Evidence for the weak coupling scenario of the Peierls transition in the blue bronze Bogdan Guster, Miguel Pruneda, and Pablo Ordejón Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC and The Barcelona Institute of Science and Technology, Campus Bellaterra, 08193 Barcelona, Spain Enric Canadell Institut de Ciència de Materials de Barcelona (ICMAB-CSIC), Campus Bellaterra, 08193 Barcelona, Spain Jean-Paul Pouget Laboratoire de Physique des Solides, CNRS UMR 8502, Université de Paris-Sud, Universié Paris-Saclay, 91405 Orsay, France On the basis of f irst-principles DFT calculations the wave vector and temperature dependen- cies of the Lindhard response function of the blue bronze K0.3 MoO3 have been calculated. The kFI + kFII inter-band component of the response, which is responsible for the Peierls instability, has been quantitatively analyzed. It is found that (i) the electron-hole coherence length of this response determines the length scale of the experimental intra-chain CDW correlations, and (ii) the intra-chain qk dependence of such response also determines the shape of the Kohn anomaly experimentally measured. These findings provide compelling evidence that the Peierls transition of the blue bronze K0.3 MoO3 follows the weak electron-phonon coupling scenario in the adiabatic approximation, something that had not yet been proved on the basis of f irst-principles calculations for a real material. It is proposed that the CDW interchain coupling occurs through a Coulomb coupling between dipolar CDWs. The nature of the phonon mode leading to the dipolar nature of the CDWs is also discussed, and the relevance of these results to rationalize the CDW instabilities in other oxides and bronzes is pointed out. These findings are also contrasted with recent results for other CDW materials like chalcogenides and tellurides. I. INTRODUCTION where fF is the Fermi function and i (k) are the band eigenvalues. Many low dimensional metallic systems exhibit a In the weak electron-phonon coupling limit and within charge density wave (CDW) ground state where the elec- the adiabatic approximation where Ω0 τeh < 1 (Ω0 be- tronic density is modulated with a wave number related ing the frequency of the bare critical phonon Kohn to the shape of their Fermi surface [1–3]. In experimen- anomaly and τeh the lifetime of the electron-hole pair), tal situations reported in the literature, the electronic the electron-hole pairs fluctuate so fastly during the modulation is stabilized by a periodic lattice distortion phonon oscillation period that they screen the coupling (PLD) [4]. Thus, electronic and lattice degrees of free- between the atoms. This leads to a phonon softening dom are coupled by the electron-phonon coupling [5, 6]. around qc (i.e. to the occurrence of a Kohn anomaly) for The electronic energy gain is caused by the fact that the lattice modes whose frozen displacements stabilize the new periodicity of the PLD opens a full gap in the band PLD below the Peierls transition. In the adiabatic ap- structure for 1D electronic systems or partial gaps in elec- proximation, the critical lattice dynamics is of the dis- tronic systems of higher dimension [7]. In the case of 1D placive type with a progressive softening of the frequency systems, the CDW/PLD modulation is accompanied by of the Kohn anomaly when approaching the Peierls tran- a metal-insulator transition (i.e. Peierls transition in the sition, following the thermal divergence of the electron- literature) and the modulation wave vector, 2kF , is sim- hole response [8]. In this limit the width of the disper- ply twice the Fermi wave vector [8]. sion of the Kohn anomaly is related to the width of the In the standard description of the Peierls transition [8] electron-hole response around qc (2/ξeh , where ξeh is the the electron-phonon coupling is assumed to be small so coherence length of the electron-hole pair). Finally, in the that the electronic wave functions are weakly perturbed standard theory of the Peierls transition, the entropy of by the lattice vibrations. In this limit, which relies on the critical phonon mode is neglected, an approximation a PLD of small amplitude, the instability towards the valid if the Kohn anomaly involves only a small fraction formation of electron-hole pairs is driven by the thermal of the Brillouin zone (i.e. if ξeh 2kF >> 1 in 1D). In the divergence at a well defined critical wave vector (qc = 2kF weak coupling limit, the theory of the Peierls transition is in the 1D case) of a sharp electron-hole (i.e. Lindhard [9]) of the BCS type [8]. Note that in the presence of a broad response function of the non-perturbed electron gas. This electron-hole response spreading over a sizeable fraction response can be written as of the Brillouin zone (i.e. if ξeh 2kF ∼ 1 in 1D) the phonon X X fF (i (k)) − fF (j (k + q)) entropy cannot be neglected [10]. In that case it is needed χ(q) = − , (1) to go beyond the weak coupling BCS formalism to de- i,j i (k) − j (k + q) scribe the Peierls instability and theoretical treatments k 2 suggest more likely strong electron-phonon coupling sce- dition to obtain a maximum of χ(q). We illustrate this narios. statement using a simple model for a 1D metal with two In the non-adiabatic approximation (Ω0 τeh > 1) the conduction bands having the Fermi wave vector ±kFI and critical qc phonon fluctuates very fastly during the life- ±kFII . If the dispersion of the two bands is parallel, the time of the electron-hole pairs so that phonon modes can- nesting process linking band dispersions with opposite not efficiently couple with the electron-hole condensate. Fermi velocities leads to maxima of χ(q) for q = 2kFI , There is no screening and thus no critical softening of a 2kFII and kFI + kFII but not for kFI − kFII (Fig. 1a). This phonon branch. The long-living electron-hole condensate situation is relevant for the blue bronze considered be- induces, via the electron-phonon coupling, a quasi-static low [14]. If the dispersion of the two bands is inverted, local PLD quite well decoupled from the phonon spec- the nesting process linking band dispersions with oppo- trum. The critical lattice dynamics of the Peierls transi- site Fermi velocities leads to maxima of χ(q) for q = 2kFI , tion is thus of relaxation or order-disorder type [11]. Note 2kFII and kFI − kFII but not for kFI + kFII (Fig. 1b). This that such dynamics is also found in the strong electron- situation is relevant for the charge transfer salts of the phonon coupling limit [12] where the PLD fluctuations TTF-TCNQ family [15]. In conclusion, the sign of the are of large amplitude. Fermi velocity of the different bands connected by the The largest contribution of the Lindhard function nesting process determines the type of inter-band con- (Eq. 1) originates from electronic energies leading to a tribution leading to maxima of the Lindhard function. vanishing denominator. This occurs for electronic states With more complex band structures it has been also re- located at the Fermi level such that ported that maxima of χ(q) could differ from the best FS nesting condition [16]. It is thus necessary to go be- EF (k + q) = EF (k) (2) yond the simple consideration of the nesting properties Eq. (2) defines k states of the Fermi surface (FS) which of the FS and to perform the direct calculation of the coincide with other states of the FS after application of Lindhard function to obtain the critical wave vectors of a q translation. If there are large portions of the FS ex- the CDW instability. In the case of the blue bronze, the hibiting such a property the nesting of the FS by q will CDW-Peierls instability has been proposed to be caused dominantly contribute to χ(q) [7]. Such nesting process by the inter-band kFI + kFII nesting proces [14, 17]. can simply account for the maxima of χ(q) and provide The Mo blue bronze A0.3 MoO3 , where A= K, Rb or Tl a simple explanation for the occurrence of charge and is a monovalent atom, exhibits a complex C-type centred spin density wave instabilities found in many experimen- monoclinic structure [18]. Fig. 2a shows that this struc- tal systems [13]. For example, in 1D systems where the ture is built from (b, a+2c) layers of MoO6 octahedra. FS is made of two planes at the +kF and -kF wave vec- In the perpendicular interlayer 2a*-c* direction vacan- tors, the translation wave vectors q = ±2kF nest com- cies incorporate an ordered sub-lattice of A atoms. The pletely the FS leading to a divergence of χ(q) and to the essential building blocks of the layers are the clusters of stabilization of a CDW-Peierls ground state [8]. How- ten MoO6 octahedra highlighted in Fig. 2b (see Sect. III ever, at finite T , because of the Fermi-Dirac distribution for a detailed structural discussion). Both tight bind- spreading over several kB T on each side of EF , the con- ing [14] and Density Functional Theory (DFT) [19] elec- tributions to the Lindhard function are not restricted to tronic structure calculations as well as ARPES measure- the Fermi level. This is the case for a single band disper- ments [20] show that the blue bronze exhibits a quasi-1D sion where FS nesting connecting also electronic states band structure with two parallel conduction bands lead- with opposite Fermi velocities preserves the divergence. ing to a slightly warped double sheet open Fermi surface. The repeat unit of the crystal structure in the 1D di- rection is the cluster of ten octahedra. As monovalent A atoms provide 3 electrons per cluster, the two bands are partially filled with ρ= 3 electrons. The blue bronze is a 1D metal with a sizeable RT conductivity of σb ∼ 103 S/cm and an anisotropy of conductivity [21] which follows the structural anisotropy σb : σa+2c : σ2a∗−c∗ ∼ 1:10−2 :10−3 . Polarized reflectance measurements show that the blue bronze is a true 1D metal exhibiting a metallic Drude behaviour only if the light is polarized in the chain direction [22]. At Tp = 180 K the blue bronze exhibits a Peierls metal-insulator transition accompanied FIG. 1. Schematic illustration of why for a 1D system with by the setting of a CDW/PLD modulation of critical two partially filled bands with different Fermi velocities only wave vector qc = (0, 1-(kFI + kFII ), 0.5) or equivalently, certain inter-band nestings may be effective. within a reciprocal wave vector, of qc = (1, kFI + kFII , 0.5). At low temperature, the value of kFI +kFII is ≈ 0.75 [21]. It However, for a more complex band structure with dif- exhibits a very sizeable regime of quasi-1D CDW fluctua- ferent Fermi velocities, FS nesting is not a sufficient con- tions [17] above Tp and the occurrence of a Kohn anomaly 3 in the phonon spectrum whose frequency continuously II. COMPUTATIONAL DETAILS softens as Tp is approached [23, 24]. DFT calculations [25, 26] were carried out using a numerical atomic orbitals approach, which was devel- oped for efficient calculations in large systems and im- plemented in the Siesta code [27, 28]. We have used the generalized gradient approximation (GGA) to DFT and, in particular, the functional of Perdew, Burke and Ernzerhof [29]. Only the valence electrons are considered in the calculation, with the core being replaced by norm- conserving scalar relativistic pseudopotentials [30] factor- ized in the Kleinman-Bylander form [31]. The non-linear core-valence exchange-correlation scheme [32] was used for all elements. We have used a split-valence double-ζ basis set including polarization functions [33]. The en- ergy cutoff of the real space integration mesh was 350 Ry. To build the charge density, the Brillouin zone (BZ) was sampled with the Monkhorst-Pack scheme [34] using grids of (21×45×21) k-points. The Lindhard response function (Eq. 1) was obtained from the computed DFT values of the band eigenvalues i (k). The integral over k-points of the BZ was approximated by a direct sum- mation over a dense, regular grid of points. As the Lindhard function is more sensitive to the accuracy of the BZ integration than the total energy, especially in very anisotropic systems, and/or in the presence of hot spots in the band structure (e.g. saddle points with the corresponding van Hove singularity in the DOS), the k- points grid used for its calculation must be more dense FIG. 2. Crystal structure of the blue bronzes. The lattice vec- than in the standard self-consistent determination of the tors shown in (a) are those of the C-centered cell of Graham charge density and Kohn-Sham energy. The calculations and Wadsley[18]. The three different types of MoO6 octahe- are done, nevertheless, using the eigenvalues obtained in dra are shown with different colors. The elementary building block of the structure (i.e. a cluster of ten octahedra) and the the DFT calculation for the coarser grid, and interpolat- octahedral chains it generates along the b direction are shown ing their values in the denser grid, using a post-processing in (b). An alternative description of the octahedral chains utility available within the Siesta package. In this work, associated with the red dashed line in (a) and the hump oc- for the calculation of the Lindhard response function, the tahedra is shown in (c). BZ was sampled using a grid of (256×256×16) k-points. The two partially filled bands were those taken into ac- count in the calculations. In this work we confirm the interband kFI + kFII nesting mechanism by a direct calculation of the Lindhard func- III. ELECTRONIC STRUCTURE OF K0.3MOO3 tion using the real DFT band structure for the undis- torted, high-temperature metallic phase of K0.3 MoO3 . K0.3 MoO3 crystallizes in a centered monoclinic struc- Our calculation of the thermal dependence of the shape ture and contains twenty formula units per unit cell, i.e. and width of the peaks of the Lindhard function for the K6 Mo20 O60 (Fig. 2)[18, 35]. In our DFT calculations metallic phase allows to quantitatively account for the we used a half-sized unit cell based on the centrosym- standard weak coupling scenario of the Peierls transi- metric nature of the crystal where the lattice parameters tion. Our results do not provide direct insight on the used are a0 = 1/2(a + b) + c, b0 = b and c0 = c where CDW phase, such as the lattice distortions or the spe- a, b and c are the lattice constants reported by Graham cific displacements of the phonon mode exhibiting the and Wadley.[18] The crystal structure consists of slabs of Kohn anomaly, but on the driving mechanism for the MoO6 octahedra with the alkali metal atoms in between Peierls transition originating from the instability of the these slabs. As mentioned, the structure contains three electron gas, and the physical regime of weak electron- inequivalent Mo atoms and thus three different types of phonon coupling and adiabatic limit. To the best of our MoO6 octahedra (see Fig. 2). Although the structure of knowledge, this type of validation of the weak coupling the blue bronzes is usually discussed on the basis of the scenario based on actual data for a real material has never clusters of ten octahedra shown in Fig. 2b, an alternative been reported in the literature. description is more convenient in order to understand the 4 FIG. 4. DFT Fermi surface for K0.3 MoO3 : (a) Representation using the rhombohedral Brillouin zone, and (b) View along the perpendicular to the (a*,b*) plane. chain direction) and with a nil dispersion along Γ−Z0 (i.e. the inter-layer direction). The band structure of Fig. 3a is in very good agreement with the results of pre- vious DFT calculations [19, 37] and different ARPES studies [20, 38–45]. For instance, the calculated val- ues of the kFI and kFII Fermi wave vectors are found to be 0.59π/b and 0.91π/b respectively, which are within the range of values determined in different ARPES stud- FIG. 3. DFT band structure (a) and density of states (DOS) ies: between 0.55 and 0.69 for kFI and between 0.86 and (b) for K0.3 MoO3 . In (a) the size of the blue, red and green 0.97 for kFII [20, 38–45]. In addition, the two calculated dots are proportional to the Mo I, Mo II and Mo III character. band dispersions are also in good agreement with the respectively. Γ= (0, 0, 0), X0 = (1/2, 0, 0), Y0 = (0, 1/2, 0) ARPES studies. The ratio of the calculated Fermi ve- and Z0 =(0, 0, 1/2) in units of the a0 *, b0 * and c0 * reciprocal locities of the two bands in the Γ-Y direction is ≈ 4.5 lattice vectors (see beginning of Sect. III and Fig. 4). (b) In which compares well with the more recent ARPES stud- (b) the total DOS and Mo I, Mo II and Mo III projected DOS are shown. The DOS is given in units of states per eV per ies, ≈ 4.6 [45]. Note also that in the Peierls theory of unit cell of 10 Mo and per spin direction. inter-band nesting [17] an effective Fermi velocity can be 1(2) defined as vF = 2vF1 vF2 /(vF1 + vF2 ), where vF is the Fermi velocity of band 1 (2). Using the ARPES results of reference [45] one obtains vF ≈ 1.9 eV.Å, which is in nature of the electronic structure [36] (Fig. 2c). The Mo nice agreement with the vF ≈ 2 eV.Å value that we cal- II and Mo III type octahedra form units of four corner- culate from the slope of the thermal dependence of the shared octahedra (marked with a dashed red segment electron-hole inverse coherence length (see Fig.7). Thus, in Fig. 2a) which lead to quadruple chains along the b in contrast with earlier tight-binding calculations [14], axis by further corner-sharing along this direction. The which led to a pair of bands with similar and consider- Mo I hump octahedra fill one every two outer holes of ably smaller dispersion, DFT provides a sound picture of these quadrupole chains in an zigzag way sharing two the electronic structure of the blue bronzes. Also shown edges with Mo II octahedra (Fig. 2c). Consequently, the in Fig. 3a is the contribution of the three different Mo chains along b are really built from a repeat unit of ten atoms to the different bands. Mo I does not contribute octahedra: two rows of the four octahedral units and two to the two partially filled bands. In fact the orbitals of Mo type I hump octahedra. The cluster of ten Mo octahe- I only participate in levels between 1.0 and 2.5 eV higher dra which is the repeat unit of this chain is structurally than the Fermi level (see Fig. 3b). The reason is that the equivalent to that highlighted in Fig. 2b. The layers of Mo I octahedra exhibit two strong O-Mo...O short/long the blue bronze result from the condensation of these bond alternations in the basal plane as a consequence of chains along the a+2c direction through edge-sharing of the fact that half of the octahedron shares edges with Mo octahedra of the different chains. II and Mo III octahedra whereas the other half does not The calculated band structure of K0.3 MoO3 along the share any edge, thus leading to a strong distortion. This three main directions of the Brillouin zone is shown in structural feature leads to a strong destabilization of the Fig. 3a. The Fermi level cuts two bands dispersive along three t2g orbitals of the Mo I atom [36]. Consequently, Γ−Y0 (i.e. not far from the intra-chain b direction, Γ-Y) the carriers associated with the two partially filled bands but only slightly dispersive along Γ-X0 (i.e. the inter- of K0.3 MoO3 are confined within the quadruple chains of 5 Mo II and Mo III octahedra. This is in agreement with detail the significance of these results to understand the the fact that the participation of the Mo I atoms in the workings of the CDW/PLD modulation at the origin of structural modulation of the blue bronzes was found to the metal to insulator transition of K0.3 MoO3 . be practically nil in the X-ray study of Schutte and De Boer [46]. It is however remarkable that, as discussed in detail by Machado-Charry et al. [47], using DFT cal- IV. QUANTITATIVE ANALYSIS OF THE culations very similar to the present ones, the brightest LINDHARD FUNCTION spots of the scanning tunneling microscope images of the modulated blue bronzes originate from oxygen atoms of A. The three individual responses these Mo I octahedra. These atoms are the uppermost oxygen atoms of the surface and the STM experiment is The Lindhard function (Fig. 5) is made of three re- mostly measuring the differences in the local density of sponses corresponding to three different nesting processes states of these O atoms as a result of the CDW. between the four sets of open Fermi surfaces. Scans along The two partially filled bands result from the mixing of the b* chain direction (see Fig. 6) show (using the la- the Mo II and Mo III orbitals (as well as the associated belling of Fig. 1a) three peaks at: (i) 1-2kFII = 0.09b* cor- O p orbitals), although Mo III dominates in the lower responding to the intra-band nesting of the blue FS, (ii) band whereas Mo II dominates in the upper one. These 1-2kFI = 0.41b* corresponding to the intra-band nesting of bands are almost exclusively built from the Mo dxz or- the red FS and (iii) 1-(kFI + kFII ) = 0.25b* correspond- bitals (here we assume a local system of axis with z along ing to the inter-band FS nesting. The strongest response the chain b direction and x along the dashed red line in is for the inter-band FS nesting. This nesting process Fig. 2a). This feature originates from the local distor- achieves the Peierls transition of the blue bronze. With tions of the Mo II and Mo III octahedra, which lead to a single modulation all four FS sheets can be connected one strong O-Mo...O alternation in the basal plane of the and thus a gap opens on the entire FS. The intra-band octahedra along the inter-layer direction. Such alterna- I response is stronger than the intra-band II response, tion strongly destabilizes the Mo dyz and dx2 −y2 orbitals which agrees with a lesser degree of warping of the cor- but does not noticeably affect the dxz orbital. The rea- responding FS. Upon cooling the intensity of these re- son is that the Mo dyz and dx2 −y2 orbitals make π-type sponses increases but does not diverge at low T . Diver- antibonding interactions with the O pz and py orbitals, gence is predicted in the case of nested planar FS [7]. respectively, and the destabilizing effect due to the short Here, the formation of residual pockets in the nesting Mo-O distance largely outweights the stabilizing effect of process between warped FS stops the divergence. Be- the long Mo-O distance [36]. The inner Mo III octahe- cause of the presence of nesting breaking pockets, the dra dominate in the lower band because the octahedral three responses exhibit a slightly anisotropic profile at distortion is weaker than in Mo II octahedra. The dxz low T . The kFI + kFII inter-band response exhibits a orbitals of the more distorted Mo II octahedra dominate cusp anomaly whereas a shoulder is exhibited by the 2kFI in the upper partially filled band. K0.3 MoO3 is thus a response and a tilted plateau by the 2kFII response (see pseudo-1D metal because the dxz orbitals of the Mo II Figs. 5 and 6). The maximum of the inter-band response and Mo III quadruple units undergo strong antibonding occurs for kFI + kFII =0.75b*, which corresponds to the π-type interactions with the O px orbitals along the chain intra-chain component of the experimental modulation direction but only weak inter-chain interactions as a re- wave vector measured at low T [17]. However, the exper- sult of the unfavourable orientation of the dxz in different imental intra-chain component decreases significantly by chains. a few percent upon heating [17] while Fig. 6 shows that The calculated Fermi surface for K0.3 MoO3 is reported the maximum of the Lindhard function stays constant in Fig. 4. It contains two pairs of slightly warped sheets at 0.25b* for whole temperature range considered. Thus, perpendicular to the b quadruple chains direction. The our calculation shows that the experimental decrease of red sheets, originating from the Mo II octahedra are kFI + kFII is not due to a shift of the q dependence of found to be slightly more warped. The computed Fermi the Lindhard function induced by thermal population ef- surface compares very well with those determined by fects of the curved conduction bands, as previously sug- ARPES [42, 45] which exhibit a weak but definite warp- gested [48]. This point will be further considered at the ing. In fact, as it will be shown in the next section, end of section IV C. the warping is practically irrelevant around the tempera- ture of the metal to insulator transition, Tp = 180 K [21] so that the four sheets may be nested by a single inter- B. Electronic parameters of the inter-band nesting band nesting vector, q. The calculated Lindhard response process and validation of the weak coupling scenario function associated with the two partially filled bands is shown in Fig. 5 where a clear cusp occurs for q= 0.25b* Except at very low T when nesting breaking effects (note that the chain direction is along Γ-Y, the green are relevant, each individual Lindhard component ex- dashed line in Fig. 5). Two weaker maxima are also visi- hibits a lorentzian profile in q. The half-width at half- ble along the Γ-Y line. In the next sections we discuss in maximum (HWHM) of the Lorentzian in the chain direc- 6 FIG. 5. DFT Lindhard response function for K0.3 MoO3 at 10 K -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 4.4 q=(0.0,b*,0.0) T= 0K 4.0 T = 50 K T = 100 K 3.6 T = 150 K T = 200 K χ(a.u.) 3.2 T = 230 K T = 270 K 2.8 T = 300 K 2.4 2.0 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 b*(a.u.) FIG. 6. Scans of the Lindhard response function along the Γ-Y chain direction for different temperatures. This figure clearly shows the individual responses of the three Fermi surface nesting processes (i), (ii) and (iii) defined in the text. tion gives the inverse coherence length of the fluctuating For the inter-band nesting process vF is an effective Fermi 1D electron-hole pair, 1/ξehk (T ), which depends on the velocity defined in ref [17]. Dividing Eq. 3 by = temperature T and on the microscopic parameters of the (kFI + kFII )/2 (= 0.31 Å−1 ) leads to the dimensionless 1D electron gas [49]. Fig. 7 gives the thermal depen- expression dence of 1/ξehk (T ) for the inter-band response compo- nent. This quantity increases linearly with the temper- 0 1/[< kF > ξehk (T )] = 1/[< kF > ξehk ] + T /Tef f (5) ature, but does not vanish at 0 K, as expected for pla- 0 nar nested FS. The intercept, which amounts to 1/ξehk which, according to the data of Fig 7, gives a Tef f ≈ −1 = 0.015Å = 1.8%b*, represents the longitudinal size of 2400 K. the pocket due to imperfect longitudinal nesting. One Using the data of Fig. 7 it is found that above TP the thus has value of < 2kF > ξehk (T ) is ≈ 16 at TP and ≈ 11 at room 0 1/ξehk (T ) = 1/ξehk T + 1/ξehk (3) temperature (RT). Thus, only a very small fraction of the Brillouin zone is affected by the Peierls critical instability. T where 1/ξehk is the thermal length associated with the This justifies the criteria for which the phonon entropy broadening of the FS at T . According to the microscopic change can be neglected at the Peierls transition. Conse- theory of refs. [49, 50] quently the weak coupling theory of the Peierls transition should apply to the blue bronzes. Using Tef f the cut-off T ξehk = ~vF /πkB T. (4) energy (Ecut−of f ) entering the standard theory of the 7 Then, using the half-optical direct gap in the Peierls ground state [22], ∆0 = 75 meV, and EF ≈ 0.65 eV (Eq. 6), a value of λ ≈ 0.28 is obtained. Using the explicit expression for the reduced electron-phonon coupling [7] λ = 2g 2 N (EF )/~Ω0 , (10) one can obtain, using the DFT density of states at the Fermi level, N (EF )= 2.36 eV−1 per cluster and spin di- rection (see Fig. 3b), together with a bare critical phonon frequency of Ω0 = 1.7 THz (determined in Fig. 9), g = 20 meV. This quantity is more than twice larger than the electron-phonon coupling g determined by the same method for molecular conductors [52]. FIG. 7. Thermal dependence of the inverse coherence lenghts along the chain (k) and the intra-layer (⊥) directions. These quantities are compared (dashed dotted lines) with the inverse experimental CDW correlation lengths reported in ref[51] Peierls transition can be obtained as [7, 15, 50] πkB Tef f = ~vF < kF >= 0.65 eV (6) This energy is comparable to the average Fermi energy of the two conduction bands determined by DFT calcu- lation (see Fig. 3) and ARPES measurements [20]. Eq. 6 gives an effective Fermi velocity of vF ≈ 2 eV·Å. For a 1D free electron gas with perfect longitudinal 2kF FIG. 8. Logarithmic dependence of the electron-hole response nesting, the Lindhard function should thermally diverge for the inter-band nesting process of the blue bronze according as [50] to Eq. 8. χ(2kF ) = N (EF ) ln(Ecut−of f /πkB T ) (7) The finding of λ<1 and of ∆0 << EF sustains the weak coupling approximation for the blue bronze. Fig. 7 where N (EF ) is the density of states at the Fermi level compares the thermal dependence of the electron-hole and where the cut-off energy Ecutof f is of the order of coherence length (ξehk (T )) with the longitudinal CDW EF , as given by Eq. 6. This divergence does not really structural correlation length (ξCDW k (T )) measured by occur at low T in the blue bronze because of the presence X-ray scattering [17, 21, 51] and whose divergence drives of residual pockets due to the imperfect longitudinal nest- the Peierls transition at TP . The CDW/PLD fluctua- ing. Under such conditions it is better to use 1/ξehk (T ), tion divergence is driven by the coupling of the 1D elec- as given by Eq. 3, instead of the scaling variable T . This tron gas with the critical phonon modes via the electron- leads to phonon coupling g [53, 54]. More precisely, Fig. 7 shows that at high temperature (i.e. above RT) ξCDW k (T ) χ(2kF ) ∝ ln[kF ξehk (T )] (8) tends asymptotically towards ξehk (T ). Thus, above ∼ 2 TP , when the coupling to the lattice is not yet critical, and, as shown in Fig. 8, this logarithmic dependence is the length scale of the CDW fluctuations amount to the fulfilled by the peak intensity of the inter-band Lindhard electron-hole coherence length, as expected in the weak function of the blue bronze for all the temperature range. coupling scenario of the Peierls transition. The BCS weak coupling relationship relates the Peierls gap at 0 K, 2∆0 , with the reduced electron-phonon cou- pling constant λ [7, 15, 50]. For a parabolic band disper- C. Relation with the dynamics of the Peierls sion agreeing with the DFT calculation (see Fig 3a) and transition ARPES measurements [20] it is found that Using the relation −1/λ ∆0 = 4EF e (9) ξehk = vF τeh , (11) 8 because the total number of electrons filling the two con- duction bands in the calculation remains constant, the wave vector of the experimental minimum frequency of the Kohn anomaly or the qb maximum of the CDW fluc- tuations [17] continuously increases upon cooling to reach 0.25b* at low temperature. The origin of such a wave vec- tor shift is unclear. Beside the simple yet unlikely expla- nation that, for some unknown reason, the total number of electrons is changing with temperature, it is possible that the reduced electron phonon coupling λ in Eq. 12 de- pends upon qb . According to Eq. 10, λ should vary if the bare phonon frequency Ω0 and/or the electron-phonon coupling constant g change with qb . The fit in Fig. 9 shows that this is indeed the case for Ω0 . The other pos- sibility, a variation of g with qb , should also be seriously considered because, as shown by Fig. 13 and discussed in the appendix, the Kohn anomaly develops in a phonon branch resulting from a strong hybridization between the FIG. 9. Fitting of the square of the frequency Kohn anomaly transverse acoustic TA2 branch and a low frequency op- measured at 230 K in ref [24] (see also Fig. 13a) with the tical branch. The two vibration modes certainly involve calculated Lindhard function. The best fit allows to determine different atomic displacements so that these lattice defor- the base line corresponding to the square of the bare critical mations should lead to a different electron-phonon cou- phonon mode Ω20 (q). pling. Thus, with a mode mixture changing substantially with qb in the vicinity of kFI +kFII , one should expect that the electron-phonon coupling g should exhibit a strong qb it is possible to obtain the lifetime of the electron-pair. dependence. It amounts to τeh ≈ 10−16 s at room temperature. Thus with Ω0 = 1.7 THz, a very small dimensionless quantity Ω0 τeh ≈ 1.7·10−4 << 1 is obtained. This means that the D. Inter-chain coupling electron-hole pairs fluctuate so fastly during the phonon oscillation period that they are able to screen the cou- In a purely 1D electron-phonon coupled system the pling constant between the atoms. The Peierls transition intra-chain correlation length diverges at 0 K because the of the blue bronze is thus located in the adiabatic limit structural fluctuations prevent any ordering at finite tem- where the screening effects lead to the formation of a perature [53, 54]. The observation of a Peierls transition well defined Kohn anomaly in the vicinity of 2kF . Such at finite T is due to the inter-chain coupling between the a phonon anomaly is experimentally observed [24]. With CDW fluctuations [54]. In general inter-chain coupling the screening effects treated in the RPA approximation can occur through three different mechanisms [11, 54]: the square of the Kohn anomaly frequency is given as [8] (i) A transverse nesting of the FS. In that case the Lindhard function should exhibit an inter-chain wave vec- 2 ωK (q, T ) = Ω20 (q)[1 − λχ(q, T )/N (EF )], (12) tor dependence (i.e. q⊥ ) along which direction the FS is warped, which involves the Lindhard function χ(q, T ). In Eq. 12 (ii) An inter-chain Coulomb coupling between CDW’s, we have explicitly included the 3D dependence of the since each individual CDW exhibits a charge modulation, Kohn anomaly (with q= (qk , q⊥ )). Fig. 9 shows that or Eq. 12 quantitatively accounts for the qk dependence of (iii) A transverse wave vector, q⊥ , dependence of the the Kohn anomaly of the blue bronze in the chain di- bare critical phonon Ω0 (q). rection at 230 K using the Lindhard function calculated It is worth considering the likeliness of the three cou- with DFT. Note that the best fit of the experimental data pling mechanisms for the blue bronze. Let us start with with the calculated Lindhard function shown in Fig. 9, mechanism (i). Fig. 10 presents thermal scans of the allows the determination of the square of the bare critical Lindhard function along the a + 2c ≈ 2a*+c* intra- phonon frequency, Ω20 (q). It is found that Ω0 (kFI +kFII ) ≈ layer transverse direction (see Fig. 2). It exhibits a very 1.7 THz, which coincides with the frequency of the am- broad response centered at q⊥ = 0 (in the a*,b* frame plitude mode at low T [24, 55]. defined in Fig. 4b) corresponding to an inter-chain cou- Note that in the fit of Fig. 9, the qb values of the pling in-phase along the a+2c direction. This wave vec- Lindhard response function have been shifted by about tor achieves the best FS nesting of band I to band II, as -0.02b*. The reason is the following: whereas the com- outlined in previous band structure calculations [14, 19]. puted electron-hole response function exhibits a maxi- Note however that mechanisms (b) and (c), which will mum at 0.25b* for the whole temperature range (Fig. 6) be considered below, give a similar phasing. However the 9 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 4.0 q=(a'*,-0.25 b'*,0.0) T = 10 K T = 30 K 3.6 T = 50 K T = 70 K T = 100 K χ(a.u.) 3.2 T = 200 K T = 300 K 2.8 2.4 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 a’*(a.u.) FIG. 10. Scans of the Lindhard response function along the intra-layer transverse direction for different temperatures. Note that the maxima at about 0.12a’* corresponds to 0a*. present calculation shows that the transverse response of the Lindhard function is quite broad. From its HWHM on gets an inverse inter-chain electron-hole coherence −1 length ξeh⊥ ≈ 0.19 Å−1 at RT which very slightly de- creases upon cooling (Fig 7 ), reaching 0.18 Å−1 at TP . The inverse of this quantity leads to a coherence length of ∼5.5 Å which is about the distance between the two outer Mo III of the cluster but twice smaller than the distance (12 Å) between the two outer Mo II of the seg- −1 ment defined in Fig. 2c. ξeh⊥ is larger than the inverse −1 CDW structural correlation length ξCDW ⊥ ∼ 0.14-0.17 −1 Å measured along the 2a*+c* direction at RT [17, 51]. In addition, the non-detection of a transverse plasma edge for light polarized in the transverse direction [51] shows that the warping effect of the FS is not thermally relevant above TP . The blue bronze thus remains a true 1D metal in the CDW fluctuating regime above TP . We thus conclude that mechanism (i) is not relevant for the blue bronze and we must concentrate on the other two mechanisms. FIG. 11. Lateral phasing of the dipolar CDWs in K0.3 MoO3 . Let us first consider mechanism (ii) which was pre- viously discussed in ref. [11]. According to the struc- tural refinement of the modulated structure of the blue CDWs in the dipole direction, Wk , leads to a uniform bronze below TP [46] there is basically a longitudinal dis- inter-chain CDW order, placement of the Mo atoms in the direction of each lin- - the inter-layer coupling along 2a*-c* between polar ear segment of four corner-sharing MoO6 octahedra (see layers of CDWs in the perpendicular direction, W⊥ , leads Fig. 2c). This in-phase displacement induces a ferroelec- to a staggered inter-layer CDW order. tric polarization of the segment, and the CDW modulates From the anisotropy of inter- and intra-layer transverse along b the amplitude of polarization of linear segments correlation lengths [4] one gets an anisotropy of couplings with the 2kF = 0.75b* wave number. In this scenario, the of Wk ≈ 8W⊥ (see Fig. 11). inter-chain electrostatic coupling mechanism occurs be- Finally, let us consider mechanism (iii) above TP . This tween dipolar CDW. Such a coupling is quite anisotropic mechanism is compatible with mechanism (ii) below TP . and as shown in Fig. 11: Mechanism (iii) relies on the presence of an incipient - the intra-layer coupling along a+2c between polar anisotropic valley of low frequency phonons whose bare 10 frequency Ω0 (q) sizably varies with q. In the blue bronze such an anisotropic dispersion along q⊥ is observed for the acoustic phonon branch polarized along a+2c mixed with a polar optical branch (see Appendix for experi- mental evidence taken from the phonon spectrum). This composite mode involves optical-like uniform off-center Mo displacements within the four MoO6 octahedra of each segment. The correlated Mo displacement between neighboring corner-sharing octahedra should induce a lo- cal polarization (Fig. 12a). When the correlated Mo displacements are in-phase between neighboring overlap- ping segments of four octahedra separated by a/2+c (see Fig. 2a), a frequency softening of the composite acous- tic/optical phonon branch should occur. This softening persists along the b* and 2a*-c* perpendicular propa- gating wave vectors starting from each 2π/ | a/2 + c | reciprocal position. In reciprocal space this should give rise to a sheet of low frequency phonons perpendicular to the polar segment direction (Fig. 12c). The trace of such sheet of low frequency phonons leads to an enhanced X- ray planar diffuse scattering (Fig. 12b) which has been detected in the blue bronze at RT (see ref. [11, 21]). The FIG. 12. Illustration of the low-frequency phonon coupling CDW/PLD instability and the associated Kohn anomaly mechanism for inter-chain coupling in the blue bronze and its develop inside this valley of soft phonons. Such a pre- detection. (a) Local polarization induced by the correlated existing valley imposes the intra-layer CDW component Mo displacements in neighboring corner sharing octahedra. q⊥ = 0. We thus suggest that mechanisms (ii) and (iii) (b) Planar diffuse scattering developed in X-ray scatering are strongly intermingled and are at the origin of the measurements due to (c) a sheet of low-frequency phonons perpendicular to the polarization direction when the corre- inter-chain coupling in the blue bronzes. lated Mo displacements are in-phase between neighboring seg- Similar low frequency planar valleys of mixed acous- ments. tic/polar phonons are observed in ferroelectric per- ovskites such as BaTiO3 KNbO3 and KTaO3 [56–58] or the chain-like ferroelectric SbSi [59]. Note that divergence of the electron-hole response function (Fig. 9). features similar to those of the blue bronze are ob- This well pronounced Kohn anomaly leads to the emer- served in the CDW monophosphate tungsten bronzes, gence below TP of two collective excitations of the ampli- (PO2 )4 (WO3 )2m , which are built from segments of m tude and phase of the incommensurate CDW modulation corner-sharing WO6 octahedra and which exhibit a ten- whose dispersion has been clearly measured by inelas- dency to ferroelectricity. Note that WO3 , which cor- tic scattering [24, 62, 63]. The phase excitation mode responds to the limit of this family when m → ∞ is whose frequency tends to zero at the 2kF wave vector and an antiferroelectric. The CDW/PLD instability in the which stiffening of the linear dispersion increases upon monophosphate tungsten bronzes develops inside a pla- cooling [62, 63], is a basic ingredient allowing to under- nar valley of low frequency phonons also perpendicular stand the sliding of the CDW of the blue bronze under to the segment directions [60, 61]. The possible link be- electric fields [1–3]. This situation should be contrasted tween the CDW/PLD instabilities of the blue bronzes, with the case of NbSe3 where analogous CDW sliding ef- γ− and η−Mo4 O11 Magnéli phases and monophosphate fects have been reported [64]. Two differences are worth tungsten bronzes suggested by our discussion is thus a noting. First, the Lindhard function of the blue bronze challenging issue in the quest for a full understanding of shows a well decoupled 2kF sharp response for the dif- microscopic origin of the CDW instabilities in oxides and ferent intra- and inter-band FS nesting processes (Fig. 6) bronzes. while NbSe3 shows a broad electron-hole response in the chain direction where the different intra- and inter-band FS nesting processes are superposed [65]. Second, up V. COMPARISON WITH OTHER CHARGE to now no Kohn anomaly has been detected in NbSe3 DENSITY WAVE SYSTEMS by neutron and X-ray inelastic scattering methods [66]. Such absence of Kohn anomaly has been interpreted as The present study shows that the CDW instability and providing evidence for a strong coupling scenario of the the Peierls transition of the blue bronze can be quantita- Peierls transition [66]. Note however that these measure- tively understood within the weak electron-phonon cou- ments report an anomalous drop of the frequency of the pling scenario. As a consequence, there is a clear soft- longitudinal acoustic (LA) branch for wave vectors tend- ening above TP of a sharp Kohn anomaly driven by the ing to the zone boundary. Thus, it is expected that in 11 the next Brillouin zone the LA branch transforms into the Peierls/CDW instability. In rare-earth tellurides and an longitudinal optical (LO) branch. This LO branch, transition metal dichalcogenides, the electron-hole re- which has the right symmetry to achieve the out-of-phase sponse is broad and leads to the formation of a broad longitudinal deformation of first neighbor coupled NbSe3 Kohn anomaly. In the latter 2D CDW metals the way chains required to form a π shifted CDW [67], could by which the critical CDW wave vector is selected re- exhibit a soft frequency on a large intra-chain wave vec- mains unclear. It has been recently proposed that a tor range around the CDW critical wave number. In wave vector dependent electron-coupling is essential for this respect note that, in agreement with this hypothesis, that purpose [71]. Another important question concerns both the Lindhard function calculation [65] and the mea- the mechanism of the CDW/Peierls instability. In the surement of the intra-chain CDW correlation length [68] presence of a broad Kohn anomaly, where the softening give a small ξeh ∼ 5-10 Å at RT which leads to a prod- involves a large number of wave vectors of the Brillouin uct 2kF ξeh ∼ 2-4, more than three times smaller than zone, the lattice degrees of freedom play a more impor- 2kF ξeh ≈ 11, estimated here for the blue bronze. This tant role (in particular through its entropy) than that indicates that the phonon entropy should certainly not be assigned in the standard theory of the Peierls transition, neglected when considering the mechanism of the Peierls where only the elastic energy cost is considered. A full transition of NbSe3 . In this respect, NbSe3 could bear account of lattice effects, considered earlier by McMil- a resemblance to the 2D CDW systems which will be lan [10], should invalidate the BCS-type theory for the considered below. Peierls transition and should lead to strong coupling sce- It is also interesting to compare these findings to those narios. for 2D CDW systems such as the RTe3 (where R is a rare Finally, let us mention that other 2D transition metal earth atom) tellurides and the transition metal dichalco- Mo and W metallic oxides and bronzes exhibit a succes- genides. In TbTe3 , the CDW lattice instability is re- sion of CDW instabilities [1, 3]. In contrast with the tran- vealed by the formation of a broad Kohn anomaly in sition metal dichalcogenides, the electronic structure of an optical branch where frequency softens and damping these oxides can be described by a 2D lattice of interpen- increases at the critical CDW wave number when ap- etrating and differently oriented chains [72]. As a con- proaching the Peierls transition (TP = 330 K) [69]. Fur- sequence, their Lindhard function, which is dominated thermore, below TP the CDW modulation develops with by the 2kF FS nesting processes of individual chains, ex- a critical wave vector slightly off the best nesting FS wave hibits in reciprocal space a collection of differently ori- vector [16, 69]. It is thus proposed [69] that a wave vector ented and well-defined chain-like electron-hole responses dependent electron-phonon coupling could be responsible similar to that calculated for the blue bronze [73, 74]. for this effect. The CDW instability occurs at the crossing point of Controversial interpretations have been proposed dur- the differently oriented chain responses, whose associ- ing many years concerning the mechanism of the CDW ated 1D-like CDW fluctuations have been detected by instability of the layered transition metal dichalco- X-ray diffuse scattering methods [75]. The CDW tran- genides. According to a recent review [70], predic- sition of these materials thus presents a Peierls charac- tions based on the mean-field model agree only semi- ter [61]. However since inelastic scattering measurements quantitatively with experimental data and it appears have not been performed, the CDW lattice dynamics is that generally there is no real dominant factor favoring still unknown. the CDW formation. A recent inelastic X-ray investi- gation of 2H-NbSe2 evidences the formation of a broad and damped Kohn anomaly in a longitudinal acoustic VI. CONCLUDING REMARKS branch whose frequency critically softens at the CDW wave number when reaching TP = 33 K (contrary to ear- The wave vector and temperature dependencies of the lier neutron scattering investigations) with a damping electron-hole (i.e. Lindhard) response function of the diverging at the CDW transition [71]. As it is the case blue bronze K0.3 MoO3 have been calculated on the basis for TbTe3 [69], it was proposed [71] that in the presence of its f irst-principles DFT electronic structure. This re- of a broad electron-phonon response, which does not re- sponse has three components corresponding to three pos- ally select a particular FS nesting mechanism [16], the sible nesting processes of its warped double sheet quasi- q-dependent electron-phonon coupling should be incor- 1D FS. We have quantitatively analyzed the kFI + kFII porated into the explanation of the CDW instability. inter-band electron-hole component which is responsible Compared to NbSe3 , rare-earth tellurides and transi- for the Peierls instability of the blue bronze. We have tion metal dichalcogenides, the blue bronze, which ex- shown that the electron-hole coherence length of this re- hibits a CDW instability that can be accounted for by the sponse determines the length scale of the experimental standard (weak electron-phonon coupling) Peierls sce- intra-chain CDW correlations. In addition, the intra- nario, appears to be unique. This is due to the fact chain qk dependence of such response also determines the that it exhibits a sharp 2kF electron-hole response, due shape of the Kohn anomaly experimentally measured. to a simple FS nesting mechanism, which drives the for- All these features prove that the Peierls transition of mation of a sharp Kohn anomaly which softening drives the blue bronze can be well accounted for by the weak 12 electron-phonon coupling theory of this transition in the elastic constants given in Table I. The relative Cij values adiabatic approximation. To the best of our knowledge follow the structural anisotropy: this is the first time that such theory, already established - for the compression deformation: C22 > C11 » C33 , in the 1970s, is quantitatively verified. Finally, the calcu- the deformation energy of the chain of clusters is stronger lation of the intra-layer transverse q⊥ dependence of this than that between clusters along the [102] direction of response shows that the warping effect of the FS above the layer and much stronger than that in the inter-layer TP does not provide a pertinent mechanism for the CDW direction incorporating alkaline atoms. interchain coupling. We propose that such inter-chain - for the shear deformation: C66 > C44 ≈ C55 , the shear coupling is achieved through the Coulomb coupling be- deformation energy of the (-201) layer of octahedra is tween dipolar CDWs. Furthermore, we show that the stronger than that associated with their relative shift. dipolar nature of the CDW modulation is due to the condensation at TP of a critical phonon mode resulting The dispersion of the phonon branch bearing the Kohn from the hybridization of an acoustic branch polarized in anomaly is not straightforward to analyze because it re- the MoO6 intra-layer segment direction, and an optical sults from the hybridization of the acoustic mode polar- polar branch similar to those found in ferroelectric per- ized in the segment direction (label 2 and red lines in ovskites. We suggest that such an inter-chain coupling Fig. 13a) with a low lying optical phonon mode (blue mechanism also occurs in CDW oxides and bronzes such lines in Fig. 13a) whose frequency is 1.4 THz at the Γ as the monophosphate tungsten bronzes. point and 1.8 THz at the A point (blue empty circles in Fig. 13a; the Γ and A points are defined in Figs. 13b and c). Although not measured from a (2ξ, 0, -ξ) scan ACKNOWLEDMENTS along ΓM, the TA2 branch should be very close to the TA1 branch because C44 ≈ C55 (see Table I). At the This work was supported by Spanish MINECO (Grant zone boundary M point one expects a mixing of the TA2 Nos. FIS2015-64886-C5-3-P and FIS2015-64886-C5-4-P, branch with the above mentioned optical branch reach- and the Severo Ochoa Centers of Excellence Program ing 0.9 THz at the M point along a different (2ξ, 0, ξ) under Grants SEV-2017-0706 and SEV-2015-0496), by scan (see Fig. 13a). Another mixing between the two Generalitat de Catalunya (Grant 2017SGR1506 and the modes occurs near the A point, very close to the posi- CERCA Programme), and by the European Union MaX tion of the Kohn anomaly. As a result, the low frequency Center of Excellence (Grant No. 824143). phonon branch sustaining the Kohn anomaly should be strongly hybridized in the MA direction. The (2ξ, 0, ξ) scan clearly shows an avoided crossing near the Γ point Appendix: Critical low frequency phonon branches between the bare LA2 and the 1.4 THz optical branches. of K0.3 MoO3 Beyond this crossing point the frequency of the optical branch stays nearly constant until the γ zone boundary The phonon spectrum of K0.3 MoO3 is quite complex crossing point. Then, the γM scan shows that its fre- because it contains many optical branches of low fre- quency drastically decreases to reach 0.9 THz at the M quency which hybridize with acoustic branches. Figure point (where it should mix with the TA2 branch). Such 13a shows the dispersion of some of these branches mea- dispersion, which recalls that depicted in Fig. 12c, is the sured by inelastic neutron scattering between 225 K and cut of a preexisting valley of soft phonons discussed in RT along the reciprocal directions defined in Figs. 13b the main text, where the frequency softening should be and 13c [24, 76]. Since previous neutron scattering stud- associated to correlated 1D Mo displacements, ξ. From ies focused on the dynamics of the Kohn anomaly, the the inverse of the half-width of the wave vector softening phonon spectrum of the blue bronze has never been an- along γM it can be inferred a correlated Mo displace- alyzed. In this respect the true polarization of the crit- ment on ξ ≈ 9 Å, which is about the distance between ical phonon branch has never been considered. This is two neighboring clusters in the transverse layer direc- the purpose of this appendix. Let us first consider the tion (Fig. 2a).The dispersion along MA of the bottom acoustic branches. The orthogonal frame used to label of the valley of soft acoustic/optic phonon is represented the acoustic modes is based on the structural anisotropy in Fig. 13a. Note that this dispersion exhibits an ex- (see Fig. 2a) according to the decrease in the bonding tra screening with the 1D electron gas near 2kF , which strength: leads to the formation of a Kohn anomaly in the phonon - the monoclinic b direction: label 1, spectrum. - the [102] direction: label 2, Another difficulty to analyze the Peierls lattice in- - the perpendicular to the (-201) layer of MoO6 octa- stability resides in the presence of four additional low- hedra: label 3. lying phonon modes at the position of the Kohn anomaly Using the slope of the acoustic branch one obtains the (empty green circles in Fig. 13a). 13 FIG. 13. (a) Dispersion of the low-lying phonon branches of the blue bronze measured between 225 and 295 K. The acoustic branches are labelled according to their polarization defined in the text. The hybridized phonon branches involved in the Peierls instability are colored in red and blue for the acoustic and optic counterparts respectively. 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