Electrodynamics of Bechgaard Salts: Optical Properties of One-Dimensional Metals

The electrodynamic properties of the quasi-one-dimensional organic conductors (TMTSF)2X are discussed, with particular emphasis on important deviations from the simple Drude model, the transition from a Luttinger-liquid to a Fermi-liquid behavior at the dimensional crossover when pressure is applied or temperature reduced, indications of a pseudogap as well as a lowfrequency collective mode. Superconductivity and spin-density-wave ground states breaking the symmetry and gaps should occur in the excitation spectra. The previous literature is summarized and the current status of our understanding presented. Novel THz experiments on (TMTSF)2PF6 and (TMTSF)2ClO4 not only shine light into some of the open questions, but also pose new ones.


Introduction
Physics in one dimension is a fascinating topic for theory and challenging for experiments.One-dimensional models are simpler compared to three-dimensional ones; in many cases, analytical solutions exist only in one dimension, while numerical approaches have to be used in higher dimensions [1].Often the reduction of dimensionality does not really matter because the essential physics remains unaffected.But there are also a number of phenomena in condensed matter which only or mostly occur in one dimension.In general, the dominance of the lattice is reduced and electronic interactions become superior.This implies that physics in reduced dimensions is physics of low energies; the relevant effects do not occur in the electron-volt range but at millielectron volts and below.Quantum mechanical effects are essential as soon as the confinement approaches the electron wavelength.Fundamental concepts of physics, like the Fermi liquid theory of interacting particles breaks down in one dimension and has to be replaced by alternative concepts based on collective excitations [2].
One-dimensional structures are intrinsically unstable for thermodynamic reasons.Hence various kinds of ordering phenomena may take place which break the translational symmetry of the lattice, charge, or spin degrees of freedom: phase transitions occur as a function of temperature or some order parameter.On the other hand, fluctuations suppress long-range order at any finite temperature in one (and two) dimension.The ordered ground state is only stabilized by the fact that real systems consist of onedimensional chains, which are coupled to some degree.The challenge now is to extract the one-dimensional physics from experimental investigations of quasi-one-dimensional systems and to check the theoretical predictions.Besides pure scientific interest, the crucial importance of these phenomena in nanotechnology might not lie ahead too far.
After a short overview of some milestones in the development of quasi-one-dimensional crystalline structures, the electrodynamic properties of the Bechgaard salts are discussed in detail, as they serve as the superior model systems of quasi-one-dimensional conductors.More than 30 years after the first optical spectra on the quasi-one-dimensional organic superconductor (TMTSF) 2 PF 6 were published by Jacobsen et al. [3], the electrodynamic properties of the Bechgaard salts are still puzzling and subject to continuous experimental and theoretical efforts.Several issues make the materials and their physical properties so interesting and challenging.
(i) The metallic properties are extremely anisotropic; this makes these organic salts the prime examples to study one-dimensional physics.
(ii) The optical reflectivity and conductivity deviate from the simple Drude model; while there is a plasma edge present like in regular metals, the low-frequency properties exhibit very unusual behavior, not all of them are well understood by now.
(iii) The conduction electrons cannot be described by the Fermi liquid theory but exhibit the scaling behavior of a Tomonaga-Luttinger liquid.
(iv) A pseudogap seems to open in the far-infrared frequency range at low temperatures; this reduced spectral weight might correspond to the extremely low density of electronic states observed by photoemission spectroscopy.
(v) The broken-symmetry ground states-such as charge density wave, spin density wave, or superconducting states-exhibit their particular fingerprint in the electrodynamic properties, single-particle gaps as well as collective excitations.
(vi) But even in the metallic state, there are indications of collective modes observed in the THz range of frequency.
1.1.One-Dimensional Metal: Pt-Chains.The low-frequency electrodynamic properties of one-dimensional metals in general and organic conductors in particular have puzzled researches for decades.Although theoretically discussed for a long time [7], it all started in the early 70s of the last century with the Krogmann salt K 2 Pt(CN) 4 Br 0.3 • H 2 O, known as KCP, that consists of a chain of platinum ions with overlapping d orbitals [8,9].In Figure 1, the optical reflectivity and conductivity of KCP are displayed for different temperatures and polarizations [4,5].The room temperature conductivity of KCP along the chain direction is very high σ = 10 2 (Ω cm) −1 ; the anisotropy ratio is σ /σ ⊥ = 10 5 .Due to the anisotropic nature, the reflectivity R(ω) shows a plasma edge only for the electric field E along the chains (E z) while it remains low and basically frequency independent perpendicular to it, as known from dielectrics and other insulators.At low temperatures, the single-particle gap around 1000 cm −1 becomes more pronounced, and an additional structure is observed in the farinfrared conductivity which is assigned to the pinned-mode resonance induced by the charge density wave (Figure 1

(b)).
A detailed investigation of the pinned-mode resonance, its center frequency and lineshape, and furthermore its dependence on temperature and impurity content turned out to be extremely difficult because it commonly occurs in the range of 3 to 300 GHz (0.1 to 10 cm −1 ); that is, it falls right into the gap between high-frequency experiments using contacts and optical measurements by freely travelling waves [10].Microwave technique based on resonant cavities and quasi-optical THz spectroscopy was advanced over the years in order to bridge this so-called THz gap [11][12][13][14][15][16].

First One-Dimensional Organic Conductor: TTF-TCNQ.
The breakthrough in the field of one-dimensional organic conductors happened around the same time with the synthesis of tetrathiafulvalene-tetracyanoquinodimethane (TTF-TCNQ), which exhibits a room temperature conductivity of 10 3 (Ωcm) −1 and an anisotropy of more than a factor of 100 [7,[17][18][19].In Figure 2, the optical properties of TTF-TCNQ are presented [6].Clear deviations from the Drude behavior of a conventional metal [10] are observed due to the onedimensional nature.The most surprising fact is the somewhat low reflectivity and conductivity in the far-infrared range.This discrepancy between the high dc conductivity and the low far-infrared conductivity seems to be a generic feature found in all one-dimensional conductors.Although the dc conductivity increases by more than an order of magnitude by cooling from room temperatures down to T = 60 K [20][21][22], the infrared conductivity remains low (Figure 2(b)).When the temperature is reduced below the T CDW ≈ 53 K, the low-frequency reflectivity drops because an energy gap opens at the Fermi level.
In their first infrared transmission and reflection studies on TTF-TCNQ films and single crystals, Tanner et al. observed that within the energy gap of 1050 cm −1 a collective mode develops in the optical conductivity at zero energy that moves to finite frequencies as T < T CDW = 53 K [20,21,[23][24][25][26][27][28][29][30].They conclude that a collective chargedensity-wave mode develops below T CDW that is centered around 80 cm −1 due to pinning by impurities and contains an oscillator strength of 500 cm −1 ; this corresponds to an effective mass of m * = 300m b with a bandmass m b = 3m 0 [10].The issue is not settled yet, but the interpretation seems questionable since the center frequency is too high, implying unreasonably strong coupling.The situation in TTF-TCNQ contains the particular complication of two stacks (TTF and TCNQ), both developing a charge density wave with some interaction.An alternative interpretation of the 80 cm −1 peak would be optical excitations across the Mott gap.
It is quite interesting that the extremely high dc conductivity in the metallic regime above the charge-densitywave transition could barely be reconciled with the optical properties.Some spectral weight is shifted toward low frequencies upon cooling, which infers improved metallic conductivity.Nevertheless, there is a positive slope in σ 1 (ω) at any temperature, indicating a pseudogap and effectively reducing the far-infrared conductivity.It was suggested that the Drude-like mode with a plasma frequency of approximately 10 4 cm −1 exhibits a width of only 1/(2πτc) = 0.5 cm −1 as depicted in Figure 3.This is confirmed by the dielectric constant (or the imaginary part of the conductivity σ 2 (ω)) that evidences considerable spectral weight at very low frequencies, that is, below 10 cm −1 .It should be noted here that the data are taken at a temperature T = 60-85 K, where such a narrow response can only be imagined if phonons scattering does not contribute considerable to the broadening, a fact rather common to one-dimensional metals.by enlarging the organic molecule from TTF to TMTTF and TMTSF, which stands for tetramethyltetrathiofulvalene and tetramethyltetraselenafulvalene, respectively [19,[31][32][33][34][35].Single crystals are usually grown by electrochemical methods and reach a size of several millimeters in length and less than a millimeter in width.All compounds of the TMTTF and TMTSF family are isostructural.Due to the triclinic symmetry, b denotes the projection of the b axis perpendicular to a, and c * is normal to the ab plane.

1.3.
Because of the stronger coupling between the stacks, the TMTSF salts are electronically more two dimensional, as seen by the appreciable warping of the Fermi surface.Metallic behavior is reached down to lower temperatures (Figure 4 at T c ≈ 1 K when cooled slowly, as presented later in Figure 7(b).(TMTSF) 2 PF 6 undergoes a spin-density-wave transition at 12 K, which can be suppressed by hydrostatic pressure [50].Since 30 years both compounds serve as prime examples of one-dimensional metals, and enormous effort was put in exploring the low-energy electrodynamics in the search for the pseudogap, the Drude-like component, the collective mode, Luttinger-or Fermi-liquid behavior, dimensional crossover, and so forth.In Figure 5, the various results are compiled by plotting the absorption A(ω) = 1 − R(ω) as a function of frequency on a double logarithmic scale, in order to highlight the low-frequency and highreflectivity part.
Even after a quarter of a century, the extended reviews by Jacobsen [51][52][53] still give the best overview on the optical properties of Bechgaard salts, the theoretical background, and thorough discussion of their implication.

Dimensional Crossover
In their seminal paper, Jacobsen et al. [3] already realized that (TMTSF) 2 PF 6 exhibits a plasma edge in the optical reflectivity not only along the stacking direction a, that becomes more pronounced as the temperatures decreases below T = 300 K.At low temperatures a plasma edge is also seen for E b , although at frequencies (2360 cm −1 ) lower by a factor of 5 compared to E a (11400 cm −1 ).Similar results are displayed in Figure 6.They conclude a dimensional crossover from a one-dimensional to a two-dimensional metal as the As the temperature is reduced, the charges become increasingly localized in (TMTTF) 2 AsF 6 and (TMTTF) 2 PF 6 , before the charge-ordered state is entered below 100 K. (TMTTF) 2 SbF 6 shows a transition from a metal-like state directly into the charge-ordered state at T CO = 157 K. (TMTSF) 2 PF 6 undergoes a SDW transition at T SDW = 12 K.Only (TMTSF) 2 ClO 4 remains metallic all the way down to approximately T c = 1.2K where it becomes superconducting (after Dressel et al. [36][37][38][39]).temperature is lowered.The transverse coupling becomes effective below 100 K, while the smearing of the Fermi-Dirac function exceeds the transverse bandwidth of approximately W b = 13 meV.These conclusions are in perfect agreement with transport measurements along all three directions [58,59] where a crossover from two-dimensional Fermiliquid to a one-dimensional Luttinger liquid behavior was identified around T ≈ 100 K (Figure 7).At exactly the same temperature, the c * -axis transport turns from an insulating to a metallic one; the systems are actually three-dimensional metals at low temperatures.Jacobsen et al. made similar observation on the optical properties of (TMTSF) 2 AsF 6 , (TMTSF) 2 ClO 4 , and (TMTSF) 2 ReO 4 [40,60], where the latter compound undergoes a metal-insulator transition at T AO = 182 K due to the anion ordering [61].The missing plasma edge in the third direction (E c * ) [60] was later confirmed by Henderson et al. [62].It is interesting to look at electronic band structure and how it changes with pressure [63]: the bands dispersion along the b-direction varies most, while basically no change is seen for the c-axis.This is in perfect agreement with the increasing plasma frequency observed perpendicular to the stacks.The results of different groups are compared in order to show the similarities in the overall behavior, but also the distinct and substantial differences.The data of (TMTSF) 2 PF 6 are taken from Jacobsen et al. [3,40], Eldridge et al. [41], and collaborators [42][43][44] at temperatures as indicated.The data of (TMTSF) 2 AsF 6 are taken from Jacobsen et al. [40], Ng et al. [45], Eldridge and Bates [46], and Schwartz et al. [47] at temperatures as indicated.The results on (TMTSF) 2 SbF 6 were obtained by Jacobsen et al. [40], Ng et al. [48] Eldridge and Bates [46], and Degiorgi at temperatures as indicated.The measurements on (TMTSF) 2 (AsF 6 ) 1−x (SbF 6 ) x (x ≈ 0.15) probe the effect of impurities on the electrodynamic response [49].
The issue of the dimensional crossover was finally explored in more detail by pressure and temperaturedependent optical investigation of (TMTSF) 2 PF 6 , (TMTTF) 2 PF 6 , and (TMTTF) 2 AsF 6 [64][65][66].The pressureinduced deconfinement transition in the Mott insulator (TMTTF) 2 PF 6 occurs at approximately 2 GPa.This critical pressure is basically temperature independent and characterized by the rapid onset of the interstack electronic transport (along the b direction).In Figure 8, the transverse hopping integral t b is depicted as a function of pressure, calculated from the transverse optical conductivity (E b )via where V c denotes the unit cell volume, b the separation of the stacks, and t a the transfer integral along the stacks.The values of t b for (TMTTF) 2 PF 6 and (TMTSF) 2 PF 6 perfectly agree with each other, taking into account a pressure offset of 3 GPa between these compounds.Above the deconfinement transition (i.e., P > 2 GPa) and up to 4 GPa, the increase in t b for (TMTTF) 2 PF 6 is almost linear with a slope of 5 meV/GPa, in accord with our earlier results for (TMTTF) 2 AsF 6 [64,65].Above 4 GPa the pressure-induced increase in t b becomes weaker (∼2 meV/GPa) for (TMTTF) 2 PF 6 and comparable to the low-pressure behavior of (TMTSF) 2 PF 6 .The quantitative criterion Δ ρ ≈ 2t * b for the deconfinement transition [67] can be verified by comparing the pressure dependence of the charge gap Δ ρ with that of 2t b (Figure 8).The onset of the coherent electronic transport normal to the stacks occurs at  [42][43][44]54].The dashed lines represent a Drude fit, respectively.The inset of panel (c) shows the schematic phase diagram of the deconfinement transition for a system of weakly coupled conducting chains as suggested by [2,55,56].The transition from a Mott insulator to a two-or three-dimensional metallic state occurs at T = 0 when t ⊥ reaches a critical value t * ⊥ .At high enough temperature, the increase in t ⊥ leads to a transition from a Mott insulating to a one-dimensional Luttinger liquid and further to a dimensional crossover into a metallic state.(Figure adopted from [57]).around 2 GPa, where Δ ρ ≈ 2t b .The size of the Mott gap, Δ ρ , rapidly decreases as the transition point is approached and stabilizes at a finite value upon further pressure increase.These findings are in accordance with theoretical predictions [2,55,56,[67][68][69][70][71][72] and earlier experiments [59,[73][74][75] which utilized the effect of chemical pressure.
A quantitative analysis of the dimensional crossover can be obtained from the degree of coherence of the charge transport.The coherence parameter of the Drude response is given by where ω p and Γ are the plasma frequency and the scattering rate, respectively.Coherent transport corresponds to κ > κ * , while incoherent transport occurs for κ < κ * , where κ * is a critical value of the dimensional crossover which should be close to unity.The evolution of the dimensionality of (TMTSF) 2 PF 6 is summarized in Figure 9(b), where the coherence parameter κ b is plotted as a function of temperature and pressure, determined from the Drude fits of the interstack reflectivity spectra.The highest degree of coherence is achieved for high pressures and low temperatures (lower right corner of the diagrams).It gradually decreases as the pressure is released and the temperature is simultaneously raised towards the upper left corner of the diagram.The lines of constant coherence level are almost linear with pressure for κ b > 1, however, for κ b < 1 they become sublinear.The critical κ * = κ b 0.85 values that characterize the dimensional crossover are defined from resistivity measurements of Moser et al. [73].The constant level line κ b 0.85, that is, the "crossover line," is depicted in Figure 9.For pressures below 1.2 GPa, the crossover line exhibits a remarkable strong slope, not expected in the naive picture of noninteracting electrons where T * ∝ t b .According to Figure 8, the interstack transfer integral t b in (TMTSF) 2 PF 6 increases only by about 30% when pressure increases by 4 GPa.Thus, one would expect the crossover temperature of about 130 K at 4 GPa for noninteracting electrons, in contrast to the experimental observation.This provides evidence that electronic correlations play a decisive role in the renormalization of the dimensional crossover in the Bechgaard salts, leading to the very fast suppression of the one-dimensional state in favor of a high-dimensional metallic state.
A corresponding temperature-pressure diagram of the coherence parameter κ b is depicted in Figure 9(a) for (TMTTF) 2 PF 6 in the deconfined state, that is, for P > 2 GPa.The coherence parameter demonstrates a temperature and pressure dependence similar to that of the (TMTSF) 2 PF 6 salt.Thus, the chemical pressure offset of 3 GPa between the two studied salts in the generic phase diagram also holds for the occurrence of the dimensional crossover.

Mott Physics
The low-frequency properties of (TMTSF) 2 PF 6 and (TMTSF) 2 ClO 4 resemble features previously observed in KCP and TTF-TCNQ (cf.Section 1).From Figures 6(c) and 10, it can be seen that a strong deviation from the simple Drude behavior is observed below 1000 cm −1 in the reflectivity for E a; for lower frequencies R(ω) actually drops to a minimum between 50 and 100 cm −1 and only then rises rapidly in order to approach unity.This behavior was first observed by Jacobsen et al. [3,40,60,76], later confirmed by Ng et al. [45], Degiorgi et al. [42,43], and eventually explained by the UCLA group [44,47] when the optical reflection experiments in the infrared spectral range could be complemented with THz measurements and microwave experiments down to 0.1 cm −1 .
As can be seen in Figure 11 for (TMTSF) 2 PF 6 , the frequency-dependent conductivity consisting of two distinct features: a narrow zero-energy mode at low energy containing a very small part of the spectral weight (approximately 1%) and a finite-energy mode centered around 200 cm −1 .Similar observations have been made for (TMTSF) 2 PF 6 , (TMTSF) 2 AsF 6 , and (TMTSF) 2 ClO 4 , as demonstrated in Figure 12.The general idea now is that these systems resemble a one-dimensional Mott system with filling slightly above one half due to internal doping as a result of the interchain coupling.The isolated stack is effectively half filled (due to the dimerization of the quarter-filled system), leading to a Mott gap Δ ρ in the charge degrees of freedom.This is still seen in the excitation maximum around 200 cm −1 where the one-dimensional physics is found.The interaction between the stacks results in the Drude-like term and is solely responsible for the finite dc conductivity; here we find the higher-dimensional physics.
These are the general characteristics of a highly anisotropic interacting electron system, with either a halfor quarter-filled band.Coulomb repulsion leads to a Mott gap and, at frequencies above the effective interchain transfer integral, to a Luttinger liquid state.The finite energy feature can be described as the absorption above the Mott gap in such a one-dimensional Luttinger liquid.In particular, above the gap, the optical conductivity behaves as a power law of the frequency σ(ω) ∼ (1/ω) ν , in a way characteristic of a Tomonaga-Luttinger liquid.As demonstrated in Figure 13, the exponent ν is determined experimentally to be ν ≈ 1.3 for (TMTSF) 2 PF 6 , (TMTSF) 2 AsF 6 , and (TMTSF) 2 ClO 4 .
Comparison of the data with the Tomonaga-Luttinger theory along with the value of the exponent ν suggests that the dominant mechanism responsible for the opening of the Mott gap is the quarter-filling of the band; n is the order of the commensurability, that is, electrons per site: n = 1 for half-filling and 2 for quarter-filling.This conclusion leads to a Luttinger liquid parameter K ρ ≈ 0.23, corresponding to very strong repulsion.In a general way, K ρ = 1 is the noninteracting point with K ρ > 1 corresponding to attraction and K ρ < 1 to repulsion.The fact that the one-dimensional theory is unable to account quantitatively for the data below the high-energy peak seems to suggest that the crossover to a two-dimensional regime occurs at relatively high energies.This is in accord to the temperaturedependent resistivity displayed in Figure 7.There we observe a linear temperature dependence of the resistivity ρ(T) at elevated temperatures, that crosses over to a ρ(T) ∝ T 2 behavior at low temperatures.It is interesting that this power-law behavior extends over about an order of magnitude in temperature if the thermal contraction is properly taken into account [59].
The remaining zero-energy mode of the itinerant electrons carries only 1% of the spectral weight and shows deviations from a simple Drude response; it can be adequately described with a frequency-dependent mass and relaxation rate with quadratic frequency-dependence, suggestive of a Fermi liquid.The fact that the resistivity is dominated by electronic scattering basically up to room temperature evidences the negligible contribution of phonons.A similar conclusion is drawn from the extremely narrow zero-energy contribution, that exhibits a width of less than 1 cm −1 at T = 20 K.As mentioned above, this seems to be a general property of low-dimensional metals.

Spin-Density-Wave State
As can be seen from the temperature-dependent resistivity (Figures 4 and 7) of (TMTSF) 2 PF 6 , also (TMTSF) 2 AsF 6 and (TMTSF) 2 SbF 6 , and other salts of the Bechgaard family, a spin-density-wave ground state develops at low temperature (T SDW ≈ 12 K), where the systems become insulating due to the gap in the density of states at the Fermi energy that is caused by the instability of the Fermi surface due to nesting [10,78].The low-field dc resistivity shows an activated behavior similar to a standard semiconductor with a gap value 2Δ/k B ≈ 45 K [42,59].In analogy to the resistivity, the gap is expected to show up also in the optical conductivity, however, it turned out to be quite challenging to unambiguously measure the spin-densitywave gap by optical means, due to the high reflectivity, augmented by the reduced spectral weight in the relevant range of frequency.
Based on far-infrared reflection measurements on (TMTSF) 2 SbF 6 , a spin-density-wave gap of 180 cm −1 was suggested by Ng et al. [48]; following the arguments previously developed for (TMTSF) 2 PF 6 [79], for T < T SDW ≈ 12-14 K, the semiconducting spin-density-wave state coexists with a metallic state, leading to a high conductivity background.The extracted ratio 2Δ/k B T SDW = 18-21 seems to be rather large.However, it is known from incommensurate Peierls transitions that the meanfield temperatures are several times the actual transition temperature due to three-dimensional ordering [78].
The Vancouver group of Eldridge tried to overcome the mentioned experimental problem by directly probing the absorptivity A(ω) = 1 − R(ω), that is, the power absorbed by the sample [41,46,82].They point out that the reflectivity of (TMTSF) 2 AsF 6 , for instance, evaluated via the directly probed absorption is almost 10% higher than previous reflection measurements on mosaics by Ng et al. [45].Applying a novel composite-bolometric technique, the far-infrared properties of (TMTSF) 2 PF 6 , (TMTSF) 2 AsF 6 , and (TMTSF) 2 SbF 6 were measured at low temperatures T < T SDW .For (TMTSF) 2 PF 6 , they find a conductivity peak at 45 ± 5 cm −1 with a gap at 2Δ = 33 cm −1 that corresponds to the thermal gap Δ obtained from magnetotransport measurements by Chaikin et al. [83].Along the chain direction (E a) an additional sharp feature was observed at 18 ± 1 cm −1 .Interestingly for the perpendicular direction also a peak is found at 35 ± 5 cm −1 that also follows the (ω−ω 0 ) −2 dependence.A gap value of 2Δ ≈ 33 cm −1 is rather close to the mean field ratio 2Δ/k B T SDW = 3.53.
It was Degiorgi et al. [43] who could clearly identify the spin-density-wave gap in (TMTSF) 2 PF 6 and later in (TMTSF) 2 ClO 4 [77,84] by looking in the perpendicular direction (E b ).Using a Fourier-transform spectrometer for infrared and far-infrared reflection measurements (15-10 5 cm −1 ) and a THz coherent source spectrometer for lower frequencies (7-20 cm −1 ), they probed an extremely broad spectral range.Below T SDW = 12 K and 6 K,   The inset in (b) shows the total integrated spectral weight and the fraction in the zero-energy mode, both as functions of temperature, demonstrating that the spectral weight is not redistributed, and that only about 1% is in the Drude-like mode (after [44,47]).
the reflectivity is significantly reduced below 70 cm −1 and 100 cm −1 , respectively, leading to an energy gap in the optical conductivity.In Figure 14, the frequency-dependent reflectivity and conductivity of (TMTSF) 2 PF 6 is plotted for different temperatures.A gaplike feature developing at about 70 cm −1 for T < T SDW , which is rather sharp, moves to slightly lower frequencies and decreases in intensity as the temperature increases.It nicely follows the mean-field behavior also extracted from magnetic measurements [85,86].This feature seems to persist at temperatures slightly above T SDW .
Watanabe et al. measured the reflection of (TMTSF) 2 PF 6 using a THz-time-domain spectroscopic system [87] that had sufficient resolution in phase shift to observe the transition from the metallic to the insulating spin-densitywave state.A distinct reduction of the conductivity below approximately 65 cm −1 gives a clear indication of the spindensity-wave gap formation in accord to [43].In addition, they were able to investigate the photoexcited dynamics of the spin-density-wave state in (TMTSF) 2 PF 6 .After the ultrashort laser-pulse excitation (90 fs at λ = 800 nm), a metallic phase is generated within 3 ps.The sudden closing of the gap (observed in the far-infrared spectrum) is followed by a recovery of the spin-density-wave gap; the recovery time is found to diverge toward T SDW , while the gap remains open at T SDW [88,89].
Compared to the two compounds discussed above, the experimental work on (TMTSF) 2 AsF 6 and (TMTSF) 2 SbF 6 is less substantial [48].For (TMTSF) 2 AsF 6 , a very narrow Γ = 1/(2πτc) ≈ 5 cm −1 Drude-like contribution is identified in the metallic state above the spin-density-wave transition (T SDW = 12.2 K), that vanishes completely as T < T SDW .Although the low-frequency conductivity (ν < 60 cm −1 ) is slightly reduced in the spin-density-wave state (according to a decrease in reflectivity below 63 cm −1 ), no clear evidence of a spin-density wave gap was identified [45].
The spin-density-wave ground state is characterized by a complete and more or less isotropic energy gap 2Δ in the single-particle excitation spectrum that is seen in the temperature-dependent dc resistivity ρ(T) in all three directions (Figure 7).In addition, collective excitations of the many-body ground state are expected, as known from the pinned-mode resonance in the case of charge density waves.In a comprehensive series of microwave experiments, Donovan et al. [42,[90][91][92][93] succeeded to probe the frequencydependent conductivity down to 3 GHz (corresponding to 0.1 cm −1 ), complemented by radio frequency and standard optical experiments.As displayed in Figure 15, a Drude-like metallic behavior was found above the spin-density wave transition together with a temperature-independent feature at higher frequencies.The later one corresponds to the excitations across the Mott gap discussed above [44,47].A Figure 13: The normalized conductivities of (TMTSF) 2 PF 6 , (TMTSF) 2 AsF 6 , and (TMTSF) 2 ClO 4 shown on a log-log scale to demonstrate the power law frequency-dependence of the conductivities above the finite energy peak.The solid line shows a fit of the form σ(ω) ∼ ω −ν with ν = 1.3 ± 0.1.(reproduced from [47]).
fit of the data by the Drude model yields a Γ = 1/(2πcτ) = 3 cm −1 , which places the material well into the clean limit.
The ac response shows a strong frequency-dependence, and most importantly, two subgap modes can be identified: a very broad one in the radio frequency range, due to internal deformations of the spin density wave and a narrow mode near 0.1 cm −1 , which is interpreted as the response of the q = 0 phason.No evidence for a single-particle gap is seen in the infrared spectral range, but this is in full accord with a material in the clean limit.Later Petukhov and Dressel performed [94][95][96] microwave experiments along all three directions of the spin-density-wave model compound (TMTSF) 2 PF 6 and could demonstrate that the pinned-mode resonance is present along the a and b axes.Since the collective transport of the spin density wave is considered to be the fingerprint of the condensate, this unambiguously proves that the density wave also slides in the perpendicular b direction.The collective response, however, is absent along the least conducting c * direction.When a magnetic field of up to 10 Tesla is applied along the a axis, no change in the 16.5 GHz microwave response is observed for (TMTSF) 2 PF 6 and (TMTSF) 2 AsF 6 [97][98][99].For H c * direction, however, striking changes are detected below T SDW in the dielectric constant and conductivity.The local maximum near T = 4 K in σ 1 (T) is accompanied by a sharp decrease in 1 (T); orbital effects might be important for the Fermi-surface nesting properties.This leads to changes in the behavior of the condensate.Based upon anomalies in 1 (H, T) and σ 1 (H, T), an ambient pressure H-T diagram is suggested by the Sherbrooke group [98].

Very Low-Frequency Behavior
The discussion on the collective mode and pseudogap in onedimensional conductors raised in the 1970 by measurements of KCP and TTF-TCNQ (cf.Section 1) was revived in the case of the Bechgaard salts.Several aspects come together are often entangled and confused; however, also the materials might not be in a well-defined ground state, but electronically phase separated between metallic, spindensity-wave and even superconducting state.It should be pointed out that the single crystals are extremely clean, with Figure 14: (a) The low-frequency optical reflectivity of (TMTSF) 2 PF 6 measured perpendicular to the chains (E b ).The spin-density-wave gap opens as the temperature is lowered below 15 K.The solid lines are optical reflectivity data, the symbols indicate the results of the THz measurements; the dashed line represents the extrapolation for the Kramers-Kronig analysis.(b) The corresponding optical conductivity of (TMTSF) 2 PF 6 perpendicular to the chain direction as obtained by the Kramers-Kronig transformation of the reflectivity data (after [43,77]).

]
Figure 15: Frequency-dependent conductivity of (TMTSF) 2 PF 6 measured above and below the spin-density-wave transition at T SDW = 12 K.The single-particle gap around 30 cm −1 obtained from dc measurements is indicated by the arrow.The Drude-like conductivity collapses in the insulating state, but there remains a finite frequency pinned-mode resonance ascribed to the collective spin-density-wave excitation (reproduced from [42]).a very low amount of impurities and defects.The coexistence on a microscopic scale is subject to intense investigations [100,101].Unfortunately, real features are also mixed with experimental artefact due to a variety of reasons, some of them discussed above.It should be noted, however, that most groups were well aware of the experimental insufficiencies and pitfalls: they present and discuss their findings with the appropriate caution.Strong claims have been rare in all those years.
Due to the ambient pressure superconductivity (TMTSF) 2 ClO 4 has drawn most attention starting from the early days [60,[102][103][104] until the nice reviews of Timusk in the 1990s [80,105,106].In order to allow the anions to order near 24 K, the sample has to be cooled down very slowly through T AO ; only then the metallic state is preserved and superconductivity reached at T c = 1.2 K. Fast cooling results in the insulating spin-density-wave ground state (Figure 7(b)0.Unfortunately the actual situation for most optical experiments will be a mixture of both states, making the interpretation of the spectra difficult [107].
The challenge of the low-frequency electrodynamic properties is presented in Figure 16, following the review of Cao et al. [80].Although the overall reflectivity obtained from various groups agrees quite well, the absolute values differ by up to 10% [80,108].As nicely demonstrated by Eldridge and Bates [46], an increase of R(ω < 100 cm −1 ) by only (b) Low-frequency reflectivity (solid line) and the fitted low-frequency extensions (data taken from [81]).Note the different axes for both panels.The sharp plasma edge at 4 cm −1 is required to match the dc conductivity; alternative extrapolations are displayed by dashed lines.(c) Calculated optical conductivity of (TMTSF) 2 ClO 4 at T = 10 K (after [80]).
1.5% doubles the peak height in conductivity by half the width.The lowest frequency extrapolation of the reflectivity is commonly done by a Hagen-Rubens assumption, with or without taking the actual dc conductivity into account.
is influenced by the assumed extrapolation, but also the amplitude of the low-lying phonon line at 7 cm −1 .
As can be seen in Figures 16(c) and 17, the far-infrared optical conductivity is rather small in contrast to the dc conductivity with no sign of a Drude absorption directly visible.Instead the optical conductivity is dominated by a broadband centered around 300 cm −1 , that can be assigned to excitations across the Mott gap [47], as discussed above.An energy gap might be identified in the spectra at 2Δ = 170 cm −1 for T = 10 K.There is an overall shift of spectral weight to lower energies as the temperature is reduced.
Challener et al. measured the far-infrared properties of (TMTSF) 2 ClO 4 between 4 and 40 cm −1 down to T = 2 K [109,110].They identified two phonon-like peaks in the reflectivity at 7 and 29 cm −1 ; the upper one does not change with magnetic fields up to 4 Tesla, but vanishes upon irradiation and temperature increase to 20 K.In contrast, the McMaster group [81,104] found this mode to be sensitive to magnetic field and thus rules out direct phonon absorption.Two far-infrared features, one at 7 and one at 25 cm −1 seem to be robust, where the later one is suppressed as a field B = 0.7 Tesla is applied and shifts by 2 cm −1 when the temperature is raised to 24 K.However, it is obvious that this feature cannot be identified with the superconducting gap nor with the spin-density-wave state [104].
It was pointed out by Eldridge et al. [111][112][113][114] that the TMTSF salts with centrosymmetric anions, such as (TMTSF) 2 ClO 4 , behave quite differently than the one with tetrahedral anions, for instance (TMTSF) 2 PF 6 , (TMTSF) 2 AsF 6 or (TMTSF) 2 SbF 6 , since the zone folding at the anion order T AO causes a sharp structure in the far-infrared spectral range.Recently a complete vibrational study was published [115] including Raman and infrared modes, intramolecular as well as lattice vibrations.The (TMTTF) 2 X salts crystallize in triclinic P1 space, but the stack dimerization can be neglected and a pseudomonoclinic unit cell approximated with C 2h symmetry [116].In the spectral region below 200 cm −1 at most, 15 lattice modes are expected: 6 due to the rigid translations and rotations of the anions and 9 due to the (TMTTF) 2 dimer degrees of freedom.The anion translations are infrared active and have A u symmetry (translation along a-axis), and B u symmetry (translations perpendicular to the a axis) while the three anion rotations are Raman active with one A g and two B g symmetry modes.When the TMTSF salts with tetrahedral anions undergo an anion order transition at T AO , the unit cell is doubled in all three directions: q = (1/2, 1/2, 1/2) [117]; implying that the there are eight times as many phonons present below T AO .Eldridge et al. interpret the 30 cm −1 feature-that actually exists already above the anion order T AO as a very broadband in their powder absorption spectra-as a transverse acoustic zone-boundary phonon which is coupled very strongly with the electrons [111].Note, however, that Ng et al. [81] found this phonon mode to be sensitive to magnetic field in (TMTSF) 2 ClO 4 , although not confirmed by the Richards' group [109].By extending the experiments up to magnetic fields of 20 Tesla, Janssen et al. confirmed that this zone-folding phonon mode actually couples to the field-induced spin-density-wave state [118].
These far-infrared spectroscopic measurements (in reflection and transmission mode) in very high magnetic fields (B < 20 Tesla and T = 0.4 K) made it possible to explore the field-induced spin-density-wave states.Perel et al. [119] identified a spin-density-wave gap at 12 cm −1 from their reflection measurements at T = 1.2 K, slightly lower than the one reported by Janssen et al..A partial gap around 14 cm −1 is most pronounced in all field-induced spin-density-wave phases.Two sharp features appear around 17 and 32 cm −1 , and a smaller one at 23 cm −1 ; none of them shift with magnetic field.A peak shows up at 2.4 cm −1 that becomes stronger with increasing field and is thought to be a collective mode excitation [118], although its eigenfrequency is considerably lower than the pinned-mode resonance observed by the UCLA group [42,90].
In order to probe the superconducting state Reedyk and collaborators [120] measured the low-temperature reflectivity (E a) of (TMTSF) 2 ClO 4 in the spectral range from 10-70 cm −1 down to 0.45 K.When T < T c they find a shift of the low-frequency reflectivity step (associated to the plasma edge of the Drude contribution) to higher frequencies.Also the 30 cm −1 mode exhibits an upward shift by about 2 cm −1 .These observations are interpreted as coupling to quasiparticle excitations across the superconducting gap.As already pointed out previously [104], no direct evidence of the superconducting energy gap can be found in optical experiments.
In Figure 18, recent transmission measurements are presented for the spectral range from 1.6 to 46 cm −1 [121] obtained by employing a coherent source THz spectrometer [16].In order to avoid the well-known difficulties with the high reflection values, the sample surface and problems due to mosaic sample, finely ground (TMTSF) 2 ClO 4 crystals were dispersed in an appropriate oil and pressed in freestanding pellets.Although this procedure mixes the different crystallographic directions and causes uncertainties in the absolute values, it is free of spurious peaks due to diffraction and leakage.The sample was cooled by approximately 0.8 K per min in the temperature range from T = 300 K down to 5 K in order to arrive at the relaxed metallic state.At low temperatures, we can clearly identify a peak in the conductivity centered around 30 cm −1 and a smaller mode at 7 cm −1 .In addition, there always remains a Drude-like contribution necessary to describe the metallic conductivity.In Figure 19, the temperature dependences of those parameters are plotted that were used to fit these three contributions.The static conductivity σ 1 (ω → 0) obtained from a zerofrequency extrapolation corresponds to a mixture of all three crystallographic directions and thus cannot be simply compared to direct four-probe dc measurements (Figure 4).If we confine ourselves to the optical data, we obtain 1/(2πτc) ≈ 2-10 cm −1 , as plotted in Figure 19(b), which is different than the very narrow Drude-like contribution as previously discussed by Timusk et al. [80,81] for the results for E a.They used values of 1/(2πτc) ≈ 0.03 cm −1 , in order to fit the dc conductivity along the chains and suggest even smaller one (Figure 16(b)).It is interesting to note, that the dielectric constant is still positive and increases as the frequency is reduced, indicating the dominance of the leastconducting direction and a gaplike feature.
Part of it comes from the low-frequency mode seen around 7 cm −1 .It was already discovered by several groups three decades ago [81,109], without a clear assignment.The fit by a Lorentzian line is not really satisfactory, implying that it might reflect a gaplike feature.From the temperature dependence of its spectral weight displayed in Figure 19(c we see a strong increase of strength below 20 K, which infers a connection to the anion order at T AO = 24 K rather than to spin-density-wave state that develops in the quenched state only below 7 K and should not be present in our sample.It should be noted here that a similar mode is not observed for centrosymmetric anions (Figure 21).This supports our assignment to an acoustic phonon mode that becomes activated by the back-folding of the Brillouin zone below T AO .
Most pronounced in the (TMTSF) 2 ClO 4 spectra (Figure 18) is the strong mode at 30 cm −1 that can be identified up to elevated temperatures but strongly decreases its width (Figure 19(f)).Below T ≤ 20 K a second peak seems to appear close by and a satisfactory fit can only be obtained by introducing an additional Lorentz term, indicated by the open symbols in Figures 19(d)-19(f).The constant frequency and decreasing line width presented in Figure 19(f) confirm previous findings by Eldridge et al. [111], also based on powder transmission measurements in the far-infrared.Finding similar modes in other Bechgaard salts with octahedral anions, such as (TMTSF) 2 BF 4 and (TMTSF) 2 ReO 4 , makes them conclude that this feature has to be assigned to a transverse-acoustic zone-boundary phonon that couples strongly with the electrons.
In Figure 20, the low-temperature (T = 5 K) optical transmission of (TMTSF) 2 PF 6 is shown for the extremely low-frequency range spanning from 1.6 to 12.6 cm −1 .The experiments are performed on a large number of ground   crystals pressed into pellets of different thickness (d = 0.98 mm and 0.115 mm) in order to optimize the transmission for low and high frequencies.In the thin sample Fabry-Pérot interferences are present due to the multireflection within the pellet.These are strongly damped for the thick sample.Analyzing the spectra at different temperatures with Fresnel's equations [10,16], we obtain the conductivity σ 1 (ω, T) and dielectric constant 1 (ω, T) as a function of frequencies for various temperatures.As an example, the temperature-dependent parameters are plotted in Figure 21.According to the metallic behavior, the low-frequency transmission (ν = 6 cm −1 ) strongly decreases as the temperature is reduced down to the spin-density-wave transition at T SDW = 12 K; in the insulating state, the transmission rapidly increases (Figure 21(a)).As seen from the overview of Figure 11, at ν = 6 cm −1 we still probe the Drudelike contribution that collapses at the metal-to-insulator transition.In Figure 21(b), we consider the parameters right between the Drude term and the band assigned to the excitations across the Mott gap.As the temperature is lowered, there is only a minor variation of σ 1 (T) around T SDW , that is also seen in the dielectric constant.This implies that the change in spectral weight is small due to the spindensity-wave transition.

Outlook
After 30 years of enormous efforts from various groups all around the globe, the low-frequency properties of onedimensional conductors remain a challenge with many open questions.In particular, the GHz and THz range is still unexplored, except some isolated points obtained from the cavity perturbation method.What is the width of the zerofrequency mode?Does it exhibit a Drude shape, or does the scattering rate follow a ω 2 dependence expected for a Fermi liquid?What is the development with temperature?Is the spectral weight conserved?Up to which frequency the spectrum is involved in a redistribution with temperature?How does the reduced spectral density observed in photoemission experiments reflect in the optical data?Can we probe the superconducting gap by optical means?Is there a coexistence of the collective excitations of the spin-densitywave and superconducting ground state?Hopefully it will not take another 30 years to clarify these issues.

Figure 1 :
Figure 1: (a) Reflectivity of K 2 Pt(CN) 4 Br 0.3 • H 2 O (abbreviated KCP) measured parallel and perpendicular to the chains at different temperatures as indicated.(b) Optical conductivity of KCP for E stacks at T = 40 K (after[4,5]).The excitations across the single-particle Peierls gap lead to a broadband in the mid-infrared while the small and sharp peak centered around 15 cm −1 is due to the pinned mode.

)Figure 2 :
Figure 2: (a) Optical reflectivity and (b) conductivity of TTF-TCNQ parallel to the stack direction b for temperatures above and below the charge-density-wave transition T CDW = 53 K (some data taken from Basista et al. [6]).The insulating state is seen by the drop in the low-frequency reflectivity.The suppression of the conductivity below 300 cm −1 indicates the opening of the charge-density-wave gap.For comparison, the thin green lines represent the simple metallic behavior according to the Drude model with a plasma frequency of ω p /(2πc) = 42000 cm −1 and a scattering rate of 1/(2πτc) = 1200 cm −1 .

− 1 )Figure 3 :
Figure3: Schematic picture of the real part of the optical conductivity of TTF-TCNQ along the chain direction, at three temperatures, after Tanner et al.[25].The experiments include microwave, far-to near-infrared and cover a wide range from 0.3 to 10 4 cm −1 .

Figure 4 :
Figure4: Temperature dependence of the dc resistivity of several Fabre and Bechgaard salts.As the temperature is reduced, the charges become increasingly localized in (TMTTF) 2 AsF 6 and (TMTTF) 2 PF 6 , before the charge-ordered state is entered below 100 K. (TMTTF) 2 SbF 6 shows a transition from a metal-like state directly into the charge-ordered state at T CO = 157 K. (TMTSF) 2 PF 6 undergoes a SDW transition at T SDW = 12 K.Only (TMTSF) 2 ClO 4 remains metallic all the way down to approximately T c = 1.2K where it becomes superconducting (after Dressel et al.[36][37][38][39]).

Figure 6 :
Figure6: Reflectivity spectra of (TMTSF) 2 PF 6 measured at different temperatures along the stacking axis a (solid black line) and perpendicular to it (E b , solid green line).The filled symbols are obtained by a coherent source THz spectrometer, and the open symbols are calculated from microwave experiments[42][43][44]54].The dashed lines represent a Drude fit, respectively.The inset of panel (c) shows the schematic phase diagram of the deconfinement transition for a system of weakly coupled conducting chains as suggested by[2,55,56].The transition from a Mott insulator to a two-or three-dimensional metallic state occurs at T = 0 when t ⊥ reaches a critical value t * ⊥ .At high enough temperature, the increase in t ⊥ leads to a transition from a Mott insulating to a one-dimensional Luttinger liquid and further to a dimensional crossover into a metallic state.(Figure adopted from[57]).

Figure 7 :
Figure 7: Temperature dependence of the dc resistivity of (a) (TMTSF) 2 PF 6 and (b) (TMTSF) 2 ClO 4 measured along the a (red dots), b (blue dots), and c * (green dots) directions (after [59]).For (TMTSF) 2 ClO 4 , the quenched and relaxed states are distinguished by solid and open circles.The ordering of the tetrahedral anions around T AO = 24 K leads to a doubling of the unit cell that suppresses the spin-densitywave transition and leads to superconductivity at T c ≈ 1 K.The Fermi-liquid and Luttinger-liquid regime can be distinguished by the slope in the resistivity ρ a (T) and the maximum in ρ c (T).

Figure 8 :
Figure 8: Pressure dependence of the Mott gap energy Δ ρ (full symbols) and the transverse hopping integral 2t b (open symbols) for (TMTTF) 2 PF 6 and (TMTSF) 2 PF 6 salts as obtained from the room-temperature data.The upper pressure scale corresponds to (TMTSF) 2 PF 6 and is shifted by 3 GPa with respect to the lower pressure scale for the (TMTTF) 2 PF 6 salt.The dashed line schematically indicates the vanishing of t b at the point of the deconfinement transition (Pashkin et al. [66]).

Figure 9 :pFigure 10 :
Figure 9: The coherence parameters of the interstack charge transport κ b as a function of temperature and pressure for (TMTTF) 2 PF 6 SF and (TMTSF) 2 PF 6 .The solid blue lines correspond to κ b = 0.85.The circles illustrate the line of the dimensional crossover determined from transport measurements.The dotted line is a guide to the eye.The asterisks mark the measurement points.(after Pashkin et al. [66]).

Figure 11 :
Figure 11: Measured (a) absorptivity A(ω) = 1 − R(ω) and (b) conductivity σ 1 (ω) of (TMTSF) 2 PF 6 at 300 K, 100 K, and 20 K, for E a.The open symbols were obtained by the cavity perturbation technique.The solid symbols are from THz reflectivity measurements.The solid lines are infrared through ultraviolet reflectivity data.The dashed lines show the spectra which were used as the input for the KK calculations.The inset in (b) shows the total integrated spectral weight and the fraction in the zero-energy mode, both as functions of temperature, demonstrating that the spectral weight is not redistributed, and that only about 1% is in the Drude-like mode (after[44,47]).

Figure 16 :
Figure16: (a) Frequency-dependent reflectivity of a (TMTSF) 2 ClO 4 mosaic measured along the a-axis at different temperatures (after[80]).(b) Low-frequency reflectivity (solid line) and the fitted low-frequency extensions (data taken from[81]).Note the different axes for both panels.The sharp plasma edge at 4 cm −1 is required to match the dc conductivity; alternative extrapolations are displayed by dashed lines.(c) Calculated optical conductivity of (TMTSF) 2 ClO 4 at T = 10 K (after[80]).

ISRN
(b), different extrapolations are plotted, where the solid line corresponds to a Drude-like term with a plasma frequency of ν p = ω p /(2πc) = 634 cm −1 and a scattering rate of only 0.034 cm −1 .The corresponding optical conductivity obtained from a Kramers-Kronig analysis is shown in Figure16(c).Not only the zero-frequency mode

Figure 19 :
Figure19, the temperature dependences of those parameters are plotted that were used to fit these three contributions.The static conductivity σ 1 (ω → 0) obtained from a zerofrequency extrapolation corresponds to a mixture of all three crystallographic directions and thus cannot be simply compared to direct four-probe dc measurements (Figure4).If we confine ourselves to the optical data, we obtain 1/(2πτc) ≈ 2-10 cm −1 , as plotted in Figure19(b), which is different than the very narrow Drude-like contribution as previously discussed by Timusk et al.[80,81] for the results for E a.They used values of 1/(2πτc) ≈ 0.03 cm −1 , in order to fit the dc conductivity along the chains and suggest even smaller one (Figure16(b)).It is interesting to note, that the dielectric constant is still positive and increases as the frequency is reduced, indicating the dominance of the leastconducting direction and a gaplike feature.Part of it comes from the low-frequency mode seen around 7 cm −1 .It was already discovered by several groups three decades ago[81,109], without a clear assignment.The fit by a Lorentzian line is not really satisfactory, implying that it might reflect a gaplike feature.From the temperature dependence of its spectral weight displayed in Figure19(c),

Figure 20 :
Figure 20: Transmission spectra through (TMTSF) 2 PF 6 powder dispersed in oil.The spectra were taken at T = 5 K.The red solid lines correspond to a fits using Fresnel's equations.(a) Transmission coefficient through a thin sample of d = 0.115 mm.Interference within the pressed pellet yields the maximum at 13 cm −1 .The strong scattering of the data below 5 cm −1 is a result of standing waves within the cryostat windows.(b) Transmission coefficient through a film with thickness d = 0.98 mm.The data are more accurate at low frequencies.

Figure 21 :
Figure 21: Optical properties of (TMTSF) 2 PF 6 as a function of temperature obtained from the pressed-powder transmission measurements displayed in Figure 20.(a) Temperature-dependent transmission coefficient measured through the d = 0.98 mm sample at ν = 6 cm −1 .(b) Dielectric constant 1 (left scale), dielectric absorption 2 and optical conductivity σ 1 (right axis) measured at v = 13 cm −1 at different temperatures.The lines connect the data points.