Charge transport in charge-ordered layered crystals $\theta \text{−}{\left(\text{BEDT-TTF}\right)}_{2}M\text{Zn}{\left(\text{SCN}\right)}_4$ $\left(M=\text{Cs},\text{Rb}\right)$: Effects of long-range Coulomb interaction and the Pauli exclusion principle

We have measured the current-voltage $\left(I\text{−}V\right)$ characteristics, dielectric properties, and magnetoresistances of insulating layered organic crystals $\theta \text{−}{\left(\text{BEDT-TTF}\right)}_{2}M\text{Zn}{\left(\text{SCN}\right)}_4$ $\left(M=\text{Cs},\text{Rb}\right)$, in which electron-electron Coulomb interactions are considered to induce charge ordering. The in-plane $I\text{−}V$ characteristics follow the power law with a large exponent that exceeds 10 in the low-temperature limit. The nonlinear $I\text{−}V$ characteristics are attributed to electric field induced unbinding of pairs of an electron and a hole that are thermally excited and attracted to each other due to two-dimensional long-range Coulomb interaction. The temperature and frequency dependences of the in-plane dielectric constant for $M=\text{Cs}$ are explained by the polarization of the electron-hole bound pairs, consistently with the $I\text{−}V$ characteristics. The large dielectric anisotropy ($\approx 100$ at 0.6 K) observed for $M=\text{Cs}$ suggests two-dimensional long-range Coulomb interaction, which is also consistent with the explanation of the nonlinear $I\text{−}V$ curves. The organic crystals have a large positive magnetoresistance ratio, e.g., $\approx 10000\mathrm{\%}$ for $M=\text{Cs}$ in a magnetic field of 10 T at 0.1 K. The magnetoresistance is nearly independent of the magnetic field orientation despite the highly two-dimensional charge transport, indicating that it is electron-spin related. The magnetoresistance may be caused by magnetic field induced parallel alignment of spins of mobile and localized electrons, both in the highest occupied molecular orbital of a BEDT-TTF molecule, and by the resulting suppression of conduction due to the Pauli exclusion principle.

Figure 1

(a) Schematic of the layered organic crystal $\theta \text{−}{\left(\text{BEDT-TTF}\right)}_{2}X\left[X=\text{CsZn}{\left(\text{SCN}\right)}_4,\text{RbZn}{\left(\text{SCN}\right)}_4\right]$. (b) Arrangement of BEDT-TTF molecules in the BEDT-TTF layer of $\theta \text{−}{\left(\text{BEDT-TTF}\right)}_{2}M\text{Zn}{\left(\text{SCN}\right)}_4$ $\left(M=\text{Cs},\text{Rb}\right)$. (c) Activation energy of $\theta \text{−}{\left(\text{BEDT-TTF}\right)}_{2}X$ plotted as a function of transfer integral $\left[=\left(\text{bandwidth}\right)/8\right]$ (Ref. 6). We obtained the activation energy of $\theta \text{−}{\left(\text{BEDT-TTF}\right)}_2{\text{Cu}}_2\left(\text{CN}\right){\left[\text{N}{\left(\text{CN}\right)}_2\right]}_2$ from the lowest-temperature part of the temperature dependence of conductivity in Ref. 7. The dashed line is a guide to the eye.

Figure 2

(Color online) In-plane low-bias conductance of a CsZn crystal (electrode spacing $L=20\mu \text{m}$; $V\parallel c$; $V=\pm 1\text{mV}$ for $T^{-1}\le 0.8{\text{K}}^{-1}$ and $\pm 0.2\text{mV}$ for $T^{-1}\ge 0.8{\text{K}}^{-1}$) and a RbZn crystal ($L=10\mu \text{m}$; $V\parallel a$; $V=\pm 1\text{mV}$) as a function of inverse temperature. The dashed (red) lines are fits to the Arrhenius law.

Figure 3

In-plane ($a$-axis) $I\text{−}V$ curves of a CsZn crystal. $T=0.78$, 0.73, 0.69, 0.63, 0.59, 0.57, 0.49, 0.456, 0.418, 0.294, 0.213, 0.138, and 0.099 K from top left to bottom right. The electrodes are attached to both sides of the crystal and the electrode spacing $L=90\mu \text{m}$. Inset: $I\text{−}V$ curve at 0.137 K on linear scale.

Figure 4

In-plane ($a$-axis) $I\text{−}V$ curves of a RbZn crystal under (a) slow-cooling and (b) rapid-cooling conditions. (a) $T=59.1$, 49.2, 44.3, 39.5, 34.6, 29.8, 24.9, 19.9, 14.9, 10.0, 6.01, 4.22, 3.51, 1.22, and 0.31 K from top left to bottom right. (b) $T=19.9$, 17.9, 15.9, 14.9, 13.9, 12.9, 11.9, 10.0, 7.99, 6.01, 3.51, 1.27, 0.55, and 0.33 K from top left to bottom right. The $I\text{−}V$ curves in (a) and (b) were measured using the same pair of electrodes (with separation $L$ of $10\mu \text{m}$) on the same crystal under different cooling conditions.

Figure 5

(Color online) $d\text{ln}\left(I\right)/d\text{ln}\left(V\right)$ of the CsZn, and RbZn salts, and $\kappa \text{−}{\left(\text{BEDT-TTF}\right)}_2\text{Cu}\left[\text{N}{\left(\text{CN}\right)}_2\right]\text{Cl}$ at $I={10}^{-10}\text{A}$ plotted as a function of (a) temperature, (b) $k_{B}T/{\Delta }_0$, and (inset of b) $k_{B}T/U_0$. $d\text{ln}\left(I\right)/d\text{ln}\left(V\right)$ of the CsZn and RbZn salts are obtained from the $I\text{−}V$ curves in Figs. 3, 4. ${\Delta }_0$ is the activation energy obtained from the temperature dependence of low-bias conductance. $U_0$ is the coefficient of the logarithmic potential (see text). The solid (orange) line in the inset of (b) shows $U_0/2k_{B}T+1$ [the power-law exponent of Eq. (3)] for $k_{B}T/U_0<1/4$ and 1 for $k_{B}T/U_0>1/4$, where $T=U_0/4k_B\equiv T_{\text{BKT}}$ is the Berezinskii-Kosterlitz-Thouless transition temperature.

Figure 6

(a) and (b) Excitation energy for a doublon-holon pair with a distance $r$ for (a) $V_{ij}=0$ and (b) $V_{ij}\ne 0$. $V_{n.n.}$ denotes the nearest-neighbor $V_{ij}$. (c) and (d) Excitation energy under the applied electric field for (c) $V_{ij}=0$ and (d) $V_{ij}\ne 0$. $\Delta \left(E\right)$ is the activation energy to unbind a doublon-holon pair.

Figure 7

Calculated $I\text{−}V$ curves for the slowly cooled RbZn salt [Fig. 4a]. $T=59.1$, 49.2, 44.3, 39.5, 34.6, 29.8, 24.9, 19.9, 14.9, 10.0, 6.01, 4.22, and 3.51 K from top left to bottom right.

Figure 8

(Color online) Temperature dependences of in-plane ($a$-axis) and out-of-plane ($b$-axis) relative dielectric constants of the CsZn salt (samples Da3 and Db3). The curves labeled CB were measured with the capacitance bridge while the others were measured with the $LCR$ meter. The ac excitation voltages are $0.1{\text{V}}_{\text{rms}}$ ($LCR$ meter), $0.1{\text{V}}_{\text{rms}}$ (CB, $T\le 1.27\text{K}$), and $0.03{\text{V}}_{\text{rms}}$ (CB, $T\ge 1.29\text{K}$) for the in-plane direction and $1{\text{V}}_{\text{rms}}$ ($LCR$ meter) and $0.75{\text{V}}_{\text{rms}}$ (CB) for the out-of-plane direction.

Figure 9

(Color online) Temperature dependence of the dissipation factor, $\text{tan}\left(\delta \right)$, of the CsZn salt (samples Da3 and Db3). The curves labeled CB were measured with the capacitance bridge while the others were measured with the $LCR$ meter. The data were obtained at the same time as those in Fig. 8.

Figure 10

(Color online) Temperature dependence of the in-plane relative dielectric constant calculated using Eq. (5).

Figure 11

(Color online) In-plane ($a$-axis) $I\text{−}V$ curves of a CsZn crystal at $T=0.135$ and 0.63 K in magnetic fields of $B=0$, 3, and 17.5 T applied parallel to the $a$ axis. The electrode spacing $L$=90 $\mu \mathrm{m}$.

Figure 12

(Color online) $d\text{ln}\left(I\right)/d\text{ln}\left(V\right)$ of the CsZn salt for $T=0.63\text{K}$ plotted as a function of (a) voltage and (b) current.

Figure 13

(Color online) (a) Magnetic field dependence of current measured while voltage was kept constant in a CsZn crystal. The current is normalized to the value obtained in zero magnetic field at each temperature. Both the voltage and magnetic field were applied along the $a$ (in-plane) axis so that orbital effects would be minimized. For $T=0.135\text{K}$, data measured with $B\parallel c$ (the other in-plane axis) are also shown. Two curves for each temperature correspond to increasing and decreasing magnetic fields. (b) The same data are plotted as a function of ${\mu }_{B}B/k_{B}T$. The curves fall on a universal curve represented by $I/I\left(B=0\right)={\left[\text{cosh}\left({\mu }_{B}B/k_{B}T\right)\right]}^{-1/2}$ (dotted line) or $I/I\left(B=0\right)=1/\left[1+{\left({\mu }_{B}B/2k_{B}T\right)}^2\right]$ (dashed line).

Figure 14

.(Color online) Magnetic field dependence of current measured while voltage was kept constant for the same electrode pairs on a RbZn crystals under (a) slow- and (b) rapid-cooling conditions. The voltage was applied along the $a$ axis and the magnetic field was applied along the $b$ axis.

Figure 15

(Color online) Dependence of $I/I\left(B=0\right)$ of the CsZn salt on the direction of the magnetic field at 0.135 K. (a) The voltage was applied along the $c$ axis, and the magnetic field was tilted by $\theta$ from $b$ toward $c$. The solid lines are fits to the formula, $I\left(B,\theta \right)/I\left(B,\theta =90°\right)=1+A\left(B\right)\left|\text{cos}\left(\theta \right)\right|$. (b) The voltage was applied along the $a$ axis, and the magnetic field was tilted by $\varphi$ from $c$ toward $a$. The solid lines are fits to the formula, $I\left(B,\varphi \right)/I\left(B,\varphi =90°\right)=1+A\left(B\right)\left|\text{cos}\left(\varphi \right)\right|$.