This is the accepted version of the journal article: Guster, Bogdan; Pruneda, Miguel; Ordejon, Pablo; [et al.]. «Fermi surface electron-hole instability of the (TMTSF)2PF6 Bechgaard salt revealed by the first-principles Lindhard response function». Journal of Physics Condensed Matter, Vol. 32, Num. 34 (May 2020), art. 345701. DOI 10.1088/1361- 648X/ab8522 This version is available at https://ddd.uab.cat/record/277520 under the terms of the license J. Phys.: Condens. Matter Fermi surface electron-hole instability of the (TMTSF)2 PF6 Bechgaard salt revealed by 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. We report the first-principles DFT calculation of the electron-hole Lindhard response function of the (TMTSF)2 PF6 Bechgaard salt using the real triclinic low-temperature structure. The Lindhard response is found to change considerably with temperature. Near the 2kF spin density wave (SDW) instability it has the shape of a broad triangular plateau as a result of the multiple nesting associated with the warped quasi-one-dimensional Fermi surface. The evolution of the 2kF broad maximum as well as the effect of pressure and deuteration is calculated and analyzed. The thermal dependence of the electron-hole coherence length deduced from these calculations compares very well with the experimental thermal evolution of the 2kF Bond Order Wave correlation length. The existence of a triangular plateau of maxima in the low-temperature electron-hole Lindhard response of (TMTSF)2 PF6 should favor a substantial mixing of q-dependent fluctuations which can have important consequences in understanding the phase diagram of the 2kF SDW ground state, the mechanism of superconductivity and the magneto-transport of this paradigmatic quasi-one-dimensional material. The first-principles DFT Lindhard response provides a very accurate and unbiased approach to the low-temperature instabilities of (TMTSF)2 PF6 which can take into account in a simple way 3D effects and subtle structural variations, thus providing a very valuable tool in understanding the remarkable physics of molecular conductors. Keywords: Bechgaard salts, spin density waves, density functional theory, Lindhard response function 2 1. Introduction between the diffusive and coherent regimes decreases in the presence of sizeable intra-chain Coulomb repulsions At the simplest level of description the electronic because the latter tend to localize the electron wave properties of quasi-one dimensional (1D) conductors packets so that the effective inter-chain tunneling are governed by two antagonist interactions: inter- process should be reduced. This leads to a downward chain transfer integrals (t⊥ ) and inter-site Coulomb renormalized TCO , at which temperature there is a repulsions (Vij ). The former tend to delocalize carriers deconfinement transition from a 1D Luttinger liquid on neighboring chains leading to the formation of 2D to a 2D or 3D Fermi liquid [12]. However, beyond or 3D anisotropic metals. In contrast, the latter this qualitative picture, there are many important tend to localize the carriers on individual chains theoretical questions which should be elucidated in leading to the formation of 1D Luttinger liquids [1]. order to quantitatively describe the deconfinement The pertinence of this basic approach is illustrated transition [1]. These considerations should particularly by the isostructural series of quarter-filled Fabre apply to quasi-1D organic metals. In (TMTSF)2 PF6 , (TMTTF)2 X and Bechgaard (TMTSF)2 X organic for instance, the 1D to 2D deconfinement temperature salts where TMTTF is tetramethyltetrathiafulvalene, TCO at ambient pressure is found to occur at ∼ TMTSF is tetramethyltetraselenafulvalene and X is 100 K on the basis of the detection of a transverse a monovalent anion which in the present work is plasma edge when an electric field is applied along PF− 6 . The structure of (TMTSF)2 PF6 is shown in b [6] as well as from the change of slope of the c* Fig. 1 [2] where slightly dimerized zig-zag stacks of transverse electrical conductivity [13]. As shown in TMTSF donor molecules run along the a direction. Fig. 2, TCO substantially increases under pressure. The organic stacks form (a,b) layers which delimit In contrast with (TMTSF)2 PF6 , which exhibits intra- cavities along c* which are filled with monovalent stack metallic conductivity, a charge localization (X− ) anions. The temperature-pressure (T-P) phase occurs in (TMTTF)2 PF6 below Tρ ∼ 230 K at ambient diagram of Fabre and Bechgaard salts incorporating pressure, as proved by the occurrence of an activated centrosymmetric anions such as PF6 (see Fig. 2) shows intra-stack conductivity [14]. Due to this fact TCO is that 1D hole localization in the donor stacks dominates renormalized to 0 K [3]. When pressurized, Tρ for in the (TMTTF)2 PF6 Fabre salt at ambient pressure, (TMTTF)2 PF6 decreases (Fig. 2) and the behavior while the intra-stack metallic hole delocalization is typical of the Bechgaard salts is recovered [3]. the relevant aspect for the (TMTSF)2 PF6 Bechgaard According to the present DFT calculations the salt [3]. The generalized phase diagram of Fig. 2 transfer integrals for (TMTSF)2 PF6 at 300 K along is also controlled by pressure since the inter-stack the a, b and c* directions are ta ≈ 210 meV, tef f ≈ b transfer integrals, t⊥ , are enhanced under applied ef f 28 meV and tc∗ ≈ 0.8 meV, respectively (tb is the pressure. [4, 5, 6]. effective transfer integral along b according to the tight- Earlier studies [7, 8] clearly pointed out the binding model of Yamaji [15, 16]). As expected from difference between the inter-chain diffusive electron the crystal structure (Fig. 1), these values imply a transfer and coherent electron transfer mechanisms. highly anisotropic system. However, these transfer In the first case there is no coherence between Bloch integrals are rather similar to those calculated for functions located on neighbouring chains. Thus, (TMTTF)2 PF6 under the same conditions: ta ≈ 180 in that case one has a true 1D metal where the meV, tefb f ≈ 24 meV and tc *≈ 0.5 meV at 300 K. warping of the Fermi surface (FS) is not relevant Thus, the differences in the phase diagram between and there is no transverse plasma edge. In the the Bechgaard and Fabre salts should be attributed second case, the electronic wave function is 2D or to the presence of stronger Coulomb repulsions in 3D, delocalized over several chains, and t⊥ is a the TMTTF vs. TMTSF stacks. Such difference is relevant interaction. More precisely, for a non- clearly evidenced by the observation of 4kF Wigner- interacting electron gas the coherent regime occurs type charge density wave (CDW) instabilities in salts for kB T< t⊥ /π = kB TCO , where TCO is the 1D to with uniform TMTTF stacks but 2kF Peierls-type 2D/3D crossover temperature [8, 9] and below TCO a CDW instabilities in salts with uniform TMTSF transverse plasma edge should be detected. It has been stacks (kF is the Fermi wave vector of the donor predicted [10, 11] that the crossover transition TCO 3 (a) H F (b) C Se P c o a b b c o Figure 1. Crystal structure of the (TMTSF)2 PF6 Bechgaard salt: (a) Quasi-planar organic molecules stack along the a axis and are separated along the c* direction by PF− 6 anions. (b) Projection of the (TMTSF)2 PF6 structure along the stack direction. stacks 1D electron gas) [17]. The strong intra-stack Coulomb repulsions in the 1D electronic regime of (TMTTF)2 PF6 at high temperature lead to a 4kF hole localisation on the bonds of the donor zig-zag chains. Note that the charge localization on the donor dimers of the Fabre salts can be viewed as being stabilized by the response of the 4kF CDW instability of the TMTTF stacks to the 4kF potential set by the anion sublattice [18, 19, 20]. Charge localization induces a spin-charge decoupling so that the spin degrees of freedom drive anti-ferromagnetic (AF) or spin-Peierls (SP) low-temperature orders in the Fabre salts as shown in Fig. 2. In contrast, (TMTSF)2 PF6 is a room temperature metallic conductor whose electronic dimensionality changes from 1D to 2D at TCO ∼ 100 K at ambient pressure. Thus, below TCO the warping along b* of the open sheets at ±kF = ±a*/2 FS should be thermally relevant. In this context, it was suggested [16] that a 2kF FS nesting process could be responsible for the stabilization of the 2kF spin density wave (SDW) Figure 2. Generalized phase diagram for the Fabre (S-X) and insulating ground state detected below TSDW = 12 K Bechgaard (Se-X) salts where X is a centrosymmetric anion. The pressure scale corresponds to (TMTTF)2 PF6 . Vertical at ambient pressure. This mechanism was sustained arrows place (TMTTF)2 Br and (TMTSF)2 PF6 at atmospheric by the NMR measurement of a SDW modulation pressure in the phase diagram. CO, SP, AF, SDW and SC refer component of 0.20-0.24b* [21, 22] which agrees with to charge ordered, Spin-Peierls, antiferromagnetic, spin density the best nesting wave vector [23] of a FS modeled wave and superconducting ground states. QCP is a quantum critical point between the SP and AF ground states. Tρ indicates according to the tight-binding pseudo-orthorhombic the onset temperature for charge localization (Loc in the Figure) approximation of Yamaji [15, 16]. After this as detected in the conductivity measurements. TCO is the agreement was noticed, FS nesting effects became the crossover temperature from a 1D (in the stack direction (a) to a 2D (in the (a, b) donor layer) delocalized electron gas according key ingredient to elucidate the low-temperature (low- to ref [6]. On the right hand scale there is also a crossover to energy) physics of the Bechgaard salts. In particular, a 3D delocalization regime along c* through the anion cavities a significant decrease of TSDW was observed under at a temperature of the order of tc∗ . The value of the nesting pressure (Fig. 2) and, when the metallic state is breaking term t0b introduced in the text is also indicated. restored above ∼ 9.5 kbar, (TMTSF)2 PF6 becomes superconducting at TS ≈ 1.2 K [24]. This effect is ascribed to the suppression of the FS nesting when pressure enhances the magnitude of the so-called FS nesting breaking or (unnesting) term along b* (t0b , estimated to be 14-22 K at ambient pressure in 4 the pseudo-orthorhombic approximation, see below 2. Computational details in Sect. 3 and Fig. 2). The drastic effect of the t0b on the SDW phase diagram was quantitatively The DFT calculations [36, 37] were carried out using analyzed using an orthorhomic approximation of the a numerical atomic orbitals approach, which was electronic structure of Bechgaard salts [15, 16, 25, 26]. developed for efficient calculations in large systems Note also that when the low-temperature metallic and implemented in the Siesta code [38, 39]. We state is restored and when the inter-layer transport have used the generalized gradient approximation becomes coherent, a 2D to 3D electronic crossover is (GGA) to DFT and, in particular, the functional expected. The detection of a weak transverse Drude of Perdew, Burke and Ernzerhof [40]. Only the edge below 10 K in metallic (TMTSF)2 ClO4 Bechgaard valence electrons are considered in the calculation, salt at ambient pressure [27] corroborates that the 3D with the core being replaced by norm-conserving electronic crossover temperature is of the order of tc∗ scalar relativistic pseudopotentials [41] factorized in and occurs above the superconducting transition. the Kleinman-Bylander form [42]. The non-linear core- Remarkably, in the 3D metallic state when valence exchange-correlation scheme [43] was used for superconductivity vanishes under a high magnetic field, all elements. We have used a split-valence double-ζ a cascade of induced SDW ground states has been basis set including polarization functions [44]. The detected when the applied magnetic field along c* energy cutoff of the real space integration mesh was exceeds a threshold related to the FS nesting breaking 350 Ry. The Brillouin zone (BZ) was sampled term along this direction, t0c . These field induced SDW with the Monkhorst-Pack scheme [45] using grids of phases (FISDW) are due to the quantization of the (45×45×18) k-points to build the charge density. The 2D orbital movement in pockets remaining because Lindhard response function, of the imperfect FS nesting along b*. Thus, the X X fF (i (k)) − fF (j (k + q)) occurrence of FS nesting breaking terms is needed χ(q) = − , (1) i,j i (k) − j (k + q) to account for the occurrence of FISDW phases. A k quantitative analysis of the FISDW instability has where fF is the Fermi function and i (k) are the band been performed from the calculation of the Lindhard eigenvalues, was obtained from the computed DFT function in the presence of a magnetic field within band eigenvalues i (k) (i.e. in this work the TMTSF the orthorhomic approximation of the electronic HOMO-based bands of the system). The integral structure [16, 28, 29, 30]. However the interpretation over k-points of the BZ was approximated by a direct of additional observations, such as the reversal of sign summation over a dense, regular grid of points. As the of the quantum Hall effect between different FISDW Lindhard function is more sensitive to the accuracy phases, requires the existence of additional FS nesting of the BZ integration than the total energy, especially breaking terms which were introduced in an ad hoc in very anisotropic systems, and/or in the presence of manner in the calculation of the Lindhard function [31] hot spots in the band structure (e.g. saddle points and whose origin needs to go beyond the orthorhombic with the corresponding van Hove singularity in the approximation. DOS), the k-points grid used for its calculation must In summary, although FS nesting effects are the be more dense than in the standard self-consistent key ingredient to elucidate the low-temperature/low- determination of the charge density and Kohn-Sham energy physics of the Bechgaard salts, this quantitative energy. The calculations are done, nevertheless, using analysis has been performed using an orthorhombic or the eigenvalues obtained in the DFT calculation for pseudo-orthorhombic model of the electronic structure the coarser grid, and interpolating their values in the because of the difficulty to handle analytic calculations denser grid, using a post-processing utility available within the real triclinic structure of these salts. within the Siesta package. In this work, for the Present-day density functional theory (DFT)-based calculation of the Lindhard response function, the approaches are reliable enough to explore many aspects BZ was sampled using a grid of (400×400×16) k- of the electronic structure of molecular conductors [32, points. The partially filled bands were those taken 33, 34, 35]. Thus, here we revisit the FS nesting into account in the calculations. The DFT calculations mechanism using an unbiased DFT approach and the reported here have been performed using three different real triclinic 3D crystal structure of the Bechgaard crystal structures: (i) protonated (TMTSF)2 PF6 -H12 salt (TMTSF)2 PF6 . However as it is not easy to at 4 K [46] and ambient pressure, (ii) protonated clearly appreciate nesting effects from the simple visual (TMTSF)2 PF6 -H12 at 1.7 K and 7 kbar [47], and inspection of the FS we calculate and analyze the first- (iii) 98 % deuterated (TMTSF)2 PF6 -D12 at 4 K principles 3D electron-hole response function of this and ambient pressure [48]. Note that the thermal paradigmatic low-dimensional material and study in dependences of the Lindhard functions reported below detail its temperature and wave vector dependence. are due to the Fermi function in Eq. 1. 5 3. Electronic Structure a) T = 200 K Y M a) 0.25 b) Y 0.00 M X E - EF (eV) -0.25 -0.50 -0.75 X -1.00 X Y M Z b) T = 200 K Figure 3. Calculated band structure (a) and Fermi surface (b) b* for (TMTSF)2 PF6 using the 4 K crystal structure. The energy a* zero corresponds to the Fermi level. Γ = (0, 0, 0), X = (1/2, 0, X 0), Y = (0, 1/2, 0), Z = (0, 0, 1/2) and M = (1/2, 1/2, 0) in units of the triclinic reciprocal lattice vectors. The calculated band structure near the Fermi level for the 4 K crystal structure of (TMTSF)2 PF6 is shown in Fig. 3a. The two bands are almost exclusively built from the HOMO of TMTSF and because of c) T = 100 K the stoichiometry they contain one hole so that the b* upper band is half-filled. The total dispersion of a* these bands is 1.23 eV and the dimerization gap at X X is 96 meV. The two bands exhibit quite different dispersion along the b* inter-chain direction, the upper one being considerably flatter. These features agree with other reported band structures [49, 23, 5, 50]. Fitting the DFT results to the tight-binding model of Yamaji [15, 16] leads to the following values of the transfer integrals along the a, b and c* directions d) T = 50 K for the PF6 salt at 4 K: ta ≈ 256 meV, tef b f ≈ b* 38 meV and tc∗ ≈ 0.5 meV, respectively. Thermal a* contraction from 300 K to 4 K leads to increases of X 23% and 33 % for ta and tef b f and a decrease of 37% for tc∗ , respectively. In the same temperature range the variation is considerably smaller for the isostructural (TMTTF)2 PF6 Fabre salt: 13% and 11% increase for ta and tef f b , respectively, and 5% decrease for tc∗ . Another interesting quantity is the so-called e) T = 12 K nesting breaking (or unnesting) parameter [16, 15, 26] b* mentioned above, a* 1 t2 X t0b = √ b (2) 2 2 ta which measures how much the actual FS differs from the perfect nesting condition within Yamaji’s tight- binding model. Using the above transfer integrals t0b for (TMTSF)2 PF6 is estimated to be -1.9 meV (21.7 K) (see Fig. 2). Figure 4. 2D (a) and 3D (b) plots of the Lindhard function of Drude analysis of infrared measurements for (TMTSF)2 PF6 in the (a*,b*) reciprocal plane at 200 K. (c)-(e) (TMTSF)2 PF6 leads to ta ≈ 0.25 eV and tb0 ≈ 3D plots at 100 K, 50 K and TSDW = 12 K, respectively. 22 meV (at 30 K) (b0 is the intra-layer transverse direction perpendicular to a) [51]. ta is in excellent agreement with our calculated value, 0.256 eV. tb0 is slightly smaller than our tef b f value. Note however that the measurement of the conductivity anisotropy 6 a) 0.25 0.5 0.75 b) -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 T= 1K 3.0 3.0 T= 5K T= 10 K q1 T= 15 K T= 20 K 2.4 2.7 T= 30 K T= 40 K T= 50 K (a.u.) (a.u.) T= 60 K 1.8 2.4 T= 70 K T= 80 K T= 90 K T= 100 K 1.2 2.1 T= 110 K T= 120 K T= 130 K T= 140 K 0.6 1.8 T= 150 K 0.25 0.5 0.75 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 T= 160 K T= 180 K a*(a.u.) b*(a.u.) Figure 5. Thermal dependence of the: (a) longitudinal a* scans across the Lindhard function of (TMTSF)2 PF6 at 0.23 b* [i.e. (xa*, 0.23b*) scans] and (b) transverse b* scans at 0.5a* [i.e. (0.5a*, xb*) scans]. The red dashed line in the temperature scale indicates the temperature at which the single maximum leads to a tilted segment of maxima in (b). In part (a) the two kinks observed at 0.35a* and 0.63a* correspond to the crossing of the singularity lines mentioned in the text which separate the light blue and the dark blue regions in Fig. 8b. in the SDW phase of (TMTSF)2 PF6 leads to tb0 ∼ 1.6 35-28 meV [52] which is close to tef f b . Finally, the 1.4 -1 eh//(X) work of Zamborszky et al. [52] led to tc∗ ≈ 0.8 meV which is comparable to our tc∗ ≈ 0.5 meV 1.2 -1 eh//(q0) a (nm ) 1.0 1/ 2k -1 value. Using an average of the first-principles tef b f F values for 300 K and 4 K (33 meV), one estimates a 0.8 -1 1D-2D deconfinement temperature for non-interacting -1 0.6 electrons (kB TCO = tef f b /kB π) of 90 K, which is close BOW to the experimental determination of 100 K [6, 13]. 0.4 Finally, the energy 4t0b of remaining pockets due to 0.2 nesting breaking effects in the SDW phase can be 0.0 obtained from the difference between the direct SDW 0 50 100 150 200 250 300 350 gap from optical measurements (2∆= 100 K) [53] and twice the activation gap of the conductivity (2∆th = 41 Temperature (K) K [52]-48 K [54]). From this difference one obtains t0b ≈ Figure 6. Comparison of the thermal dependence of the inverse 14 K, which is not far from the 21.7 K value estimated −1 electron-hole coherence length ξehk calculated at the q0 and from Eq. 2. −1 X points with the inverse BOW correlation length ξBOW of The reasonable agreement between calculated and (TMTSF)2 PF6 measured in ref. [55]. experimentally estimated transfer integrals shows that DFT calculations provide reliable electronic structure parameters for (TMTSF)2 PF6 so that the Lindhard Lindhard function for (TMTSF)2 PF6 and later function calculations for this salt should be meaningful. (Sect. 5) these results will be compared with those The calculated FS for (TMTSF)2 PF6 at 4 K is shown based on the orthorhombic or pseudo orthorhombic in Fig. 3b. In the absence of anion ordering transitions approximation as well as with the experimental the shape of this FS is common to all Bechgaard salts information. Shown in Fig. 4 are intensity plots of with the only difference of slight variations in their the Lindhard function at 200 K (a) and (b), 100 warping. However these subtle differences are at the K(c), 50 K(d) and TSDW = 12 K (e) in the (a*, b*) heart of their different low-temperature physics and the reciprocal plane. As shown in these plots the response calculation of the 3D Lindhard function (see Sect. 4) is very anisotropic. In addition, we have verified that is the best quantitative approach to understand it. the response does not vary with the inter-layer c* wave vector component (this is not the case in other Bechgaard salts like the ClO4 and NO3 salts. [56]). 4. Analysis of the Lindhard function of At 200 K (Fig. 4a) the intensity of the response, (TMTSF)2 PF6 which varies significantly along the a* direction (see In this section we will explore the temperature also the a* scans in Fig. 5a) is centered around q1D = and wave vector dependencies of the first-principles 0.5a*. This chain component amounts to the 2kF wave 7 a) M number of the quasi-1D electron gas in a dimerized r r 3 2 r 1 T= 1K T= 80 K quarter-filled (hole) band structure. Fig. 4a shows that T= 5K T= 90 K the 200 K response is slightly modulated along the b* q2 T= 10 K T= 100 K T= 15 K T= 110 K direction. The magnification along b* in Fig. 4b reveals q1 T= 20 K T= 120 K T= 30 K T= 130 K the presence of two broad maxima at about ± 0.23b* q3 T= 40 K T= 140 K which are more clearly visualized on the b* scans of T= 50 K T= 150 K T= 60 K T= 160 K Fig. 5b. T= 70 K T= 180 K Upon cooling down to 100 K (Fig. 4c) there is a LEGEND X net increase in the intensity maxima of the Lindhard b) -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 response. Figs. 5a and b show that the intensity 3.0 increase concentrates around a broad maximum in q1 q2 ±q0 = (0.5, ±0.23). In addition, when T decreases 2.4 the global shape of the Lindhard response becomes (a.u.) sinusoidally modulated along a* (Fig. 4c). As a 1.8 consequence, the response along a* remains broader at the X (0.5, 0) and M points (0.5, 0.5) and sharpens in- 1.2 between these points near q0 . This is quantitatively 0.6 illustrated by Fig. 6 giving the thermal dependence -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 of the half-width at half-maximum (HWHM) of the r1(a.u.) response function along a* at the X and q0 points. -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 This quantity corresponds to the inverse electron-hole c) 3.0 coherence length in the chains direction (1/ξeh ) which q3 q1 will be discussed and confronted with the experimental 2.4 information in Sect. 5. Upon cooling 1/ξeh (X) stays (a.u.) roughly constant while 1/ξeh (q0 ) gently decreases but 1.8 does not vanish at low temperature. Finally, note 1.2 also that the HWHM of the q dependence of the Lindhard function along b* (Fig. 5b) leads to an inverse 0.6 electron-hole inter-stack transverse coherence length of -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 about 0.12-0.09 Å−1 between 200 K and 100 K. The r3(a.u.) inverse of this quantity, which is nearly constant when 0 0.1 0.2 0.3 0.4 0.5 varying the temperature, is of the order of the inter- d) 3.0 q3 q2 stack distance b. This means that, since the electron- 2.4 hole pairs remain basically confined on the stack, it can be reasonably considered that the high-temperature (a.u.) 1.8 electron-hole response is essentially 1D. However, this picture must be modified below 100 K because the shape 1.2 of the Lindhard function drastically changes. Fig. 4d shows a 3D plot of the intensity of 0.6 0 0.1 0.2 0.3 0.4 0.5 the Lindhard function at 50 K. The asymmetric r2(a.u.) development of arcs of intensity can be clearly distinguished. More precisely, the maximum of the Figure 7. Thermal dependence of the scans across q1 -q2 (b), q3 -q1 (c) and q2 -q3 (d) lines of the Lindhard function electron-hole response adopts a boomerang-like shape for (TMTSF)2 PF6 . In (a) we present a legend for the scans limited by the q1 ∼ q0 , q2 and q3 wave vectors whose directions where ~ q1 − q~2 |))a∗ + b∗ , ~ r1 = (1 − tan(|~ r2 = −(1 + components, schematically represented in Fig. 7a, will ∗ ∗ q3 − q~2 |))a + b and ~ tan(|~ r3 = −(1 + tan(|~ q3 − q~1 |))a∗ + b∗ . be carefully scrutinized below. The change of shape of Positive values are designated for q points above the Γ-X direction. Note the tilted plateau which continuously develops the Lindhard response is illustrated by the diagonal q1 - upon cooling in (b) and (c). The red dashed line in the q2 and q1 -q3 scans crossing q1D shown in Figs. 7b and temperature scale indicates the temperature at which the single c, respectively. It can be clearly seen the progressive maximum leads to a tilted segment of maxima. The kinks growth of intensity singularities at q2 and q3 , away from observed for q1 -q2 ∼ 0 and q3 -q1 ∼ 0 and 0.4 correspond to the crossing of singularity lines mentioned in the text which separate 0.5a*, which upon cooling give rise to a continuum the light blue to dark blue regions in Fig. 8b. of intensity maxima transforming at low-temperature into a tilted segment joining q1 . The q2 -q3 scans (Fig. 7d), which are out of the q1D direction, clearly illustrate the progressive development of the q2 and q3 8 Y M Y Y M Y Y M Y a) b) c) q2 (a.u.) q1 q3 X X X Figure 8. Magnified section of the Lindhard functions at 5 K for 98% deuterated (TMTSF)2 PF6 -D12 at Patm (a), protonated (TMTSF)2 PF6 -H12 at Patm (b), and protonated (TMTSF)2 PF6 -H12 at 7 kbar pressure (c). The X→ Γ and M→Y arrows are only used to indicate the associate directions. In (b) the q1 , q2 and q3 points used in the text are labelled. maxima which limit the arc of intensity depicted in to 2.72 a.u. in (TMTSF)2 PF6 -H12 at Patm (i.e.∼ 3%) Fig. 7a. These maxima begin to be detected below and then to 2.64 a.u, in (TMTSF)2 PF6 -H12 at 7 kbar 100 K (Fig. 7d). Upon further cooling the q2 and q3 (i.e. an additional ∼ 3%). maxima transform into singularities at q2 = (0.52, 0.30) and q3 = (0.53, 0.14) reciprocal positions. As a result, a 5. Discussion tilted plateau of intensity maxima develops around 15- 20 K and the Lindhard function exhibits well defined 5.1. Shape of the Lindhard function and its relation singularities at the q0 , q2 and q3 wave vector positions with the topology of the Fermi surface when the SDW transition occurs (TSDW = 12 K, see Fig. 4d). At TSDW the maximum of the Lindhard The main result of the present work is the first- response occurs at the q0 wave vector. However around principles determination of the Lindhard response of 10 K the q0 maximum shifts to q1 = (0.48, 0.23) the (TMTSF)2 PF6 Bechgaard salt based directly on (Fig. 5a) and the absolute maximum of intensity of the real triclinic low-temperature crystal structure. At the Lindhard response shifts to q2 (see Figs. 7b). high temperature (i.e. above ∼100 K) this function Fig. 8b represents the curved and tilted triangu- basically exhibits a line maximum at q1D = 0.5a* lar plateau of maxima of the Lindhard response de- (Figs. 4a, 4b and 5a), which corresponds to the limited by the q1 , q2 and q3 wave vectors in proto- 2kF critical wave vector of the 1D band structure, nated (TMTSF)2 PF6 -H12 at 5 K and ambient pres- as expected for an electron-hole instability of 1D sure. This is compared with the 5 K Lindhard func- conductors [16, 26]. However, this line of maxima tion calculated under the same conditions for deuter- is found to be slightly modulated in intensity along ated (TMTSF)2 PF6 -D12 (Fig. 8a) and protonated b* and exhibits a broad maximum at q0 (Fig. 5b) (TMTSF)2 PF6 -H12 at 7 kbar (Fig. 8c) [47] respec- which corresponds to the best nesting wave vector of tively. The main differences between these Lindhard the thermally broadened warped FS (see below). The functions (see also Fig. 9 where we directly plot the shape of the Lindhard response changes appreciably differences) are: below ∼100 K (Fig. 4) with the onset of a q2 -q0 -q3 (a) At ambient pressure the intensity of the curved arc of maxima, where the a* component of triangular plateau maximum decreases by a few the q2 and q3 singularities differs from q1D = 0.5a*. per cent from deuterated (TMTSF)2 PF6 -D12 to The intensity of the q2 and q3 singularities grows protonated (TMTSF)2 PF6 -H12 , and upon cooling (Fig. 7d) and below 20-15 K leads to (b) The intensity of the triangular plateau max- the appearance of a tilted plateau of maxima with imum for protonated (TMTSF)2 PF6 -H12 decreases a triangular or, more accurately, a boomerang shape from ambient pressure to 7 kbar. (Fig. 8). A triangular-like maxima of the Lindhard However, Figs. 8 and 9 show that the position of function at 0 K was previously obtained analytically the q1 , q2 and q3 singularities do not appreciably shift through inclusion of a nesting breaking term t0b in within 0.01a* and 0.01b*. Nevertheless, the intensity the quasi-1D electron dispersion built from single of the q2 singularity for which the response is maximal sinusoidal intra- (ta ) and inter-chain (tb ) couplings and at 5 K decreases from 2.81 a.u. in (TMTSF)2 PF6 -D12 assuming an orthorhombic lattice (when tb = 0 the 9 protonated- deuterated pressure_7kbar- deuterated pressure_7kbar- protonated Y M Y Y M Y Y M Y a) b) c) X X X Figure 9. Magnified section of the differences in the Lindhard functions at 5 K between 98% deuterated (TMTSF)2 PF6 -D12 at Patm and protonated (TMTSF)2 PF6 -H12 at Patm (a), 98% deuterated (TMTSF)2 PF6 -D12 at Patm and protonated (TMTSF)2 PF6 -H12 at 7 kbar pressure (b), protonated (TMTSF)2 PF6 -H12 at Patm and 7 kbar pressure (c). The X→ Γ and M→Y arrows are only used to indicate the associate directions. These sections clearly show the sinusoidal singularity lines whose intersection occurs at the q1 , q2 and q3 singularity wave vectors. in the Lindhard response below 100 K is consistent with the 1D to 2D electronic dimension crossover observed by optics and transport measurements at TCO ∼ 100 K [6, 13]. In this context the warping of the open FS becomes progressively relevant upon cooling below TCO . In the case of a mainly sinusoidal component of the dispersion of the open FS, the best nesting condition should lead to a main singularity in the Lindhard response. However in our case the best nesting wave vector q0 is already detected above TCO so that TCO appears to be more likely at the temperature below which two additional nesting wave vectors q2 and q3 begin to be detected. Fig. 10 presents the first-principles FS surface of (TMTSF)2 PF6 calculated using the 4 K crystal structure. This figure shows that the Figure 10. Schematic illustration of the three nesting vectors warped open FS exhibits three good nesting wave at the origin of the arcs of the Lindhard function. vectors for the q1 , q2 and q3 wave vectors which are associated with singularities in the Lindhard function. Lindhard function diverges at 0 K) [25, 29, 57]. To have The occurrence of several FS nesting processes means the boomerang shape additional inter-chain couplings that the FS has definitely not a regular sinusoidal must be introduced for either the orthorhomic [58] shape. This is the reason why several additional inter- and triclinic [59] lattices. These calculations also show chain interactions having different kb dependencies that the q0 (which shifts to q1 upon cooling), q2 and must sizeably contribute to the band dispersion. q3 singularities are the crossing point of sinusoidal In other words to account for the first-principles lines of singularities whose analytical expression can be results one must go beyond the pseudo-orthorhombic obtained [58, 59]. Such sinusoidal lines of singularities approximation [15, 16] which is based on a sinusoidally are clearly revealed in our low-temperature calculations shaped FS. This process is carried out in an unbiased (Figs. 4e and Fig. 8) and in the intensity differences of way in the DFT calculation. Fig. 9. However, no ad hoc FS nesting breaking terms are introduced in the first-principles Lindhard function which only relies on the actual crystal structure of the material. The progressive growth of a plateau of singularities 10 5.2. Relationship with the 2kF density wave incoherent). An interesting question is to understand instabilities why ξeh remains small for a quite large temperature range. This could be due to the presence of many The electron-hole Lindhard response of a quasi- deviations to a single FS nesting process (Fig. 10). 1D metal exhibits a maximum for the 2kF chain When the electron-phonon drives the Peierls component wave vector of the associated 1D electron instability, as found in the blue bronze [65] and in the gas. Divergence of this maximum upon cooling transition metal trichalcogenides [66], it is possible to is the prerequisite to achieve a 2kF density wave observe the critical growth of ξBOW with respect to a instability. In the presence of a sizeable electron- sizeably T dependent ξeh which exceeds λ2kF [67] for phonon coupling a 2kF charge density wave (CDW) a large temperature range. The fact that ξBOW ≈ ξeh instability is generated together with, when the phonon in the data of Fig. 6 proves that the electron-phonon field modulates the bond distances, a 2kF bond order coupling is not critical for (TMTSF)2 PF6 . In contrast, wave (BOW) instability. In the presence of exchange the on-site electron-electron coupling should be critical interactions due to sizeable on-site electron-electron to achieve the SDW ground state of (TMTSF)2 PF6 repulsion, a 2kF SDW instability can occur [15]. if one expects a critical growth of ξSDW over ξeh for Surprisingly, (TMTSF)2 PF6 concomitantly exhibits temperatures lower than 50 K. This is apparently what 2kF CDW/BOW and 2kF SDW instabilities at high- is observed by NMR below ∼ 25 K with the critical temperature. growth of gaussian fluctuations [61, 60]. Apparently Weak 1D SDW fluctuations have been detected the enhancement of ξSDW over ξeh with temperature by NMR below 200 K in (TMTSF)2 PF6 [60]. These is small because the SDW pseudo-gap is only detected fluctuations, which are less correlated than those a few K above TSDW [63]. measured in the Fabre salts, grow upon cooling. NMR measurements show also the development of a (3D?) regime of critical fluctuations below ∼ 25 K [60, 61]. 5.3. Relationship with the 2kF SDW ground state However, specific heat measurements provide evidence In the weak coupling limit and random phase for a very narrow regime of 3D fluctuations starting approximation (RPA) the maximum of the Lindhard 0.3 K above TSDW = 12 K [62]. Note also that optical function selects the critical wave vector of the SDW measurements provide evidence for the development modulation which develops at TSDW [16, 68]. Our of a (3D ?) pseudo-gap below 15 K, few K above first-principles calculation gives a maximum at 12 K for TSDW [63]. Quasi-1D 2kF BOW fluctuations are q0 = (0.5, 0.23), which accounts for the experimental detected by X-ray diffuse scattering measurements determination. The SDW modulation wave vector below about 175 K [64]. In contrast with the SDW has been experimentally determined by fitting the fluctuations, the intensity of the 2kF BOW fluctuations NMR line-shape for (TMTSF)2 PF6 assuming that the saturate below about 100 K (i.e. TCO ) and vanish a* component of the modulation is q1D = 2kF = 0.5 below 50 K [20, 55]. This vanishing can be understood a*. Using this assumption it is found that the b* as resulting from the increase of the lattice rigidity transverse SDW modulation component lies in the caused by the strengthening of the linkage of the PF6 range 0.20± 0.05 b* [21] / 0.24± 0.03 b* [22]. Both anions with the methyl terminal groups of the TMTSF determinations correspond, within experimental errors, through hydrogen bonding [48]. to the present Lindhard function maximum q0 = (0.5, X-ray diffuse scattering experiments allow to 0.23) at TSDW = 12 K. measure the inverse of the correlation length of the However, the first-principles calculation brings −1 BOW fluctuations in the chain direction, ξBOW . The more information. In particular, it shows that for thermal dependence of this quantity is plotted in temperatures below 20-15 K (i.e. slightly above −1 Fig. 6. This figure clearly shows that ξBOW behaves as TSDW ) the Lindhard response of (TMTSF)2 PF6 the inverse intra-chain electron-hole coherence length exhibits a sizable q2 - q0 / q1 - q3 triangular maximum. of the Lindhard function measured at the q0 point, Furthermore, the maximum of the response shifts from −1 ξeh (q0 ). This quantitative agreement means that the q0 to q1 and then to q2 around 10 K. This means that first-principles Lindhard response brings very relevant within the RPA scenario the SDW modulation wave information concerning the density wave instability vector should shift from q0 to q1 and then to q2 when of the Bechgaard salts. Fig. 6 also shows that TSDW decreases under pressure. The shift from q0 −1 the inverse electron-hole coherence length ξeh (q0 ) to q1 has been predicted from calculations performed decreases weakly upon cooling. It is only below 50 within the orthorhombic approximation [16, 25, 29]. K that the inverse of this quantity exceeds λ2kF = 2a Our calculation of the Lindhard response at 5 K for and thus that the 2kF electron-hole modulation wave is (TMTSF)2 PF6 under 7 kbar predicts a q2 absolute spatially defined (above this temperature when ξeh (q0 ) maximum and thus, within the RPA approximation, < λ2kF the electron-hole excitations are thermally the stabilization of a q2 SDW modulation at 7 kbar 11 (where TSDW = 4 K). contributing to the triangular shape of the maxima of In addition, the presence of a wide range of the Lindhard function [25, 58] do not significantly vary close maxima means that different sets of modulations under pressure. Another argument corroborating this could be simultaneously (locally) present without any finding is that the first-principles calculation leads to prohibitive cost in magnetic energy. In other words, the estimation of a quite weak increase of the nesting the SDW ground state could be very fallible, which breaking term t’b as given by Eq. 2, i. e. from 35.6 meV provides a suitable explanation for the origin of the in (TMTSF)2 PF6 -D12 to 36.8 meV in (TMTSF)2 PF6 - glassy behavior observed inside the SDW ground H12 at Patm and then to 37.3 meV in (TMTSF)2 PF6 - state [69]. Along this line it is interesting to remark H12 at 7kbar. Thus, the present data suggests that that the commensurate modulation wave vector qc = the strong reduction of the SDW conductivity gap (1/2, 1/4) belongs to the triangular plateau close under pressure [52] should be primarily attributed to to the q1 -q2 singularity line of intensity maximum. the decrease of the optical direct SDW gap related to Thus the finding in ref. [70] of a local commensurate intra-site Coulomb repulsion more than to the increase lock-in modulation in the 4 K sub-SDW phase of of the nesting breaking term t’b , as previously assumed (TMTSF)2 PF6 can be simply rationalized by the in the literature [15, 16, 53, 25]. present calculation. The presence of a triangular q2 - q0 / q1 - q3 6. Concluding remarks plateau of maxima in the electron-hole response due to a continuous range of FS nesting wave vectors, means The thermal dependence of the electron-hole Lindhard that (TMTSF)2 PF6 should be subject to a large panel response of the Bechgaard salt (TMTSF)2 PF6 has been of 2D magnetic spatial fluctuations. Considering them calculated from first-principles using its real triclinic on the same footing means that one must go beyond structure. This calculation can be contrasted with a simple RPA approximation of the standard Peierls earlier calculations based on the quasi-orthorhombic scenario which selects a single 2kF critical wave vector approximation of the electronic structure. Contrary for which the Lindhard function diverges in order to to the assumptions in the literature concerning the describe the SDW ground state. In particular a more existence of a single q singularity, our calculation elaborate mechanism, taking into account the q2 -q1 - reveals the presence of a low temperature triangular q3 triangular set of fluctuations, could be responsible plateau of maxima of the Lindhard response (Fig. 8). for the 1st order nature of the SDW transition of This plateau originates from a continuum of FS nesting (TMTSF)2 PF6 [71], since it has been established that wave vectors (Fig. 10) which in the first-principles enhanced fluctuations [72, 73] should induce a first calculation only depend on the details of the low- order transition. temperature crystal structure. Indeed our calculation Another feature which can be discussed in the predicts that at Patm the absolute maximum of the context of the present calculation concerns the (T,P) response at TSDW accounts for the experimental SDW phase diagram. In this respect note that (i) observation of the q0 SDW modulation. However the (TMTSF)2 PF6 -D12 is equivalent to the application of calculation also predicts a shift of the SDW modulation a negative pressure of 5 kbar in (TMTSF)2 PF6 -H12 wave vector from q0 to q1 and then to q2 . The at Patm [48] and, (ii) there is a strong reduction of calculated Lindhard function at 7 kbar shows that TSDW to about 4 K and coexistence between the SDW the q2 SDW modulation should be stabilized and modulation and superconductivity in (TMTSF)2 PF6 - thus could coexist with superconductivity instead of H12 at 7 kbar [74]. The finding in our calculations q0 -q1 modulations as previously assumed [74]. The that the q2 maximum of the Lindhard response at existence of a large plateau of maxima implicates 5 K decreases by 3% from (TMTSF)2 PF6 -D12 to that the (TMTSF)2 PF6 Bechgaard salt is subject to (TMTSF)2 PF6 -H12 at Patm and by the same amount considerable q2 -q1 /q0 -q3 dependent SDW fluctuations in (TMTSF)2 PF6 -H12 from Patm to 7 kbar shows at low-temperature through all the pressure range of that within the RPA approximation, the decrease of the experiments, a feature that was previously ignored TSDW under pressure can be simply rationalized by in the literature. the decrease of the Lindhard response. However at Although the mechanism of superconductivity the presence stage of our investigation we cannot still in organic conductors is still the object of a provide a clear-cut explanation of the weak decrease long debate [75], experimental evidence has been of the Lindhard response because the triangular accumulated in favor of a pairing mechanism due to shaped maxima is not substantially modified from incipient SDW fluctuations in the Bechgaard salts [76]. (TMTSF)2 PF6 -D12 to (TMTSF)2 PF6 -H12 at 7 kbar Theoretical analysis has recently pointed out that (Figs. 8 and 9). This means that nesting breaking superconductivity should emerge from an extended effects and additional components to the FS warping quantum critical region of the phase diagram which 12 is a function of nesting deviation of the FS [77, 78]. this work. In this context the real shape of the electron-hole instability in the P-T region near the superconducting References transition TS , together with its relationship with the FS nesting properties, as directly given in the first- [1] Giamarchi T 2004 Chem. Rev. 104 5037–5056 [2] Thorup N, Rindorf G, Soling H and Bechgaard K 1981 Acta principles approach, should be incorporated to improve Crystallogr. B 37 1236–1240 theoretical treatments. [3] Bourbonnais C and Jérome D 2008 Interacting Electrons The modification of the Lindhard response under in Quasi-One-Dimensional Organic Superconductors The magnetic field, and the presence of FS breaking Physics of Organic Superconductors and Conductors ed Lebed A (Springer, Berlin, Heidelberg) chap 12, p effects are important ingredient to calculate a FISDW 357–412 ISBN 978-3-540-76667-4 diagram [16, 57]. The FISDW phase diagram of the [4] Jacko A C, Feldner H, Rose E, Lissner F, Dressel M, Valenti literature has been constructed for an orthorhombic R and Jeschke H O 2013 Phys. Rev. B 87 155139 model taking into account one nesting breaking term [5] Auban-Senzier P, Jérome D, Doiron-Leyraud N, de Cotret S R, Sedeki A, Bourbonnais C, Taillefer L, Alemany P, t’b . Our results show that one must go beyond Canadell E and Bechgaard K 2011 J. Phys.: Condens. this approximation. Only a recent publication builds Matter 23 345702 the FISDW phase diagram by taking into account [6] Pashkin A, Dressel M, Hanfland M and Kuntscher C A 2010 the various inter-chain interaction of the triclinic Phys. Rev. B 81 125109 [7] Jérome D and Schulz H J 1982 Adv. Phys. 31 299–490 lattice [59]. Note that these inter-chain components, [8] Weger M 1978 J. Phys. Colloques 39 C6–1456 – C6–1465 which are responsible of subtle modification of the [9] Emery V J 1983 J. Phys. Colloques 44 C3–977 – C3–982 shape of the FS, must be taken into account to [10] Bourbonnais C 1985 Mol. Cryst. Liq. Cryst. 119 11–18 explain subtle physical effects such as the sign reversal [11] Caron L G and Bourbonnais C 1988 Synth. Met. 27 A67–A74 of the quantum Hall effect [31]. The shape of the [12] Biermann S, Georges A and Lichtenstein A 2001 Phys. Rev. FS of the Bechgaard salts also induces a quite large Lett. 87 276405 range of additional physical phenomena which can be [13] Moser J, Gabay M, Auban-Senzier P, Jérome D, Bechgaard K and Fabre J M 1998 Eur. Phys. J. B 16 39–46 revealed by the so-called rapid magnetic oscillation [14] Coulon C, Delhaes P, Flandrois S, Lagnier R, Bonjour E phenomena [16]. Such effects have been recently and Fabre J M 1982 J. Phys. France 43 1059–1067 reviewed and analyzed within the orthorhombic band [15] Yamaji K 1983 J. Phys. Soc. Jpn. 51 2787 dispersion approach for which nesting deviations can [16] Yamaji K, Ishiguro T and Saito G 1998 Organic Superconductors (2nd edition, Springer-Verlag, Berlin) be simply modeled [79]. Our study shows that the [17] Pouget J P 2015 Physica B 460 45–52 FS of the real triclinic structure is considerably more [18] Emery V J, Bruinsma R and Barisic S 1982 Phys. Rev. Lett. complex than that previously considered so that how 48 1039–1043 such effects can modify the present theory of rapid [19] Medjanik K, Chernenkaya A, Nepijko S A, Ohrwall G, Foury-Leylekian P, Alemany P, Canadell E, Schonhense magnetic oscillations is a challenging question. G and Pouget J P 2015 Phys. Chem. Chem. Phys. 17 Although there is no doubt that the model 19202–19214 calculations will continue to provide important insights [20] Pouget J P, Alemany P and Canadell E 2018 Mater. Horiz. into the remarkable low-temperature physics of the 5 590–640 [21] Delrieu J M, Roger M, Toffano Z, Moradpour A and Bechgaard salts, the first-principle approach provides Bechgaard K 1986 J. Phys. France 47 839–861 an accurate and unbiased complement to these [22] Takahashi T, Maniwa Y, Kawamura H and Saito G 1986 J. approaches. It can also take into account in a simple Phys. Soc. Jpn. 55 1364–1373 way the 3D effects [56] which are considerably more [23] Ducasse L, Abderraba M and Gallois B 1985 J. Phys. C: Solid State Phys. 18 L947–L951 difficult to model in analytical approaches. [24] Jérome D, Mazaud A, Ribault M and Bechgaard K 1980 J. Physique Lett. 41 95–98 [25] Hasegawa Y and Fukuyama H 1986 J. Phys. Soc. Jpn. 55 Acknowledments 3978 [26] Montambaux G 1988 Phys. Rev. B 38 4788–4795 This work was supported by Spanish MINECO [27] Henderson W, Vescoli V, Tran P, Degiorgi L and Gruner G (the Severo Ochoa Centers of Excellence Program 1999 Eur. Phys. J. B 11 365–368 under Grants No. SEV-2017-0706 and SEV-2015- [28] Gor’kov L P and Lebed A G 1984 J. Phys. Lett. 45 433–440 [29] Montambaux G, Héritier M and Lederer P 1985 Phys. Rev. 0496), Spanish MICIU, AEI and EU FEDER (Grants Lett. 55 2078–2081 No. PGC2018-096955-B-C43 and No. PGC2018- [30] G. Montambaux, Ph. D. Thesis, Université de Paris-Sud 096955-B-C44), Generalitat de Catalunya (Grant No. (1985). 2017SGR1506 and the CERCA Programme), and the [31] Zanchi D and Montambaux G 1996 Phys. Rev. Lett. 77 366–369 European Union MaX Center of Excellence (EU- [32] Alemany P, Pouget J P and Canadell E 2014 Phys. Rev. B H2020 Grant No. 824143). C. Bourbonnais and G. 89 155124 Montambaux are thanked for a critical reading of the [33] Altmeyer M, Valentí R and Jeschke H 2015 Phys. Rev. B manuscript and D. Jérome for useful discussions before 91 245137 [34] Lazić P, Pinterić M, Rivas Góngora D, Pustogow A, 13 Treptow K, Ivek T, Milat O, Gumhalter B, Doslic̆ N, [69] Monceau P 2012 Adv. Phys. 61 325–581 Dressel M and Tomić 2018 Phys. Rev. B 97 245134 [70] Nagata S, Misawa M, Ihara Y and Kawamoto A 2013 Phys. [35] Zhou B, Ishibashi S, Ishii T, Sekine T, Takehara R, Rev. Lett. 110 167001 Miyagawa K, Kanoda K, Nishibori E and Kobayashi A [71] Clark W G, Hanson M E, Wong W H and Alavi B 1994 2019 Chem. Comm. 55 3327–3330 Physica B 194–196 285–286 [36] Hohenberg P and Kohn W 1964 Phys. Rev. 136 B864–B871 [72] Bak P 1980 Theory of elastic phase transitions and [37] Kohn W and Sham L J 1965 Phys. Rev. 140 A1133–A1138 soft acoustic modes Modern Trends in the Theory of [38] Soler J M, Artacho E, Gale J D, García A, Junquera Condensed Matter ed Pekalski A and Przystawa J A J, Ordejón P and Sánchez-Portal D 2002 J. Phys.: (Lecture Notes in Physics vol. 115, Springer-Verlag, Condens. Matter 14 2745–2779 Berlin Heidelberg) p 414–431 [39] Artacho E, Anglada E, Diéguez O, Gale J D, García A, [73] More explicitly, if for simplicity one only considers Junquera J, Martin R M, Ordejón P, Pruneda J M, fluctuations associated to the nearly equal maxima Sánchez-Portal D and Soler J M 2008 J. Phys.: Condens. q0 and q1 of the Lindhard function at TSDW for Matter 20 064208 (TMTSF)2 PF6 , the free energy Landau development [40] Perdew J P, Burke K and Ernzerhof M 1996 Phys. Rev. associated to the complex order parameters η1 and η2 , Lett. 77 3865–3868 corresponding to the q0 and q1 modulations, respectively, [41] Troullier N and Martins J L 1991 Phys. Rev. B 43 can be schematically written as F = a1 | η1 |2 + a2 | η2 |2 1993–2006 + u1 | η1 |4 + u2 | η2 |4 + w | η1 |2 | η2 |2 . The [42] Kleinman L and Bylander D M 1982 Phys. Rev. Lett. 48 treatment of F using renormalization group methods 1425–1428 lead to a regime where the fourth order terms are no [43] Louie S G, Froyen S and Cohen M L 1982 Phys. Rev. B 26 longer positive definite, which conducts to a first order 1738–1742 transition. [44] Artacho E, Sánchez-Portal D, Ordejón P, García A and [74] Vuletic T, Auban-Senzier P, Pasquier C, Tomic S, Jérome Soler J M 1999 Phys. Stat. Solidi (b) 215 809–817 D, Héritier M and Bechgaard K 2002 Eur. Phys. J. B 25 [45] Monkhorst H J and Pack J D 1976 Phys. Rev. B 13 319–331 5188–5192 [75] Ardavan A, Brown S, Kagoshima S, Kanoda K, Kuroki M, [46] Gallois B, Gaultier J, Hauw C, Lamcharfi T and Filhol A Mori H, Ogata M, Uji S and Wosnitza J 2012 J. Phys. 1986 Acta Crystallogr. B 42 564–575 Soc. Jpn. 81 011004 [47] Gallois B, Gaultier J, Bechtel F, Filhol A and Vettier C [76] Jérome D and Yonezawa S 2016 C. R. Physique 17 357–375 1987 Mol. Cryst. Liq. Cryst. 148 279–293 [77] Sedeki A, Bergeron D and Bourbonnais C 2012 Phys. Rev. [48] Foury-Leylekian P, Petit S, Mirebeau I, André G, de Souza B 85 165129 M, Lang M, Ressouche E, Moradpour A and Pouget J P [78] Shahbazi M and Bourbonnais C 2016 Phys. Rev. B 94 2013 Phys. Rev. B 88 024105 195153 [49] Grant P M 1983 J. Phys.(Paris) 44 C3–847 – C3–857 [79] Montambaux G and Jérome D 2016 C. R. Physique 17 [50] Nakamura K, Nohara Y, Yosimoto Y and Nomura Y 2016 376–388 Phys. Rev. B 93 085214 [51] Jacobsen C S, Tanner D B and Bechgaard K 1983 Phys. Rev. B 28 7019–7032 [52] Zámborszky F, Szeghy G, Abdussalam A, Forró L and Mihály G 1999 Phys. Rev. B 60 4414–4417 [53] Degiorgi L, Dressel M, Schwartz A, Alavi B and Grüner C 1996 Phys. Rev. Lett. 76 3838–3841 [54] Pethukov K and Dressel M 2005 Phys. Rev. B 71 073101 [55] Pouget J P 2012 Crystals 2 466–520 [56] B Guster, M Pruneda, P Ordejón, E Canadell, and J-P Pouget 2019 work in progress [57] Montambaux G 1987 Low dimensional conductors and superconductors Low Dimensional Conductors and Superconductors ed Jeróme D and Caron L G (NATO ASI B 155, Plenun Press) p 233–242 [58] Hasegawa Y and Kishigi K 2008 Phys. Rev. B 78 045117 [59] Hasegawa Y and Kishigi K 2010 Phys. Rev. B 81 235118 [60] Wzietek P, Creuzet F, Bourbonnais C, Jérome D, Bechgaard K and Batail P 1993 J. Phys. I France 3 171–201 [61] Creuzet F, Bourbonnais C, Caron L G, Jérome D and Moradpour A 1987 Synth. Met. 19 277–282 [62] Coroneus J, Alavi B and Brown S E 1993 Phys. Rev. Lett. 70 2332–2335 [63] Vescoli V, Degiorgi L, Dressel M, Schwartz A, Henderson W, Alavi B, Gruner G, Brinckmann J and Virosztek A 1999 Phys. Rev. B 60 8019–8027 [64] Pouget J P, Moret R, Comès R, Bechgaard K, Fabre J M and Giral L 1982 Mol. Cryst. Liq. Cryst. 79 485–499 [65] Guster B, Pruneda M, Ordejón P, Canadell E and Pouget J P 2019 Phys. Rev. Mater. 3 055001 [66] B Guster and M Pruneda and P Ordejón and E Canadell and J-P Pouget 2019 Phys. Rev. Mater. submitted for publication [67] Pouget J P 2016 C. R. Phys. 17 332–356 [68] Grüner G 1994 Rev. Mod. Phys. 60 1–24