Phase coexistence at the first-order Mott transition revealed by pressure-dependent dielectric spectroscopy of $\kappa -(\mathrm{BEDT}-{\mathrm{TTF})}_2\text{−}{\mathrm{Cu}}_2{\left(\mathrm{CN}\right)}_3$

The dimer Mott insulator $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2{\mathrm{Cu}}_2{\left(\mathrm{CN}\right)}_3$ can be tuned into metallic and superconducting states on applying pressure of 1.5 kbar and more. We have performed dielectric measurements (7.5 kHz to 5 MHz) on $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2{\mathrm{Cu}}_2{\left(\mathrm{CN}\right)}_3$ 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 ${\varepsilon }_1\left(T\right)$ 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.

Figure 1

Three-dimensional plot of ${\varepsilon }_1(p,T,f=380$ kHz) and phase diagram of $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2\text{−}{\mathrm{Cu}}_2{\left(\mathrm{CN}\right)}_3$. Strikingly, the dielectric permittivity is strongly enhanced at the first-order Mott transition (blue line around 1.8 kbar and $T<T_{\mathrm{crit}}\simeq 16$ K), ascribed to metal-insulator phase coexistence, as predicted by dynamical mean-field theory [12]. Above $T_{\mathrm{crit}}$, the first-order IMT becomes a gradual crossover (quantum Widom line) [9, 10, 13]. A relaxor-ferroelectric response in ${\varepsilon }_1\left(T\right)$ is observed in the Mott-insulating phase ($p<1.5$ kbar) [14, 15]. The green circles represent the bifurcation temperature $T_B$ that indicates a change in the relaxation mechanism as discussed in Sec. 3a1. The black solid circles correspond to $T_{\mathrm{FL}}$ [16].

Figure 2

Temperature-dependent dielectric constant ${\varepsilon }_1\left(T\right)$ of $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2\text{−}{\mathrm{Cu}}_2{\left(\mathrm{CN}\right)}_3$ 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, shoulderlike 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 $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2\text{−}{\mathrm{Cu}}_2{\left(\mathrm{CN}\right)}_3$ 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 ${\varepsilon }_1\left(T\right)$ occurs at low temperatures, which is a result of spatial phase separation between metallic and insulating regions on the first-order Mott IMT. [(d) and (e)] ${\varepsilon }_1$ acquires negative values for $p\ge 3.37$ kbar indicating the onset of metallic transport. Note the different ordinates in panels [(d) and (e)].

Figure 4

The real and imaginary parts of the dielectric constant of $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2\text{−}{\mathrm{Cu}}_2{\left(\mathrm{CN}\right)}_3$, ${\varepsilon }_1\left(f\right)$ and ${\varepsilon }_2\left(f\right)$, 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).

Figure 5

Arrhenius plots of (a) the dielectric strength $\mathrm{\Delta }{\varepsilon }^{\text{mode}\text{1,2}}\left(T\right)$, (b) the distribution of relaxation times $1-\alpha \left(T\right)$, and (c) the mean relaxation time $\tau \left(T\right)$ for both modes obtained from the fits of dielectric data of $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2\text{−}{\mathrm{Cu}}_2{\left(\mathrm{CN}\right)}_3$ measured at ambient pressure. The solid purple symbols refer to mode 1 while the open orange squares indicate the data of mode 2.

Figure 6

Arrhenius plots of the Debye parameters of the low-frequency mode 1 in $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2\text{−}{\mathrm{Cu}}_2{\left(\mathrm{CN}\right)}_3$ for pressures up to 1.22 kbar. (a) Dielectric strength $\mathrm{\Delta }{\varepsilon }^{\text{mode}\text{1}}\left(T\right)$, (b) distribution of relaxation times $1-{\alpha }_1\left(T\right)$ and (c) mean relaxation time ${\tau }_1\left(T\right)$. The black and red lines represent fits with Eq. (3) above and below the kink in ${\tau }_1\left(T\right)$ at $T_B$, respectively.

Figure 7

(a) Pressure dependence of the bifurcation temperature $T_B$ and the positions of the high and low-temperature peak, $T_{\mathrm{HT}}$ and $T_{\mathrm{LT}}$, in $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2\text{−}{\mathrm{Cu}}_2{\left(\mathrm{CN}\right)}_3$; here $p_{\mathrm{IMT}}=1.45$ kbar denotes the critical pressure of the insulator-to-metal transition [9, 13]. (b) For the example of $p=0.86$ kbar and $f=100$ kHz the temperature dependence of the dielectric constant ${\varepsilon }_1\left(T\right)$ is plotted together with the fits of the HT and LT peak by two Gaussian functions (blue and green lines). (c) The relaxation time ${\tau }_1\left(T\right)$ with the fits from Fig. 6 according to Eq. (3) for the temperature range above (black line) and below (red line) the kink at $T_B$. The crossover from the HT to the LT peak is located at $T_B$.

Figure 8

Pressure dependence of the activation energy $\mathrm{\Delta }$ and relaxation time $\tau$ of $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2\text{−}{\mathrm{Cu}}_2{\left(\mathrm{CN}\right)}_3$ obtained from fits of the corresponding temperature behavior of ${\tau }_1\left(T\right)$ shown in Fig. 6. The upper panels [(a) and (b)] correspond to the high-temperature regime; the lower panels [(c) and (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) and (c) represent mean-field fits in analogy to Eq. (2). Most importantly, ${\mathrm{\Delta }}_{\mathrm{LT}}$ is rather constant and abruptly decreases around 1.1 kbar, reminiscent of a first-order phase transition with $p_{\mathrm{IMT}}=1.45$ kbar.

Figure 9

Temperature dependence of the parameters describing mode 2 in $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2\text{−}{\mathrm{Cu}}_2{\left(\mathrm{CN}\right)}_3$ at various pressures. (a) Dielectric strength $\mathrm{\Delta }{\varepsilon }^{\text{mode}\text{2}}$, (b) distribution of relaxation times $1-{\alpha }_2$ and (c) mean relaxation time ${\tau }_2$ in an Arrhenius plot versus inverse temperature. On increasing pressure, a pronounced minimum in ${\tau }_2$ develops, indicating a nonmonotonic relaxation dynamics in $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2\text{−}{\mathrm{Cu}}_2{\left(\mathrm{CN}\right)}_3$.

Figure 10

(a) Pressure dependence of ${\varepsilon }_2$ of $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2\text{−}{\mathrm{Cu}}_2{\left(\mathrm{CN}\right)}_3$ 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.

Figure 11

(a) Dielectric constant ${\varepsilon }_1$ of $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2\text{−}{\mathrm{Cu}}_2{\left(\mathrm{CN}\right)}_3$ 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 $T_{\mathrm{crit}}$ to the change from the first-order insulator-metal transition to a crossover at higher temperatures. For clarity reasons, we also include $T_{\mathrm{crit}}=16$ K (dashed red line) and the predictions for $q$ according to the BEMA model (dashed blue line).

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 $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2\text{−}{\mathrm{Cu}}_2{\left(\mathrm{CN}\right)}_3$ and our hybrid DMFT calculations. [(b)–(e)] The Mott-insulating state yields thermally activated resistivity and small, postive values of the permittivity. [(f) and (g)] While ${\rho }_1$ indicates a reduction with cooling when metallic clusters percolate, ${\varepsilon }_1$ is strongly increased on metal-insulator phase coexistence. [(j)–(m)] The correlated metallic state below the Brinkman-Rice temperature $T_{\mathrm{BR}}$ 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).

Figure 13

Frequency dependence of the real part of the dielectric constant, ${\varepsilon }_1\left(\omega \right)$, for different power-law behavior of the optical conductivity ${\sigma }_1\left(\omega \right)\propto {\omega }^{\beta }$ as plotted in the inset.

Figure 14

Plot of the dielectric permittivity ${\varepsilon }_1\left(T\right)$ of $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2\text{−}{\mathrm{Cu}}_2{\left(\mathrm{CN}\right)}_3$ sample 1 for several frequencies on 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) and (c)] A shoulderlike feature is already present at ambient pressure and develops into a second peak on pressurizing. [(d) and (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) and (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 $\mathrm{\Delta }{\varepsilon }^{\text{mode}\text{1}}\left(T\right)$, (b) distribution of relaxation times $1-{\alpha }_1\left(T\right)$, and (c) mean relaxation time ${\tau }_1\left(T\right)$. The black and red lines represent fits with Eq. (3) above and below the kink in ${\tau }_1\left(T\right)$ at $T_B$, respectively.

Figure 16

Arrhenius plot of the fit parameters of mode 2 for $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2\text{−}{\mathrm{Cu}}_2{\left(\mathrm{CN}\right)}_3$ sample 1 at different pressures as indicated. (a) Dielectric strength $\mathrm{\Delta }{\varepsilon }^{\text{mode}\text{2}}\left(T\right)$, (b) distribution of relaxation times $1-{\alpha }_2\left(T\right)$, and (c) mean relaxation time ${\tau }_2\left(T\right)$.

Figure 17

(a) Temperature dependence of ${\varepsilon }_1\left(T\right)$ 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. (B2), as shown for the example of $f=380\mathrm{kHz}$ (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.