arranged in form of chains weakly linked via Br-N contacts, (Springer Series in Solid-State Sciences vol 88) (Germany:
Springer)
the anions Cu2 (CN)− 3 are arranged in a 2D network fashion, [13] Jérome D 1991 The physics of organic superconductors
so that the cyanide groups are confined between the Cu Science 252 1509
atoms and therefore the vibration modes associated with [14] McKenzie R H 1998 Comment. Condens. Matter Phys. 18
the cyanide groups cannot propagate along the structure 309–37
as it occurs, for example, in fully deuterated salts of κ- [15] Wosnitza J 2000 Studies of High Temperature
Superconductors vol 34 (New York: Nova Science
(BEDT-TTF)2 Cu[N(CN)2 ]Br, see the discussion above. A
Publishers) pp 97–131
negative thermal expansion is observed in the temperature [16] Singleton J 2002 J. Solid State Chem. 168 675
range 14 < T < 50 K. In addition, the thermal expansion [17] Miyagawa K, Kanoda K and Kawamoto A 2004 Chem. Rev.
coefficient reveals a huge anomaly, indicative of a phase 104 5635–53
transition at T = 6 K, coinciding nicely with a hump in the [18] Fukuyama H 2006 J. Phys. Soc. Japan 75 051001
specific heat, which has been assigned to a crossover to the [19] Mori H 2006 J. Phys. Soc. Japan 75 051003
[20] Kanoda K 2006 J. Phys. Soc. Japan 75 051007
quantum spin-liquid state. Last but not least, during the [21] Powell B J and McKenzie R H 2006 J. Phys.: Condens.
preparation of the final version of this article, a theoretical Matter 18 R827–66
treatment for the thermal expansivity of the charge-transfer [22] Toyota N, Lang M and Müller J 2007 Low-Dimensional
salts of the κ-(BEDT-TTF)2 X family based on electronic Molecular Metals (Germany: Springer)
correlations was proposed by Kokalj and McKenzie [144]. [23] Kanoda K and Kato R 2011 Annu. Rev. Condens. Matter
Phys. 2 167
These authors consider both the dependence of the system [24] Powell B J and McKenzie R H 2011 Rep. Prog. Phys.
entropy on the Hubbard parameters U , t, t  and the dependence 74 056501
with temperature of the double occupancy and bond-orders to [25] Müller J, Lang M, Steglich F, Schlueter J A, Kini A M and
describe the thermal expansion of these materials. Sasaki T 2002 Phys. Rev. B 65 144521
[26] Sasaki T, Yoneyama N, Kobayashi N, Ikemoto Y and
Kimura H 2004 Phys. Rev. Lett. 92 227001
Acknowledgments [27] Salameh B 2005 PhD Thesis University of Stuttgart, Germany
[28] Kandpal H C, Opahle I, Zhang Y-Z, Jeschke H O and
We would like to thank A Brühl, M Dressel, R M Fernandes, Valenti R 2009 Phys. Rev. Lett. 103 067004
M Garst, A A Haghighirad, K Kanoda, S Köhler, P Kopietz, [29] Nakamura K, Yoshimoto Y, Kosugi T, Arita R and Imada M
2009 J. Phys. Soc. Japan 78 083710
R S Manna, R E L Monaco, J Müller, R Paupitz, J-P Pouget, [30] Schlueter J A, Geiser U, Whited M A, Drichko N, Salameh B,
B Powell, T Sasaki, J A Schlueter, J Schmalian, D Schweitzer, Petukhov K and Dressel M 2007 Dalton Trans. 2007 2580
A C Seridonio, C Strack, R Urbano, B Wolf, M Zacharias and in [31] Schlueter J A, Wiehl L, Park H, de Souza M, Lang M, Koo
particular M Lang for valuable discussions and collaborations H J and Whangbo M H 2010 J. Am. Chem. Soc. 132 16308
along the years. MdS acknowledges financial support from [32] Kini A M 1990 Inorg. Chem. 29 2555
[33] Geiser U et al 1991 Inorg. Chem. 30 2586
the DFG via SFB/TRR 49, São Paulo Research Foundation—
[34] Geiser U et al 1991 Physica C 174 475
Fapesp (Grant No. 2011/22050-4) and National Counsel of [35] Kanoda K 1997 Hyperfine Interact. 104 235
Technological and Scientific Development—CNPq (Grant No. [36] Mckenzie R H 1997 Science 278 820
308977/2011-4). [37] Dagoto E 2005 Science 309 257

25
J. Phys.: Condens. Matter 27 (2015) 053203 M de Souza and L Bartosch

[38] Taylor O J, Carrington A and Schlueter J A 2007 Phys. Rev. [70] Mizuno M, Garito A and Cava M 1978 J. Chem. Soc. Chem.
Lett. 99 057001 Commun. 1978 18
[39] Lefebvre S, Wzietek P, Brown S, Bourbonnais C, Jérome, [71] Gärtner S, Schweitzer D and Keller H J 1991 Synth. Met.
Méziere C, Fourmigué M and Batail P 2000 Phys. Rev. 44 227
Lett. 85 5420 [72] Grießhaber E 2000 PhD Thesis University of Stuttgart,
[40] Nam M-S, Ardavan A, Blundell S J and Schlueter J A 2007 German
Nature 449 584 [73] Barron T H K, Collins J G and White G K 1980 Thermal
[41] Limelette P, Wzietek P, Florens S, Georges A, Costi T A, expansion of solids at low temperatures Adv. Phys.
Pasquier C, Jérome D, Mézière C and Batail P 2003 Phys. 29 609–730
Rev. Lett. 91 016401 [74] de Souza M, Foury-Leylekian P, Moradpour A, Pouget J-P
[42] Fournier D, Poirier M, Castonguay M and Truong K D 2003 and Lang M 2008 Phys. Rev. Lett. 101 216403
Phys. Rev. Lett. 90 127002 [75] de Souza M, Brühl A, Strack C, Wolf B, Schweitzer D and
[43] Lang M, de Souza M, Brühl A, Strack C, Wolf B, Lang M 2007 Phys. Rev. Lett. 99 037003
Schlueter J A, Müller J and Schweitzer D 2007 Proc. of [76] Manna R S, de Souza M, Bruehl A, Schlueter J A and Lang
the 8th Int. Conf. on Materials and Mechanisms of M 2010 Phys. Rev. Lett. 104 016403
Superconductivity: High Temperature Superconductors [77] Manna R, Wolf B, de Souza M and Lang M 2012 Rev. Sci.
(Dresden, Germany, 1 September 2007) (Physica C) vol Instrum. 83 085111
460–2 p 129 [78] Pregelj M et al 2010 Phys. Rev. B 82 144438
[44] Kagawa F, Itou T, Miyagawa K and Kanoda K 2004 [79] Jesche A, Krellner C, de Souza M, Lang M and Geibel C
Phys. Rev. B 69 064511 2010 Phys. Rev. B 81 134525
[45] Kagawa F, Miyagawa K and Kanoda K 2005 Nature 436 534 [80] Jesche A et al 2012 Phys. Rev. B 86 020501
[46] Papanikolaou S, Fernandes R M, Fradkin E, Phillips P W, [81] de Souza M, Hofmann D, Foury-Leylekian P, Moradpour A,
Schmalian J and Sknepnek R 2008 Phys. Rev. Lett. Pougetb J-P and Lang M 2010 Physica B 405 S92
1 026408 [82] Jesche A, Krellner C, de Souza M, Lang M and Geibel C
[47] Lunkenheimer P et al 2012 Nat. Mater. 11 755 2009 New J. Phys. 11 103050
[48] Kurosaki Y, Shimizu Y, Miyagawa K, Kanoda K and Saito G [83] de Souza M, Haghighirad A-A, Tutsch U, Assmus W and
2005 Phys. Rev. Lett. 95 177001 Lang M 2010 Eur. Phys. J. B 77 101
[49] Strack C et al 2005 Phys. Rev. B 72 054511 [84] de Souza M and Pouget J-P 2013 J. Phys.: Condens. Matter
[50] Kawamoto A, Miyagawa K, Nakazawa Y and Kanoda K 25 343201
1995 Phys. Rev. B 52 15522 [85] de Souza M, Brühl A, Strack C, Schweitzer D and Lang M
[51] Hassan S R, Georges A and Krishnamurthy H R 2005 Phys. 2012 Phys. Rev. B 86 085130
Rev. Lett. 94 036402 [86] Kund M, Mtiller H, Kushch N D, Andres K and Saito G 1995
[52] Merino J and McKenzie R H 2000 Phys. Rev. B 62 16442 Synth. Met. 70 951
[53] Singleton J, Goddard P A, Ardavan A, Harrison N, [87] Pott R and Schefzyk R 1983 J. Phys. E 16 445
Blundell S J, Schlueter J A and Kini A M 2002 Phys. Rev. [88] Fietz W H, Grube K and Leibrock H 2000 High Pressure
Lett. 88 037001 Res. 19 373
[54] Welp U, Fleshler S, Kwok W K, Crabtree G W, Carlson K D, [89] Köppen M et al 1999 Phys. Rev. Lett. 82 4548
Wang H H, Geiser U, Williams J M and Hitsman V M [90] Lang M 1991 PhD Thesis Darmstadt Technical University,
2000 Phys. Rev. Lett. 69 840 Darmstadt, German
[55] Pinterić M, Miljak M, Bis̃kup N, Milat O, Aviani I, Tomić S, [91] Müller J 2001 PhD Thesis Max-Planck-Institute, Dresden,
Schweitzer D, Strunz W and Heinen I 2000 Eur. Phys. J. B Germany
11 217 [92] Brühl A 2007 PhD Thesis University of Frankfurt, Frankfurt,
[56] Miyagawa K, Kawamoto A, Nakazawa Y and Kanoda K Germany
1995 Phys. Rev. Lett. 75 1174 [93] Loram J W, Mirza K A, Joyce C P and Osborne A J 1989
[57] Kagawa F, Itou T, Miyagawa K and Kanoda K 2004 Phys. Europhys. Lett. 8 263
Rev. Lett. 93 127001 [94] Chasseau D 1991 Synth. Met. 42 2039
[58] Smith D F, De Soto S M, Slichter C P, Schlueter J A, Kini [95] Toyota N, Watanabe Y and Sasaki T 1993 Synth. Met.
A M and Daugherty R G 2000 Phys. Rev. B 68 024512 55 2536
[59] Smith D F, Slichter C P, Schlueter J A, Kini A M and [96] Lakes R 1987 Sience 235 1038
Daugherty R G 2004 Phys. Rev. Lett. 93 167002 [97] Wolter A U B, Feyerherm R, Dudzik E, Süllow S, Strack C,
[60] Kawamoto A, Yamasita M and Kumagai K-I 2004 Lang M and Schweitzer D 2007 Phys. Rev. B 75 104512
Phys. Rev. B 70 212506 [98] Goodwin A L, Calleja M, Conterio M J, Dove M T,
[61] Miyagawa K, Kawamoto A and Kanoda K 2002 Phys. Rev. Evans J S O, Keen D A, Peters L and Tucker M G 2008
Lett. 89 017003 Science 319 794
[62] Anderson P W 1973 Mater. Res. Bull. 8 153 [99] Goodwin A L and Kepert C J 2005 Phys. Rev. B 71 140301
[63] Anderson P W 1987 Science 235 1196 [100] de Souza M et al unpublished results
[64] Ramirez A P 2008 Nat. Phys. 4 442 [101] Evans J S O 1999 J. Chem. Soc. Dalton Trans. 19 3317
[65] Lee S-S, Lee P A and Senthil T 2007 Phys. Rev. Lett. Barrera G D, Bruno J A O, Barron T H K and Allan N L 2005
98 067006 J. Phys.: Condens. Matter 17 R217
[66] Yamashita M, Nakata N, Kasahara Y, Sasaki T, Yoneyama N, [102] Mary T A, Evans J S O, Vogt T and Sleight A W 1996
Kobayashi N, Fujimoto S, Shibauchi T and Matsuda Y Science 272 90
2009 Nat. Phys. 5 44 [103] Roth S 1995 One-Dimensional Metals (Germany: VCH)
[67] Qi Y, Xu C and Sachdev S 2009 Phys. Rev. Lett. 102 176401 [104] Müller J et al 2000 Phys. Rev. B 61 11739
[68] Pratt F L, Baker P J, Blundell S J, Lancaster T, [105] Pouget J-P and Ravy S 1984 J. Phys. I 6 L393
Ohira-Kawamura S, Baines C, Shimizu Y, Kanoda K, [106] Elsinger H, Wosnitza J, Wanka S, Hagel J, Schweitzer D and
Watanabe I and Saito G 2011 Nature 471 612 Strunz W 2000 Phys. Rev. Lett. 84 6098
[69] Hartke K, Kissel T, Quante J and Matusch R 1980 Chem. Ber. [107] Nakazawa Y, Taniguchi H, Kawamoto A and Kanoda K 2000
113 1898 Phys. Rev. B 61 R16295

26
J. Phys.: Condens. Matter 27 (2015) 053203 M de Souza and L Bartosch

[108] Lang M, de Souza M, Brühl A, Strack C, Wolf B and Ginzburg V L 1960 Sov. Phys.–Solid State 2 1824 (Engl.
Schweitzer D 2008 Physica B 403 1384 transl.)
[109] Taniguchi H, Kawamoto A and Kanoda K 1999 Phys. Rev. B [127] Limelette P, Georges A, Jerome D, Wzietek P, Metcalf P and
59 8424 Honig J M 2003 Science 302 89
[110] Blundell S 2001 Magnetism in Condensed Matter (Oxford: [128] LaBella V P, Bullock D W, Anser M, Ding Z, Emery C,
Oxford University Press) Bellaiche L and Thibado P M 2000 Phys. Rev. Lett.
Pankhurst Q A, Johnson C E, Jones D H and Thomas M E 84 4152
1988 Hyperfine Interact. 41 505 [129] Lipa J A, Swanson D R, Nissen J A, Chui T C P and
[111] Ramirez A P, Hayshi A, Cava R J, Siddharthan R and Israelsson U E 1996 Phys. Rev. Lett. 76 944
Shastry B S 1999 Nature 399 333 [130] Pelissetto A and Vicari E 2002 Phys. Rep. 368 549
[112] Taniguchi H et al 2003 Phys. Rev. B 67 014510 [131] Campostrini M, Hasenbusch M, Pelissetto A and Vicari E
[113] Zhang S-C 1997 Science 275 1089 2006 Phys. Rev. B 74 144506
[114] Georges A et al 1996 Rev. Mod. Phys. 68 13 [132] Campostrini M, Hasenbusch M, Pelissetto A, Rossi P and
[115] Smith D F 2000 Phys. Rev. B 68 024512 Vicari E 2002 Phys. Rev. B 65 144520
[116] Pauling L 1935 J. Am. Chem. Soc. 57 2680 [133] Onsager L 1944 Phys. Rev. 65 117
[117] Pankhurst Q A, Johnson C E, Jones D H and Thomas M F [134] Kadanoff L P, Götze W, Hamblen D, Hecht R, Lewis E A S,
1988 Hyperfine Interact. 41 505 Palciauskas V V, Rayl M and Swift J 1967 Rev. Mod. Phys.
[118] Su X, Zuo F, Schlueter J A, Kini A M and Williams J M 1998 39 395–431
Phys. Rev. B 58 R2944 [135] Imada M 2005 Phys. Rev. B 72 075113
[119] Griesshaber E, Schiller M, Schweitzer D, Heinen I and [136] Imada M, Misawa T and Yamaji Y 2010 J. Phys.: Condens.
Strunz W 1999 Physica C 317 421 Matter 22 164206
[120] Huang K 1987 Statistical Mechanics (New York: Wiley) [137] Castellani C, Di Castro C, Feinberg D and Ranninger J 1979
[121] Goldenfeld N 1992 Lectures on Phase Transitions and Phys. Rev. Lett. 43 1957
the Renormalization Group (Reading, MA: [138] Fonseca P and Zamolodchikov A 2003 J. Stat. Phys. 110 527
Addison-Wesley) [139] Zhu L, Garst M, Rosch A and Si Q 2003 Phys. Rev. Lett.
[122] Cardy J L 1996 Scaling and Renormalization in Statistical 91 066404
Physics (Cambridge: Cambridge University Press) [140] Kotliar G, Lange E and Rozenberg M J 2000 Phys. Rev. Lett.
[123] Kardar M 2007 Statistical Physics of Fields (Cambridge: 84 5180
Cambridge University Press) [141] Fujita A, Suzuki T, Kataoka N and Fukamichi K 1994 Phys.
[124] Kopietz P, Bartosch L and Schütz F 2010 Introduction to the Rev. B 50 6199
Functional Renormalization Group (Berlin: Springer) [142] Schlesinger Z, Rosen J A, Hancock J N and Ramirez A P
[125] Levanyuk A P 1959 Zh. Eksperim. i Teor. Fiz. 36 810 2008 Phys. Rev. Lett. 101 015501
Levanguk A P 1959 Sov. Phys.—JETP 9 571 (Engl. transl.) [143] Sémon P and Tremblay A-M S 2012 Phys. Rev. B 85 201101
[126] Ginzburg V L 1960 Fiz. Tverd. Tela. 2 2031 [144] Kokalj J and McKenzie R H 2014 arXiv:1411.1085

27