Comprehensive transport study of anisotropy and ordering phenomena in quasi-one-dimensional ${\left(\mathbf{TMTTF}\right)}_{2}X$ salts ($X={\mathbf{PF}}_6,{\mathbf{AsF}}_6,{\mathbf{SbF}}_6,{\mathbf{BF}}_4,{\mathbf{ClO}}_4,{\mathbf{ReO}}_4$)

The temperature-dependent dc resistivity of the quasi-one-dimensional organic salts ${\left(\mathrm{TMTTF}\right)}_{2}X$ ($X={\mathrm{PF}}_6,{\mathrm{AsF}}_6,{\mathrm{SbF}}_6,{\mathrm{BF}}_4,{\mathrm{ClO}}_4,{\mathrm{ReO}}_4$) [tetramethyltetrathiafulvalene (TMTTF)] has been measured in all three crystal directions in order to investigate anisotropy, localization effects, and charge- (CO) and anion-ordering (AO) phenomena at low temperatures. For all compounds and directions, we extract the transport mechanisms in different regimes. The data are thoroughly analyzed, related to structural properties, and extensively discussed in view of previous studies and latest theories. The most important observation is the opening of a gap in the density of states below the CO temperature $T_{\mathrm{CO}}$ that influences the transport in all three directions. The effect becomes stronger as the distance between the anions and the sulfur atoms decreases. The coupling between the anions and the TMTTF stack also has a strict influence on the anomaly in the resistivity at the AO $T_{\mathrm{AO}}$. At very low temperatures, the activation law is replaced by three-dimensional variable-range hopping.

Figure 1

Temperature dependence of the dc resistivity in ${\left(\mathrm{TMTTF}\right)}_{2}X$ crystals with octahedral anions $X={\mathrm{PF}}_6,{\mathrm{AsF}}_6$, and ${\mathrm{SbF}}_6$ along the three crystal directions $a$, $b^{\prime }$, and $c^{*}$.

Figure 2

Temperature-dependent dc resistivity of ${\left(\mathrm{TMTTF}\right)}_{2}X$ crystals with thetrahedal anions $X={\mathrm{BF}}_4,{\mathrm{ClO}}_4$, and ${\mathrm{ReO}}_4$ along the three directions $a$, $b^{\prime }$, and $c^{*}$.

Figure 3

Enlarged view of the temperature-dependent resistivity ${\rho }_{b^{\prime }}\left(T\right)$ of ${\left(\mathrm{TMTTF}\right)}_{2}X$ for $X={\mathrm{BF}}_4,{\mathrm{ClO}}_4$, and ${\mathrm{ReO}}_4$ in the range of the AO. The slight shift of the transition temperature $T_{\mathrm{AO}}$ upon cooling down and warming up evidences the first-order phase transition. For clarity reasons, the curves are vertically displaced with respect to each other.

Figure 4

Parallel-stack dc resistivity ${\rho }_a^{\left(V\right)}\left(T\right)$ of various Fabre salts. The constant-pressure data are corrected according to Eq. (1) to account for the thermal expansion. The upper panels (a) and (b) summarize the results for the octahedral anions ${\left(\mathrm{TMTTF}\right)}_2{\mathrm{PF}}_6$, ${\left(\mathrm{TMTTF}\right)}_2{\mathrm{AsF}}_6$, and ${\left(\mathrm{TMTTF}\right)}_2{\mathrm{SbF}}_6$; while in the lower panels (c) and (d), the data for the tetrahedral anions ${\left(\mathrm{TMTTF}\right)}_2{\mathrm{BF}}_4$, ${\left(\mathrm{TMTTF}\right)}_2{\mathrm{ClO}}_4$, and ${\left(\mathrm{TMTTF}\right)}_2{\mathrm{ReO}}_4$ are shown. On the left side (a) and (c), the high-temperature data are plotted as a function of $T^{-1}$, and on the right side (b) and (d), the low-temperature regime is presented as a function of $T^{-1/4}$.

Figure 5

Direct-current resistivity ${\rho }_c^{\left(V\right)}\left(T\right)$ along the $c^{*}$ direction of ${\left(\mathrm{TMTTF}\right)}_2{\mathrm{AsF}}_6$, ${\left(\mathrm{TMTTF}\right)}_2{\mathrm{PF}}_6$, and ${\left(\mathrm{TMTTF}\right)}_2{\mathrm{SbF}}_6$ as a function of (a) $T^{-1}$ and (b) $T^{-1/4}$. In the lower panels (c) and (d), the same representation for the results of ${\left(\mathrm{TMTTF}\right)}_2{\mathrm{BF}}_4$, ${\left(\mathrm{TMTTF}\right)}_2{\mathrm{ClO}}_4$, and ${\left(\mathrm{TMTTF}\right)}_2{\mathrm{ReO}}_4$. The constant-pressure data are corrected according to Eq. (1) to account for the thermal expansion.

Figure 6

Energy gaps of ${\left(\mathrm{TMTTF}\right)}_{2}X$ as a function of temperature along the $a$, $b^{\prime }$, and $c^{*}$ axes. The activation energy was determined from the constant-volume data of the resistivity according to Eq. (4), choosing an appropriate value of $\rho \left(T\right)$. The small green arrows indicate $T_{\mathrm{AO}}$, the larger black arrows denote the CO temperature $T_{\mathrm{CO}}$.

Figure 7

(a) Energy gaps of ${\left(\mathrm{TMTTF}\right)}_2{\mathrm{SbF}}_6$ as a function of temperature along the $a$, $b^{\prime }$, and $c^{*}$ axes. The contribution by the hopping conduction to the transport [${\rho }_{c^{*}}^{\mathrm{hop}}\left(T\right)=(4.31\times {10}^{-5}\Omega \mathrm{cm})\exp (72.45\mathrm{K}/T^{1/4})$, ${\rho }_a^{\mathrm{hop}}\left(T\right)=(39.8\times {10}^{-3}\Omega \mathrm{cm})\exp (74.5\mathrm{K}/T^{1/4})$] was subtracted, and the remaining curve was fitted by Eq. (5) (dashed line), shown here, exemplary only for two crystal directions. (b) To eliminate the hopping conduction for ${\left(\mathrm{TMTTF}\right)}_2{\mathrm{ReO}}_4$, the dotted curves are obtained by a fit with Eq. (5). Below the AO transition, an additional gap opens, and the complete behavior follows Eq. (6) (gray dashed line). The fit for a temperature-independent gap ${\Delta }_{\mathrm{AO}}$ is indicated by the dashed-dotted curve (cyan). Activation energies obtained from the fits are listed in Table for the limit $T\to 0$.

Figure 8

The phase diagram of ${\left(\mathrm{TMTTF}\right)}_{2}X$ shows the different phases (AFM, spin-Peierls, CO, charge localization, and one-dimensional metal) as a function of pressure. While some of the boundaries are clear phase transitions, the ones indicated by dashed lines are better characterized as a crossover. The position in the phase diagram can be tuned by chemical pressure or external pressure with the approximate scale given. For the different compounds with octahedral and spherical anions, the ambient-pressure position in the phase diagram is indicated by the arrows on the top axis.

Figure 9

In the left panel, the dependence between the resistivity ${\rho }_a$ and the structural dimerization along the $a$ axis, both taken at room temperature, is plotted. The structural data are taken from Refs. 34, 35, 36, 37, 38, 39. The right panels show the room-temperature resistivities ${\rho }_a$, ${\rho }_{b^{\prime }}$, and ${\rho }_{c^{*}}$ for all investigated compounds ${\left(\mathrm{TMTTF}\right)}_{2}X$; from left to right, the volume of the unit cell increases, i.e., $X={\mathrm{BF}}_4,{\mathrm{ClO}}_4,{\mathrm{ReO}}_4,{\mathrm{PF}}_6,{\mathrm{AsF}}_6$, and ${\mathrm{SbF}}_6$.

Figure 10

The CO transition temperature $T_{\mathrm{CO}}$ as a function of the shortest distance between the ligands of the anions (F or O) and sulfur atoms in TMTTF. For ${{\left(\mathrm{TMTTF}\right)}_2\mathrm{ClO}}_4$, the distance is much larger compared to the other compounds: It develops no CO. The structural data are taken from Refs. 34, 35, 36, 37, 38, 39.

Figure 11

Unit-cell and dimerization parameters for the six investigated compounds of ${\left(\mathrm{TMTTF}\right)}_{2}X$. The compounds are ordered from left to right by increasing unit-cell volume ($X={\mathrm{BF}}_4,{\mathrm{ClO}}_4,{\mathrm{ReO}}_4,{\mathrm{PF}}_6,{\mathrm{AsF}}_6$, and ${\mathrm{SbF}}_6$). The structural data are taken from Refs. 34, 35, 36, 37, 38, 39 and are tabulated in Table .