Low-temperature dielectric anomaly arising from electronic phase separation at the Mott insulator-metal transition

Coulomb repulsion among conduction electrons in solids hinders their motion and leads to a rise in resistivity. A regime of electronic phase separation is expected at the first-order phase transition between a correlated metal and a paramagnetic Mott insulator, but remains unexplored experimentally as well as theoretically nearby T = 0. We approach this issue by assessing the complex permittivity via dielectric spectroscopy, which provides vivid mapping of the Mott transition and deep insight into its microscopic nature. Our experiments utilizing both physical pressure and chemical substitution consistently reveal a strong enhancement of the quasi-static dielectric constant ε1 when correlations are tuned through the critical value. All experimental trends are captured by dynamical mean-field theory of the single-band Hubbard model supplemented by percolation theory. Our findings suggest a similar ’dielectric catastrophe’ in many other correlated materials and explain previous observations that were assigned to multiferroicity or ferroelectricity.


INTRODUCTION
Insulator-metal transitions (IMTs) remain one of the unresolved science problems of condensed-matter physics, which are rather general and of fundamental importance. Especially intriguing are those IMTs not associated with static symmetry changes, where paradigms for conventional phase transitions provide little guidance. Early examples of such behavior are found in certain disorder-driven IMTs 1 . In recent years, IMTs with no symmetry breaking were also identified around the Mott transition [2][3][4][5] , which bears close connection to exotic states of strongly correlated electron matter, such as unconventional superconductivity and quantum spin liquids (QSLs). From a theoretical point of view, the single-band Hubbard model is well understood 6,7 , and is found to be in excellent agreement with experiments 5,[8][9][10] . While commonly concealed by antiferromagnetism, recent development in the field of organic QSLs enabled investigation of the lowtemperature Mott IMT in absence of magnetic order [11][12][13][14] , revealing finite-frequency precursors of the metal already on the insulating side 15 .
At half filling, the Mott insulator and the correlated metal converge at the critical endpoint T crit (Fig. 1). The former is bounded by a quantum-critical region along the quantum Widom line (QWL) 5,[15][16][17][18] . On the metallic side, resistivity maxima at the Brinkman-Rice temperature T BR signal the thermal destruction of resilient quasiparticles 19,20 and the crossover to semiconducting transport. Below T crit , the IMT is of first order and comprises an insulator-metal coexistence regime 6,16,17 . Figure 1 displays the phase diagram of genuine Mott systems at half filling without magnetic order 15 , as realized in triangular-lattice organic compounds subject to strong geometrical frustration 5,[11][12][13][14]21 . Note, many oxides are charge-transfer insulators 22 and involve a pronounced coupling of the IMT to the lattice 10,23 , whereas the molecular materials studied here are ideal model systems to elucidate single-band Mott insulators without significant structural modifications at the transition to the metallic state. It is currently debated whether the electrodynamics of the genuine Mott IMT is dominated by closing of the Mott gap or by spatial inhomogeneity, fueled by recent low-temperature transport studies 14 .
Although the high-quality crystals we investigate typically feature uniform insulating and metallic phases, the situation is more subtle within the IMT phase-coexistence region at T < T crit . Similarly to most other systems close to first-order phase transitions, here one generally expects spatial segregation into domains of the coexisting phases, and the associated metastability and hysteresis phenomena. Seminal transport and susceptibility experiments indeed found a pronounced hysteresis in the Mott systems V 2 O 3 and κ-(BEDT-TTF) 2 Cu[N(CN) 2 ]Cl featuring magnetically-ordered ground states 8,9,24 . Unfortunately, analogue measurements with continuous pressure tuning are not feasible on QSL compounds, such as κ-(BEDT-TTF) 2 Cu 2 (CN) 3 , due to the low temperatures (T < 20 K) and high pressures (p > 1 kbar) required to cross the first-order IMT. Alternative methods are, therefore, required in these systems in order to reveal and mapout regimes where such inhomogeneous electronic states arise. While different microscopic mechanisms can, in principle, contribute to stabilizing such inhomogeneous phase separation (see discussion below), in electronic systems it proves to dramatically modify the electrodynamic response leading to robust percolation phenomena that depend only weakly on the system-dependent details. This observation, as we shall demonstrate below, allows us to clearly recognize and understand the regime dominated by phase coexistence, resolving much of the controversy associated with the nature of the Mott point in QSL compounds.
A more direct insight into the coexistence region was provided by spatially resolved optical spectroscopy 25 . The most compelling results came from near-field infrared experiments on VO 2 by Basov and collaborators 26 where a spatial separation of metallic and insulating regions upon heating could be visualized, in accord with X-ray studies 10,27 ; the range of hysteresis in ρ(T) coincides with the observed phase coexistence. Although recent developments in cryogenic near-field instrumentation are rather promising [28][29][30] , they fall short of covering the regime T < T crit ≈ 15 K required here and do not allow for pressure tuning. For this reason, we suggest dielectric spectroscopy as a powerful bulksensitive method in order to unravel the coexistence regime, distinguish the individual phases and obtain a deeper understanding of the dynamics around the IMT. The complex conductivity σ 1 + iσ 2 not only reveals the closing of the Mott gap but yields insight into the growth of metallic regions and the formation of quasiparticles as correlation effects decrease.
Here we tackle the fundamental question whether the electrodynamic response around the Mott IMT is overwhelmed by the gradual decrease of the Mott-Hubbard gap within a homogeneous insulating phase, or whether the effects of phase coexistence dominate. Furthermore, is it possible to distinguish on the metallic side between the coherent (quasiparticle) low-T regime and incoherent transport at high-T? To answer these questions, we present temperature-and frequency-dependent dielectric measurements on a genuine Mott compound that is bandwidth-tuned across its first-order IMT. In addition to hydrostatic pressure we developed an innovative approach of chemically substituting the organic donor molecules. The experimental findings are complemented by dynamical mean-field theory (DMFT) calculations, incorporating spatial inhomogeneities in a hybrid approach. We conclude that electronic phase segregation plays a crucial role, leading to percolative phenomena due to the separation of insulating and metallic regions, also allowing clear and precise mapping of different dynamical regimes around the IMT.

Mott transition through pressure and chemical substitution
We have chosen κ-(BEDT-TTF) 2 Cu 2 (CN) 3 single crystals for our investigations because this paradigmatic QSL candidate is well characterized by electric, optical and magnetic measurements 3,14,15,21,31,32 . Although the dimerized charge-transfer salt possesses a half-filled conduction band, strong electronic interaction U ≈ 250 meV stabilizes a Mott-insulating state below the QWL (T QWL ≈ 185 K at ambient pressure 5,15 ) with no magnetic order observed down to mK temperatures 3 . The effective correlation strength U/W can be reduced by increasing the bandwidth W; for pressure p > 1.4 kbar the metallic state is reached at low temperatures, with T crit ≈ 15-20 K. In addition, we exploited an alternative route of partially replacing the S atoms of the donor molecules by Se, where more extended orbitals lead to larger bandwidth (see sketches in Fig. 2h, k). The substitutional series spans the interval ranging from a Mott insulator to a Fermiliquid metal. Details on the sample characterization and experimental methods are given in Supplementary Note 1. Here we focus on the out-of-plane dielectric response measured from f = 7.5 kHz to 5 MHz down to T = 5 K. Both physical pressure and BEDT-STF substitution allow us to monitor the permittivity while shifting the system across the first-order IMT. Figure 2 displays the temperature-dependent conductivity and permittivity data of κ-(BEDT-TTF) 2 Cu 2 (CN) 3

when p rises (a-e) and
x increases in κ-STF x (f-j). The insulating state (p < 1.4 kbar, x < 0.1), characterized by dσ 1 /dT > 0, generally features small positive ε 1 ≈ 10. The relaxor-like response previously observed in the parent compound below 50 K has been subject of debate 33,34 . The metallic state (p > 3 kbar, x > 0.2) is defined by dσ 1 /dT < 0 and, concomitantly, ε 1 < 0 that becomes very large at low T as itinerant electrons increasingly screen. Comparison of the results in Fig. 2j with optical data measured on the same substitution yields fair agreement of the metallic values ε 1 < 0 (see Supplementary Fig. 7). This onset of metallic transport identifies T BR 19 ; while thermal fluctuations prevail at higher T, the quasiparticle bandwidth is the dominant energy scale for T < T BR . Below T FL the resistivity ρ(T) ∝ T 2 indicates the Fermi-liquid state (see Fig. 1).
Right at the first-order IMT, however, the dielectric behavior appears rather surprising. When approaching the low-temperature phase boundary, ε 1 rapidly increases by several orders of magnitude. This colossal permittivity enhancement is more pronounced in the quasi-static limit, ε 1 ≈ 10 5 at f = 7.5 kHz, and the peak value approximately follows a f −1.5 dependence. The overall range in T and p/x of the divergency is robust and does not depend on the probing frequency (see Supplementary Figs. 4 and 8).
In Fig. 3a, b the pressure evolution of σ 1 and ε 1 is plotted for fixed T. At T = 10 K, σ 1 (p) rises by six orders of magnitude in the narrow range of 1 kbar. This behavior flattens to a gradual transition above 20 K, associated with the quantum-critical crossover at the QWL. The inflection point shifts to higher p, in accord with the positive slope of the phase boundary 15 associated with the rising onset of metallicity at T BR . The κ-STF x series exhibits similar behavior (Fig. 3c, d): around the critical concentration of x ≈ 0.12 a drastic increase in σ 1 is observed at low T that smears out as T rises. The maximum in ε 1 (x) is reached for x = 0.16 but broadens rapidly upon heating. Fig. 1 Electronic transport regimes around the genuine Mott insulator-metal transition at half filling. a Tuning the bandwidth W, for instance by chemical or physical pressure, transforms a Mott insulator to a correlated metal. Dynamical mean-field theory predicts a first-order transition with phase coexistence (delimited by solid lines) up to the critical endpoint 6 , and a quantum-critical regime associated with the quantum Widom line (QWL, dashed black line) above T crit 16 . The metallic state is confined by the Brinkman-Rice temperature T BR (dashed cyan) at T > T crit , the coherent Fermi-liquid regime by T FL (dashed blue). When interactions U are comparable to W, and T ≫ T crit , semiconducting behavior prevails; neither a gap nor a quasiparticle peak are stabilized. b-d Resistivity signatures of the crossovers in dc transport. b The steepest slope of ln fρðpÞg indicates the crossing of the QWL upon a pressure sweep. c T BR indicates the transition from an insulating (dρ/dT < 0) to a metallic (dρ/dT > 0) temperature dependence. d The ρ(T) ∝ T 2 behavior extends up to T FL .

Modeling phase coexistence by hybrid DMFT
The Mott IMT is based on the physical picture that a reduction of electronic correlations (i.e., the rise of W/U) gradually closes the Mott-Hubbard gap: a coherent charge response develops, causing a finite metallic conductivity. Pressure-dependent optical studies on several organic Mott insulators indeed observe this behavior 13,35 . Concerning dielectric response, it was pointed out 36 that even in the case of certain continuous IMTs, the reduction of the charge gap might induce an enhancement in ε 1 , perhaps leading to a divergence at low T. Probing the optical response at THz frequencies is actually a convenient method to monitor the gap contribution to the permittivity. From p and T sweeps across the Mott IMT of several different materials, an increase by a factor of 10 is consistently reported 13,26 ; in the case of κ-STF x we find it to be even smaller (see Supplementary Fig. 7). Hence, the dielectric catastrophe of ε 1 ≈ 10 5 observed in our pressure and substitutiondependent dielectric experiments clearly signals a different mechanism at play.
To understand the physical origin of this anomaly, we recall the phase diagram of the Mott IMT at half filling, as obtained from DMFT 6,7,37,38 , which has been found to successfully describe many experimental features of QSL materials 4,15 such as ours. It features a first-order IMT at low temperatures (k B T crit ≈ 0.02W), and an associated phase-coexistence region over an appreciable range of U/W. As in most other systems displaying phase coexistence, here we expect spatial phase segregation of the respective metallic and insulating domains, with a relative volume fraction that smoothly varies with U/W. Such a picture resembles composite materials, such as microemulsions, composites or percolating metal films 39-43 .
Here we expect percolation-a phenomenon that has been long studied in a number of spatially inhomogeneous systems-to distinctly affect all physical quantities. One of the prominent consequences is a strong enhancement of ε 1 around the percolation transition, in addition to dramatic changes in the dc resistivity [44][45][46][47] .
While percolation theory is by now well understood, its specific experimental manifestation may be very sensitive to the physical properties of the coexisting phases, especially in strongly correlated systems. To theoretically investigate the physical consequences of such percolative effects for our Mott system, we set up a "Hybrid DMFT" simulation across the entire phase diagram. Here we utilized DMFT to calculate ε 1 + iε 2 for a singleband Hubbard model at half filling, and obtained the respective responses for both the insulating and the metallic phase as a function of T/W, U/W and frequency ω/W. Using this information on the homogeneous phases as input to percolation theory, we  3 were measured as a function of temperature and frequency for various applied a-e pressures and f-j chemical substitutions (introduction of Se-containing BEDT-STF molecules illustrated in h and k) that drive the system across the Mott transition. a, f Starting from the insulator, σ 1 (T) grows with increasing p or x; a metallic phase is stabilized below T BR , in accord with dc transport 5,14,21 . b, c, g, h In the Mott-insulating state ε 1 (T) exhibits relaxor-ferroelectric behavior similar to the parent compound 33,34 . Extrinsic high-temperature contributions are subtracted. d, i The strong enhancement of ε 1 (T) at the transition is a hallmark of a percolative IMT, as sketched in l-o. e, j When screening becomes dominant in the metal, ε 1 turns negative; σ 1 exhibits Fermi-liquid behavior below T FL .
then set up an appropriate random resistor network describing the dielectric properties within the phase-coexistence region, and solved it using standard effective-medium methods 48 , which suffices for our purposes.
We stress here that our Hybrid DMFT method provides a fully microscopic modeling for the physical properties of the respective phases across the phase diagram, a task that required a large-scale computational effort which we carried out in detail. In contrast, here we do not address the microscopic mechanism leading to the assumed phase separation, but instead adopt a phenomenological model (see Methods) describing how the relative volume fractions of the percolating species vary with T and U/W. In principle, spatial phase separation we envision may originate either from disorder effects 49,50 , or due to inherent ("Coulomb limited") phase separation [51][52][53] . In either case, one expects the relative volume fraction of the respective metallic/insulating components to smoothly vary with pressure, doping or temperature, and the resulting percolative effects should be very similar. We do not expect any of the resulting anomalies to depend on the details of our phenomenological model, e.g., the strong enhancement of the dielectric permittivity or the abrupt drop of the resistivity around the percolation point (see Supplementary Note 4 for a more detailed discussion of percolation phenomena in general).
Our hybrid DMFT simulation yields excellent agreement with experiment-both pressure tuning and chemical substitution-as illustrated in Fig. 3e, f and in the false-color plots in Fig. 4. We find that the colossal peak in ε 1 is confined to the spatially inhomogeneous coexistence regime, exactly as observed in experiments. As correlation effects diminish, the dynamical conductivity (upper panels) increases from the Mott insulator to the Fermi liquid. The step in σ 1 and drop in ε 1 appear abruptly in the model but more smoothly in experiment, most likely due to inhomogeneities which broaden the coexistence regime in real materials by providing nucleation seeds for the incipient phase. Note, the percolative transition region narrows for T → T crit and vanishes above that; the metallic fraction m of the simulation is indicated in Fig. 4f. The inset of panel f clearly demonstrates that the colossal permittivity enhancement appears exclusively for a percolating mixture of metallic and insulating regions, and not for the pure phases. These results render the gap closing irrelevant for the electrodynamics at the low-temperature Mott IMT. We further point out that the discussed mechanism of a percolative enhancement of ε 1 may also apply to other related organic compounds, where similar dielectric anomalies at or nearby a firstorder IMT were previously assigned to ferroelectricity 54 and multiferroicity 55 .

DISCUSSION
While previous transport and optical studies 5,15 already provided hints favoring the DMFT scenario, they do not map-out the predicted dynamical regimes, especially regarding a well-defined coexistence region at T < T crit . Our pressure-and substitutiondependent dielectric data, however, reveal all phases in vivid detail and in remarkable agreement with the respective crossover lines obtained from dc transport. Indeed, we recognize the gapped Mott insulator by essentially constant ε 1 (light red) bounded precisely by the QWL 16 , while also below T BR 19,20 the response clearly follows the dielectric behavior expected for a metal (ε 1 < 0, blue). Most remarkably, these two crossover lines converge towards T crit , which marks the onset of the coexistence region, just as anticipated from Fig. 1. The emergence of phase segregation is evidenced by the huge peak of ε 1 in excellent agreement with our current DMFT-based modeling. The sharply defined boundaries of this dielectric anomaly imply that the corresponding inhomogeneities are the consequence and not the cause of phase separation, the latter resulting from strong correlation effects inherent to Mottness. Our findings substantiate that the DMFT scenario offers a rather accurate picture of the Mott IMT, in contrast to other theoretical viewpoints 56 , which focus on the spin degrees of freedom in the QSL. This also confirms recent experimental and theoretical results 57,58 suggesting that such gapless spin excitations, while dominant deep within the We demonstrated an expedient method of partial chemical substitution that triggers a similar Mott transition in κ-(BEDT-TTF) 2 Cu 2 (CN) 3 as physical pressure. The pronounced divergency in ε 1 , occuring for both types of bandwidth tuning, evidences a spatial coexistence of metallic and insulating electronic phases around the first-order IMT that can be circumstantially described by DMFT blended with percolation theory. Our results yield that the Mott gap has a minor effect on the dielectric properties while the effects of phase coexistence dominate.

METHODS Experimental
High-quality single crystals of κ-(BEDT-TTF) 2 Cu 2 (CN) 3 were grown by the standard electrochemical synthesis method 59,60 at Stuttgart and at Argonne. Plate-like single crystals of κ-[(BEDT-TTF) 1−x (BEDT-STF) x ] 2 Cu 2 (CN) 3 with a typical size of 1 × 1 × 0.3 mm 3 were prepared according to ref. 61 ; here BEDT-TTF stands for bis(ethylenedithio)tetrathiafulvalene. Substituting two of the inner sulfur atoms by selenium leads to bis-ethylenedithiodiselenium-dithiafulvalene, abbreviated BEDT-STF. Energy-dispersive X-ray spectroscopy was employed to determine the chemical composition of the alloys 0 ≤ x ≤ 1 61 . The complex impedance of the crystals was measured as a function of temperature and frequency to obtain the permittivityε ¼ ϵ 1 þ iϵ 2 and the conductivityσ ¼ σ 1 þ iσ 2 . The crystals are contacted outof-plane (E∥a) by attaching thin gold wires with carbon paint to the opposite crystal surfaces. The measurements were performed with two contacts in a pseudo four-point configuration using an Agilent 4294 Impedance Analyzer. The applied ac voltage was set to 0.5 V, which was still in the Ohmic regime.
For the pressure-dependent experiments on the κ-(BEDT-TTF) 2 Cu 2 (CN) 3 crystals, we used a piston-type pressure cell ranging up to 10 kbar with a self-made electrical feedthrough for coaxial cables designed for pressuredependent dielectric spectroscopy measurements as described in detail elsewhere 62 . Daphne oil 7373 was used as liquid pressure-transmitting medium because it is inert to molecular solids, has a good hydrostaticity and stays fluid at room temperature for all applied pressures. The inherent pressure loss upon cooling was recorded continuously in situ by an InSb semiconductor pressure gauge 63 , which shows a negligible pressure gradient below 50 K. The main temperature range discussed here is collected from the same pressure cycles; and the pressure reading at 10 K is given throughout the manuscript, unless stated otherwise. A continuous flow cryostat was utilized to cool down the pressure cell. A total cable length of only 50 cm enables measurements up to 5 MHz. The rather steep thermal gradient limits the lowest reachable temperature to 8 K, hence the superconducting state at T ≈ 4 K 14,21 was not accessible to our dielectric experiments. No dependence on the cooling rate was observed, which was kept below 0.4 K/min for all measurements. Additional data, including analysis on the frequency dependence, can be found in Supplementary Notes 2 and 3.

Dynamical mean-field theory modeling
We performed standard 6,20 DMFT calculations of the optical conductivity σ 1 of a single-band Hubbard model at 1/2-filling, using a semicircular model density of states and iterated perturbation theory as an impurity solver, which suffices 16,17 for our purposes. The imaginary part of conductivity σ 2 was then obtained via the Kramers-Kronig transformation, and the dielectric function ϵ ω ð Þ ¼ 1 þ i 4π ω σ ω ð Þ was obtained across the entire phase diagram, for both uniform phases. To describe percolative effects within the phase-coexistence region, we assumed spatial segregation of metallic and insulating regions with a smoothly varying metallic filling fraction. We used the standard effective-medium theory 64 for the corresponding random conductor/dielectric network, with a binary distribution of the local dielectric functions as calculated from DMFT for given temperature T and Coulomb repulsion U. All energies are expressed in units of the bandwidth W, i.e., T/W and U/W. For simplicity, we assign a smooth hyperbolic tangent function, as routinely used to describe phase coexistence near any first-order phase transition 49 , to model the volume fraction of the metallic phase: mðT=W; U=WÞ ¼ 1 2 tanh c½ðT=WÞ crit À T=W= È ½ðU=WÞ crit À U=Wg þ 1 2 , which is centered around the middle of the coexistence region: (U/W) crit = (0.20 − T/W)/0.14. In order to match the coexistence region with the experimental data, we selected c = 0.1 for U/W > (U/W) crit and c = 0.3 for U/W < (U/W) crit . More details and a visualization of the assumed metallic fraction m can be found in Supplementary Note 4. Fig. 4 Phase diagram of κ-(BEDT-TTF) 2 Cu 2 (CN) 3 when tuned through the Mott IMT. Both conductivity and permittivity exhibit similar behavior upon varying correlation strength U/W by a, b physical pressure or c, d chemical substitution, which is reproduced by e, f our hybrid DMFT calculations; the respective color codes are given to the right. In all cases, the colossal enhancement of ε 1 reveals a sharply defined insulator-metal phase-coexistence region around the first-order IMT, which is not seen in σ 1 . The permittivity clearly distinguishes the Mottinsulating and metallic states via small positive and large negative values, which line up perfectly with the quantum Widom line (QWL) and the Brinkman-Rice temperature T BR determined from σ 1 (T, p, x), respectively. Inset of f: metallic and insulating DMFT solutions are contrasted to the mixed phase (metallic fractions m indicated in main panel) that reproduces the strong peak in ε 1 . Note, T BR is indicated by a dashed cyan line at T > T crit in all panels; the resistivity maxima below T crit (solid cyan line) arise due to percolative effects.