Electronic properties of Fabre charge-transfer salts under various temperature and pressure conditions

Using density functional theory, we determine parameters of tight-binding Hamiltonians for a variety of Fabre charge transfer salts, focusing, in particular, on the effects of temperature and pressure. Besides relying on previously published crystal structures, we experimentally determine two new sets of structures: (TMTTF)${}_2$SbF${}_6$ at different temperatures and (TMTTF)${}_2$PF${}_6$ under various hydrostatic pressures. We find that a few trends in the electronic behavior can be connected to the complex phase diagram shown by these materials. Decreasing temperature and increasing pressure cause the systems to become more two dimensional. We analyze the importance of correlations by considering an extended Hubbard model parameterized using Wannier orbital overlaps and show that while charge order is strongly activated by the intersite Coulomb interaction, the magnetic order is only weakly enhanced. Both orders are suppressed when the effective pressure is increased.

Figure 1

Temperature-pressure phase diagram for the TMTTF and TMTSF charge transfer salts, as first suggested in Ref. 1 and refined by many others. Position in the phase diagram can be tuned by physical (in this case hydrostatic), or chemical (changing anion) pressure. The ambient pressure position for each salt is indicated with an arrow above the diagram. An increase of pressure (external or chemical) causes the system to be less one dimensional. The possible phases are charge ordered (CO), Mott insulating (MI), antiferromagnetic (AF), spin Peierls (SP), spin density wave (SDW), superconducting (SC), and 1D, 2D, or 3D metal. The positions of (TMTTF)${}_2$ClO${}_4$ and (TMTTF)${}_2$BF${}_4$ are not well known. In the phase diagram, their approximate positions have been indicated with dashed arrows. These two anions break the inversion symmetry of the system, allowing for an anion ordering transition. After they go through this transition (between 40 and 75 K), the shown phase diagram is no longer relevant.[4] More work is needed to understand the physics of these two systems at low temperatures, but it seems that just considering effective pressure will not be sufficient.

Figure 2

Crystal structure of the Fabre CT salts projected into the $ac$ plane. The organic molecules form $\pi$-stacked 1D chains along the crystal $a$ direction (with a slight zigzag pattern) and form layers parallel to the $ab$ plane. These organic layers alternate with anion layers (with the anions centered on the pink As sites) stacked in the $c$ direction. Grey atoms are carbon, yellow are sulfur, while hydrogen atoms are shown in white.

Figure 3

Electronic properties of (TMTTF)${}_2$PF${}_6$ ($T=4$ K structure). (a) Band structure and density of states. (b) Path through $k$ space considered for the band structure plotting. (c) Fermi surface in the $k_z=0$ plane. The total density of states is shown in black and the partial density of states of the anions (increased by a factor of 100) is shown in orange (dashed). The partial density of states shows that within this energy window, all of the bands in this energy window have predominantly TMTTF character, and the two bands at the Fermi level are nearly purely TMTTF. We find that these general features of the bands and density of states are consistent across the TMTTF salts studied here with the exception of (TMTTF)${}_2$Br, which has three additional bands (of anion origin) near its Fermi level; the variations in the bands at the Fermi level are discussed further in Sec. 5.

Figure 4

The two TMTTF molecules of a unit cell, along with one of the Wannier orbitals of (TMTTF)${}_2$PF${}_6$ for the bands near the Fermi level. It is clear that this Wannier orbital has the structure of the HOMO of a TMTTF molecule in the gas phase. The other Wannier orbital needed to describe the two organic bands corresponds to the HOMO of the other TMTTF molecule in the unit cell (and is related to the first one by inversion symmetry). This is in agreement with the information in the partial density of states [see Fig. 3a]; the orbitals near the Fermi energy are predominantly of TMTTF nature.

Figure 5

Band structures in the energy window $[-0.8\mathrm{eV},0.2\mathrm{eV}]$ of the TMTTF salts with crystal structures measured at ambient pressure and room temperature. In this energy range, all of the materials shown here have two bands arising from TMTTF HOMO orbitals. The (TMTTF)${}_2$Br salt additionally has three anion bands within this window. The common TMTTF bands differ in details; between $\Gamma$ and $X$ (corresponding to the in-chain direction) the bands are very similar. There is more variation in the inter-chain direction, indicating the differing degrees of interchain coupling.

Figure 6

Visualization of the strength of the hopping between the sites of the tight-binding model, the centers of mass of the TMTTF molecules (gray spheres); shown for (a) (TMTTF)${}_2$AsF${}_6$ and (b) (TMTTF)${}_2$ClO${}_4$ (both at room temperature). The diameter of the bonds is proportional to the tight-binding parameter strength $|t_{\alpha }|$. $|t_{\alpha }|$ above 0.1 eV are each a different shade of blue, $|t_{\alpha }|$ between 0.1 eV and 0.01 eV are a shade of red/orange, while $|t_{\alpha }|$ less than 0.01 eV are a shade of green. See Table for $t_{\alpha }$ values.

Figure 7

Band structure of (TMTTF)${}_2$SbF${}_6$ calculated from the crystal structures obtained at several temperatures between $T=100$ and 300 K. At $T=140$ and 300 K, structures from two different samples were used; the additional bands at those temperatures are plotted with dashed lines. As the temperature is decreased, the bandwidth increases and the dispersion between the $X$ and $V$ points becomes steeper; this indicates that the electronic structure becomes more two dimensional with decreasing temperature.

Figure 8

Evolution of tight-binding parameters of (TMTTF)${}_2$SbF${}_6$ with temperature. As temperature is lowered to $T=100$ K, the dominant hoppings $t_0$ and $t_1$ become nearly equal, making the TMTTF chain nearly isotropic. The sizable 2D couplings $t_2$, $t_4$, and $t_5$ show a complicated temperature dependence, with $t_2$ and $t_5$ changing sign and $t_4$ increasing considerably as temperature is lowered.

Figure 9

Dimensionality vs bandwidth of (TMTTF)${}_2$SbF${}_6$ with temperature (diamonds) and (TMTTF)${}_2$PF${}_6$ under pressure (circles); the bandwidth is the energy difference between the highest and lowest energies in the TMTTF bands around the Fermi energy, and the dimensionality is defined by the ratio of the hopping integrals in the intra and interchain directions [see Eq. (5)]. We see the expected positive correlation between bandwidth and pressure (indicated by the arrow): as the pressure increases, so does the intermolecular hopping and therefore the bandwidth. We also see a strong positive correlation between pressure and dimensionality. There is a negative correlation between temperature and dimensionality and bandwidth; increasing the temperature has a similar effect to decreasing the pressure. The 140- and 300-K points for (TMTTF)${}_2$SbF${}_6$ are averaged over the two structures available at those temperatures.

Figure 10

Band structures of (TMTTF)${}_2$AsF${}_6$ (red) at $T=4$ K (dashed) and room temperature (solid line), (TMTTF)${}_2$PF${}_6$ (blue) at $T=4$ K (dashed) and room temperature (solid line). The black lines are the bands resulting from the model Hamiltonian (2), parametrized by the Wannier orbital overlaps for the eight shortest hops. In the model used there is no dispersion in the $k_z$ direction since the eight shortest hops are all in the same plane, i.e., we have a two-dimensional model. It is clear from the DFT bands that the interplanar coupling is small, which is why the 2D model fits so well. Note that $T=4$ K is below both the charge ordering and spin Peierls transitions of (TMTTF)${}_2$AsF${}_6$ and (TMTTF)${}_2$PF${}_6$.

Figure 11

Band structure of (TMTTF)${}_2$PF${}_6$ at various pressures. As the pressure is increased, the bandwidth increases, and the dispersion becomes steeper between the $X$ and $V$ points; the system becomes more two dimensional. These trends are made obvious in Fig. 9 (b).

Figure 12

Tight-binding parameters of (TMTTF)${}_2$PF${}_6$ at various pressures as a function of distance. Note that as the pressure is increased, the trend is for the $t$'s to become larger. In Fig. 9b, we see that the increases are such that the system becomes more two dimensional.

Figure 13

Schematic representation of the TMTTF molecules in the conducting plane for the Fabre salts studied here. (a) Balls correspond to the TMTTF molecules and ellipses indicate the dimers. The site and dimer labeling is shown. (b) A sketch of a possible charge- and magnetically ordered state; the size of the circles represent the hole density, and the arrows the net magnetism.

Figure 14

Structure factors as a function of temperature for (TMTTF)${}_2$SbF${}_6$. As the temperature is decreased the dimer charge and magnetic orders are somewhat suppressed. Simultaneously, the bandwidth and dimensionality increase. Thus decreasing temperature has the same effects as increasing pressure (see Fig. 15); the charge order is strongly activated by $V_0$, and the magnetic order is only weakly enhanced. The squares correspond to results with $V_0=2t_0$ while the triangles correspond to $V_0=0.5t_0$. The two points at 140 and 300 K are the results based on the two crystal structures we have at those temperatures.

Figure 15

Structure factors as a function of pressure for (TMTTF)${}_2$PF${}_6$. As the pressure is increased the dimer charge and magnetic orders are somewhat suppressed. Simultaneously, the bandwidth and dimensionality increase [seen in Fig. 9b]. The charge order is strongly activated by the value of $V_0$, while the magnetic order is only weakly enhanced, and is even finite for $V_0=0$ (not shown). The circles correspond to results with $V_0=2t_0$, while the diamonds correspond to $V_0=0.5t_0$.

Figure 16

Electronic dimerization for (a) (TMTTF)${}_2$SbF${}_6$ as a function of temperature and (b) for (TMTTF)${}_2$PF${}_6$ under pressure.