# Phase coexistence at the first-order Mott-transition revealed by
pressure-dependent dielectric spectroscopy of $\kappa$-(BEDT-TTF)2Cu2(CN)3

R. Rösslhuber  1\. Physikalisches Institut, Universität Stuttgart,
Pfaffenwaldring 57, 70569 Stuttgart, Germany  A. Pustogow  1\. Physikalisches
Institut, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
Department of Physics and Astronomy, UCLA, Los Angeles, California 90095,
U.S.A.  Institute of Solid State Physics, Vienna University of Technology,
1040 Vienna, Austria.  E. Uykur  1\. Physikalisches Institut, Universität
Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany  A. Böhme  1\.
Physikalisches Institut, Universität Stuttgart, Pfaffenwaldring 57, 70569
Stuttgart, Germany  A. Löhle  1\. Physikalisches Institut, Universität
Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany  R. Hübner  Institute
for Functional Matter and Quantum Technology, Universität Stuttgart, 70569
Stuttgart, Germany  J. A. Schlueter  Material Science Division, Argonne
National Laboratory, Argonne, Illinois 60439-4831 and
National Science Foundation, Alexandria, Virginia 2223, U.S.A.  Y. Tan
National High Magnetic Field Laboratory, Florida State University,
Tallahassee, U.S.A.  V. Dobrosavljević  National High Magnetic Field
Laboratory, Florida State University, Tallahassee, U.S.A.  M. Dressel  1\.
Physikalisches Institut, Universität Stuttgart, Pfaffenwaldring 57, 70569
Stuttgart, Germany

###### Abstract

The dimer Mott insulator $\kappa$-(BEDT-TTF)2Cu2(CN)3 can be tuned into
metallic and superconducting states upon applying pressure of 1.5 kbar and
more. We have performed dielectric measurements (7.5 kHz to 5 MHz) on
$\kappa$-(BEDT-TTF)2Cu2(CN)3 single crystals as a function of temperature
(down to $T=8$ K) and pressure (up to $p=4.3$ kbar). In addition to the
relaxor-like dielectric behavior seen below 50 K at $p=0$, that moves toward
lower temperatures with pressure, a second peak emerges in
$\varepsilon_{1}(T)$ around $T=15$ K. When approaching the insulator-metal
boundary, this peak diverges rapidly reaching $\varepsilon_{1}\approx 10^{5}$.
Our dynamical mean-field theory calculations substantiate that the dielectric
catastrophe at the Mott transition is not caused by closing the energy gap,
but due to the spatial coexistence of correlated metallic and insulating
regions. We discuss the percolative nature of the first-order Mott insulator-
to-metal transition in all details.

###### pacs:

71.30.+h, 74.70.Kn, 72.90.+y, 77.22.-d

††preprint: k-CuCN dielectrics pressure

##  I Introduction and Motivation

The quasi two-dimensional organic charge-transfer salts (BEDT-TTF)2$X$ became
model compounds for investigating the interplay of strongly correlated
electrons, reduced dimensionality, spin-charge interactions and ordering
phenomena [1, 2, 3, 4]. Their molecular composition enables fine-tuning of
physical properties by modifying the donor molecules bis-ethylenedithio-
tetrathiafulvalene (BEDT-TTF), varying the monovalent anions $X$, or by
applying comparably low pressures – usually a few kbar induce drastic changes
[5, *ToyotaBook, *MoriBook]. The dimerized $\kappa$-phase compounds have been
established as prime examples for the bandwidth-tuned Mott insulator-metal
transition (IMT) and the quantum critical region above [8, 9, 10, *Kagawa09,
12, *Furukawa18, 14, 15, *Vucicevic15].

Figure 1: 3D plot of $\varepsilon_{1}(p,T,f=380$ kHz) and phase diagram of
$\kappa$-(BEDT-TTF)2Cu2(CN)3. Strikingly, the dielectric permittivity is
strongly enhanced at the first-order Mott transition (blue line around 1.8
kbar and $T<T_{\rm crit}\simeq 16$ K), ascribed to metal-insulator phase
coexistence, as predicted by dynamical mean-field theory [17]. Above $T_{\rm
crit}$, the first-order IMT becomes a gradual crossover (quantum Widom line)
[18, 12, 14]. A relaxor-ferroelectric response in $\varepsilon_{1}(T)$ is
observed in the Mott-insulating phase ($p<1.5$ kbar) [19, 20, *Pinteric15,
*Pinteric18]. The green circles represent the bifurcation temperature $T_{\rm
B}$ that indicates a change in the relaxation mechanism as discussed in Sec.
III.1.1. The black solid circles correspond to $T_{\rm FL}$ [23].

Since two decades, the dimer Mott insulator $\kappa$-(BEDT-TTF)2Cu2(CN)3
attracted most attention as it has been considered the prime candidate for a
quantum spin liquid [24, *Kurosaki05, 26]. The nature of the low-temperature
spin state remains subject of intense studies and controversial discussions
[27, 28], fueled by the recent finding of a spin gap [29]. The absence of
long-range magnetic order even at lowest temperatures provides the opportunity
to investigate the $genuine$ Mott transition [13, 30, *Pustogow18a, 14], which
is solely driven by Coulomb interactions, without breaking any symmetry.
Nevertheless, recent studies revealed that for these layered BEDT-TTF
compounds the lattice properties play an important role – in addition to
disorder – and the interaction with the anions can be decisive [32, 33, 34].

Besides these fundamental issues, there are some more peculiarities observed
in $\kappa$-(BEDT-TTF)2Cu2(CN)3, which are far from being understood. Around
$T\approx 6$ K the thermal expansion exhibits a pronounced anomaly [35,
*Manna18] with related features observed in specific heat [37], thermal
conductivity [38], ultrasound propagation [39], magnetic susceptibility [40],
and microwave dielectric properties [41]. Here, we add another striking
phenomenology, presented in Fig. 1, namely a collossal enhancement of the
dielectric permittivity at the Mott IMT, reaching up to
$\varepsilon_{1}(T,p)\approx 10^{5}$ at the lowest frequencies (see Fig. 3).
This ‘dielectric catastrophe’ is assigned to phase coexistence of spatially
separated metallic and insulating regions at the first-order transition [23].
Note, this feature is distinct from the relaxor-type ferroelectric response
(see Fig. 2) that was observed at ambient pressure in the audio- and radio-
frequency range below $T\approx 50$ K [19, 20, *Pinteric15, *Pinteric18].
Previous attempts to link the latter to charge disproportionation within the
dimers due to intersite Coulomb repulsion, dubbed quantum-electric dipole or
paired-electron crystal [42, *Hotta12, 44, 45, *Dayal11, *Clay12, 48], could
not be verified in experiment as various spectroscopic methods have
unanimously proven homogeneous charge distribution on the molecules [49, 50,
51, 52, 53, 54].

Figure 2: Temperature-dependent dielectric constant $\varepsilon_{1}(T)$ of
$\kappa$-(BEDT-TTF)2Cu2(CN)3 at low pressure values ($p=0$–1.22 kbar) plotted
for frequencies 7.5 kHz–5 MHz. (a) At ambient pressure and $T<50$ K, the
dielectric permittivity exhibits a relaxor-type ferroelectric behavior with a
peak that diminishes in amplitude and shifts to higher $T$ as frequency
increases. (b-e) In addition, we identify another, shoulder-like feature at
lower temperature. While traces are revealed around $T=15$ K already at $p=0$,
it forms a second peak with distinct temperature dependence with increasing
pressure. Note the different ordinates. Figure 3: (a-c) The dielectric
permittivity of $\kappa$-(BEDT-TTF)2Cu2(CN)3 is shown from $f=7.5$ kHz to 5
MHz in the coexistence regime between pressure values $p=1.45$ and 2.23 kbar.
A strongly frequency-dependent, colossal enhancement of $\varepsilon_{1}(T)$
occurs at low temperatures, which is a result of spatial phase separation
between metallic and insulating regions upon the first-order Mott IMT. (d,e)
$\varepsilon_{1}$ acquires negative values for $p\geq 3.37$ kbar indicating
the onset of metallic transport. Note the different ordinates in panels (d,e).

In order to motivate our detailed experimental investigations and analysis,
theoretical calculations and discussions, let us first give an overview on the
temperature-dependent dielectric response of $\kappa$-(BEDT-TTF)2Cu2(CN)3 in
the various regimes [23]. In Fig. 2 we plot the real part of the dielectric
permittivity $\varepsilon_{1}(T)$ for selected frequencies and pressures as
indicated. At $p=0$ and low pressures, the $T$ dependence of the permittivity
is dominated by a pronounced peak. The maximum exhibits the characteristic
frequency dependence of a relaxor ferroelectric and moves to lower
temperatures with increasing pressure, as indicated by the dotted lines in
Fig. 2(a-e). Most prominent, however, are the dramatic changes of the
dielectric response close to the IMT ($p_{\rm IMT}=1.45$ kbar) [13]. At low
temperatures, $T<20$ K, the dielectric constant is strongly enhanced up to 2.2
kbar and acquires a frequency-dependent amplitude even exceeding
$\varepsilon_{1}\approx 10^{5}$ at the lowest measured frequency [Fig.
3(a-c)]. We ascribe this observation to a phase coexistence around the Mott
IMT, where metallic regions grow in an insulating matrix and eventually form
percolating clusters through the sample. For $p\geq 3.37$ kbar [Fig. 3(d,e)],
$\varepsilon_{1}$ acquires large negative values as metallic behavior sets in.

After explaining the experiments, we analyze our observations on the
insulating phase in detail, followed by the colossal permittivity enhancement
in the transition region. In a next step, we present our theoretical modelling
of the percolative Mott IMT, using a hybrid DMFT approach. Finally our
findings and new insight on the Mott transition are discussed comprehensively.

##  II Experimental details

We synthesized high-quality $\kappa$-(BEDT-TTF)2Cu2(CN)3 single crystals by
standard electrochemical synthesis [18, 55]; sample 1 was grown at the
Universität Stuttgart while sample 2 was prepared at Argonne National
Laboratory. In this study we measured the complex electrical impedance as a
function of pressure, temperature and frequency in order to obtain the complex
conductivity $\hat{\sigma}=\sigma_{1}+{\rm i}\sigma_{2}$ or, equivalently, the
permittivity $\hat{\varepsilon}=\varepsilon_{1}+{\rm i}\varepsilon_{2}$. To
that end, the crystals are contacted with thin gold wires that are attached by
carbon paste to opposite surfaces of a single crystal, such that the
measurements were performed out-of-plane with $E\perp bc$. The experiments
were carried out by measuring through two contacts in a pseudo four-point
configuration [56] with an Agilent 4294 impedance analyzer. To make sure that
we operate in the Ohmic regime a small voltage of 0.5 V was applied. In order
to characterize the crystals, we have measured the low-frequency resistivity
as a function of temperature and pressure.

Our pressure-dependent dielectric experiments were performed utilizing a
piston-type pressure cell as described in detail in Refs. 57, 23. Using a
self-made electrical feedthrough for coaxial cables, we could reach up to
approximately 10 kbar. Daphne oil 7373 serves as the liquid pressure-
transmitting medium, because it has a good hydrostaticity, is inert to
molecular solids, and stays fluid at 300 K for pressures applied in this
study. An InSb semiconductor pressure gauge with negligible pressure gradient
below $T=50$ K was employed for in-situ determination of the inherent pressure
loss upon cooling. As a consequence, in the temperature range of particular
interest here, the data are collected in the same pressure cycles; this is
important for comparison. Unless indicated otherwise, throughout the
manuscript we state the pressure reading at the lowest temperature $T=10$ K.

The pressure cell was cooled down in a custom-made continuous-flow helium
cryostat that allows us to reduce the total cable length to 50 cm enabling
reliable measurements at frequencies up to 5 MHz. The compact cryostat design
results in a rather steep thermal gradient limiting the lowest reachable
temperature to about 8 K. We kept the cooling rate below $0.4$ K/min for all
measurements and observed no cooling-rate dependence. Since good agreement
between the results of both samples was obtained, we present here the data of
sample 2. The results obtained on sample 1 are added in Appendix A.

##  III Results and analysis

###  III.1 Dielectric response in the Mott insulating phase

At reduced temperatures the ambient-pressure dielectric constant
$\varepsilon_{1}(T)$ of $\kappa$-(BEDT-TTF)2Cu2(CN)3 reveals a peak, as first
reported by Abdel-Jawad et al. [19] and later confirmed by Pinterić and
collaborators [20, 21, 22]; here, we label this feature as high-temperature
(HT) peak. When probing with a frequency of $f=100$ kHz, for instance, the
maximum appears at $T=28$ K, in other crystals up to 40 K, in agreement with
previous reports. The observed sample dependence [20] is confirmed by disorder
studies, which reveal a shift of the maximum in $\varepsilon_{1}(T)$ to lower
temperatures upon x-ray irradiation [58]. Fig. 2(a) illustrates how the
maximum moves to low temperatures when probed at smaller frequencies; at the
same time, however, it gets more pronounced. This behavior resembles the well-
known phenomenology of relaxor ferroelectrics [59].

A closer look reveals a shoulder-like feature around $T=15$ K, which we denote
as low-temperature (LT) mode; it evolves into a small second peak for
frequencies between $f=53$ and 200 kHz. As pressure rises, this LT mode
becomes a well-defined peak, it grows in amplitude and eventually dominates
the spectrum at $p=0.86$ kbar, as seen in Fig. 2(c). Although the HT feature
seems to maintain its amplitude and width, it becomes secondary. Both modes
shift to lower temperatures with pressure (cf. Fig. 7).

Figure 4: The real and imaginary parts of the dielectric constant of
$\kappa$-(BEDT-TTF)2Cu2(CN)3, $\varepsilon_{1}(f)$ and $\varepsilon_{2}(f)$,
as a function of frequency for $T=14$ K at ambient pressure. The dashed lines
represent two Cole-Cole modes; the full lines correspond to their sum of both,
according to Eq. (1).

For a better understanding of the physical background, we analyze the
frequency-dependence of the dielectric response, plotted in Fig. 4 for the
example of $T=14$ K and $p=0$ kbar. We should note that even at these low
temperatures the resistivity of $\kappa$-(BEDT-TTF)2Cu2(CN)3 remains at
moderate values; implying that the dc-conductivity $\sigma_{dc}$ gives a
considerable contribution to the imaginary part of the permittivity. Following
the common procedure, we subtract this part:
$\varepsilon_{2}(\omega)=\left[\sigma_{1}(\omega)-\sigma_{dc}\right]/\omega\varepsilon_{0}$
[60, 20]. In Fig. 4 we can distinguish two relaxation modes as roll-offs in
the real part $\varepsilon_{1}(f)$ and broad maxima in the imaginary part
$\varepsilon_{2}(f)$. Hence we fit our data by the sum of two Cole-Cole modes:

$\hat{\varepsilon}(\omega)-\varepsilon_{\infty}=\frac{\Delta\varepsilon^{\text{mode
1}}}{1+({\rm
i}\omega\tau_{1})^{1-\alpha_{1}}}+\frac{\Delta\varepsilon^{\text{mode
2}}}{1+({\rm i}\omega\tau_{2})^{1-\alpha_{2}}}\quad,$ (1)

wherein $\tau_{1,2}$ are the relaxation times, $\omega=2\pi f$ the angular
frequency of the applied electric ac-field, $(1-\alpha_{1,2})$ are real-valued
and the parameters describing the symmetric broadening of the relaxation time
distribution functions, $\Delta\varepsilon^{\text{mode 1}}$ and
$\Delta\varepsilon^{\text{mode 2}}$ are real-valued and denote the dielectric
strengths of the corresponding modes, with $\Delta\varepsilon^{\text{mode
1}}+\Delta\varepsilon^{\text{mode 2}}=\varepsilon_{\rm
static}-\varepsilon_{\infty}$, wherein $\varepsilon_{\rm static}$ and
$\varepsilon_{\infty}$ are the real values for low and high frequencies,
respectively. Ambient-pressure studies [20] previously identified only a
single mode; similar to the related compounds $\kappa$-(BEDT-TTF)2Ag2(CN)3
[53].

Figure 5: Arrhenius plots of (a) the dielectric strength
$\Delta\varepsilon^{\text{mode 1,2}}(T)$, (b) the distribution of relaxation
times $1-\alpha(T)$ and (c) the mean relaxation time $\tau(T)$ for both modes
obtained from the fits of dielectric data of $\kappa$-(BEDT-TTF)2Cu2(CN)3
measured at ambient pressure. The solid purple symbols refer to mode 1 while
the open orange squares indicate the data of mode 2.

The parameters obtained from fitting the ambient-pressure data are plotted in
Fig. 5 as a function of inverse temperature. The strength of the first mode
$\Delta\varepsilon^{\text{mode 1}}(T)$ shows a peak around $T=22$ K,
resembling $\varepsilon_{1}(T)$ for low frequencies. The second contribution
$\Delta\varepsilon^{\text{mode 2}}$ is approximately one order of magnitude
smaller, but increases monotonously upon cooling. With reducing the
temperature, $1-\alpha_{1}(T)$ decreases and the relaxation time $\tau_{1}(T)$
increases, providing evidence for significant broadening and slowing down of
the dielectric relaxation; in relaxor ferroelectrics this is usually ascribed
to cooperative motion and glassy freezing [59]. Interestingly, we observe a
kink in $\tau_{1}$ and a concomitant slight increase of $1-\alpha_{1}$ around
$T=20$ K. For the second mode, $\tau_{2}$ becomes continuously larger as the
temperature is reduced, and $1-\alpha_{2}$ decreases; again indicating that
the corresponding relaxation freezes out.

Figure 6: Arrhenius plots of the Debye parameters of the low-frequency mode 1
in $\kappa$-(BEDT-TTF)2Cu2(CN)3 for pressures up to 1.22 kbar. (a) Dielectric
strength $\Delta\varepsilon^{\text{mode 1}}(T)$, (b) distribution of
relaxation times $1-\alpha_{1}(T)$ and (c) mean relaxation time $\tau_{1}(T)$.
The black and red lines represent fits with Eq. (3) above and below the kink
in $\tau_{1}(T)$ at $T_{\rm B}$, respectively.

####  III.1.1 Analysis of Low-Frequency Mode 1

When turning to the pressure dependence, Fig. 6 displays the fit parameters
corresponding to mode 1 as a function of inverse temperature; the presentation
is limited to the insulating state with $p\leq 1.2$ kbar. The peak in
dielectric strength $\Delta\varepsilon^{\text{mode 1}}(T)$ shifts to lower
temperatures and increases in amplitude as pressure is applied. This
corresponds to the evolution of $\varepsilon_{1}(T)$ plotted in Fig. 2, where
the appearance of the LT feature broadens the relaxation. The resulting lower
values of $1-\alpha_{1}$ do not indicate more cooperativity or glassy behavior
compared to ambient pressure. The kink in $\tau_{1}(T)$ broadens with pressure
and shifts to lower temperatures, the corresponding relaxation time gets
shorter. A similar feature was previously observed at ambient pressure around
$T=17$ K and attributed to a bifurcation temperature $T_{\rm B}$ [20]; here
free charge carriers start to freeze out and hopping-like conduction sets in.

Figure 7: (a) Pressure dependence of the bifurcation temperature $T_{\rm B}$
and the positions of the high and low-temperature peak, $T_{\rm HT}$ and
$T_{\rm LT}$, in $\kappa$-(BEDT-TTF)2Cu2(CN)3; here $p_{\rm IMT}$=1.45 kbar
denotes the critical pressure of the insulator-to-metal transition [18, 13].
(b) For the example of $p=0.86$ kbar and $f=100$ kHz the temperature
dependence of the dielectric constant $\varepsilon_{1}(T)$ is plotted together
with the fits of the HT and LT peak by two Gaussian functions (blue and green
lines). (c) The relaxation time $\tau_{1}(T)$ with the fits from Fig. 6(c)
according to Eq. (3) for the temperature range above (black line) and below
(red line) the kink at $T_{\rm B}$. The crossover from the HT to the LT peak
is located at $T_{\rm B}$.

In Fig. 7(b,c), $\varepsilon_{1}(T)$ and $\tau_{1}(T)$ for $f=100$ kHz are
plotted as a function of temperature at $p=0.86$ kbar. The HT and LT features
in $\varepsilon_{1}(T)$ are well described by two Gaussian functions. Now it
becomes clear that the kink in $\tau_{1}(T)$ at $T_{\rm B}$ (indicated by
solid green circles in Fig. 1) corresponds to the transition from the HT peak
to the LT peak with a concomitant change in the relaxation mechanism. We apply
this procedure to all pressures and plot the pressure evolution of $T_{\rm B}$
in panel (a) of Fig. 7 together with the positions of the peaks at $f=100$
kHz. The dashed lines represent extrapolations to $p_{\rm IMT}=1.45$ kbar [18,
13] according to

$\frac{T_{i}(p)}{T_{i}^{0}}=\left(\frac{p_{\rm IMT}-p}{p_{\rm
IMT}}\right)^{z}\quad,$ (2)

with $i=\left\\{\mathrm{HT,B,LT}\right\\}$ the three characteristic
temperatures, $T_{i}^{0}$ giving a intercept at $p=0$ and $z$ the critical
exponent. The obtained parameters are listed in Table 1.

Table 1: Parameters of $\kappa$-(BEDT-TTF)2Cu2(CN)3 obtained from fits of the pressure evolution of the high-temperature peak $T_{\rm HT}$, the bifurcation temperatures $T_{\rm B}$ and the low-temperature peak $T_{\rm LT}$ according to Eq. (2) with $p_{\rm IMT}$=1.45 kbar. $T_{i}^{0}$ gives the respective ambient-pressure temperature and $z$ is a unitless exponent.  | $T_{\rm LT}$ | $T_{\rm B}$ | $T_{\rm HT}$
---|---|---|---
$T_{i}^{0}$ (K) | 13.0$\pm$ 0.1 | 19.9 $\pm$ 0.2 | 27.7 $\pm$ 0.5
$z$ | 0.18 $\pm$ 0.01 | 0.26 $\pm$ 0.02 | 0.41 $\pm$ 0.08

For $T>T_{\rm B}$, we can describe the dependence of the relaxation time
$\tau_{1}$ on temperature by an activated behavior

$\tau_{1}=\tau_{\rm HT}\exp\left\\{{\Delta_{\rm HT}}/{T}\right\\}\quad,$ (3)

which is represented by the black line in Fig. 6(c). Also the low-temperature
regime can be fitted with an activated behavior in analogy to Eq. (3), which
is illustrated by the red line in Fig. 6(c).

Figure 8: Pressure dependence of the activation energy $\Delta$ and relaxation
time $\tau$ of $\kappa$-(BEDT-TTF)2Cu2(CN)3 obtained from fits of the
corresponding temperature behavior of $\tau_{1}(T)$ shown in Fig. 6(c). The
upper panels (a,b) correspond to the high-temperature regime; the lower panels
(c,d) to the low-temperature regime. The black symbols correspond to sample 1,
the red symbols to sample 2 while the solid lines in (a,c) represent mean-
field fits in analogy to Eq. 2. Most importantly, $\Delta_{\rm LT}$ is rather
constant and abruptly decreases around 1.1 kbar, reminiscent of a first-order
phase transition with $p_{\rm IMT}=1.45$ kbar.

The extracted fit parameters for the HT mode ($T>T_{\rm B}$) and the LT mode
($T<T_{\rm B}$) are plotted in Fig. 8 as a function of pressure; here we also
include the qualitatively similar results obtained from sample 1 while the
corresponding plots are presented in the Appendix A. Our findings are in line
with previous ambient-pressure studies [20]. $\Delta_{\rm HT}(p)$ decreases as
pressure rises following a mean-field behavior in analogy to Eq. 2. Best fits
were obtained by using fixed $p_{\rm{IMT,2}}=1.45$ kbar and
$p_{\rm{IMT,1}}=1.05$ kbar for sample 2 and sample 1, respectively. The fit
parameters are summarized in Table 2 and show good agreement between the two
samples for the critical exponents.

Table 2: Mean field parameters $\Delta_{i}^{0}$ and $z$ of $\Delta_{i}(p)$ in analogy to Eq. (2) using fixed $p_{\rm{IMT,2}}$=1.45 kbar for sample 2 and $p_{\rm{IMT,1}}=1.05$ kbar for sample 1. sample 2 | $\Delta_{\rm{HT}}$ | $\Delta_{\rm{LT}}$
---|---|---
$\Delta_{i}^{0}$ (K) | 354 $\pm$ 19 | 84 $\pm$ 2
$z$ | 0.50 $\pm$ 0.09 | 0.30 $\pm$ 0.04
sample 1 | $\Delta_{\rm{HT}}$ | $\Delta_{\rm{LT}}$
$\Delta_{i}^{0}$ (K) | 355 $\pm$ 5 | 69 $\pm$ 4
$z$ | 0.46 $\pm$ 0.03 | 0.29 $\pm$ 0.05

$\Delta_{\rm LT}(p)$ [Fig. 8(c)] is basically pressure-independent up to $1.1$
kbar, followed by a strong decrease when $p$ is further increased. The
behavior is reminiscent of a first-order phase transition and extrapolates to
$\Delta_{\rm LT}=0$ at $p_{\rm IMT}=1.45$ kbar; in excellent agreement with
the findings in Fig. 7. For sample 1, it is difficult to pin down the pressure
dependence of $\Delta_{\rm{LT}}$ because the number of data points is limited.
Since $\Delta_{\rm{HT}}$ and $T_{\rm{B}}$ both follow mean-field behavior, we
also apply rough tests for mean-field behavior on $\Delta_{\rm LT}$,
represented by the solid lines.

Apart from a rather strong sample dependence, the time scale $\tau_{\rm HT}$
[Fig. 8(b)] becomes longer upon applying pressure. On the other hand,
$\tau_{\rm LT}$ [Fig. 8(d)] decreases upon applying pressure and sample
dependence is less pronounced.

####  III.1.2 Analysis of High-Frequency Mode 2

In the Arrhenius plot of Fig. 9 we show the parameters of the second mode:
$\Delta\varepsilon_{2}(T)$, $1-\alpha_{2}(T)$ and $\tau_{2}(T)$, as obtained
from the Cole-Cole fits. Throughout the whole pressure-temperature range
wherein mode 2 is observed, its strength $\Delta\varepsilon^{\text{mode 2}}$
stays an order of magnitude below the one of the low-frequency mode 1.
$\Delta\varepsilon^{\text{mode 2}}(T)$ increases steadily as the temperature
is reduced. With rising pressure the mode shifts to lower temperatures (Fig.
2) and correspondingly does the enhancement in $\Delta\varepsilon^{\text{mode
2}}(T)$. A drop in $1-\alpha_{2}(T)$ indicates a considerable broadening of
the mode. The temperature dependence of $\tau_{2}$ is strongly influenced by
pressure. While at ambient pressure $\tau_{2}(T)$ monotonically increases, at
$p=0.52$ kbar a minimum is observed that shifts to lower temperatures and gets
more pronounced at $p=0.86$ kbar. In other words, $\tau_{2}(T)$ depends on
pressure in a non-monotonic way. A similar relaxation dynamics has been widely
observed for confined systems as well as in the relaxor ferroelectric
KTa0.65Nb0.35O3 when doped with Cu by approximately 0.1% [61].

Figure 9: Temperature dependence of the parameters describing mode 2 in
$\kappa$-(BEDT-TTF)2Cu2(CN)3 at various pressures. (a) Dielectric strength
$\Delta\varepsilon^{\text{mode 2}}$, (b) distribution of relaxation times
$1-\alpha_{2}$ and (c) mean relaxation time $\tau_{2}$ in an Arrhenius plot
versus inverse temperature. Upon increasing pressure, a pronounced minimum in
$\tau_{2}$ develops, indicating a non-monotonic relaxation dynamics in
$\kappa$-(BEDT-TTF)2Cu2(CN)3.

###  III.2 Dielectric response at the insulator-metal transition

While the relaxor-ferroelectric response from Fig. 2 has been subject to much
controversy and debate[19, 20], the most striking observation of our present
study is the colossal enhancement of the dielectric constant around the first-
order Mott IMT, displayed in Figs. 1 and 3(a-c). At low temperatures,
$\varepsilon_{1}$ acquires values of several hundred up to $10^{5}$ –
increasing towards low frequencies – in the pressure region 1.45–2.23 kbar,
with a peak around 1.8 kbar at $T=10$ K. Our thorough analysis of the
dielectric relaxation in Sec. III.1 clearly shows that this behavior has a
distinct origin. We interpret this phenomenology as a result of percolating
metallic clusters embedded in an insulating matrix [23] – a situation similar
to other systems subject to a metal-insulator transition [62, 63]. On these
grounds, we now estimate the pressure dependence of the filling fraction from
our experimental data.

If the pressure-driven IMT in our Mott system is a first-order transition for
$T<T_{\rm crit}$, two phases are thermodynamically stable in the coexistence
region [64, 17]: One represents the metallic state while the other corresponds
to the insulating phase. In the presence of weak disorder, such a region will
feature a mixture of randomly distributed metallic and insulating domains,
with respective volume fractions that vary with pressure. The IMT takes place
when the volume fraction of the metallic phase approaches the percolation
threshold. The dielectric properties of such a mixture can be modeled by
Bruggeman’s effective medium approximation (BEMA) [65, 66, 67, 68]:

$x\,\frac{\varepsilon_{m}-\varepsilon_{\rm eff}}{\varepsilon_{\rm
eff}+L(\varepsilon_{m}-\varepsilon_{\rm
eff})}+(1-x)\,\frac{\varepsilon_{i}-\varepsilon_{\rm eff}}{\varepsilon_{\rm
eff}+L(\varepsilon_{i}-\varepsilon_{\rm eff})}=0\quad,$ (4)

where $x$ is the volume fraction of the metallic inclusions, $L$ is the shape
factor, $\varepsilon_{i}$ and $\varepsilon_{\rm m}$ are the complex
permittivities of the insulating and metallic phases, respectively, and
$\varepsilon_{\rm eff}$ is the effective permittivity of the composite. In
Fig. 10(a) the imaginary part of the dielectric constant of $\kappa$-(BEDT-
TTF)2Cu2(CN)3 is plotted as a function of pressure measured at various
frequencies at the lowest accessible temperature. We find the step in
$\varepsilon_{2}(p)=\sigma_{1}(p)/(\varepsilon_{0}\omega)$ becoming less
pronounced with increasing frequencies.

Figure 10: (a) Pressure dependence of $\varepsilon_{2}$ of $\kappa$-(BEDT-
TTF)2Cu2(CN)3 measured at $T=10$ K using different frequencies as indicated.
The solid lines represent fits by Bruggeman’s effective medium approximation
[Eq. (4)] for spherical inclusions with the metallic filling fraction as free
parameter. (b) Consistent for all frequencies, the metallic fraction $x$ in
dependence of pressure exhibits a rapid change around percolation; the grey
line is a guide to the eye.

In order to estimate the metallic filling fraction $x$ from these data, we
assume that for $p=0$ the specimen is completely in the insulating phase, the
properties resemble $\varepsilon_{i}$; for $p=4.3$ kbar the metallic state is
fully established, corresponding to $\varepsilon_{m}$. For each particular
pressure, we can now obtain a value of $x$ that best describes the
experimental data by the effective permittivity $\varepsilon_{\rm eff}$
calculated via Eq. (4) assuming spherical inclusions: $L=1/3$. In Fig. 10(a)
the respective fits are shown by solid lines for the various frequencies. The
resulting filling fraction $x$ is plotted in Fig. 10(b) as a function of
pressure. The gradual increase follows a $\tanh$-like behavior around the IMT,
supporting our assumption for the theoretical simulations [23].

We expect this crude method to severely overestimate the extent of the
coexistence regime because the pressure dependence of the conduction
properties is neglected. A narrower region of around 1 to 2 kbar is more
likely. In Section III.3 we address this issue by a phenomenological model
utilizing an approach of hybrid dynamical mean-field theory.

Figure 11: (a) Dielectric constant $\varepsilon_{1}$ of $\kappa$-(BEDT-
TTF)2Cu2(CN)3 measured at $f=100$ kHz for several temperatures in dependence
of the relative pressure $(p-p_{c})$. The solid lines represent fits according
to Eq. 5. (b) Temperature dependence of the percolation threshold $p_{c}$, as
obtained from the fits in (a). (c) Temperature dependence of the exponents
$q$. We attribute the drop of $q$ above $T_{\rm crit}$ to the change from the
first-order insulator-metal transition to a crossover at higher temperatures.
For clarity reasons, we also include $T_{\rm crit}=16$ K (dashed red line) and
the predictions for $q$ according to the BEMA model (dashed blue line).

Now that we obtained the pressure-dependence of the metallic filling fraction
$x(p)$, we are in the position to analyze $\varepsilon_{1}(x)$. The
corresponding plot for the 100 kHz data is shown in Fig. 11(a). The static
dielectric constant of a percolating system is a function of $x$ and its
divergence at the percolation threshold $x_{c}$ can be described by [69]

$\varepsilon_{1}(\omega\rightarrow 0,T\rightarrow
0,x)\propto(x_{c}-x)^{-q}\quad,$ (5)

where the critical exponent $q$ depends on the dimension of the system; in
three dimensions we expect $q$ ranging from 0.8 to 1 [69, 70, 71, 63,
*Hovel11], while $q=1.3$ is calculated for two dimensions [69]. The BEMA model
predicts $q=1$ independent of the dimensionality of the percolating system
[71].

The temperature dependence of the percolation threshold $x_{c}$ and the
exponent $q$ obtained from the fits (solid lines) in Fig. 11(a) is presented
in panels (b) and (c), respectively. Percolation is established at a critical
filling fraction $x_{c}\approx 0.35$, in excellent agreement with the
prediction of ${1}/{3}$ for a three-dimensional system. From our pressure-
dependent dielectric measurements we find $q\approx 1$ up to $T_{\rm crit}$ as
expected within the BEMA framework.

###  III.3 Theoretical analysis of the dielectric permittivity at the IMT

Figure 12: (a) Phase diagram of $genuine$ Mott insulators with first-order
insulator-metal coexistence indicated at low temperatures. The bottom panels
are grouped to illustrate the responses of the respective regimes, comparing
our experiments on $\kappa$-(BEDT-TTF)2Cu2(CN)3 and our hybrid DMFT
calculations. (b-e) The Mott-insulating state yields thermally activated
resistivity and small, postive values of the permittivity. (f-g) While
$\rho_{1}$ indicates a reduction with cooling when metallic clusters
percolate, $\varepsilon_{1}$ is strongly increased upon metal-insulator phase
coexistence. (j-m) The correlated metallic state below the Brinkman-Rice
temperature $T_{\rm BR}$ exhibits Fermi-liquid properties with a quadratic
temperature dependence of the resistivity at low temperatures, accompanied by
large negative values of $\varepsilon_{1}$. The dielectric permittivity
$\varepsilon_{1}$ was measured at $f=380$ kHz and calculated for $hf/W=5\times
10^{-9}$. Note, there is no residual resistivity in the theory data plotted in
panel (k).

We have just established a way to estimate $\varepsilon_{\rm eff}$ of this
percolative first-order insulator-metal transition (IMT) provided we know the
properties of the constituting phases. Now we want go this avenue on purely
theoretical ground. In particular, we determine the complex permittivity of
the mixed phase by calculating first the pure phases, together with their full
correlation dependence, and then embedding them in a percolating network
according to Eq. (10). The technical details and the results of our single-
site dynamical mean-field theory (DMFT) calculations of the optical
conductivity [64, 73, 74] are described in full length in Ref. 23. The
resulting resistivities and permittivities in the dielectric range
($hf/W=5\cdot 10^{-9}$) are compared to exemplary experimental data in Fig.
12. Panel (a) summarizes the three main regimes of the phase diagram, with the
corresponding $\rho_{1}(T)$ and $\varepsilon_{1}(T)$ curves from experiment
(left) and hybrid DMFT (right) placed below each other.

Fig. 12(b-e) illustrate the dielectric response in the homogeneous (i.e.
$x=0$) Mott-insulating state. While the resistivity exhibits a monotonic
increase upon cooling, the permittivity acquires small values between 1–10
typical of a charge-gapped state. In the coexistence regime around the IMT,
plotted in panels (f-i), the temperature dependence of $\rho(T)$ becomes
metal-like as the filling fraction exceeds the percolation threshold, forming
a conducting path through the entire sample [see bottom sketches of frame
(a)]. This is accompanied by a collossal enhancement of $\varepsilon_{1}$ by
several orders of magnitude at low temperatures; the ‘dielectric catastrophe’
is well reproduced by our hybrid DMFT calculations. Also the response of the
pure metallic state ($x=1$) in Fig. 12(j-m) is captured well by theory. Here,
Fermi-liquid behavior with $\rho_{1}\propto AT^{2}$ below $T_{\rm FL}$ is
followed by a bad metallic state extending up to the Brinkman-Rice temperature
$T_{\rm BR}$ where quasiparticles are ultimately destroyed [75, 76, 77].
Accordingly, the dielectric constant turns negative for $T<T_{\rm BR}$ as
expected for a metal.

Remarkably, our hybrid DMFT approach succeeds in modeling the main features of
low-frequency complex transport properties throughout the Mott IMT, supporting
our conclusions based upon optical conductivity work that the main physics of
$\kappa$-(BEDT-TTF)2Cu2(CN)3 is captured by the single-band Hubbard model
[14]. Having said that, we note that the measured relaxor-ferroelectric
behavior in the Mott-insulating state is not reproduced by theory, indicating
that it does not originate from intrinsic Mott physics. This is corroborated
by a pronounced sample dependence indicative of the relevance of impurities
[20].

##  IV Discussion

###  IV.1 Dielectric relaxation in the insulating state

####  IV.1.1 High-temperature peak

Our pressure-dependent investigations unveil that the dielectric response of
$\kappa$-(BEDT-TTF)2Cu2(CN)3 contains two dielectric contributions. The high-
temperature (HT) peak was first observed by Abdel-Jawad et al. [19], but there
is no consensus on its origin. The dimer approach takes the BEDT-TTF dimer as
an entity, neglecting the intra-dimer degrees of freedom; this leads to a
quasi two-dimensional electron system with a half-filled conduction band,
where on-site Coulomb repulsion dominates, making $\kappa$-(BEDT-TTF)2Cu2(CN)3
a prime example to study the physics of genuine Mott insulators [14].
Alternatively, one considers a single BEDT-TTF molecule with certain inter-
and intra-dimer interactions. As a consequence, $\kappa$-(BEDT-TTF)2Cu2(CN)3
is regarded as a $\frac{3}{4}$-filled system, making it unstable towards a
charge-ordered state, which is competing with the dimer-Mott state. Starting
from the quarter-filled extended Hubbard model, some theories predict
fluctuating charge disproportionation within a dimer, resulting in quantum
electric dipoles [42, 44]. The dielectric response in the audio- and radio-
frequency range was interpreted as a consequence of these electric dipoles
[19, 42, *Hotta12, 44, 45, *Dayal11, *Clay12, 48]; their collective optical
excitations should show up in the THz region.

However, infrared and vibrational spectroscopy clearly discards a sizable
charge disproportionation on the dimers [50]; a mixture of lattice and
molecular vibrations perfectly explains all the observed optical modes even
below 1 THz [32]. In a recent theoretical study [78] Fukuyama et al.
considered the crossover from a quarter-filled system with charge-ordered
ground state to a dimer-Mott insulator due to strong dimerization. At high
energy (in the range of eV, i.e. optical frequencies) the latter is stable,
whereas at very low energy ($10^{-10}{\rm eV}\approx 10$ kHz) extended domains
of different charge polarities arise. As a consequence, domain walls form in
the system, giving rise to the HT peak. Very recently, Pouget and
collaborators [34] thoroughly investigated the crystal structure of
$\kappa$-(BEDT-TTF)2Cu2(CN)3 and discovered a triclinic symmetry with two
inequivalent dimers in the unit cell. This implies a rather weak charge
imbalance between dimers in the whole temperature range.

During the last years evidence has accumulated that the interaction between
the cationic (BEDT-TTF)${}^{+}_{2}$ and the anionic Cu2(CN)${}_{3}^{-}$ layer
is crucial for the understanding of these charge-transfer salts [32, 20, 34].
The ambiguity in the arrangement of the polar CN- linking the triangular
coordination of Cu atoms results in intrinsic disorder. Density functional
theory calculations estimate that flipping a CN link, which is mainly oriented
along the b-direction, costs 174 meV, whereas flipping one that is mainly
oriented along the c-direction is only 10-15 meV [32]. The interaction via
hydrogen bonds maps the domains onto the BEDT-TTF layer, leading to long-range
charge inhomogeneities that are detected by low-frequency probes in the kHz
and MHz range [20, 21, 22].

The extension of the relaxation time $\tau_{\rm HT}$ and decrease of
$\Delta_{\rm HT}$ as pressure rises indicate that the domains increase in size
and move more easily. On the other hand, a decrease of $\tau_{\rm HT}$ and
barrier energy $E_{\rm VFT}$ upon x-ray irradiation, reported by Sasaki et al.
[58], infer more domains of smaller size. This can be explained by a larger
number of charged defects in the anion layer upon irradiation which act as
pinning centers. The qualitatively similar but quantitatively slightly
different behavior found in our sample 1 (Fig. 14) corroborates these
observations.

It is interesting to recall that also x-ray irradiation leads to a shift of
the HT peak to lower temperatures [58]. High-energy irradiation produces
crystal defects mainly in the anion layer and is supposed to increase the
number of charge carriers. Similar to the rising pressure, the conductivity of
the sample is enhanced compared to the pristine case. This clearly indicates
that the relaxor-ferroelectric HT peak is influenced by screening due to free
charge carriers, disentangling it from an intrinsic origin related to the
conduction electrons themselves, thus rendering a scenario of intra-dimer
dipoles unlikely.

####  IV.1.2 Low-temperature peak

Let us come to the LT peak, which was not observed in previous ambient
pressure studies of $\kappa$-(BEDT-TTF)2Cu2(CN)3[19, 20, 58, 22].
Consistently, in both samples under inspection, we see a remarkable growth of
the LT peak with pressure and a shift in its position as shown in Fig. 7. The
clearly distinct pressure evolution of the LT parameter compared to the HT
peak (Fig. 8) indicates a different origin. As can be seen from Fig. 7, the
peak appears only below the critical temperature $T_{\rm crit}$, which
establishes the upper bound of the coexistence regime. Note that spatially
separated metallic inclusions can persist in an insulating host well before
percolation sets in. Starting from the LT peak at low $p$, the pressure
evolution of $\varepsilon_{1}(p)$ upon approaching the phase boundary can be
well fitted by Eq. (5) for $T<T_{\rm crit}$, as expected for a percolating
system (Fig. 11). Hence it is tempting to assign this feature to ‘metallic
quantum fluctuations’ previously concluded from optical spectroscopic studies
[14]. The LT peak grows towards the phase transition and eventually becomes
the dominant peak at the percolation threshold, as seen in Fig. 2.

The energy $\Delta_{\rm LT}$ stays constant up to the IMT, where
$\varepsilon_{1}(p)$ follows the BEMA model (Fig. 10) – the latter does not
consider a capacitive coupling of the metallic inclusions. Hence, we attribute
the drop in $\Delta_{\rm LT}$ close to the percolation threshold to an
increased coupling between the metallic inclusions. With values of $\tau_{\rm
LT}$ decreasing from $10^{-6}$ to $10^{-8}$ s, the LT relaxation is clearly
slower than the HT peak; it hardens upon rising $x$. The origin of this
intriguing behavior has yet to be clarified, i.e. whether this can be assigned
to changes in size and/or shape of the metallic inclusions. Such a behavior
was revealed by ellipsometric studies of VO2 films [79, *Voloshenko19].

At this point it is worth to mention that the LT peak is reminiscent of the
dielectric response in the related compound $\kappa$-(BEDT-TTF)2Cu[N(CN)2]Cl
[81, 82, 52], which is very close to the metal insulator transition already at
0 kbar. This enables investigations of the LT peak and the nature of the
metallic inclusions at ambient pressure via a broad spectrum of experimental
techniques, such as scanning near-field infrared microscopy [62]. Deuterating
$\kappa$-(BEDT-TTF)2Cu[N(CN)2]Br crystals is another way to approach the
metal-insulator transitions. In these systems Sasaki et al. succeeded to
spatially map micrometer size domains using infrared spectroscopy [83].
However, the insulating ground state is an antiferromagnet, and hence the
phase boundary has opposite slope due to the Clausius-Clapeyron relation.

###  IV.2 Dielectric catastrophe

The transition from insulating to metallic conduction properties manifests in
numerous forms in condensed-matter physics [84]. Doping silicon with
phosphorous, for instance, turns it metallic as the amount of electron donors
exceeds a critical concentration $N_{c}=3.5\times 10^{18}~{}{\rm cm}^{-3}$
[85, 86, *Lohneysen00, *Lohneysen11]. The transition between localized and
metallic phases in disordered electronic systems is known as Anderson
transitions [89, 90]. Random systems or networks form long-range connectivity
when crossing the percolation threshold; extended scaling theories have been
developed but the details strongly depend on the particular lattice and
dimension [67, 91, 92]. The formation of density waves due to Fermi surface
nesting – in particular in low-dimensional solids – leads to the opening of a
gap in the density of states [93, 94]. In the present case of a Mott
transition, electron-electron interaction causes the metal to become
insulating. The transition can be driven by either varying the electron
density $N$ or the interaction strength $U$ [95, 96, 97].

Besides the usual thermodynamic signatures of phase transitions, the
experimental hallmark of all of these insulator-to-metal transitions is a drop
in resistivity, often by many orders of magnitude, upon changing the order
parameter or temperature. In addition, a divergency of the static dielectric
constant is predicted by classical percolation theory when approaching the
transition from either side, with some characteristic scaling behavior [98,
69, 99, *Bergman78, 101, 71]. Experimentally, Castner et al. first observed a
strong increase of the static dielectric constant at a critical concentration
$N_{c}$ when they measured $n$-doped silicon in the kHz range at low
temperatures [102, *Castner79]; systematic studies of P:Si [104, 105] revealed
$\varepsilon_{1}-\varepsilon_{\rm host}\propto(N_{c}/N-1)^{1.2}$. Most
investigations deal with material mixtures, such as microemulsions [106, 107,
108, *Clarkson88b, 110], composites [111, *Pecharroman01, 113] or percolating
metal films [114, 63, *Hovel11, 115, *deZuani16].

In many cases, the metal-insulator transition is only crossed by lowering the
temperature; here thermal fluctuations and inhomogeneities may occur. The
situation is distinct from investigations of the phase coexistence by tuning
the effective correlation strength via pressure in the limiting case of
$T\rightarrow 0$. Tanner and collaborators [117], for instance, analyzed their
temperature-dependent measurements on the charge-density-wave transition in
TTF-TCNQ using a self-consistent effective-medium approximation [65,
*Stroud78]. Temperature-dependent near-field infrared microscopy of Qazilbash
et al. [62, 118, *Huffman18] on VO2 films actually maps the spatial phase
separation, and they extract a divergence of the dielectric constant at the
transition temperature. In the case of the high-temperature Mott transition of
V2O3, Limelette et al. concluded the coexistence region close to the critical
endpoint at $T_{\rm crit}=458$ K from the hysteresis in the pressure-dependent
conductivity curves [120]. From transport measurements on $\kappa$-(BEDT-
TTF)2Cu2[N(CN)2]Cl similar conclusions were drawn on the coexistence of Mott
insulator and correlated metal [9]. In addition, NMR experiments revealed the
coexistence of antiferromagnetism and superconductivity, as well as a
hysteresis in susceptibility [8].

Considering the IMT in doped semiconductors, Sir Nevill Mott termed the
divergence of the electric susceptibility ‘dielectric catastrophe’ only in the
second edition of his seminal monograph on the IMT [84]. Beyond percolation
and polarization issues, he pointed out the importance of the localization
length $\xi$ and electronic interaction. Aebischer et al. came back to this
idea and theoretically analyzed the $\varepsilon_{1}\propto\xi^{2}$ behavior
when approaching the Mott transition from the insulating side [121]. These
situations are distinct from the present case, where phase coexistence causes
the permittivity enhancement at the first-order IMT.

Record-high dielectric constants are observed when low-dimensional metals
undergo a spin- or charge-density wave transition [93]. Due to nesting of the
Fermi surface, a gap $\Delta_{0}$ opens in the density of states, leading to
$\varepsilon\propto\Delta_{0}^{-2}$ on the order of $10^{6}$ to $10^{8}$
[122]. In general, this mechanism also holds for opening a Mott gap [121] and,
therefore, we estimate the impact of reducing the gap on $\varepsilon_{1}$ in
the following.

####  IV.2.1 Reducing the Mott gap

Let us assume that the Mott gap is already closed. Then the real part of the
conductivity follows $\sigma_{1}(\omega)\approx
A\left|\omega\right|{}^{\beta}$ at low frequency, as shown in the inset of
Fig. 13, where $A$ is a normalization factor. Any non-analytic behavior of the
dielectric function $\varepsilon_{1}(\omega\rightarrow 0)$ can only result
from a sufficiently singular form of $\sigma_{1}$ also at low frequency. The
corresponding $\varepsilon_{1}(\omega)$ can be directly obtained from the
Kramers-Kronig relations [122], yielding

$\varepsilon_{1}(\omega)=1+4\int_{-\infty}^{+\infty}\frac{\sigma_{1}(\omega^{\prime})/\omega^{\prime}}{\omega^{\prime}-\omega}{\rm
d}\omega^{\prime}\quad.$ (6)

For a linear-in-frequency increase of the optical conductivity ($\beta=1$), it
is straightforward to analytically calculate $\varepsilon_{1}(\omega)$:

$\varepsilon_{1}(\omega)=-2A\log\left\\{\omega\right\\}+B(\omega)\quad,$ (7)

where $B(\omega)$ is an analytic function that remains finite at $\omega=0$.
Fig. 13 displays the behavior of $\varepsilon_{1}(\omega)$ for different
values $\beta$ of the power law. For $\beta=1$, the dielectric constant
$\varepsilon_{1}(\omega)$ assumes a logarithmic divergence at zero frequency,
which is a very weak singularity. If $\beta<1,$ then
$\varepsilon_{1}(\omega=0)$ diverges more strongly; the static dielectric
constant is just a finite number as $\beta$ gets larger than unity.

Figure 13: Frequency dependence of the real part of the dielectric constant,
$\varepsilon_{1}(\omega)$, for different power-law behavior of the optical
conductivity $\sigma_{1}(\omega)\propto\omega^{\beta}$ as plotted in the
inset.

Within DMFT, the precise exponent $\beta$ is still under debate [123], but it
is at most of order unity, consistent with our numerical results. This means
that the permittivity varies only little by reducing the Mott gap as the
transition is approached from the insulating side and cannot cause the
enormous dielectric enhancement found in our experiments. The latter is well
described by microscopic dynamical mean-field theory combined with macroscopic
percolation theory [23]. In the present case of $\kappa$-(BEDT-TTF)2Cu2(CN)3,
optical investigations at ambient pressure unambiguously show that no clear-
cut gap exists, despite the strongly insulating behavior [124, 125]. In the
THz and far-infrared range the conductivity is rather well described by a
linear frequency dependence, corresponding to $\beta\approx 1$ [30]. The low-
energy spectral weight actually increases upon lowering the temperature, in
contrast to the quantum spin liquid compounds $\kappa$-(BEDT-TTF)2Ag2(CN)3 and
$\beta^{\prime}$-EtMe3Sb[Pd(dmit)2]2, which are all far away from the IMT
boundary [14]. Low-temperature optical investigations on $\kappa$-[(BEDT-
TTF)1-x-(BEDT-STF)x]2Cu2(CN)3 in the far-infrared range yield that around 1
THz $\varepsilon_{1}(x)$ increases only slightly when spectral weight is
transferred to low frequency as the IMT is approached by substitution (see
Supplementary Fig. 7 in Ref. [23]). Hence, we conclude that the observed
enhancement of the permittivity upon approaching the Mott transition
originates from the first-order nature and the concomitant percolating phase
coexistence, and is not caused by closing the Mott gap.

####  IV.2.2 Phase coexistence at the genuine Mott transition in organic spin
liquids

The enhancement of $\varepsilon_{1}(U/W)$ – respectively $\varepsilon_{1}(p)$
– is sharply confined to the coexistence region, where it exceeds the values
of the (homogeneous) Mott insulator by orders of magnitude. In related studies
on $\kappa$-[(BEDT-TTF)1-x-(BEDT-STF)x]2Cu2(CN)3 chemical substitution was
utilized to increase the bandwidth in order to tune the system across the Mott
transition [23, 77]. The dielectric permittivity exhibits a similar maximum
when approaching the phase boundary as correlations decrease. In the
coexistence regime, percolation and correlation effects both contribute to the
dielectric properties. It is interesting to note that this peak appears in an
abrupt fashion, very distinct from the smooth and gradual increase in
disorder-driven IMTs [126]. Moreover, around the first-order IMT below $T_{\rm
crit}$, the permittivity exhibits a pronounced frequency dependence beyond
standard percolation theory [23].

Finally, we draw attention to recent DMFT calculations [123] revealing a
crossover (non-asymptotic) power-law behavior in the spectral function
$A(\omega)$ and the self-energy $-{\rm Im}\left\\{\Gamma(\omega)\right\\}$
extending from low to elevated temperatures. Concomitantly, consistent scaling
of the resistivity is found above and below $T_{\rm crit}$ as well. Both are
traced back to the metastable insulating phase in the coexistence region,
suggesting local quantum criticality of the Mott transition below $T_{\rm
crit}$, which eventually is also responsible for its well-known counterpart at
elevated temperatures. Besides these fundamental findings, this study reveals
a peculiar low-frequency behavior of $A(\omega)$ in proximity to the Mott
transition. Whether this can be connected to the intriguing dielectric
response observed here has yet to be clarified, but might provide a route to
place the various features in $\varepsilon_{1}(p,T,\omega)$ on the same
footing, such as the pressure evolution of the HT peak and the anomalous
power-law decrease of $\varepsilon_{1}(\omega)$ in the coexistence region. We
remind at this point that the HT peak is observed in several other organic
dimer Mott insulators [127, 128, 53] with triangular lattice, showing that its
emergence is independent of details in the crystal structure.

####  IV.2.3 Applicability of percolation theory

Percolation theory is applicable for systems consisting of two distinct types
of domains. One should keep in mind, however, that in real materials domain
walls are always present with properties distinct from either of the
coexisting phases. The precise characteristics of the domain walls reflect the
specifics of the clean system displaying phase coexistence and, as such, have
particular dimensions (thickness) and occupy a finite volume fraction. The
effects of such domain walls can be expected to be negligible if their
dimension (thickness) is much smaller than the characteristic domain size,
which is what we expect for weak disorder. In contrast, when disorder is
sufficiently strong, it is expected [129, 130] to produce nucleation centers
for more and more droplets, leading to the reduction of the domain size, which
eventually becomes comparable to the thickness of the domain walls. When this
happens, a simplistic two-component percolation picture is no longer of direct
relevance, and one may expect more gradual variations of all observables
compared to simple percolation theory. We believe that finite disorder is the
main reason why the peak of $\epsilon_{1}(p)$ appears more narrow in
theoretical results than in experiments (cf. Fig. 4 in [23]).

We want to recall that in contrast to most other examples of percolative
behavior, here we do not have different materials mixed, not even the crystal
structure or symmetry changes between insulating and metallic regions. In the
ideal case, the domains are distinct by the effect of correlations on their
physical properties. From thermal expansion studies on the related compound
$\kappa$-(BEDT-TTF)2Cu[N(CN)2]Cl in the vicinity of the critical endpoint, we
know that metallic and insulating phases exhibit a slightly different volume
and distinct expansion coefficients [131]. The particular arrangement is
susceptible to strain, impurities, etc. leading to domain boundaries with an
intermediate lattice constant on a local scale.

###  IV.3 Phase diagram

The main results of our pressure- and temperature-dependent dielectric
spectroscopic studies on $\kappa$-(BEDT-TTF)2Cu2(CN)3 are summarized in Figs.
1 and 12. The former displays a three-dimensional plot of
$\varepsilon_{1}(p,T)$ measured at $f=380$ kHz. The bottom area contains a
sketch of the phase diagram constructed on the projection of the
$\varepsilon_{1}(p,T)$ values with the corresponding color code; the intense
dark red area indicates the enhanced values in the coexistence phase when
spatially separated metallic regions grow in the insulating matrix. The
percolative behavior softens as temperature increases: the maximum diminishes
and eventually a gradual crossover remains above $T_{\rm crit}$. Additionally,
we include the quantum Widom line from the data of Ref. 12 that agrees with
the results presented here. The bifurcation temperature $T_{\rm B}$ marks the
change from the HT to the LT peak and the concomitant modification in the
relaxation dynamics. The Fermi-liquid temperature $T_{\rm FL}$ was extracted
from the resistivity [23]; our findings are in accordance with previous
reports [25, 13].

Fig. 12 displays the phase diagram of a genuine Mott insulator around the
first-order phase transition to the metallic state. For each of the three
ranges we compare experimental and theoretical results of the conductivity and
dielectri permittivity. Our data provide first experimental evidence for the
coexistence of the Mott-insulating and the metallic phases, as predicted by
DMFT calculations on a disordered Hubbard model for half-filling [132, 17].
This regime of metal-insulator coexistence emerges from the insulating phase
and partially overlaps with the Fermi-liquid regime. When the phase boundary
is crossed, the metallic fraction grows and quickly forms a continuous path
through the specimen; the capacitive coupling of remaining metallic puddles in
the insulating regions leads to the large values of $\varepsilon_{1}$. The
percolative behavior is strongly suppressed for higher temperatures where the
first-order Mott IMT becomes a smooth crossover and the contrast in
conductivity between metallic and insulating fraction diminishes.

Approaching the phase boundary from the insulating side, we determine a
critical exponent $q\approx 1$ in Eq. (5) that is consistent with Bruggeman’s
effective medium approximation that allowed us to extract the metallic volume
filling fraction as a function of pressure.

##  V Summary

Our dielectric measurements as a function of frequency, temperature and
pressure in the insulating state of $\kappa$-(BEDT-TTF)2Cu2(CN)3 reveal that
the relaxor-ferroelectric peak below $T=50$ K shifts to lower temperatures as
pressure increases because the screening by free charge carriers becomes
pronounced with increasing bandwidth. A second peak emerges at lower
temperatures and grows in amplitude with applying pressure. We attribute this
behavior to the sparse occurrence of metallic puddles in the insulating host
phase. Upon moving deeper into the phase coexistence region of the first-order
transition from the Mott-insulating to the metallic phase, we discover a
strong enhancement of $\varepsilon_{1}(p)$ up to $10^{5}$ at lowest
frequencies ($f=7.5$ kHz), which resembles percolating behavior. We apply
Bruggeman’s effective medium approximation to determine the metallic filling
fraction and obtain a critical exponent $q\approx 1$ upon approaching the
threshold $x_{c}$ from the insulating side:
$\varepsilon_{1}(x)\propto(x_{c}-x)^{-q}$. Calculations by dynamical-mean-
field theory on a single-band Hubbard model reproduce our comprehensive
experimental findings in full breadth. The divergency of the dielectric
permittivity is mainly caused by classical percolation physics of a strongly
correlated electron system close to the Mott transition. Our results provide
compelling evidence for the coexistence of metallic and insulating regions,
and we demonstrate the capabilities of dielectric spectroscopy as a ‘smoking
gun’ to probe phase coexistence and spatial inhomogeneities at metal-insulator
transitions.

##  VI acknowledgments

We thank G. Untereiner for contacting the crystals. We acknowledge support by
the Deutsche Forschungsgemeinschaft (DFG) via DR228/52-1. A.P. acknowledges
support by the Alexander von Humboldt-Foundation through the Feodor Lynen
Fellowship. E.U. acknowledges the support of the European Social Fund and by
the Ministry of Science Research and the Arts of Baden-Württemberg. Work in
Florida was supported by the NSF Grant No. 1822258, and the National High
Magnetic Field Laboratory through the NSF Cooperative Agreement No. 1644779
and the State of Florida.

##  Appendix A Dielectric properties of sample 1

Most data presented in this article have been obtained from dielectric
measurements on sample 2; the conclusions on phase separation at the Mott IMT
are fully supported by similar findings on sample 1. Fig. 14 gives an overview
on the pressure evolution of its dielectric response by plotting
$\varepsilon_{1}(T)$ for selected frequencies and pressures as indicated. The
position of the HT feature at ambient pressure, for instance at 40 K when
probed at $f=100$ kHz, is in good agreement with previous ambient pressure
reports [19, 21]. Unfortunately, only a few measurements in the Mott-
insulating phase could be performed before the sample broke. Nevertheless,
despite the low pressure resolution, it is obvious that the shift of the HT
peak upon increasing pressure resembles the findings for sample 2. In
contrast, the LT peak for sample 1 is barely visible up to $p=0.45$ kbar and
becomes evident only at 0.7 kbar around $T=10$ K. The strong enhancement of
$\varepsilon_{1}$ by several orders of magnitude upon entering the coexistence
regime is already observed at $p=1.0$ kbar and extends up to 1.9 kbar [Fig.
14(f)]. We emphasize that this is in accordance with the shift of the IMT in
sample 1. Interestingly, the plateau-like shape of $\varepsilon_{1}(T)$ is
observed only for $f\leq 400$ kHz whereas for higher frequencies a slight
downturn occurs. Upon further pressure increase, metallic behavior with
$\varepsilon_{1}<0$ sets in.

Figure 14: Plot of the dielectric permittivity $\varepsilon_{1}(T)$ of
$\kappa$-(BEDT-TTF)2Cu2(CN)3 sample 1 for several frequencies upon increasing
pressure. The results are qualitatively identical to those obtained on sample
2, displayed in Figs. 2 and 3. (a) At ambient pressure the relaxor-type
ferroelectric peak starts already around $T=70$ K and reaches the maximum at
low frequencies at 30 K. (b-c) A shoulder-like feature is already present at
ambient pressure and develops into a second peak upon pressurizing. (d-f) In
the coexistence phase between $p=1.0$ and 1.9 kbar, an enormous increase of
$\varepsilon_{1}$ is observed due to percolation. (g,h) Above $p=3.0$ kbar,
$\varepsilon_{1}<0$ for nearly all measured frequencies indicating metallic
behavior. Figure 15: Arrhenius plot of the fitting parameters of mode 1 for
sample 1 at different pressures as indicated. (a) Dielectric strength
$\Delta\varepsilon^{\text{mode 1}}(T)$, (b) distribution of relaxation times
$1-\alpha_{1}(T)$ and (c) mean relaxation time $\tau_{1}(T)$. The black and
red lines represent fits with Eq. (3) above and below the kink in
$\tau_{1}(T)$ at $T_{\rm{B}}$, respectively. Figure 16: Arrhenius plot of the
fit parameters of mode 2 for $\kappa$-(BEDT-TTF)2Cu2(CN)3 sample 1 at
different pressures as indicated. (a) Dielectric strength
$\Delta\varepsilon^{\text{mode 2}}(T)$, (b) distribution of relaxation times
$1-\alpha_{2}(T)$, and (c) mean relaxation time $\tau_{2}(T)$.

In order to analyze the frequency-dependent permittivity of these data, we
have applied the same procedure as presented in Section sec:insulator on
sample 2. The obtained fit parameters $\Delta\varepsilon^{\text{mode 1}}$,
$1-\alpha_{1}$ and $\tau_{1}$ for mode 1 are plotted in Fig. 15 as a function
of inverse temperature. The peak in $\Delta\varepsilon^{\text{mode 1}}(T)$
shifts to lower $T$ and increases in amplitude as pressure is applied. This
behavior is reminiscent to what is observed for sample 2, however, it is less
pronounced for sample 1, for which the LT peak is weaker and becomes apparent
only at $p\geq 1.0$ kbar. This also explains the higher values of
$1-\alpha_{1}$ and their restrengthening upon cooling, the latter most
pronounced at ambient pressure, which indicate less broadening in sample 1. In
$\tau_{1}(T)$, a kink at $T_{\rm{B}}$ is observed, which shifts to lower
temperatures while the corresponding relaxation time gets shorter, reproducing
the behavior observed in sample 2. The parameters obtained by fitting
$\tau_{1}(T)$ with the activated behavior [Eq. (3)] for the HT mode
($T>T_{\rm{B}}$) and the LT mode ($T<T_{\rm{B}}$) are already presented and
discussed in Table 2 and Fig. 8 above.

Fig. 16 displays the parameters $\Delta\varepsilon^{\text{mode 2}}(T)$,
$1-\alpha_{2}(T)$ and $\tau_{2}(T)$ of mode 2 for sample 1 of $\kappa$-(BEDT-
TTF)2Cu2(CN)3. The dielectric strength $\Delta\varepsilon^{\text{mode 2}}$
exhibits a maximum around $T=30$ K at ambient pressure, which shifts towards
lower temperatures and diminishes upon increasing $p$. This is in contrast to
sample 2, for which $\Delta\varepsilon^{\text{mode 2}}$ monotonously grows
upon cooling (cf. Fig. 9). Throughout the entire pressure range
$\Delta\varepsilon^{\text{mode 2}}$ is smaller than the one of mode 1 by
approximately a factor of 2. The mode shifts towards lower temperatures with
rising pressure. A drop in $1-\alpha_{2}(T)$ indicates a considerable
broadening of the mode upon cooling, which gets less pronounced for increasing
pressure. The temperature dependence of $\tau_{2}$ is strongly influenced by
pressure and shows sample dependence. For sample 1, the monotonic increase is
much steeper and observed down to 15 K at ambient pressure which saturates
into a plateau upon increasing pressure. In contrast to sample 2, only the
onset of a shallow minimum around $T=12$ K is revealed indicating non-
monotonic relaxation dynamics.

##  Appendix B Spurious effects

###  B.1 Contacts

Owing to the two-point configuration usually applied in dielectric
spectroscopy, the obtained data may include contributions from polarization
effects at the contacts, which have to be conscientiously ruled out or
determined. Since the contacts are produced by amorphous carbon (carbon paste)
with metallic properties, Schottky contacts may form at the sample-contact
interfaces, resulting in a depletion layer at the interface with thickness

$d_{\textrm{depl}}\ =\
\left[\frac{2\varepsilon_{1}\varepsilon_{0}}{eN_{\textrm{c}}}(\Phi_{\textrm{m}}-\Phi_{\textrm{s}}\pm
eU)\right]^{1/2}\quad,$ (8)

wherein $\Phi_{\textrm{m}}$ and $\Phi_{\textrm{s}}$ the distances between the
vacuum level and the chemical potential of the metal and semiconductor,
respectively, and $U$ indicates the voltage of the applied ac signal. The
charge-carrier density
$N_{\textrm{c}}(T)\propto\exp\left\\{-\Delta/k_{\textrm{B}}T\right\\}$ is
determined by thermal excitations across the charge gap $\Delta$. The modified
charge density in the depletion zone gives rise to an additional capacitance

$C_{\textrm{depl}}\ \propto\ \frac{1}{d_{\textrm{n}}}\ \propto\
C\exp\left\\{\frac{-\Delta}{2k_{\textrm{B}}T}\right\\}\quad,$ (9)

which eventually is responsible for the spurious effects attributed to the
contact contribution. On the right side of Eq. (9) we estimated the
temperature dependence of the contact contribution, which is governed by
$N_{\textrm{c}}(T)$. In Fig. 17(a) we exemplarily plot $\varepsilon_{1}(T)$ up
to room temperature, probed at various frequencies as indicated. In the
temperature range $75~{}{\rm K}<T<300$ K, we observe a decrease of
$\varepsilon_{1}(T)$ upon cooling which is very well described by Eq. (9)
(orange line) and hence is attributed to the contact contribution.

Figure 17: (a) Temperature dependence of $\varepsilon_{1}(T)$ plotted in the
entire temperature range, for an applied pressure of $p=0.52$ kbar measured at
various frequencies $f$. In addition, we determine the contact contribution by
fitting the high-temperature part with Eq. (9), as shown for the example of
$f=380~{}$kHz (orange line). (b) Detailed view of the relaxor-ferroelectric
relaxation at low temperatures including the contact contribution. (c)
Relaxor-ferroelectric response after the contact contribution has been
subtracted: the contact contribution is negligible below $T=60$ K.

Most importantly, the contact contribution is negligible below 60 K [Fig.
17(b)] and does not influence the analysis of the relaxor like dielectric
response [Fig. 17(c)]. We also note, that this effect would be too weak to
explain the huge enhancement of $\varepsilon_{1}(T)$ close to the phase
boundary. If we inspect $\rho_{1}(T)$ and $\varepsilon_{1}(T)$ at 1.91 kbar
[23], for instance, we find that
$\rho_{1}(p={1.91~{}\text{kbar}},T=10~{}\text{K})\approx\rho_{1}(p={1.91~{}\text{kbar}},T=300~{}\text{K})$,
such that the spurious contact contribution to $\varepsilon_{1}$ should be the
same at 10 K and 300 K. On the other hand, we see that
$\varepsilon_{1}(p={1.91~{}\text{kbar}},T=10~{}\text{K})\gg\varepsilon_{1}(p={1.91~{}\text{kbar}},T=300~{}\text{K})$
(cf. Fig. 2), which cannot be explained by such a contact contribution,
corroborating our phase coexistence scenario.

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