This is the accepted version of the journal article: Guster, Bogdan; Pruneda, Miguel; Ordejon, Pablo; [et al.]. «Anion ordering transition and Fermi surface electron-hole instabilities in the (TMTSF)2ClO4and (TMTSF)2NO3Bechgaard salts analyzed through the first-principles Lindhard response function». Journal of Physics Condensed Matter, Vol. 33, issue 8 (Feb. 2021), art. 85705. DOI 10.1088/1361-648X/abc406 This version is available at https://ddd.uab.cat/record/282553 under the terms of the license J. Phys.: Condens. Matter Anion ordering transition and Fermi surface electron-hole instabilities in the (TMTSF)2 ClO4 and (TMTSF)2 NO3 Bechgaard salts analyzed through the first-principles Lindhard response function Bogdan Guster1 , Miguel Pruneda1 , Pablo Ordejón1 , Enric Canadell2 , Jean-Paul Pouget3 1 Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC and The Barcelona Institute of Science and Technology, Campus Bellaterra, 08193 Barcelona, Spain 2 Institut de Ciència de Materials de Barcelona (ICMAB-CSIC), Campus Bellaterra, 08193 Bellaterra, Spain 3 Laboratoire de Physique des Solides, CNRS UMR 8502, Université de Paris-Sud, Universié Paris-Saclay, 91405 Orsay, France Abstract. The first-principles electron-hole Lindhard response function has been calculated and analyzed in detail for two (TMTSF)2 X (X= ClO4 and NO3 ) Bechgaard salts undergoing different anion-ordering (AO) transitions. The calculation was carried out using the real triclinic low-temperature structures. The evolution of the electron-hole response with temperature for both relaxed and quenched salts is discussed. It is shown that the 2kF response of the quenched samples of both salts display a low temperature curved and tilted triangular continuum of maxima. This is not the case for the relaxed samples. (TMTSF)2 ClO4 in the AO state exhibits a more quasi-1D response than in the non AO state and relaxed (TMTSF)2 NO3 shows a sharp maximum. The curved triangular plateau of the quenched samples results from multiple nesting of the warped quasi-1D Fermi surface which implies the existence of a large q range of electron-hole fluctuations. This broad maxima region is around 1 % of the Brillouin zone area for the X= ClO4 salt (and X = PF6 ) but only 0.1 % for the X = NO3 salt. It is suggested that the strong reduction of associated SDW fluctuations could explain the non detection of the SDW-mediated superconductivity in (TMTSF)2 NO3 . The calculated maxima of the Lindhard response nicely account for the modulation wave vector experimentally determined by NMR in the SDW ground state of the two salts. The critical AO wave vector for both salts is located in regions where the Lindhard response is a minimum so that they are unrelated to any electron-hole instability. The present first-principles calculation reveals 3D effects in the Lindhard response of the two salts at low temperature which are considerably more difficult to model in analytical approaches. Keywords: Bechgaard salts, anion ordering, spin density waves, density functional theory, Lindhard response function. 2 1. Introduction detection of a transverse plasma edge when the electric field is applied in the inter-stack direction Since the synthesis of the Fabre [(TMTTF)2 X] and b’ of (TMTSF)2 PF6 [7]. The deconfinment from Bechgaard [(TMTSF)2 X] salts at the end of the 1970s the high temperature 1D Luttinger liquid to a 2D (TMTTF is tetramethyltetrathiafulvalene, TMTSF is Fermi liquid behavior, which can be probed by the tetramethyltetraselenafulvalene and X is a monovalent opposite rate of thermal variation of the inter-layer c* anion such as PF− − − 6 , ClO4 and NO3 ), a lot of conductivity, is curiously found also at about TL ∼ attention has been devoted to the establishment of 100 K in (TMTSF)2 PF6 [8]. However the study the complex phase diagram of these prototypical of (TMTSF)2 ClO4 suggests that the deconfinement quasi-one dimensional (1D) quarter-filled organic salts transition occurs at a higher temperature than the [1, 2]. The competition between numerous ground 1D-2D dimension crossover. In this salt the opposite states resulting from the coupling of charge, spin rate of variation of the thermal dependence of the and structural degrees of freedom are at the origin c* conductivity suggests an inter-stack deconfinement of the longstanding interest on these paradigmatic around room temperature (RT) [9], while polarized low-dimensional correlated materials. Basically, the reflectance measurements along b’ do not reveal any physics of the Fabre and Bechgaard salts is governed evidence of a plasma edge at this temperature [10]. by the interplay of electron-electron repulsions and This suggests that the 1D-2D dimension crossover inter-stack interactions which are very anisotropic. should occur at a lower temperature TL which can The former are dominant in the (TMTTF)2 X series be estimated between 200 K and 100 K from the inducing a Luttinger liquid physics, charge localization measurement of the dielectric function for an electric phenomena and spin-charge decoupling. These two field polarized along b’ [11]. Finally, the detection types of interaction compete in the (TMTSF)2 X series of a small Drude component below ∼ 10 K in the so that a progressive deconfinement of carriers to optical conductivity measured along the inter-layer c* neighboring stacks as well as an associated dimensional direction of (TMTSF)2 ClO4 indicates the occurrence crossover is induced upon cooling: first from 1D to 2D of a low temperature 2D to 3D dimension crossover and then to 3D [3]. At high temperature (TMTSF)2 X setting a coherent inter-layer coupling [11]. behaves as a collection of uncoupled 1D Luttinger Below TL (TMTSF)2 PF6 undergoes a metal- liquids located in the stack direction (a), then at insulator transition towards a SDW ground state in the low temperature as a 2D Fermi liquid located in the 2D electronic regime at TSDW = 12 K. (TMTSF)2 ClO4 (a,b) donor layers which is subject to a spin density and pressurized (TMTSF)2 PF6 become superconduct- wave (SDW) instability and, finally as a 3D Fermi ing at TS ∼ 1 K in the 3D electronic regime. There liquid which is subject to superconductivity [4], when is also accumulating evidence that superconductivity the delocalization along the inter-layer direction c* in the Bechgaard salts is achieved by SDW fluctua- becomes thermally relevant. tions [12] whose theoretical treatment relies, in the The 1D-2D electronic crossover is revealed by spirit of quasi-1D physics, upon interference between the thermal dependence of the electron-hole response. electron-hole and hole-hole pairing processes [13]. For a layer of chains coupled by t⊥ the electron- The SDW instability and the SDW phase diagram hole response continuously evolves upon cooling from under pressure and/or magnetic field have been the response of a 1D metal (1D Luttinger liquid in successfully analyzed within the RPA approximation case of sizeable electron-electron interactions), above in terms of Fermi surface nesting since more than TU , to the response of a 2D metal, below TL [5]. thirty years [1]. Within this framework the bare For a free electron gas: TU = 2t⊥ and TL = electron-hole or Lindhard function is appropriately t⊥ /π. In the presence of electron-electron repulsions used. There is however a questionable approximation t⊥ are downward re-normalized and TU and TL are concerning the modeling of the band structure. Until smaller [3, 5]. The continuous crossover from TU to now the electron-hole instability of the Bechgaard salts TL is clearly revealed by the q dependence of the has been analyzed via a model electron dispersion first-principles Lindhard response of (TMTSF)2 PF6 [6] based on a pseudo-orthorhombic lattice [1]. It is (note that DFT calculation neglects electron-electron only recently that the real electron-hole (or Lindhard) correlations). As expected, TL coincides with the response was calculated from first-principles and using 3 the real triclinic structure of (TMTSF)2 PF6 [6]. This principles Lindhard function for both their quenched calculation showed the occurrence below TU of a and AO ground states. Thus, the present study 2kF = a*/2 quasi-1D electron-hole instability (kF is completes our first-principles analysis of the electron- the Fermi wave vector of the incipient 1D electron hole response of the Bechgaard salts by considering gas) which is modulated along b* by the inter- how the AO interferes with the basic scenario reported stack interaction. However, the b* dependence of in ref. [6]. this electron-hole response varies substantially below TL making clear the presence of several competing 2. Computational details FS nesting processes. As a consequence, it was found a triangular continuum of maximal response The DFT calculations [24, 25] were carried out using which includes the SDW modulation wave vector a numerical atomic orbitals approach, which was experimentally observed below TSDW . The purpose developed for efficient calculations in large systems of the present work is to report and analyze the and implemented in the Siesta code [26, 27]. We first-principles electron-hole response for different have used the generalized gradient approximation Bechgaard salts incorporating non-centrosymmetric (GGA) to DFT and, in particular, the functional anions of different shape as ClO4 and NO3 . An of Perdew, Burke and Ernzerhof [28]. Only the important goal of this work is to examine the impact valence electrons are considered in the calculation, of their anion ordering (AO) transitions [14] on with the core being replaced by norm-conserving the electron-hole response. This study is all the scalar relativistic pseudopotentials [29] factorized in more interesting since, below TAO , the X = NO3 the Kleinman-Bylander form [30]. The non-linear core- salt keeps the insulating SDW ground state whereas valence exchange-correlation scheme [31] was used for the X = ClO4 salt remains metallic and becomes all elements. We have used a split-valence double-ζ superconducting [15] although, if samples of the latter basis set including polarization functions [32]. The are quenched through the AO transition, the ground energy cutoff of the real space integration mesh was state changes to a SDW one. Thus, in order to 350 Ry. The Brillouin zone (BZ) was sampled rationalize the phase diagram of the Bechgaard salts with the Monkhorst-Pack scheme [33] using grids of and in particular the occurrence of the SDW ground (45×45×18) k-points to build the charge density. The state, it is essential to determine how the AO process Lindhard response function, modifies the electron-hole Lindhard response. (TMTSF)2 ClO4 exhibits a staggered order of X X fF (i (k)) − fF (j (k + q)) the ClO− 4 anions in the inter-stack b-direction below χ(q) = − , (1) i (k) − j (k + q) TAO = 24 K [16, 17]. The resulting (0, 1/2, 0) AO i,j k structure is represented in Fig. 1a. However, as where fF is the Fermi function and i (k) are the mentioned, when samples are slowly cooled through band eigenvalues, was obtained from the computed the AO transition, (TMTSF)2 ClO4 remains metallic. DFT band eigenvalues i (k) (i.e. in this work the Upon quenching, which prevents the establishment TMTSF HOMO-based bands of the system). The of the long range anion superstructure below TAO , integral over k-points of the BZ was approximated (TMTSF)2 ClO4 exhibits a metal-insulator transition by a direct summation over a dense, regular grid of towards a SDW ground state at TSDW = 6.5 K points. As the Lindhard function is more sensitive (this temperature decreases with the speed of the to the accuracy of the BZ integration than the total cooling rate at TAO ). (TMTSF)2 NO3 exhibits a energy, especially in very anisotropic systems, and/or staggered order of the NO− 3 anions in the stack a- in the presence of hot spots in the band structure direction below about TAO = 41-49 K [18, 19]. The (e.g. saddle points with the corresponding van Hove resulting (1/2, 0, 0) AO structure is represented in singularity in the DOS), the k-points grid used for Fig. 1b. When samples are slowly cooled through its calculation must be more dense than in the the AO transition (TMTSF)2 NO3 exhibits a SDW standard self-consistent determination of the charge ground state at TSDW = 9 K. Note that upon rapid density and Kohn-Sham energy. The calculations cooling through TAO (TMTSF)2 NO3 does not exhibit are done, nevertheless, using the eigenvalues obtained the 2a AO superstructure [19]. Apparently, the SDW in the DFT calculation for the coarser grid, and features probed by ESR are independent of the cooling interpolating their values in the denser grid, using rate of the samples [20]. Although the reconstructed a post-processing utility available within the Siesta FS due to the AO in the two salts has been the package. In this work, for the calculation of the subject of a few recent studies [17, 21, 22, 23] we Lindhard response function, the BZ was sampled using present here a more quantitative estimation of their a grid of (400×400×16) k-points. The partially filled nesting properties through the calculation of the first- bands were those taken into account in the calculations. 4 Figure 1. (a) (0, 1/2, 0) anion ordered structure of (TMTSF)2 ClO4 . (b) (1/2, 0, 0) anion ordered structure of (TMTSF)2 NO3 . For simplicity hydrogen atoms have not been shown. Note that the thermal dependencies of the Lindhard as in the present case, or not) are build from the functions reported below are due to the Fermi function same orbital (the HOMO in the Bechgaard salts [1], in Eq. 1. The experimental 7 K average and anion the Mo dxz in the blue bronzes [40]). It is because ordered structures of Le Pévelen et al. [17, 34] were of the joint occurrence of these two features that the used in the calculations for (TMTSF)2 ClO4 . The nesting properties of the FS are the driving force for calculations for (TMTSF)2 NO3 were based on the 12 K their electronic instabilities and physical behaviour [1]. average and anion ordered crystal structures reported First-principles calculations for either simple model by Hebrard-Bracchetti et al. [19, 35]. All values of 1D systems [39] as well as for complex pseudo-1D the Lindhard response are given in units of eV−1 per metallic bronzes like the blue bronzes [40] clearly crystallographic repeat unit, i.e. one (TMTSF)2 X for prove it. Eq. 1 can not be directly used for 2D the structures without AO but two (TMTSF)2 X units or 3D systems without caution. In these cases the for the structures with AO. The thermal dependence of important bands often result from the hybridization of the Lindhard responses arises from the thermal factor dif f erent types of orbitals (the dz2 and dx2 −y2 /dxy in in the Fermi function fF . NbSe2 [37], the s, dt2g and deg orbitals in Cr [38]) and The electronic structure of the Bechgaard salts is consideration of the different matrix elements clearly completely determined by the two pseudo-1D HOMO- influences the calculated response [38, 39]. Thus, as based bands. Note that Eq. 1, which has been in our previous study of the electron-hole response of successfully used to rationalize the physics of the these (TMTSF)2 PF6 [6] our first-principles calculations are salts [1], is strictly valid for plane waves. In the case of based on Eq. 1. Bloch wave functions each numerator of this equation incorporates the squared matrix element |< i, k | 3. Electronic Structure exp(iqr) | j, k + q >|2 [36].The matrix element |< i, k | exp(iqr) | j, k + q >|2 takes into account the spatial The low temperature first-principles electronic struc- overlap of the | i, k > and | j, k + q > Bloch functions ture of (TMTSF)2 ClO4 and (TMTSF)2 NO3 both in of bands i and j respectively. Recently, there has their quenched (without AO) and relaxed (with AO) been some discussion concerning the need to include phases has been discussed recently [22]. However, our such matrix elements in the calculation of the response previous analysis of the FS of the (TMTSF)2 PF6 Bech- function [37, 38, 39], specially in the context of the gaard salt [6] has shown that three different nesting puzzling physics of dichalcogenides. As a matter of processes occur (see Fig. 2). Two of them mostly nest fact, Eq. 1 should only be used for 1D or pseudo- the zones around the inflection points of the warped 1D systems. In that case the Fermi surface contains lines. Because of the triclinic geometry they are differ- large sheet-like contributions. In addition, these bands ent and since these zones around the inflection points (being folded as a result of a weak external potential, are relatively flat they both lead to a linear segment 5 Pocket 2 a) b) Y Y' Band 2 X' Band 1 X Pocket 1 Figure 3. Calculated Fermi surface for: (a) (TMTSF)2 ClO4 in the (0, 1/2, 0) AO phase , and (b) (TMTSF)2 NO3 in the (1/2, 0, 0) AO phase. In (a) Y’ refers to (0, 1/2, 0) in units of the Figure 2. Schematic illustration of the three nesting processes triclinic reciprocal lattice vectors of the a × 2b × c superlattice. occurring in the real Fermi surface of the triclinic Bechgaard In (b) X’ refers to (1/2, 0, 0) in units of the triclinic reciprocal salts, which are at the origin of the occurrence of arcs in the lattice vectors of the 2a × b × c superlattice. Lindhard function (see ref [6] and discussion in section 4). (TMTSF)2 NO3 becomes a semi-metal below TAO with with high values of the Lindhard function having op- a FS made of a series of elongated hole and electron posite slopes and different intensities. A third nest- cylinders (pocket 1 and pocket 2 respectively in ing process occurs around the crossing point of these Fig. 3b) such that the AO gaps develop in regions of the two linear fragments. The occurrence of several FS original FS where the nesting is not very good. Thus nesting processes is simply a consequence of the non- it is expected that AO should exert a small influence regular sinusoidal shape of the FS and consequently, on the nature of the SDW instability of this salt. they should be expected for all Bechgaard salts in- cluding the present ones before the AO transition (i.e. 4. Analysis of the Lindhard function the quenched samples). However, note that they will certainly differ in subtle details of the warping, a fea- 4.1. (TMTSF)2 ClO4 above TAO as well as in the ture that, as it will be discussed in the next section, quenched ground state has strong consequences for their low temperature be- haviour. In order to make the link with the calculated Fig. 4 presents selected 2D intensity plots of the Lindhard function discussed in the next sections, we Lindhard function of (TMTSF)2 ClO4 at 200 K (a), only succinctly discuss here the FS of these salts after 50 K (c) and 6 K (e) (the latter temperature being the AO transition (i.e. the relaxed samples). close to TSDW in the quenched structure) in the (a*, Relaxed (TMTSF)2 ClO4 and (TMTSF)2 NO3 b*) reciprocal plane, as well as magnified 3D intensity salts crossing slowly their AO transition double plots at the same temperatures in the vicinity of their periodicity along b and a, respectively (see the M-X-M Brillouin zone boundary ((b), (d) and Fig. 1). Consequently, their appropriate FS below (f), respectively). The 200 K plots show that the TAO basically result from the folding and hybridization high temperature response is very anisotropic. The of the original (i.e. quenched) FS along ±b*/2 and longitudinal scan along a* in Fig. 5a shows that the ±a*/2, respectively. Since the folding occurs along response is sharply peaked at 0.5a*, corresponding different crystallographic directions the calculated FSs to the 2kF wave vector of the quasi-1D electron gas. exhibit different shapes (see Fig. 3). The DFT However Figs. 4a and b show that the 200 K response AO gaps (∆0 ) are calculated to be 14 meV and exhibits a significant sinusoidal modulation along b*. 8.9 meV for (TMTSF)2 ClO4 and (TMTSF)2 NO3 , A more quantitative analysis of the half-width at half- respectively. The different direction of the AO leads to maximum (HWHM) of this modulation (Fig. 6) will the development of AO gaps at different zones of the be discussed later. The transverse b* scan of Fig. 5b original FS, a feature that has important consequences also shows that the response exhibits a very broad for the physical behavior of these salts below TAO . maximum at the X point which splits into two broad For instance the folded FS for (TMTSF)2 ClO4 below maxima at q0 ≈ (0.5, ±0.14) upon cooling below about TAO is made of two warped planes separated by 140 K. Also note that two secondary maxima, q2 and small AO gaps (band 1 and band 2 in Fig. 3a) q3 , which are a bit off the 2kF line shown in Fig. 5b which develop at the regions of the original FS with appear below ∼140 K and sizeably grow in intensity better nesting. Thus, it is expected that AO should upon cooling (see Fig. 5b). At 50 K 2D and 3D exert a considerable influence on the nature of the intensity plots (Figs. 4c and d) show that, due to SDW instability of this Bechgaard salt. In contrast, the presence of these secondary maxima, the Lindhard 6 response exhibits a curved shape at low temperatures. Upon cooling the broad curved maximum is progressively transformed into a curved triangular plateau (Figs. 4e and f). The progressive formation of a plateau of maxima is more precisely revealed by the q1 -q2 , q2 -q3 and q1 -q3 scans of Figs. 7a, b and c respectively (the different qi are given in Table. 1. These figures show that the intensity of the q2 and q3 peaks reaches the intensity of the q0 peak around 20 K, temperature below which the curved triangular plateau is well developed. Also Fig. 5a shows that a continuum of intensity with a width of ∼ 0.02a* develops between q0 and another q1 maximum below about 15-20 K (as a consequence the b* scan of Fig. 5b exhibits a tilted maximum). The low temperature curved triangular plateau is thus delimited by the q0 /q1 , q2 and q3 wave vectors already introduced in ref. [6] and indicated in Table 1, as well as in Fig. 7d. At about 15 K, Fig. 7c shows that the maximum of the Lindhard response shifts from q0 to q3 . Figs. 4e and f present respectively 2D and 3D intensity plots of the Lindhard response in the (a*,b*) reciprocal plane at TSDW of the quenched samples (6 K). The intensity of the q0 /q1 , q2 and q3 responses at 1 K is given in Table 1. As these temperatures are located below TAO = 24 K, the curved triangular shaped Lindhard response is representative of the electron-hole instability of quenched samples. At temperatures around 15-17 K we also found that the shape of the curved triangular plateau slightly depends of the c* component of the Lindhard response. Thus the Lindhard response exhibits a 3D wave vector dependence due to the thermal relevance of the electronic dispersion along c*. Fig. 8 gives a superposition of the 0c* and 0.5c* cuts of the Lindhard function calculated at 1 K. One observes in particular a splitting along b* of the q1 -q3 line. A similar effect somewhat enhanced is observed in quenched (TMTSF)2 NO3 , a feature that will be considered in more detail in Sect. 4.3. However, we did not observe this c*-dependence for (TMTSF)2 PF6 [6]. Note that in the quenched samples there is a quenched anion disorder so that a distribution of transfer integrals really occurs. Strictu sensu this must result with some blurring of the electronic structure and thus some damping of the singularities of the response function. However, even if TSDW of the quenched (TMTSF)2 ClO4 samples (6.5 K) is lower than for (TMTSF)2 PF6 (12 K), where there is no anion disorder, the SDW state is not suppressed. Consequently, disorder effects do not strongly alter the nesting features. Figure 4. 2D plot of the Lindhard function of (TMTSF)2 ClO4 in the (a*,b*) reciprocal plane at 200 K (a), 50 K (c) and 6 K (e). 3D sections of the Lindhard function of (TMTSF)2 ClO4 around the X-M direction at 200 K (b), 50 K (d) and 6 K (f). 7 Figure 5. Thermal dependence of a longitudinal a* scan at 0.14b* (a) and a transverse b* scan at 0.5 a* across the Lindhard function of (TMTSF)2 ClO4 . The red dashed line in the temperature scale indicates the temperature at which the single maximum leads to a tilted segment of maxima. Figure 6. Comparison of the thermal dependence of the −1 inverse electron-hole coherence length ξehk calculated at the q0 , −1 X and M points with the BOW correlation length ξBOW of (TMTSF)2 ClO4 obtained from the data in ref. [41] 4.2. (TMTSF)2 ClO4 in the AO phase Below the (0, 1/2, 0) AO transition the electronic structure of (TMTSF)2 ClO4 is significantly modified since an avoided crossing appears in the Fermi surface, Figure 7. Thermal dependence of the scans across: q1 -q2 (a), as shown in Fig. 3a. As a consequence, the total q2 -q3 (b), q3 -q1 (c) lines of the Lindhard function for quenched (TMTSF)2 ClO4 . The red dashed line in the temperature scale Lindhard response in the AO phase represented in indicates the temperature at which the single maximum leads Figs. 9a and b for the (a*, b’*) plane (with b’* = b*/2) to a tilted segment of maxima. In (d) we present a legend for is sizeably modified. However it still exhibits a 1D the scans directions where ~ r1 = (1 − tan(|~ q1 − q~2 |))a∗ + b∗ , anisotropy. This response does not vary significantly respectively ~ r3 = −(1 + tan(|~ q3 − q~1 |))a∗ + b∗ . Positive values are designated for q points above the Γ-X direction. 8 Figure 8. Overlap of the Lindhard response function sections for quenched (TMTSF)2 ClO4 calculated for planes parallel to (a*,b*) for qc∗ = 0 and qc∗ = 0.5 at 1 K. Note the black arrows indicating the effect of 3D interactions. Table 1. Location of the q0 /q1 , q2 and q3 maxima of the low temperature Lindhard response of the (TMTSF)2 X Bechgaard salts with X = PF6 , ClO4 and NO3 . The table indicates the intensity of these maxima at 1 K. Data for X = PF6 taken from ref. [6]. X PF6 ClO4 NO3 q0 (0.50,0.23) (0.50,0.14) (0.50,0.27) q1 (0.48,0.23) (0.48,0.14) (0.50,0.27) χeh (q1 ) [eV−1 ] 2.66 2.69 2.62 q2 (0.52,0.30) (0.56,0.21) (0.52,0.29) χeh (q2 ) [eV−1 ] 2.76 2.65 2.64 q3 (0.53,0.14) (0.57,0.07) (0.52,0.24) Figure 9. 3D plot of the Lindhard function of (TMTSF)2 ClO4 χeh (q3 ) [eV−1 ] 2.58 2.72 2.54 in the AO phase at 1 K (∼TS ) (a). 2D plots of the decomposition T* (onset q2 ,q3 ) 100 K 140 K 70 K of this Lindhard function (b) into the inter-band (c), first band Fraction of (a*,b*) Brillouin zone 0.7 % 1.2 % 0.1 % (d), and second-band (e) components. The Linhard function values are given in units of eV−1 per two (TMTSF)2 ClO4 units. between 24 K (TAO ) and 1 K, temperature at which (TMTSF)2 ClO4 becomes a superconductor. modulation which can be fitted by two Lorentzians As the Fermi surface contains the contributions centered at ±0.29b’*, whose maximum is close to the of two warped quasi-1D bands (contributions noted q0 maximum observed above TAO (remind that b* = band 1 and 2 in Fig. 3a), one should decompose the 2b’*). Figs. 10b, c and d show different a* cuts of total Lindhard response into intra-band and interband this response for b’*= 0 (i.e. through the X point) components as shown in Figs. 9c to e. Each intra- (b), b’*= 0.3 (c) and b’* = 0.5 (i.e. through the M’ band response (Figs. 9d and e) appears as a pair point) (d). Figures 10b and d exhibit a truncated broad of contributions slightly split around a*/2 which are Lorentzian profile in the scan direction. The saturation warped and modulated in intensity along b’*. The of the response between 0.5 ± 0.04a* at the X point inter-band response (Fig. 9c) is more 1D, located at and 0.5 ± 0.03a* at the M’ point is due to the fact that a*/2 and weakly modulated along b’*. Figs. 9d and e the nesting process for these q wave vectors is stopped show that each quasi-1D intra-band response exhibits by the formation of sizeable pockets which suppress arcs of intensity maxima. They are located around the divergence of the electron-hole response. The (+0.6, +0.25) and (+0.4, -0.25) reciprocal positions for HWHM of the underlying Lorentzian profile of 0.05 band 1 and (+0.4, -0.15) and (+0.6, +0.15) reciprocal Å−1 leads to an electron-hole coherence length of 20 positions for band 2. The 1D inter-band response Å, slightly larger than that found in the non AO phase shown in Fig. 9c is the strongest response and largely (Fig. 6). The cut of Fig. 10c around the best nesting determines the shape of the response function. wave vector (0.5a*, 0.3b’*) shows a more pronounced Figure 10 gives full details about the quasi-1D divergence. This scan reveals also the superposition inter-band response. Figure 10a presents a b’* scan of of the previously described broad component and a this response for 0.5a*. It exhibits a weak and broad weaker and sharper response which saturates at 0.5 9 ± 0.015a*. This sharper response represents ∼ 40 % of the intensity of the total response. This intensity is modulated along b* (see Fig. 10a). If one assumes that this sharp component has a Lorentzian profile along a*, one can estimate from its HWHM of 0.02 Å−1 an electron-hole coherence length of 48 Å (i.e. ∼ 7a). In addition, using the HWHM value of ∼ 0.045 Å−1 of each Lorentzian fitting the b’* scan of Fig. 10b one obtains an electron-hole coherence length of 22 Å along b* which is slightly larger than the unit cell parameter (b’ = 2b = 15.3 Å) of the superstructure. This means that the two nonequivalent chains of the unit cell, between which there is a charge transfer, are consistently coupled by the inter-band nesting mechanism. 4.3. (TMTSF)2 NO3 above TAO and in the quenched ground state The shape of the Lindhard function for (TMTSF)2 NO3 bears some resemblance with that of (TMTSF)2 ClO4 (Sect. 4.1). Figs. 11a and b give respectively the 2D and magnified 3D intensity plots of the Lindhard response at 120 K. As shown in more detail in the scans of Fig. 12a, the Lindhard response is centered at 2kF = 0.5a*. The b* scan of Fig. 12b makes clear the existence of two broad maxima for q0 = (0.5, ±0.27) which grow in intensity upon cooling. The shape of the maxima transform into curved arcs below around 70 K. These arcs are defined by the growth of the q2 and q3 secondary maxima whose wave vectors are indicated in Table 1. Figs. 11c and d give the 2D and 3D intensity plots of the Lindhard response at 40 K which is close to the anion ordering transition (TAO ∼ 41-49 K). It is interesting to remark that the (1/2, 0, 0) AO critical wave vector is located at the X point where the Lindhard response is minimum (see Fig. 11d). Thus the (1/2, 0, 0) AO is unrelated with any electron-hole Figure 10. Different scans of the inter-band response (in units of eV−1 per two (TMTSF)2 ClO4 units) for (TMTSF)2 ClO4 in instability. the AO phase at 1 K. (a) b’* scan of the response for 0.5a*. (b)- The Lindhard response calculated below TAO (d): a* cuts of the response for 0b’* (X point) (b), 0.3b’* (c), as for instance that shown in Figs. 11e and f, is and 0.5b’* (M’ point) (d) representative of the response of quenched samples. A well defined tilted plateau with curved triangular shape appears around 15-20 K. A magnified 2D plot of this ular a splitting in the region of q3 marked with black region as calculated at 1 K is shown in Fig. 13a. Note arrows. The inner of these contributions corresponds that because of the tilting of the triangular plateau at to the plane with qc∗ = 0 and the outer, which is more these temperatures, the q0 maxima is located on the elongated, to that with qc∗ = 0.5. In the region of side of an asymmetric peak. Also at about 12 K the q2 the two contributions remain on top of each other. strongest maxima of the Lindhard response shifts from This change originates from the 3D interactions, i.e. q0 to q2 . The intensity of the maxima of the Lindhard warping of the FS along c* in the regions leading to response at 1 K is given in Table 1. the triangles. This warping varies along the FS and As for the (TMTSF)2 ClO4 but not (TMTSF)2 PF6 , is non negligible for quenched samples in the regions we have found that at low temperatures the response leading to the q3 nesting but much less in the regions also depends upon c*. In Fig. 13b we plot the overlap leading to the q2 nesting. The result is that the re- of two sections of the Lindhard function, those for the gion of the triangular plateau around q3 is spread along qc∗ components of 0.0 and 0.5. One observes in partic- the direction of the initial linear component of the arc 10 Figure 12. Thermal dependence of the transverse a* scans across the Lindhard function of (TMTSF)2 NO3 at 0.27b* (a) and b* scans at 0.5a* (b). whereas that around q2 does not. Consequently the triangular plateau spreads somewhat in this direction and becomes more inclined. Thus the prevalence of q2 at low temperature is enhanced by the inter-layer interactions. This effect starts to be visible in our cal- culations around 12-15 K. 4.4. (TMTSF)2 NO3 in the AO phase Below the (1/2, 0, 0) AO transition the electronic structure of (TMTSF)2 NO3 is slightly modified: the periodicity along the donor stacks doubles (a’ = 2a) so that the donor stacks become weakly tetramerized [19, 35]. (TMTSF)2 NO3 is a semimetal below TAO holding an electron pocket (in red in Fig. 3b) and a hole pocket (in blue in Fig. 3b) per unit cell. Thus the Lindhard response in the AO phase should incorporate several components. 2D and 3D plots of the full Lindhard function for this semi-metallic state shown in the (a’*,b*) plane (with a’* = a*/2) are presented in Figs. 14a and b, respectively. This response may be decomposed into the inter-pocket and two intra-pocket Figure 11. 2D plot of the Lindhard function of components shown in Figs. 14c, d and e respectively. (TMTSF)2 NO3 in the (a*,b*) reciprocal plane at 120 K (a), 40 K (c) and 9 K (e). 3D sections of the Lindhard function of Only the inter-pocket component exhibits a strong (TMTSF)2 NO3 around the X-M direction at 120 K (b), 40 K response under the form of an intense broad line around (d) and 9 K (f). q1D = 0a’*. The intensity of this q1D response, 11 Figure 13. (a) Magnified section of the Lindhard function for (TMTSF)2 NO3 without AO at 1 K. (b) Overlap of the Lindhard response function sections calculated at planes parallel with (a*,b*) for qc∗ = 0 and qc∗ = 0.5 at 1 K in the non-ordered phase of (TMTSF)2 NO3 . Note the black arrows indicating the effect of 3D interactions. strongly modulated along b*, as shown in Fig. 15, exhibits a maximum at 0.29b*. A 3D intensity plot of this response at 9 K (i.e. at the TSDW critical temperature of relaxed samples) is shown in Fig. 16. In contrast with the Lindhard response for quenched Figure 14. 3D plot of the Lindhard response of samples which exhibited a curved triangular plateau of (TMTSF)2 NO3 in its AO phase at 1 K (TSDW ) (a). 2D maxima (Figs. 11 and 13) the Lindhard response in the plots of the decomposition of this Lindhard function (b) into its inter-pocket (c), pocket 1 (d) and pocket 2 (e) components. AO phase exhibits a sharp maximum at (0.01, 0.29). See Fig. 3b. The Linhard function values are given in units of eV−1 per two (TMTSF)2 NO3 units. 5. Discussion 5.1. Shape of the Lindhard response above TAO and in -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 the quenched ground state 2.7 q = ( 0.00 a'*, xb*, c*) In the absence of an AO transition, the Lindhard 2.4 χ(eV-1) response of (TMTSF)2 ClO4 and (TMTSF)2 NO3 behaves qualitatively as the (TMTSF)2 PF6 one which 2.1 we analyzed in detail in ref [6]. At high temperature 1.8 (below TU which is above RT) it consists of a quasi-1D -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 response at 2kF = 1/2a*, whose intensity is slightly b*(a.u.) modulated along b*. This gives rise to a broad maximum at a q0 whose b* component depends upon Figure 15. Scan of the inter-pocket response of (TMTSF)2 NO3 the anion (see table 1). This component increases in its AO phase along b* (values in units of eV−1 per two (TMTSF)2 NO3 units.). 12 the c* dependence of the Lindhard response for X = ClO4 and NO3 salts could help fixing the qc component of the SDW modulation. Although at first glance the FS of all Bechgaard salts exhibits quite similar warping along the b*- direction, there are in fact subtle differences relying on the magnitude and the phase of the different inter- stack interactions for the different salts. As the q0 /q1 , q2 and q3 wave vectors (see table 1) correspond to different nesting processes of the FS (Fig. 2), it Figure 16. 3D plot of the inter-pocket Lindhard function of turns out that the phase of the different FS warping (TMTSF)2 NO3 in its AO phase at 9 K (=TSDW ). Values given components differs among the salts. In addition, note in units of eV−1 per two (TMTSF)2 NO3 units. The maximum also that T *(≈ TL ) below which the q2 and q3 nesting response occurs at (0.01,0.29). processes become thermally relevant, varies among the salts. Consequently, the magnitude of the warping substantially from the X = ClO4 salt to the PF6 salt components also varies. Basically, they increase from then to the NO3 salt. The shape of the Lindhard the X = NO3 salt to the PF6 salt then to the ClO4 response is continuously modified below a temperature salt. T * by the development of q2 and q3 secondary maxima In summary, all indicators qualifying the cal- out of the 2kF location. As shown in table 1, T * culated response of (TMTSF)2 PF6 and quenched decreases from the X = ClO4 salt to the PF6 salt then (TMTSF)2 ClO4 and (TMTSF)2 NO3 , such as the T * to the NO3 salt. T * apparently coincides with the 1D- onset and the f fraction, clearly reveal a considerably 2D electronic dimension crossover temperature TL at stronger 1D character of the (TMTSF)2 NO3 salt im- which a transverse plasma edge is detected in the X = plying weaker inter-stack interactions and a narrower PF6 [7] and ClO4 [11] salts. Upon cooling, the intensity range of 2D SDW fluctuations. As discussed in the of q2 and q3 maxima grow leading to the formation of a next sections we believe that this result is important boomerang shaped maximum which transforms into a to understand the peculiar low temperature behavior tilted curved triangular region of maxima around 15-20 of this salt K (in the quenched state) when the intensity of the q2 Finally, let us remark that the calculation of the and q3 maxima become comparable to the q0 intensity. Lindhard response assumes free electrons. Electron- At about 15 K (in the quenched state) q0 shifts to q1 in electron repulsions which in 1D systems give rise the X = ClO4 salt as previously observed for the PF6 to a Luttinger liquid type behaviour, are ignored. salt. There is no sizable shift for the NO3 salt. Strictly speaking the calculation of the Lindhard The area of the q1 -q2 -q3 low temperature triangu- response should be valid for a Fermi liquid below the lar region of maxima per (a*,b*) Brillouin zone is given deconfinement temperature. As this temperature is by around 300 K the calculation is validated for the X = ClO4 salt. It could be questionable in the PF6 − q−→ −−→ ~∗ × b~∗ . 1 q3 × q1 q2 = f a (2) salt where the deconfinement temperature is around 100 K. However since above 100 K the calculated Such equation defines a fractional occupancy f which quantities such as the electron-hole coherence length is indicated in table 1. f is a quantitative measure perfectly agree with experimental results [6], one of the size of the maximal electron-hole fluctuations expects that the Lindhard response should be only per Brillouin zone : f ∼ 1.2 % for the X = ClO4 marginally modified. This is also true for the X salt, decreases to ∼0.7 % for the PF6 salt and then = NO3 salt where the deconfinement temperature is to ∼ 0.1 %, one order of magnitude smaller, for the estimated as T*=70K. However, in this temperature NO3 salt. These results concern the 2D dispersion of range the physics is perturbed by pretransitional AO the Lindhard response in the (a*, b*) reciprocal plane. fluctuations [18]. However, note that low temperature calculations reveal that for temperatures around 15 K and below, the Lindhard response is slightly c* dependent for the 5.2. Shape of the Lindhard response in the AO ground X = ClO4 and NO3 salts (see Figs. 8 and 13b). state However such dependence has not been detected for AO changes the electronic band dispersion and the the Lindhard response of (TMTSF)2 PF6 because of FS of the X = ClO4 and NO3 salts (see Figs. 3a the extremely weak band dispersion along the inter- and b, respectively). As the Lindhard response is layer c*-direction in the region of the Fermi surface very sensitive to the details of the band dispersion nested by the q1 , q2 and q3 wave vectors. Note that in the vicinity of the Fermi energy, the electron-hole 13 response calculated for the AO phase of these salts accompany its ordering, and with the reinforcement should somewhat differ from those calculated above of hydrogen bonds with one donor out of two, one TAO . For (TMTSF)2 NO3 below TAO both the stack stack out of two of the 2b super-cell is substantially tetramerization and the (1/2, 0, 0) anion potential deformed [17, 22]. As a consequence, the Lindhard perturb the electronic dispersion. As the Lindhard response (Fig. 9) is modified with respect to that of response above and at TAO is small at the X point the quenched sample (Figs. 4e and f). Note also (Fig. 12b) one expects a weak response of the stack that the Lindhard response calculated for the real AO to the (1/2, 0, 0) AO potential. Indeed the structural phase differs from the response analytically calculated refinement shows that the 2kF -tetramerization of the by the introduction of an anion gap in the band stack below TAO is weak [19]. Because of this structure [42]. According to the electronic structure observation and the fact that the first-principles of Fig. 3a, the Lindhard response has one inter-band calculation leads also to a small value of the AO and two intra-band components (Fig. 9). The two gap [22], one could argue that the semi-metallic state intra-band responses are weak, warped and shifted of (TMTSF)2 NO3 should not be so different from the from the average <2kF >= 0.5a* reciprocal position metallic one above TAO . The total Lindhard response (note that as the two donor stacks of the unit cell calculated below TAO (Figs. 14a and b) bears indeed are non equivalent there is a charge transfer between a resemblance with that calculated at TAO in the them). The inter-band response is the strongest. It non AO structure (Figs. 11c and d) if we take into has a more 1D type anisotropy than the response of account the effect of the folding due to the doubling the quenched sample (Figs. 5e and f). Note that of periodicity along the a-direction. As mentioned, even if one could expect that the band splitting due the Lindhard response of the AO phase is the sum to the AO suppresses the quasi-1D high temperature of three components corresponding to the inter-pocket electron-hole response, our calculation shows that this and two intra-pocket nesting processes. The intensity this is not really the case. The intensity maxima of the inter-pocket Lindhard response, which still per (TMTSF)2 ClO4 formula unit for the AO and non exhibits a pronounced quasi-1D anisotropy, is by far AO phases are found to be 2.24 and 2.72 eV−1 so the dominant one (Figs. 14c and 15). However, in that AO induces a 17 % decrease of the intensity contrast with the response for quenched samples this maxima. AO definitely has an effect on the intensity response does not exhibit a curved triangular plateau maxima although lower than for the (TMTSF)2 ClO4 of maxima (Figs. 11e and f) but a sharp peak located and, nevertheless, the calculated anion gap is larger, at (0.01, 0.29) in the (a’*, b*) reciprocal plane of the 2a ∼ 14 meV vs 8.9 meV. The HWHM values of the sharp superstucture (Fig. 16). Note however that the wave response lying on top of the broad one of Fig. 10c vector position of this peak is quite close to that of lead to inverse electron-hole coherence lengths in the the absolute q2 maximum of the response for quenched stack and inter-stack directions of 0.024a* and 0.11b’*, samples. Yet when the peak intensity of the maxima respectively. From these values it is estimated that are compared, 1.76 eV−1 per (TMTSF)2 NO3 formula the area of this response is a fraction f =2.1% of the unit for the total response in the AO phase vs 2.64 for area of the Brillouin zone of the superstructure, i.e. a the q2 maximum (table 1) for the quenched structure, fraction of ∼ 1 % of the high temperature Brillouin a noticeable decrease of 34 % due to the AO is found. zone area. This value is comparable to the fraction This means that although the anion gap is opened in f =1.2% calculated for quenched (TMTSF)2 ClO4 (see a region of the FS which is not associated with the table 1). best nesting [22], AO has a noticeable effect on the In summary, AO brings substantial changes into absolute value of the response. In fact although the the Lindhard response of both (TMTSF)2 NO3 and maximum response is found at practically the same (TMTSF)2 ClO4 . In both cases the system acquires wave vector, the whole response along this value is a stronger 1D character and the intensity maximum depleted: the triangular plateau practically disappears of the response decreases. Nevertheless, in contrast and the maximum decreases. The occurrence of these with commonly accepted ideas the magnitude of the changes are very difficult to appreciate by simple visual induced AO gap does not correlate with the decrease inspection of the Fermi surfaces and point out the need of the intensity maximum of the response. The for the Lindhard response study for fully understanding reason is that the wave vector of the AO modulation the FS nesting properties. determines the region of the FS where the gap is going The effect of the AO transition on the electronic to be opened but this region is not necessarily the structure of (TMTSF)2 ClO4 was described in early region with the better nesting properties. Finally, times as resulting from a simple band folding due to it is also worth noting that the three components the doubling of the b periodicity. This situation is in of the Lindhard response of relaxed (TMTSF)2 ClO4 fact more subtle because a shift of the ClO4 anions bear a resemblance with those of the blue bronze 14 which undergoes an inter-band charge density wave in contrast with (TMTSF)2 PF6 the inverse electron- −1 (CDW)/Peierls transition [40]. hole coherence length ξeh (q0 ) of (TMTSF)2 ClO4 becomes comparable to the inverse wavelength 1/λ2kF 5.3. Relationship with the SDW ground state = 1/2a (= 0.07Å−1 ) of the 2kF fluctuations only below ∼ 25 K in the quenched state. Above In contrast with AO transitions, the Lindhard response this temperature the 2kF fluctuations are thermally allows to analyze (within the RPA approximation) the incoherent. The 2kF SDW, 2kF CDW and 2kF bond SDW modulation observed below TSDW in quenched order wave (BOW) fluctuations, whose occurrence can (TMTSF)2 ClO4 and (TMTSF)2 NO3 . In this analysis be probed by different experimental techniques depend the SDW modulation wave vector is that bearing on the existence of 2kF electron-hole fluctuations. an absolute maximum of the Lindhard response. In In fact, three types of 2kF fluctuations have been quenched (TMTSF)2 ClO4 , the analysis of the NMR detected for (TMTSF)2 ClO4 . First, pretransitional data [43, 44] assuming a 0.5a* longitudinal component 2kF SDW fluctuations probed by NMR [47] which for the SDW modulation, leads to a 0.12b* transverse although enhanced between 30 K and 10 K never modulation component, which is close to the q0 = become critical (superconductivity is the ground state (0.50, 0.14) maximum of the Lindhard response that of relaxed samples). Second, quasi-1D 2kF BOW we find although it is slightly away from the q3 = fluctuations are detected by X-ray diffuse scattering (0.57, 0.07) absolute maximum at TSDW . The same below 150-200 K [41]. Furthermore, as shown in type of analysis of the NMR data for the AO phase of Fig. 6 the experimental inverse intra-chain correlation (TMTSF)2 NO3 [45, 46] gives an approximate 0.25b* −1 length of these fluctuations, ξBOW , nicely amounts to transverse modulation component which is close to the calculated inverse electron hole coherence length the position of the single maximum (0.01, 0.29) of −1 ξeh (q0 ). Although detected until low temperature [41, the Lindhard function that we calculate for the 2a 48], the 2kF BOW fluctuations, whose ξBOW ≈ superstructure. Note that the NMR modulation wave ξeh (q0 ) never exceeds λ2kF , always remain thermally vector is also close to the q2 = (0.52, 0.29) maximum incoherent. Note that the low temperature growth of the Lindhard response in quenched (TMTSF)2 NO3 . of 2kF BOW fluctuations could be inhibited by The Lindhard response calculation brings also the onset below 40 K of the (0, 1/2, 0) AO valuable information by showing that quenched fluctuations [16]. In this respect, the intensity of samples at low temperature exhibit a triangular 2kF BOW fluctuations at low temperatures when the plateau of maxima instead of a sharp maximum. (0, 1/2, 0) AO instability is suppressed is enhanced This feature implies the existence of a broad region for [(TMTSF)1−x (TMTTF)x ]2 ClO4 alloys when x of q SDW fluctuations which was already discussed increases [48, 49]. Third, 2kF CDW fluctuations have in ref. [6] for (TMTSF)2 PF6 . This finding can be also been detected by optical means [50]. quantified through the calculated q area of the plateau An intriguing result is that AO (TMTSF)2 ClO4 which we present in table 1 for the X = PF6 , ClO4 and exhibits a 2kF electron-hole response whose inter-band NO3 salts. Basically, there is one order of magnitude component shows a pronounced 1D character. At difference in this area when comparing the two salts first glance it is thus surprising that superconductivity X = PF6 and quenched X = ClO4 salts with the develops in this phase instead of a 2kF density wave quenched X = NO3 one. This difference qualitatively instability. This is all the more surprising that remains between relaxed X = ClO4 and NO3 samples. quenched (TMTSF)2 ClO4 , whose Lindhard response Below we point out that this order of magnitude has a less pronounced 1D character (see Figs. 11e and difference could have some significant consequences if f), exhibits a SDW ground state. Another puzzling but the Cooper pairing mechanism for superconductivity related observation is that (TMTSF)2 NO3 apparently involves SDW fluctuations. exhibits a SDW both for AO and non AO samples despite a substantial reduction of the response intensity 5.4. Relationship with the 2kF density wave instability maxima. (TMTSF)2 NO3 is already a very 1D system The scan of the quasi-1D Lindhard response in in the absence of AO as shown by the very small value the a-direction makes possible to obtain the inverse of f (0.1). The AO exacerbates such 1D character by electron-hole coherence length in the chain direction, transforming the small plateau of the non AO phase −1 ξeh . Fig. 6 presents thermal variations of this into a cusp anomaly in the AO phase so that the SDW quantity for non relaxed (TMTSF)2 ClO4 for several formation can be understood even if the maximum b* components. The smallest ξeh−1 values are obtained response intensity is reduced. Note also that the inter- −1 for the q0 and X points. ξeh (q0 ) of (TMTSF)2 ClO4 stack component for both the AO and non AO phases has a weaker thermal dependence than ξeh −1 (q0 ) of is very near to a commensurate value. The case of (TMTSF)2 PF6 (see Fig. 5 in ref. [6]). Additionally, (TMTSF)2 ClO4 is different because in the non AO 15 phase there is a very sizeable range of fluctuations fraction of electron-hole fluctuations. However since (f = 1.2). Even if the system becomes more 1D, a non superconductivity results from interference between negligible range of fluctuations is kept in contrast with the Peierls and the Cooper channels this picture is the case of the (TMTS)2 NO3 salt. It is possible that certainly oversimplified. A complete treatment can under such circumstances the decrease of the maximum be found in ref. [13] although the FS used was response intensity brought by the AO is enough to modelled using the orthorhombic approximation with suppress the SDW. a single nesting breaking term which according to our work is a simplified representation. Work including 5.5. Relationship with the mechanism of a multi-nesting scenario may provide important superconductivity hints to unravel the superconducting mechanism in Bechgaard salts. Finally, an important output of When the metallic state is restored by the suppression the present calculation is that the 2kF electron-hole under pressure of either the insulating SDW or the SDW fluctuations mediating the Cooper appear to be insulating AO ground states or by slowly crossing the spatially incoherent. temperature range of the AO transition of the X = ClO4 salt at ambient pressure, Bechgaard salts become 5.6. Some comments concerning the 2kF modulated superconducting with a TS slightly larger than 1 K [15]. AO transitions The only exception to this observation is pressurized (TMTSF)2 NO3 . When the SDW ground state of Finally let us remark that the electron-hole Lindhard this salt is removed, superconductivity is not detected response could play a role through the electron-anion down to 50 mK [51]. There is now accumulating coupling to settle a response of the organic stack to experimental evidence that superconductivity of the the anion potential. This should be especially the Bechgaard salts could be mediated by inter-stack case when the anion potential is inherently disordered attractive SDW fluctuations [4, 12]. One strong due to the location of non-centrosymmetric anions, argument in favor of such mechanism is that the such as X = ClO4 , ReO4 and NO3 , on inversion highest observed TS is close to the SDW boundary centres of the structure. Looking at the coupling of critical pressure and then when an extra pressure is this potential with the electron gas via the Lindhard applied TS decreases when SDW fluctuations diminish. response thus appears to be a pertinent way to Keeping in mind the relevance of quasi-1D physics analyse coupled lattice/electronic instabilities involved for the Bechgaard salts, interference between hole-hole in AO processes stabilizing the 2a periodicity or 2kF superconducting and electron-hole SDW fluctuations wavelength along the stack direction. In principle, should be treated on the same footing. This leads to an the enhanced electron-hole response at 2kF could help attractive SDW mediated d-wave pairing mechanism stabilizing a 2kF stack modulation as a response to a between holes located on neighboring stacks [13]. In staggered AO potential as observed for instance for the this context the intensity of the inter-chain exchange 180 K (1/2, 1/2, 1/2) metal-insulator AO transition in mechanism which determines TS should necessarily (TMTSF)2 ReO4 [52, 53]. However, the AO transition depend upon the importance of 2kF SDW fluctuations. of (TMTSF)2 ReO4 appears to be more subtle than The amount of such 2kF SDW fluctuations is related expected from a simple RPA analysis of the electron- to the f fraction of the dominant electron-hole q hole response because there is no absolute maximum fluctuations forming the triangular maxima of the of the Lindhard function at the M point (a*/2, b*/2), Lindhard response described in this work (see table 1). and no significant low temperature enhancement of the This fraction which is of the order of 0.7% in response at M to drive an electron-hole instability (see (TMTSF)2 PF6 at ambient pressure does not vary for example Figs. 5b and 12b). In this particular case significantly under 7 kbar pressure where SDW and the gain of electronic energy due to the gap opening superconductivity coexist (see ref. [6]). For the at TAO following the 2kF stack bond deformation is superconducting AO phase of (TMTSF)2 ClO4 we achieved through a first order phase transition and estimate an f value of 1%, which is the largest not through the second order Peierls one expected among the three salts. These fractions are about from an RPA analysis of the q dependent response. one order magnitude larger than that determined The (1/2, 1/2, 1/2) AO transition of (TMTSF)2 ReO4 in the present study in (TMTSF)2 NO3 . It is thus thus seems to be more likely driven by the gain of tempting to suggest that the magnitude of SDW anions entropy caused by their ordering below TAO , fluctuations fixed by the maximum of the electron- accompanied by an electronic energy gain caused by a hole response of (TMTSF)2 NO3 is not important 2kF stack BOW deformation which follows in a subtle enough to achieve superconductivity above 50 mK. way the staggered AO process via the anion-donor If this scenario is correct the presence of a multi- interaction (for more details see ref. [53]). Given the nested FS should be a prerequisite to promote a large dominant role played by the AO process, the Peierls 16 mechanism which assumes to be driven by the 2kF mediated superconductivity in (TMTSF)2 NO3 . The divergence of the electron-hole response and neglects maxima of the Lindhard response nicely account for the role of the lattice entropy, cannot be considered the modulation wave vector experimentally determined as the primary force behind the AO transition. With by NMR in the SDW ground state of these salts. It this ideas in mind the AO transition of (TMTSF)2 NO3 is noteworthy that the strength of the AO gap in provides useful hints concerning the coupling of these the (TMTSF)2 ClO4 and (TMTSF)2 NO3 salts does not instabilities. In this case the anion order-disorder correlate with the decrease of the maximum response transition is of second order and since the salt becomes intensity because the gap opening does not necessarily a semi-metal at TAO , no true electronic gap is opened. occurs in the region of the FS with the best nesting The 2kF Lindhard response of (TMTSF)2 NO3 does not properties. Finally, our first-principles calculation of exhibit an absolute maximum at the X point (a*/2, the Lindhard response of the X = ClO4 and NO3 0b*). One can argue that the difference between the salts reveals 3D effects at low temperature which X = ReO4 and NO3 salts may rely on the strength of are considerably more difficult to model in analytical the electron-anion coupling. The anion-donor coupling approaches. In a more general vein, this work and through O-Se interactions, reinforced by an anion shift the accompanying one on (TMTSF)2 PF6 [6] suggest towards the core of the donor in (TMTSF)2 ReO4 , is that modern DFT approaches provide an accurate and stronger than in (TMTSF)2 NO3 where the coupling unbiased tool to analyze the Lindhard response of is achieved through the establishment of hydrogen low-dimensional molecular conductors which may be bonds between the anion and the methyl groups at extremely useful when considering salts with complex the periphery of the donor. The presence of a strong crystal structures where analytical approaches are anion-donor coupling in (TMTSF)2 ReO4 is certainly difficult. responsible of the 1st order character of the (1/2, 1/2, 1/2) AO transition (this is a general feature Acknowledments of all Bechgaard salts exhibiting the (1/2, 1/2, 1/2) superstructure [14, 48]).This observation as well as This work was supported by Spanish MINECO the fact that the calculated Lindhard response never (the Severo Ochoa Centers of Excellence Program exhibits a maximum for the wave vector of the AO under Grants No. SEV-2017-0706 and SEV-2015- distortion strongly suggests that the AO mechanism 0496), Spanish MICIU, AEI and EU FEDER (Grants of the X = ReO4 salt should be interpreted in the No. PGC2018-096955-B-C43 and No. PGC2018- framework of a strong coupling theory [54]. 096955-B-C44), Generalitat de Catalunya (Grant No. 2017SGR1506 and the CERCA Programme), and the 6. Conclusions European Union MaX Center of Excellence (EU-H2020 Grant No. 824143). We thank C. Bourbonnais, P. The first-principles Lindhard electron-hole response of Foury-Leylekian, S. Tomic and W. Kang for useful the (TMTSF)2 X (X = ClO4 and NO3 ) Bechgaard discussions. salts, both in their quenched and relaxed (i.e. AO) ground states, has been calculated using their real References triclinic structure. This study, which complements a previous work for (TMTSF)2 PF6 [6] which does not [1] Yamaji K, Ishiguro T and Saito G 1998 Organic Superconductors (2nd edition, Springer-Verlag, Berlin) undergo an AO transition, shows that despite clear- [2] Lebed A (ed) 2008 The Physics of Organic Superconductors cut differences the evolution with temperature of the and Conductors (Springer, Berlin, Heidelberg) Lindhard response of the three salts has many points in [3] Giamarchi T 2004 Chem. Rev. 104 5037–5056 common. 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