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

Infrared reflection measurements of the half-filled two-dimensional organic conductors $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2\mathrm{Cu}\left[\mathrm{N}{\left(\mathrm{C}\mathrm{N}\right)}_2\right]{\mathrm{Br}}_x{\mathrm{Cl}}_{1-x}$ [BEDT-TTF denotes bis-(ethylenedithio)tetrathiafulvalene] were performed as a function of temperature $(5\mathrm{K}<T<300\mathrm{K})$ and Br substitution ($x=0\%$, 40%, 73%, 85%, and 90%) in order to study the metal-insulator transition. We can distinguish absorption processes due to itinerant and localized charge carriers. The broad midinfrared absorption has two contributions: transitions between the two Hubbard bands and intradimer excitations from the charges localized on the ${\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2$ dimer. Since the latter couple to intramolecular vibrations of BEDT-TTF, the analysis of both electronic and vibrational features provides a tool to disentangle these contributions and to follow their temperature and electronic-correlation dependence. Calculations based on the cluster model support our interpretation.

Figure 1

(Color online) (a) Structural arrangement of the BEDT-TTF molecules in the $\kappa$ phase (looking along the molecular axis); the size of the unit cell is approximately $12.9\mathrm{\AA }$ in the $a$ direction, $8.5\mathrm{\AA }$ along the $c$ axis, and $30.0\mathrm{\AA }$ in the third direction. The overlap integrals are labeled according to Mori et al. (Ref. 6). The interdimer overlap integral $b1$ is around 0.027, along the dimer chains $b2\approx 0.010$, while $p$ and $q$ link orthogonal molecules with approximately 0.011 and 0.004, respectively. (b) Triangular lattice for the dimer model of $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_{2}X$. There is hopping along the stacks $t_1$ and along the diagonals $t_2$.

Figure 2

(Color online) Schematic phase diagram of the $\kappa$-phase salts ${\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_{2}X$. Instead of tuning the external pressure, the same ambient-pressure ground state can be achieved by modifying the anions $X$. The arrows indicate the approximate position of $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2\mathrm{Cu}\left[\mathrm{N}{\left(\mathrm{C}\mathrm{N}\right)}_2\right]\mathrm{Cl}$, $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2\mathrm{Cu}\left[\mathrm{N}{\left(\mathrm{C}\mathrm{N}\right)}_2\right]\mathrm{Br}$, $\kappa {\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2\mathrm{Cu}{\left(\mathrm{N}\mathrm{C}\mathrm{S}\right)}_2$, and $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2{\mathrm{I}}_3$ at ambient pressure, respectively. At room temperature, all of the compounds are moderately good conductors $({\rho }_{\mathrm{dc}}\approx 1\Omega \mathrm{cm})$. While $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2\mathrm{Cu}\left[\mathrm{N}{\left(\mathrm{C}\mathrm{N}\right)}_2\right]\mathrm{Cl}$ becomes increasingly insulating and exhibits a weak magnetic order below $35\mathrm{K}$, $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2\mathrm{Cu}\left[\mathrm{N}{\left(\mathrm{C}\mathrm{N}\right)}_2\right]\mathrm{Br}$ crosses over to a metallic behavior below $100\mathrm{K}$ and eventually enters a superconducting state at $T_c=12\mathrm{K}$. The phase transition between the (antiferromagnetic) Mott insulator and the metal/superconductor can be explored by gradually replacing Cl by Br in $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2\mathrm{Cu}\left[\mathrm{N}{\left(\mathrm{C}\mathrm{N}\right)}_2\right]{\mathrm{Br}}_x{\mathrm{Cl}}_{1-x}$. Here, $c$ and $a$ are the lattice parameters; $b1$ and $p$ indicate the transfer integral according to Fig. 1a. $U∕t$ is the on-site Coulomb repulsion with respect to the hopping integral $t$.

Figure 3

(Color online) dc resistivity versus temperature of $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2\mathrm{Cu}\left[\mathrm{N}{\left(\mathrm{C}\mathrm{N}\right)}_2\right]{\mathrm{Br}}_{0.23}{\mathrm{Cl}}_{0.77}$ and $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2\mathrm{Cu}\left[\mathrm{N}{\left(\mathrm{C}\mathrm{N}\right)}_2\right]{\mathrm{Br}}_{0.85}{\mathrm{Cl}}_{0.15}$. While the latter compound becomes metallic below $100\mathrm{K}$ and eventually superconducting at $T_c=12\mathrm{K}$, the former one gradually turns insulating.

Figure 4

(Color online) Reflectivity spectra of $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2\mathrm{Cu}\left[\mathrm{N}{\left(\mathrm{C}\mathrm{N}\right)}_2\right]{\mathrm{Br}}_x{\mathrm{Cl}}_{1-x}$ for the polarization $E\Vert a$ measured at various temperatures: $T=300$, 150, 90, 50, and $20\mathrm{K}$. Panels (a)–(e) correspond to different Br concentrations: $x=90\%$, 85%, 73%, 40%, and 0%. The insets show the room-temperature and low-temperature spectra for $x=90\%$, 73%, and 0% on a logarithmic frequency scale.

Figure 5

(Color online) Optical conductivity spectra $(E\Vert a)$ of $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2\mathrm{Cu}\left[\mathrm{N}{\left(\mathrm{C}\mathrm{N}\right)}_2\right]\text{−}{\mathrm{Br}}_x{\mathrm{Cl}}_{1-x}$ at different Br concentrations $x$ and temperatures, obtained by a Kramers-Kronig analysis from the data of Fig. 4.

Figure 6

(Color online) Reflectivity spectra of $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2\mathrm{Cu}\left[\mathrm{N}{\left(\mathrm{C}\mathrm{N}\right)}_2\right]{\mathrm{Br}}_x{\mathrm{Cl}}_{1-x}$ for light polarized $E\Vert c$ measured at various temperatures as indicated. The different panels (a)–(e) correspond to Br concentrations: $x=90\%$, 85%, 73%, 40%, and 0%. The insets show the room-temperature and low-temperature spectra on a logarithmic frequency scale.

Figure 7

(Color online) Optical conductivity spectra $(E\Vert c)$ for $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2\mathrm{Cu}\left[\mathrm{N}{\left(\mathrm{C}\mathrm{N}\right)}_2\right]\text{−}{\mathrm{Br}}_x{\mathrm{Cl}}_{1-x}$ of different Br concentrations: (a) $x=90\%$, (b) 85%, (c) 73%, (d) 40%, and (e) 0% measured at various temperatures: $T=300$, 150, 90, 50, and $20\mathrm{K}$. The midinfrared band clearly has two contributions.

Figure 8

(Color online) Fit of the frequency dependent conductivity of $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2\mathrm{Cu}\left[\mathrm{N}{\left(\mathrm{C}\mathrm{N}\right)}_2\right]\text{−}{\mathrm{Br}}_{0.85}{\mathrm{Cl}}_{0.15}$ at $20\mathrm{K}$ for the electric field polarized parallel to the $c$ axis. $L_{\text{dimer}}$ and $L_{\text{Hubbard}}$ show the two Lorentz oscillators required to fit the midinfrared peak. The narrow lines describe the vibrational features. In addition, an extended Drude model is included for the low-frequency increase.

Figure 9

(Color online) Temperature dependence of the midinfrared peak $L_{\text{dimer}}$ in $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2\mathrm{Cu}\left[\mathrm{N}{\left(\mathrm{C}\mathrm{N}\right)}_2\right]{\mathrm{Br}}_x{\mathrm{Cl}}_{1-x}$, with $x=0\%$, 73%, and 90% obtained by a Lorentzian fit for the polarization $E\Vert c$. The lines are guides to the eye.

Figure 10

(Color online) Detailed view of the optical conductivity of $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2\mathrm{Cu}\left[\mathrm{N}{\left(\mathrm{C}\mathrm{N}\right)}_2\right]{\mathrm{Br}}_x{\mathrm{Cl}}_{1-x}$ along the $a$ and $c$ directions (left and right panels, respectively) for different Br concentrations $x=90\%$, 85%, 73%, 40%, and 0%, (a)–(e) and (f)–(j), respectively. The spectra for the different temperatures (bottom to top: $T=20$, 50, 90, 150, and $300\mathrm{K}$) are offset by $400{\left(\Omega \mathrm{cm}\right)}^{-1}$ for clarity.

Figure 11

(Color online) Temperature dependence of the mode frequencies of the intramolecular vibrations ${\nu }_4\left(A_g\right)$ and ${\nu }_6\left(A_g\right)$ for $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2\mathrm{Cu}\left[\mathrm{N}{\left(\mathrm{C}\mathrm{N}\right)}_2\right]{\mathrm{Br}}_x{\mathrm{Cl}}_{1-x}$ with Br concentrations 0%, 73%, and 90%. (a) The ${\nu }_4\left(A_g\right)$ mode along the $a$ direction was fitted by a Lorentzian and the center frequency plotted. (b) In the case of ${\nu }_6\left(A_g\right)$, the frequencies are defined as the minima in the respective optical conductivity, i.e., the strongest of the quadruple. In frames (c) and (d), the equivalent data are presented for the polarization $E\Vert c$. The lines correspond to spline fits.

Figure 12

(Color online) Conductivity of $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2\mathrm{Cu}\left[\mathrm{N}{\left(\mathrm{C}\mathrm{N}\right)}_2\right]{\mathrm{Br}}_{0.85}{\mathrm{Cl}}_{0.15}$ for the two polarizations (a) $E\Vert c$ and (b) $E\Vert a$ at $T=20\mathrm{K}$. The solid red curves represent the conductivity for the intradimer transition and the emv-coupled modes calculated within the cluster model. The dotted blue curves show the difference between the experimental data (thick black line) and the red curve. Fit parameters for $E\Vert c$ are $U_{\mathrm{mol}}=0.5\mathrm{eV}$ and $t\left(b1\right)=0.21\mathrm{eV}$. The vibrational frequencies ${\nu }_i$ and respective coupling constants $Q_i$ are ${\nu }_4=1460{\mathrm{cm}}^{-1}$, $Q_4=610$; ${\nu }_6=1287{\mathrm{cm}}^{-1}$, $Q_6=95$; ${\nu }_{10}=878{\mathrm{cm}}^{-1}$, $Q_{10}=120$; and ${\nu }_{13}=445{\mathrm{cm}}^{-1}$, $Q_{13}=240$. For $E\Vert a$, we used $U_{\mathrm{mol}}=0.5\mathrm{eV}$ and $t\left(b1\right)=0.19\mathrm{eV}$; the frequencies and coupling constants are the same except $Q_4=680$.

Figure 13

(Color online) Peak position of the intradimer excitation and the emv-coupled molecular vibration ${\nu }_4\left(A_g\right)$ as a function of Br concentration $x$ as obtained from the fit of the optical spectra of $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2\mathrm{Cu}\left[\mathrm{N}{\left(\mathrm{C}\mathrm{N}\right)}_2\right]{\mathrm{Br}}_x{\mathrm{Cl}}_{1-x}$ by the dimer model. (a) and (b) show the results for the polarization $E\Vert a$, while (c) and (d) are for the $c$ direction. In the upper frames, the frequency the intradimer absorption are displayed; in the lower frames, the mode position of the emv activated vibration ${\nu }_4\left(A_g\right)$ is plotted. The experimental data are given as open squares, and the points from fit by closed dots. The lines connect the points.