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 IMTs1. In recent years, IMTs with no symmetry breaking were also identified around the Mott transition2,3,4,5, 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 understood6,7, and is found to be in excellent agreement with experiments5,8,9,10. While commonly concealed by antiferromagnetism, recent development in the field of organic QSLs enabled investigation of the low-temperature Mott IMT in absence of magnetic order11,12,13,14, revealing finite-frequency precursors of the metal already on the insulating side15.

At half filling, the Mott insulator and the correlated metal converge at the critical endpoint Tcrit (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 TBR signal the thermal destruction of resilient quasiparticles19,20 and the crossover to semiconducting transport. Below Tcrit, the IMT is of first order and comprises an insulator-metal coexistence regime6,16,17. Figure 1 displays the phase diagram of genuine Mott systems at half filling without magnetic order15, as realized in triangular-lattice organic compounds subject to strong geometrical frustration5,11,12,13,14,21. Note, many oxides are charge-transfer insulators22 and involve a pronounced coupling of the IMT to the lattice10,23, 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 studies14.

Fig. 1: Electronic transport regimes around the genuine Mott insulator-metal transition at half filling.
figure 1

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 endpoint6, and a quantum-critical regime associated with the quantum Widom line (QWL, dashed black line) above Tcrit16. The metallic state is confined by the Brinkman-Rice temperature TBR (dashed cyan) at T > Tcrit, the coherent Fermi-liquid regime by TFL (dashed blue). When interactions U are comparable to W, and TTcrit, semiconducting behavior prevails; neither a gap nor a quasiparticle peak are stabilized. bd Resistivity signatures of the crossovers in dc transport. b The steepest slope of \(\mathrm{ln}\,\{\rho (p)\}\) indicates the crossing of the QWL upon a pressure sweep. c TBR indicates the transition from an insulating (dρ/dT < 0) to a metallic (dρ/dT > 0) temperature dependence. d The ρ(T) T2 behavior extends up to TFL.

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 < Tcrit. 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 V2O3 and κ-(BEDT-TTF)2Cu[N(CN)2]Cl featuring magnetically-ordered ground states8,9,24. Unfortunately, analogue measurements with continuous pressure tuning are not feasible on QSL compounds, such as κ-(BEDT-TTF)2Cu2(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 map-out 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 spectroscopy25. The most compelling results came from near-field infrared experiments on VO2 by Basov and collaborators26 where a spatial separation of metallic and insulating regions upon heating could be visualized, in accord with X-ray studies10,27; the range of hysteresis in ρ(T) coincides with the observed phase coexistence. Although recent developments in cryogenic near-field instrumentation are rather promising28,29,30, they fall short of covering the regime T < Tcrit ≈ 15 K required here and do not allow for pressure tuning. For this reason, we suggest dielectric spectroscopy as a powerful bulk-sensitive 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 σ1 + iσ2 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.

Results

Mott transition through pressure and chemical substitution

We have chosen κ-(BEDT-TTF)2Cu2(CN)3 single crystals for our investigations because this paradigmatic QSL candidate is well characterized by electric, optical and magnetic measurements3,14,15,21,31,32. 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 (TQWL ≈ 185 K at ambient pressure5,15) with no magnetic order observed down to mK temperatures3. 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 Tcrit ≈ 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 κ-[(BEDT-TTF)1−x(BEDT-STF)x]2Cu2(CN)3 (0 ≤ x ≤ 1, abbreviated κ-STFx) spans the interval ranging from a Mott insulator to a Fermi-liquid 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.

Fig. 2: Dielectric properties upon tuning through the Mott transition.
figure 2

The dielectric conductivity and permittivity of κ-(BEDT-TTF)2Cu2(CN)3 were measured as a function of temperature and frequency for various applied ae pressures and fj 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, σ1(T) grows with increasing p or x; a metallic phase is stabilized below TBR, in accord with dc transport5,14,21. b, c, g, h In the Mott-insulating state ε1(T) exhibits relaxor-ferroelectric behavior similar to the parent compound33,34. Extrinsic high-temperature contributions are subtracted. d, i The strong enhancement of ε1(T) at the transition is a hallmark of a percolative IMT, as sketched in lo. e, j When screening becomes dominant in the metal, ε1 turns negative; σ1 exhibits Fermi-liquid behavior below TFL.

Metal-insulator transition mapped by dielectric permittivity

Figure 2 displays the temperature-dependent conductivity and permittivity data of κ-(BEDT-TTF)2Cu2(CN)3 when p rises (a–e) and x increases in κ-STFx (f–j). The insulating state (p < 1.4 kbar, x < 0.1), characterized by dσ1/dT > 0, generally features small positive ε1 ≈ 10. The relaxor-like response previously observed in the parent compound below 50 K has been subject of debate33,34. The metallic state (p > 3 kbar, x > 0.2) is defined by dσ1/dT < 0 and, concomitantly, ε1 < 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 ε1 < 0 (see Supplementary Fig. 7). This onset of metallic transport identifies TBR19; while thermal fluctuations prevail at higher T, the quasiparticle bandwidth is the dominant energy scale for T < TBR. Below TFL the resistivity ρ(T) T2 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, ε1 rapidly increases by several orders of magnitude. This colossal permittivity enhancement is more pronounced in the quasi-static limit, ε1 ≈ 105 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 σ1 and ε1 is plotted for fixed T. At T = 10 K, σ1(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 boundary15 associated with the rising onset of metallicity at TBR. The κ-STFx series exhibits similar behavior (Fig. 3c, d): around the critical concentration of x ≈ 0.12 a drastic increase in σ1 is observed at low T that smears out as T rises. The maximum in ε1(x) is reached for x = 0.16 but broadens rapidly upon heating.

Fig. 3: Electrodynamics at the percolative Mott transition.
figure 3

a The Mott IMT of κ-(BEDT-TTF)2Cu2(CN)3 appears as a rapid increase of σ1(p) that smoothens at higher T; above Tcrit a gradual crossover remains. b ε1(p) exhibits a sharp peak below Tcrit. The results at f = 380 kHz are plotted on a logarithmic scale. c, d Similar behavior is observed for chemical BEDT-STF substitution x. e, f Fixed-temperature line cuts of our hybrid DMFT simulations (see text) as a function of correlation strength U/W and T/W15,16 resemble the experimental situation in minute detail, including the shift of the IMT with T. The lack of saturation of σ1(T → 0) seen in DMFT modeling reflects the neglect of elastic (impurity) scattering in the metal (outside the coexistence region).

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 behavior13,35. Concerning dielectric response, it was pointed out36 that even in the case of certain continuous IMTs, the reduction of the charge gap might induce an enhancement in ε1, 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 reported13,26; in the case of κ-STFx we find it to be even smaller (see Supplementary Fig. 7). Hence, the dielectric catastrophe of ε1 ≈ 105 observed in our pressure and substitution-dependent 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 DMFT6,7,37,38, which has been found to successfully describe many experimental features of QSL materials4,15 such as ours. It features a first-order IMT at low temperatures (kBTcrit ≈ 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 films39,40,41,42,43. 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 ε1 around the percolation transition, in addition to dramatic changes in the dc resistivity44,45,46,47.

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 ε1 + iε2 for a single-band 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 then set up an appropriate random resistor network describing the dielectric properties within the phase-coexistence region, and solved it using standard effective-medium methods48, 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 effects49,50, or due to inherent (“Coulomb limited”) phase separation51,52,53. 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 ε1 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 σ1 and drop in ε1 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 → Tcrit 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 ε1 may also apply to other related organic compounds, where similar dielectric anomalies at or nearby a first-order IMT were previously assigned to ferroelectricity54 and multiferroicity55.

Fig. 4: Phase diagram of κ-(BEDT-TTF)2Cu2(CN)3 when tuned through the Mott IMT.
figure 4

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 ε1 reveals a sharply defined insulator-metal phase-coexistence region around the first-order IMT, which is not seen in σ1. The permittivity clearly distinguishes the Mott-insulating 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 TBR determined from σ1(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 ε1. Note, TBR is indicated by a dashed cyan line at T > Tcrit in all panels; the resistivity maxima below Tcrit (solid cyan line) arise due to percolative effects.

Discussion

While previous transport and optical studies5,15 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 < Tcrit. Our pressure- and substitution-dependent 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 ε1 (light red) bounded precisely by the QWL16, while also below TBR19,20 the response clearly follows the dielectric behavior expected for a metal (ε1 < 0, blue). Most remarkably, these two crossover lines converge towards Tcrit, 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 ε1 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 viewpoints56, which focus on the spin degrees of freedom in the QSL. This also confirms recent experimental and theoretical results57,58 suggesting that such gapless spin excitations, while dominant deep within the low-temperature Mott-insulating phase, are quickly damped away by the onset of charge fluctuations close to the IMT.

We demonstrated an expedient method of partial chemical substitution that triggers a similar Mott transition in κ-(BEDT-TTF)2Cu2(CN)3 as physical pressure. The pronounced divergency in ε1, 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)2Cu2(CN)3 were grown by the standard electrochemical synthesis method59,60 at Stuttgart and at Argonne. Plate-like single crystals of κ-[(BEDT-TTF)1−x(BEDT-STF)x]2Cu2(CN)3 with a typical size of 1 × 1 × 0.3 mm3 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-ethylenedithio-diselenium-dithiafulvalene, abbreviated BEDT-STF. Energy-dispersive X-ray spectroscopy was employed to determine the chemical composition of the alloys 0 ≤ x ≤ 161. The complex impedance of the crystals was measured as a function of temperature and frequency to obtain the permittivity \(\hat{\epsilon }={\epsilon }_{1}+{\rm{i}}{\epsilon }_{2}\) and the conductivity \(\hat{\sigma }={\sigma }_{1}+{\rm{i}}{\sigma }_{2}\). The crystals are contacted out-of-plane (Ea) 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)2Cu2(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 pressure-dependent dielectric spectroscopy measurements as described in detail elsewhere62. 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 gauge63, 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 K14,21 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 standard6,20 DMFT calculations of the optical conductivity σ1 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 suffices16,17 for our purposes. The imaginary part of conductivity σ2 was then obtained via the Kramers–Kronig transformation, and the dielectric function \(\epsilon \left(\omega \right)=1+i\frac{4\pi }{\omega }\sigma \left(\omega \right)\) 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 theory64 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 transition49, to model the volume fraction of the metallic phase: \(m(T/W,U/W)=\frac{1}{2}\tanh \left\{c[{(T/W)}_{{\rm{crit}}}-T/W]/[{(U/W)}_{{\rm{crit}}}-U/W]\right\}+\frac{1}{2}\), 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.