Bandwidth-controlled Mott transition in $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2\mathrm{Cu}\left[\mathrm{N}{\left(\mathrm{CN}\right)}_2\right]{\mathrm{Br}}_x{\mathrm{Cl}}_{1-x}$: Optical studies of correlated carriers

In the two-dimensional organic charge-transfer salts $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2\mathrm{Cu}\left[\mathrm{N}{\left(\mathrm{CN}\right)}_2\right]{\mathrm{Br}}_x{\mathrm{Cl}}_{1-x}$ a systematic variation in the Br content from $x=0$ to 0.9 allows us to tune the Mott transition by increasing the bandwidth. At temperatures below $50\mathrm{K}$, an energy gap develops in the Cl-rich samples and grows to approximately $1000{\mathrm{cm}}^{-1}$ for $T\to 0$. With increasing Br concentration spectral weight shifts into the gap region and eventually fills it up completely. As the samples with $x=0.73$, 0.85, and 0.9 become metallic at low temperatures, a Drude-type response develops due to the coherent quasiparticles. Here, the quasiparticle scattering rate shows a ${\omega }^2$ dependence and the effective mass of the carriers is enhanced in agreement with the predictions for a Fermi liquid. These typical signatures of strong electron-electron interactions are more pronounced for compositions close to the critical value $x_c\approx 0.7$, where the metal-to-insulator transition occurs.

Figure 1

(Color online) Schematic phase diagram of $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_{2}X$. The on-site Coulomb repulsion with respect to the hopping integral $U∕t$ can be tuned either by external pressure or modifying the anions $X$. The arrows indicate the approximate position of $\kappa$-phase salts with $X=\mathrm{Cu}\left[\mathrm{N}{\left(\mathrm{CN}\right)}_2\right]\mathrm{Cl}$, $\mathrm{Cu}\left[\mathrm{N}{\left(\mathrm{CN}\right)}_2\right]\mathrm{Br}$, $\mathrm{Cu}{\left(\mathrm{NCS}\right)}_2$, and ${\mathrm{I}}_3$ at ambient pressure, respectively. The bandwidth-controlled phase transition between the insulator and the Fermi liquid/superconductor can be explored by gradually replacing Cl by Br in $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2\mathrm{Cu}\left[\mathrm{N}{\left(\mathrm{CN}\right)}_2\right]{\mathrm{Br}}_x{\mathrm{Cl}}_{1-x}$. Here $c$ and $a$ are the lattice parameters; $b1$ and $p$ indicate intradimer and interdimer transfer integrals, respectively.

Figure 2

(Color online) Left frames: experimental spectra of the real part of the complex conductivity ${\sigma }_1$ of $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2\mathrm{Cu}\left[\mathrm{N}{\left(\mathrm{CN}\right)}_2\right]{\mathrm{Br}}_x{\mathrm{Cl}}_{1-x}$ for the in-plane $E\parallel c$ measured at different temperatures; right frames: spectral weight as a function of cut-off frequency ${\omega }_c∕\left(2\pi c\right)$ calculated according to Eq. (1). From the top to the bottom panels, (a)–(e) and (f)–(j), respectively, the Br content is reduced from 0.9 to 0.

Figure 3

(Color online) Frequency dependence of the conductivity ${\sigma }_1(E\parallel c)$ due to correlated charge carriers after the contributions from intradimer transitions and vibrational modes are subtracted. The frames (a)–(e) show spectra of $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2\mathrm{Cu}\left[\mathrm{N}{\left(\mathrm{CN}\right)}_2\right]{\mathrm{Br}}_x{\mathrm{Cl}}_{1-x}$ with different Br concentrations $x$ at various temperatures as indicated.

Figure 4

(Color online) (a) Anisotropic triangular lattice used in theoretical calculations of the electronic properties; the ratio $t_1∕t_2=0.8$. (b) A schematic view of conductivity spectrum predicted by DMFT for a metallic compound close to the Mott transition.

Figure 5

(Color online) Shift of the center of gravity in the conductivity spectra of the correlated charge carriers with temperature for different $x$ in $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2\mathrm{Cu}\left[\mathrm{N}{\left(\mathrm{CN}\right)}_2\right]{\mathrm{Br}}_x{\mathrm{Cl}}_{1-x}$ for $E\parallel c$. The lines correspond to spline fits.

Figure 6

(Color online) Density of states at $T=150$, 50, and $20\mathrm{K}$ as proposed for the low Br-doped compounds $(x<0.7)$, the crystal with $x\approx 0.7$, and the highly Br-doped samples $(x<0.7)$ from the optical conductivity. The zero-energy and finite-energy contributions are distinguished by different colors (shadings). The temperatures represent the metallic or Mott-insulating region $\left(20\mathrm{K}\right)$, the narrow-gap semiconducting high-temperature region $\left(150\mathrm{K}\right)$, and the crossover regime between the former which is often referred to as “bad-metal” or “bad-semiconductor” region.

Figure 7

(Color online) Comparison of the development of the optical gap with changing the Br concentration and temperature for $E\parallel c$. The panels show the frequency-dependent conductivity ${\sigma }_1$ after subtracting contributions due to localized charge carriers and vibrational modes. (a) and (b): ${\sigma }_1\left(\omega \right)\text{at}T=90$, 50, 35, and $20\mathrm{K}$ for (a) $x=0$ and (b) $x=0.4$. (c) and (d): dependence of ${\sigma }_1\left(\omega \right)$ on Br concentration for lowest $T=20\mathrm{K}$ ($x=0$ and 0.4) and (c) $T=5\mathrm{K}$ ($x=0.73$, 0.85, and 0.9) and (d) $T=50\mathrm{K}$.

Figure 8

(Color online) Temperature dependence of the spectral weight $I_{\sigma }$ of the conduction electrons in $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2\mathrm{Cu}\left[\mathrm{N}{\left(\mathrm{CN}\right)}_2\right]{\mathrm{Br}}_x{\mathrm{Cl}}_{1-x}$ with $x=0$, 0.4, 0.73, 0.85, and 0.9 calculated via integration of ${\sigma }_1$ plotted in Fig. 3 up to ${\omega }_c∕\left(2\pi c\right)=700{\mathrm{cm}}^{-1}$ for $E\parallel c$. The lines are guides to the eye. The inset shows the Drude conductivity ${\sigma }_{1,\text{Drude}}$ obtained after subtraction of all finite-frequency modes at lowest temperatures ($T=20\mathrm{K}$ for $x=0$ and 0.4 and $T=5\mathrm{K}$ for $x=0.73$, 0.85, and 0.9).

Figure 9

(Color online) Ratio of the low-temperature Drude spectral weight $D\left(x\right)$ to the total spectral weight of the correlated charge carriers $D_0\left(x\right)$ at $20\mathrm{K}$ ($x=0$ and 0.4) and $T=5\mathrm{K}$ $(x=0.73,0.85,0.9)$. The solid lines are guides to the eyes and help to identify the Mott-insulating and metallic regimes (grey area) with a boundary around $x_c=0.7$.

Figure 10

(Color online) Spectral-weight distribution for $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2\mathrm{Cu}\left[\mathrm{N}{\left(\mathrm{CN}\right)}_2\right]{\mathrm{Br}}_x{\mathrm{Cl}}_{1-x}$ at low temperature. The spectral weight is redistributed as one approaches the transition to the Mott-insulating phase. Typical error bars for the experimental data are shown (Ref. 42).

Figure 11

(Color online) Frequency dependence of the low-temperature $(T=5\mathrm{K})$ scattering rate ${\Gamma }_1\left(\omega \right)$ (left panel) and effective mass $m^{*}∕m_{b,\mathrm{opt}}$ (right panel and inset) of $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2\mathrm{Cu}\text{−}\left[\mathrm{N}{\left(\mathrm{CN}\right)}_2\right]{\mathrm{Br}}_x{\mathrm{Cl}}_{1-x}$ for different $x$ as indicated. Typical error bars for the experimental data are shown.

Figure 12

(Color online) Frequency-dependent part of the scattering rate ${\Gamma }_1\left(\omega \right)-{\Gamma }_1\left(0\right)$ at $T=5\mathrm{K}$ of $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2\mathrm{Cu}\left[\mathrm{N}{\left(\mathrm{CN}\right)}_2\right]{\mathrm{Br}}_x{\mathrm{Cl}}_{1-x}$ as function of the squared frequency. We determined for the frequency-independent dc limit of the scattering rate ${\Gamma }_1\left(0\right)$ the following values: $315{\mathrm{cm}}^{-1}$ $(x=0.73)$, $48{\mathrm{cm}}^{-1}$ $(x=0.85)$, and $280{\mathrm{cm}}^{-1}$ $(x=0.9)$. Note the different vertical scales for both frames.