Bond and charge ordering in low-dimensional organic conductors

Physica B 407 (2012) 1762–1770

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Bond and charge ordering in low-dimensional organic conductors Jean-Paul Pouget n Laboratoire de Physique des Solides, UMR 8502, CNRS, Universite´ Paris-Sud, 91405 Orsay, France

a r t i c l e i n f o

a b s t r a c t

Available online 13 January 2012

We review 35 years of structural studies of quasi-1D organic conductors during which the concepts of 2kF and 4kF BOW and CDW have been elaborated. In strongly correlated quarter filled band systems these instabilities give rise to SP, DM and CO ground states. We relate these structural features to the instabilities of the 1D electron gas. To stabilize the different ground states the nature of the electronphonon coupling has to be considered together with the coupling of the organic stacks with the anion sublattice. New results concerning the classification of the SP phase in connection with the adiabatic or antiadiabatic phonon field and its competition with the CO are also introduced. & 2012 Elsevier B.V. All rights reserved.

Keywords: Organic conductors Charge density wave Charge ordering Spin-Peierls transition

Contents 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The 2kF Peierls intability . . . . . . . . . . . . . . . . . . . . . . . . . The DA charge transfer salts . . . . . . . . . . . . . . . . . . . . . . The 4kF modulation and the generalized Wigner lattice Electronic phase diagrams . . . . . . . . . . . . . . . . . . . . . . . . Electron–phonon coupling . . . . . . . . . . . . . . . . . . . . . . . . Lattice instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The D2X and A2Y salts . . . . . . . . . . . . . . . . . . . . . . . . . . . Role of the anions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The spin-Peierls transition. . . . . . . . . . . . . . . . . . . . . . . . Coupling between the SP and CO transitions . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1. Introduction Collective behaviors trigged by electronic instabilities are one of the main issues of modern condensed matter physics. Among them a major role is played, for non interacting fermions, by the so-called Fermi surface (FS) nesting mechanism. This process drives a density wave ground state (at the 2kF reciprocal wave vector for a one dimension, 1D, electron gas) stabilized by the opening of a gap (metal-to-insulator transition) or a partial gap (metal-to-semimetal transition) at the Fermi level. The collective state which thus results is either a charge density wave (CDW), accompanied by a periodic lattice distortion (PLD) in presence of sizeable electron-phonon coupling (Peierls transition in 1D) or a spin density wave (SDW) in presence of sizeable electron–electron repulsions such as those

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provided by the intra-site term U (Slater transition in 1D). In the opposite way, strong long range coulomb repulsions (Vi) stabilize a Wigner lattice of localized charges r (at the 4kF ¼ r  1 reciprocal wave vector in 1D). The Wigner charge localization is generally accompanied by a PLD. Depending upon the incommensurate or commensurate period of the charge modulation the localization can either slide or be pinned to the lattice. In the case of one charge every two sites one distinguishes between charge localization on the sites (charge ordering – CO) or on the bonds (dimer Mott–DM). In the pinned case the charge degrees of freedom are gapped. In this process the spin degrees of freedom, which become decoupled from the charge ones, remain available to undergo at lower temperature (T) an antiferromagnetic (AF) or spin-Peierls (SP) phase transition in presence of magnetic interchain coupling or magneto-elastic coupling, respectively. Both organic and inorganic materials are concerned by these instabilities. FS nesting leads to density wave instabilities in quasi1D electronic systems but also in systems of higher electronic

J.-P. Pouget / Physica B 407 (2012) 1762–1770

dimension exhibiting either hidden 1D FS (2D transition metal oxides and bronzes, rare earth tellurides) or a FS exhibiting flat portions (transition metal dichalcogenides, Cry). Wigner localization is observed whatever the electronic dimension; for example among the 3D systems in Fe3O4 (Verwey transition) and the manganites, among the 2D systems in the cobaltate, NaXCoO2, and the quarter filled (BEDTTTF)2X salts and among the quarter filled 1D systems a-NaV2O5 and D2X and A2Y salts; these latter salts being built, respectively on the derivatives of the TTF donor (D) and of the TCNQ acceptor (A) with X and Y being monovalent anionic and cationic entities. This lecture will be focused on organic conductors experiencing both long range and sizeable electron–electron interactions and an important electron-phonon coupling due to their soft lattice. Among these systems we shall consider mainly quasi-1D salts exhibiting more divergent electronic instabilities. The temperature-pressure phase diagram of quarter filled D2X series such as (TMTTF)2X [1], d-(EDT-TTF-CONMe2)2X [2] and (o-DMTTF)2X [3] illustrates quite well the subtle interplay between the various instabilities previously considered. At low pressure these salts are 1D metal. They undergo upon cooling DM and/or CO localization, followed by a transition to an AF or SP ground state. Under pressure the increase of the interchain hopping induces an electronic deconfinement towards a 2D or 3D metal exhibiting a warped FS. Its nesting drives the system to a SDW or CDW ground state. Eventually superconductivity occurs when the FS nesting process is destroyed at higher pressure.

2. The 2kF Peierls intability It was shown by Peierls in 1955 that a 1D metal is unstable at T¼ 0 K towards the establishment of a PLD, of amplitude u, which provides a new (2kF)  1 lattice periodicity perturbing the electronic states at 7kF in the vicinity of the Fermi level, EF. kF, the Fermi wave vector of the 1D electron gas expressed in reciprocal chain unit, amounts to r/4, with r being the charge per chain repeat unit. The PLD opens a gap D ¼gu in the band structure at EF, with g being the electron phonon coupling constant. For r incommensurate the order parameter is a complex quantity u expij, where j is the phase of the PLD with respect to the origin of the lattice. The PLD leads also to a modulation of the electronic density forming an electronic CDW r(x). For molecular conductors the phase shift between the PLD and the CDW depends upon the nature of the lattice modulation (Fig. 1). If the inter-molecular bond distances are modulated by u(x), there is a p/2 phase shift between the PLD (thus called bond order wave, BOW) and the electronic CDW. If the intra-molecular coordinates d(x) are modulated, the PLD is in phase with the electronic CDW; both waves are thus simply named CDW. The PLD is generally observed by diffraction techniques giving information in reciprocal space (72kF satellite lines from each side of the layers of main Bragg reflections perpendicular to the 1D direction) while r(x) is probed in direct space by STM. X-ray diffuse scattering technique is a very convenient tool to study the 1D BOW fluctuations precursors at the Peierls transition [4]. These structural aspects as well as their associated dynamics have already been considered in an earlier lecture [5].

Fig. 1. Spatial variations of the inter-molecular, u(x), and intra-molecular, d(x), components of the PLD and of the electronic CDW r(x).

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3. The DA charge transfer salts The charge transfer salts are built from segregated stacks of D and A molecules. An incommensurate charge transfer r from D to A establishes a common Fermi level between the 1D electronic structures of each stack. Typical charge transfers are r ¼0.63 in TSF-TCNQ and r ¼0.55 0.59 (T dependant) in TTF-TCNQ. TSF-TCNQ exhibits below 230 K a 2kF instability [4] corresponding to a BOW modulation of the TSF stack caused by a sliding displacement of the donor along its long direction [6]. This sliding displacement, which does not change appreciably the interplanar distance, modulates however the inter-molecular transfer integral t. Surprisingly, the isostructural TTF-TCNQ salt exhibits two kinds of instability [7,8]: a 2kF instability which develops below 150 K on the TCNQ and, at twice this critical wave vector, a 4kF instability already present at 300 K on the TTF stack. These instabilities correspond to differently polarized BOW [7,8] due to stack modulation of different nature. The dynamical aspect of the 2kF BOW fluctuations consists in a Kohn anomaly in the TA mode polarized along c* [9]. Complementary, the 2kF electronic CDW located on the TCNQ stack has been observed by STM [10], and the growth of its structural CDW counterpart has been followed below 150 K by CDW induced Raman modes consisting in out-of-plane intra-molecular deformations of the TCNQ [11].

4. The 4kF modulation and the generalized Wigner lattice The unexpected result of the X-ray diffuse scattering investigation of TTF-TCNQ was the finding of a structural instability at twice the 2kF wave vector. Such instability can be understood as the precursor of a Peierls transition in a spinless fermion gas, where because of the loss of the spin degree of freedom the band of pseudo-fermions is now filled from  2kF to þ2kF. (the definition of band filling between 7kF for a non interacting electron gas is kept). The concept of spinless fermions relies on the presence of large Coulomb repulsions. This interpretation is sustained by the finding of a 4kF instability for the 1D Hubbard model in the U-N limit [12]. In addition, as the wave length associated to the 4kF modulation, r  1, corresponds to the average distance between charges, this modulation can be also view as the 1st Fourier component of a Wigner lattice of localized charged dressed by a lattice distortion. The stabilization of a Wigner lattice [13] requires the consideration, beyond the intrasite U repulsion term, of mth neighbor Coulomb repulsion terms Vm in an extended Hubbard model: H1D ¼ Si eðiÞni þ Si tðiÞðciþ ci þ 1 þ h:c:Þ þU Si nim nik þ Si,m 4 0 V m ni ni þ m

ð1Þ

Using this Hamiltonian with no kinetic energy (t¼0) and for spinless fermions (U-N), J. Hubbard performs an exact determination of the ground state energy. If the potential is convex (Vm 1 þVm þ 1 42Vm) and if Vm tends to 0 when m-N, the ground state is a generalized Wigner lattice (GWL). Fig. 2 illustrates this finding for r ¼1/2. The 4kF CDW (or site CO) consisting in a periodic occupancy of one site out of two (101010 order) is the ground state

Fig. 2. 4kF GWL (left) and 2kF PEC (right) ground states of the half filled band of spinless fermions in function of the intersite coulomb repulsions Vm in the limit t¼0.

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for the convex potential V1 þV3 42V2. It is however, interesting to notice that for a concave potential (V1 þV3 o2V2) the ground state is a 2kF CDW or a paired electronic crystal (PEC) formed of pairs of singly occupied sites separated by pairs of unoccupied sites (11001100 order). The effect of t is to spread the localized charge on neighboring sites and thus to reduce the harmonic content of the modulation [14]. For a concave potential and r ¼1/2, t induces a new 4kF BOW ground state which coexists with the PEC (see Fig. 5). Recent quantum chemistry calculations of Vm show that the extended Hubbard is particularly relevant to describe the organic systems. In particular it has been calculated for clusters of TTF or TMTTF molecules [15a] that Vm decreases as 1/m. For Coulomb interactions screened by the molecular polarisability it is found [15b], with U/t  13 and 9, respectively for the TTF and TCNQ stacks of TTF-TCNQ, that the Coulomb repulsions are the dominant interaction, with for each stack: U  2V1, V1 2V2 and V3 2/3V2. The Coulomb interaction potential is found convex, but we shall see in Section 9 salts that the anion potential could modify the convexity of this potential.

5. Electronic phase diagrams The 1D interacting electron gas, is atypical because it does not exhibit quasiparticle excitations, but only CDW, SDW, singlet and triplet superconducting collective fluctuations [16,33]. The most divergent fluctuations depend upon the strength and range of the Coulomb interactions (parameterized by the quantity Kr 40) and the sign of the 2kF Fourier transform of the Coulomb interaction (g1). In the charge sector (Ks ¼1) and for r incommensurate Fig. 3 gives the phase diagram of the dominant (subdominant) fluctuations. For repulsive interactions (Kr o1) density wave fluctuations, promoting an electron-hole periodic order, dominate while for attractive interaction (Kr 41) singlet or triplet Cooper pairing dominates. Kr ¼1 corresponds to the non interacting electron gas. If the extended Hubbard Hamiltonian given by (1) is truncated at the order m: Kr Z1/(mþ1)2. For the simple Hubbard model (m¼0), where Kr Z1/2, the 4kF CDW divergence is only approached in the limit U-N when Kr ¼1/2. For the 1st neighbor Hubbard model

Fig. 3. (g1, Kr) phase diagram of the 1D interacting electron gas for an incommensurate band filling in the charge sector. The dominant (sub-dominant) instabilities are indicated (adapted from Ref. [16a]).

Fig. 4. Electronic ground states of the half filled repulsive chain.

(m¼1), where Kr Z1/4, only the phase boundary of the CO ground state of the quarter filled system (see below) is reached when Kr ¼1/4, The 2nd neighbor Hubbard model where Kr Z1/9 allows exploring nearly all the phase diagram. For repulsive g1 40, the 1D metal is a Tomonaga–Luttinger liquid (TLL), with a spin–charge separation, while for attractive g1 o0, the 1D metal is a Luther– Emery liquid (LEL) where the spin excitations are gapped. The 1st neighbor Hubbard model, with g1 ¼Uþ2V1 cos pr, allows illustrating these different liquids for the half filled (r ¼1) case where g1 ¼U 2V1 (Fig. 4): – for g1 o0 (Uo2V1) one site out of two is occupied by two electrons of opposite spin, forming a 2kF CDW without magnetism, – for g1 40 (U42V1) each site is occupied by an electron, leading to a Mott–Hubbard charge localization, with available spin degrees of freedom for an additional low T instability. Fig. 3 shows that for very repulsive interactions the 4kF CDW is dominant for the TLL (one recovers the GWL limit), while the 2kF CDW is dominant for the LEL. For an incommensurate band filling, the order parameter is, as for the Peierls transition, a complex quantity. In 1D the quantum fluctuations of j lead to an algebraic decay of the correlation function preventing the establishment of a long range order (LRO) at T¼0 K. For a commensurate band filling (r ¼2/n) umklapp electron–electron scattering processes have to be added [16,33]. They strongly modify the electronic phase diagram. For n even (odd) and for Kr o4/n2(3/n2) there is a 4kF charge localization which is pinned on the lattice. The order parameter is an Ising variable (1/0 if a given site is occupied/empty) and there is LRO at T¼0 K. For n even the LRO consists of equidistant charges: in the n ¼4 case one has the 4kF 101010 CDW shown Fig. 2. The same exchange interaction connects all the localized spins, and the magnetism is that of a uniform Heisenberg AF chain. For n odd the LRO consists of non equidistant charges: in the n ¼3 case, where 110110, the 4kF CDW is equivalent to the 2kF CDW. The exchange interaction alternates between the localized spins: The AF chain can be considered as dimerized and a spin gap is open. The extended Hubbard model at half filling (n ¼2) and quarter filling (n ¼4) has been particularly studied. For the half-filled repulsive chain, Fig. 4 illustrates the three ground states of different symmetry which are stabilized for Kr o1. The 2kF CDW/neutral–ionic chain is stabilized for Uo2V1, while the Mott–Hubbard AF chain is stabilized for U42V1. For U 2V1 and for U and V1 below a bicritical point, the resonant 2kF p-BOW/SP, which is also the 1D analog of the resonant valence bond (RVB) ground state, is stabilized [17]. Fig. 5 presents the phase diagram of the quarter filled chain in the limit U-N (half filled spinless fermion band) [18]. For t ¼0, the 4kF CDW and 2kF CDW phases considered Fig. 2 are recovered. But for t finite and for V1 comparable to 2V2 charge transfer processes favor an equal occupancy of all the sites in order to

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Fig. 5. Upper part: (2t/V1, V2/V1) phase diagram of the quarter filled repulsive chain in the limit U-N. The inset gives the dependence of the order parameters of phases II and III in function of t. The lower part gives a schematic representation of the electronic modulations of phases I, II and III (adapted from Ref. [18]).

minimize on equal footing the 1st and 2nd neighbor Coulomb repulsions. This resonant process stabilizes the 4kF BOW. 4kF CDW and 4kF BOW presenting different inversion symmetries are separated by a phase boundary. The 2kF CDW undergoes a separate phase transition where it coexists with the 4kF BOW whose amplitude decreases with t decreasing. All these phase transitions occur for Kr r1/4. When V1 ¼2V2 in the m¼2 extended Hubbard model the TLL liquid is recovered for t finite because of the frustration of the Coulomb repulsions [19]. However, when V3 is considered frustration effects are removed and the 4kF CDW is further stabilized. For U finite the same phase diagram is obtained by numerical methods [20,21].

6. Electron–phonon coupling Lattice modulations occur when the Hamiltonian (1) is coupled to a phonon field. To calculate the electron–hole response of the electron–phonon coupled system, one must distinguish between: – Intra-molecular (optical) modes inducing a local deformation of the molecule. They modify the site energy e(i) in (1), and thus induces a modulation of the charge on the sites. This mechanism is at the origin of the CDW. The same feature occurs for a displacement of the anion X towards the D molecule in D2X salts. These phonon modes modulate also U in (1). – Inter-molecular (acoustic) modes changing the inter-molecular distances. They modify the single particle hopping integral t(i) in (1), and thus modulates the charge on the bonds. This mechanism is at the origin of the BOW. These phonon modes modulate also the Vm’s in (1). Via the electron–phonon coupling the PLD modes induce an electronic CDW on the molecular stacks, which at its turn screens the interatomic force constants. This leads to a softening of the phonon frequencies, which for a classical phonon field in the adiabatic limit is given, in the RPA approximation, by [22]:

oðqÞ2 ¼ OðqÞ2 f1½29g a 92 =OðqÞN a ðqÞg,

ð2Þ

Fig. 6. Thermal dependence of Ne(q) and Nt(q) for a quarter filled band system with U¼ 2V1 and U¼ 4t (adapted from Ref. [22b]).

where O(q) is the bare phonon frequency and ga is the electron– phonon coupling of the phonon modulating the term a in the hamiltonian (1). In the expression (2): – for a ¼ e, with ge ¼@e/@d, Ne(q) is the charge density polarizability whose divergence leads to a CDW, – for a ¼t, with gt ¼@t/@u, Nt(q) is the charge transfer susceptibility whose divergence leads to a BOW. For the non interacting electron gas, Ne(q)¼Nt(q) is the Lindhard function. In presence of Coulomb interactions these two response functions differs both by their q and T dependences, especially for commensurate fillings. The CDW or BOW nature of the dominant lattice instability as well as its 2kF or 4kF critical wave vector is given by the strongest divergence of Na(q). This leads to the formation of a Kohn anomaly in the phonon spectrum which frequency vanishing, given by the expression (2), drives (in the mean field approximation) the system to a CDW or BOW ground state. For half band filling, the calculation of Ne(q) shows that U suppresses the 2kF CDW while V1 tends to restore it. The calculation of Nt(q) shows that, for V1 ¼ 0 at intermediate U (  4t), the dominant fluctuations are mixed 2kF BOW-SP accompanied by the tendency to form a magnetic singlet [22c]. V1 enhances the 2kF BOW divergence of Nt(q). Fig. 6 shows for quarter band filling the T dependence of Ne(q) and Nt(q) [22(a) and (b)]. Both Ne(q) and Nt(q) exhibit an enhanced 4kF response which reflects the instability towards CDW/CO and BOW/DM orders, respectively. At low T Nt(q) exhibits also a strong 2kF divergence towards a BOW/SP instability. These results are complemented by the determination [23] of the most stable ground state in function of the electron-phonon coupling parameters ge(q) and gt(q). The intra-molecular phonons (together with the coupling to the anions) stabilize the CDW, while the

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coupling to acoustic phonons is required to stabilize BOW or SP ground states. V1 enhances the CDW and destabilizes the BOW. If one assumes that the CO/DM transition corresponds a Peierls transition exhibited by a spinless fermion gas coupled to internal/ external phonons, the non-adiabaticity of the phonon field (when the phonon frequency becomes larger than the mean field Peierls gap) should suppress the structural instability [48], as found for the SP transition (see Section 10). The correct calculation of CDW or BOW structural fluctuations requires going beyond the mean field/RPA approximation. This implies the consideration of electron–electron and electronphonon fluctuations on the same footing. Very little work is available on that topic in the literature [24]. However, the calculation has been achieved for the SP instability in the adiabatic limit [25].

7. Lattice instabilities The 2kF and/or 4kF electronic instabilities coupled to the lattice give rise to a 1D critical X-ray diffuse scattering precursor at the Peierls transition. The analysis of their structure factor shows that the associated modulation is mainly of acoustic BOW type. It includes: – the 2kF instability located on the TSF and HMTSF stacks of TSFTCNQ [6] and of HMTSF-TCNQ [26a] respectively, – the 2kF instability located on the TCNQ stack of NMP-TCNQ [26b], – the 2kF and 4kF instabilities both located on the TMTSF stack of TMTSF-DMTCNQ [27], – 2kF and 4kF instabilities located on the TCNQ and TTF stacks of TTF-TCNQ , respectively [7,8]. For the BOW located on the D stack, Fig. 7 shows that when the molecular polarizability increases the instability changes from a single 4kF BOW on the TTF to a dominant 2kF BOWþweaker 4kF BOW on the HMTTF then to a single 2kF BOW on the TSF and HMTSF. The increase of molecular polarizability leads to a decrease of electron repulsions and thus the increase of Kr. The crossover between the most divergent CDW instabilities is in agreement with the phase diagram shown Fig. 3. For TTF-TCNQ one expects Kr r1/3 from the observation of single 4kF BOW

Fig. 7. Schematic variation of the 2kF and 4kF BOW instability in function of the molecular polarizability of the D molecule in selected D-TCNQ charge transfer salts.

instability on the TTF stack. Accordingly the fit of the thermal divergence of the 4kF fluctuations gives Kr 0.3 [24a]. A shift from a dominant 2kF instability to a dominant 4kF instability located on the TCNQ stack is observed when x decreases in the solid solution NMPxPhen1  x TCNQ [28]. This behavior has been associated to the crossover from a two chain metallic system (with r ¼1/3 mobile holes on the NMP and r ¼2/ 3 mobile electrons on the TCNQ for x ¼1) to a single chain metallic system with r  0.5 electrons on the TCNQ when x  0.5. The crossover to the 4kF instability has been interpreted as being the signature of the reduction of interchain screening. The NMPxPhen1  xTCNQ series shows also that the 4kF instability develops when, with x decreasing, the quarter band filling limit is approached. The quarter band filling (r ¼0.5) is realized for x¼0.5 and by stoichiometry in the D2X and A2Y salts. Generally the 4kF BOW instability is observed in salts with a regular stacking and a band filling r equal or very close to 1/2 (an underlying stack dimerization freezes the incipient 4kF BOW instability of quarter filled systems). Fig. 8 shows two examples of 4kF BOW taken from Qn(TCNQ)2 [29] and (DI-DCNQI)2Ag [30]. Another remarkable example is TMTSF-DMTCNQ which, in spite of an enhanced TMTSF molecular polarizability, undergoes a dominant (high T) 4kF instability on the D stack [27]. In this respect, the fit of the thermal dependence of the 4kF and 2kF instabilities located on the same D stack [27] leads to Kr  0.25 [24b]. The special value r ¼0.5 for which the 4kF correlations are enhanced can be easily explained because only U and V1 are required to form a GWL in this limit. The evolution with r of the nature of the instability which develops on the TCNQ stack illustrates quite well this interpretation. At large r, NMP-TCNQ (r ¼2/3) and TTF-TCNQ (r ¼0.59  0.55) develops a dominant 2kF instability which changes into a dominant 4kF instability when r decreases with x decreasing in NMPxPhen1  x TCNQ (r o0.57 for xo0.65) and in Qn(TCNQ)2 (r ¼0.5). In this scenario the 2kF instability on the TCNQ stack of TTF-TCNQ appears to be located

Fig. 8. X-ray diffuse scattering patterns taken from Qn(TCNQ)2 at 25 K [29b] (top) and (DI-DCNQI)2Ag at room temperature [30b] (bottom) showing (arrows) the 4kF BOW instability. The stack direction is horizontal.

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on the crossover line between the two regimes. This explain the enhanced divergence of the 2kF BOW instability of TTF-TCNQ [7] compared to that of NMP-TCNQ [4]. The 4kF BOW instability has not yet been detected in the (TMTTF)2X series probably because of the donor stack dimerization. It has however, been recently observed in the non dimerized (o-DMTTF)2X series [3]. The non detection of a 4kF BOW by X-ray diffuse scattering techniques does not mean that the system does not undergo a 4kF instability. The detection of a 4kF CDW modulation generally requires a complete structural refinement. In this case NMR provides a more direct way to detect the 4kF CDW. In all the materials considered in the section the BOW structural instability occurs in the adiabatic limit: the mean field gap or even the mean field transition temperature (temperature at which the 1D structural fluctuations begin to be detected) being larger than the frequency of the acoustic phonon modes [9].

8. The D2X and A2Y salts Most of the D2X salts present a stack dimerization. It opens a gap DD in the band structure. With one hole every two molecules the non dimerized salt is quarter filled, while in the dimerized situation the upper band is half filled. Thus the classification of these salts as quarter or half filled band systems is not obvious. To resolve this point one must deepen the role of the dimerization on electron–electron scattering processes. In quarter filled systems the dimerization induces a 4kF potential allowing 2nd order umklapp scattering processes which in presence of large electron–electron repulsions localize the charges [31]. The gap of charge Dr which thus results depends upon the relative magnitude of the dimerization DD and of the Coulomb repulsions [32]. If the gap of charge Dr is larger than pkBT the salt can be considered as a half filled band system. This is the case if Dr 41000 K in the T range of study (o300 K;RT). In this limit of strongly dimerized system the charge is localized in the bonding (or anti-bonding) state of the dimer. There is no available intradimer charge degree of freedom. Thus, a charge disproportion inside the dimer is not possible: the 4kF CDW or CO ground state specific to quarter filled band systems cannot be achieved. The low T instabilities, AF or SP, are those of the AF Heisenberg chain (one spin 1/2 per dimer). Effective half filled band systems (Dr 41000 K) isostructural to the (TMTTF)2X’s are (DIMET)2SbF6 and (t-TTF)2Br which are AF at TN ¼12 K and 35 K, respectively, and (BCPTTF)2PF6 and AsF6 undergoing a SP transition at TSP ¼32.5 K and 36 K, respectively. The (TMTTF)2X’s exhibiting a minimum of resistivity at Tr  200 K below RT have to be considered as quarter filled band systems (Dr  pTr o1000 K). (TMTTF)2X’s undergo below Tr a CO transition followed at lower temperature by either an AF or SP transition [1]. Note that in the quarter band filling limit the 4th order umklapp scattering are essential in the electron localization process [33]. In pure quarter filled band systems the 4kF instability is either of the BOW or of the CDW type. It leads, respectively to DM and CO ground states. The prototypal example of a DM ground state, where the molecules remains identical while the bonds are different, is achieved by MEM(TCNQ)2 below 335 K [34]. The 4kF BOW instability of Qn(TCNQ)2, shown Fig. 8, is of the same type. Up to now there is no prototypal example of a pure CO ground state where there is only a charge disproportion on the molecular sites while the bonds remain identical. Many D2X and A2Y salts present an important CO component in their 4kF ground state as revealed by NMR studies (see the lecture of T. Takahashi). To my knowledge the first evidence of CO is provided by the (TMP)2X-CH2Cl2 (X¼PF6,AsF6) system [35] which could not be studied in detail because of the solvent disorder. (DI-DCNQI)2Ag exhibits at 220 K a transition to a 4kF

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localized state whose structural refinement shows a mixture of CO and DM [36]. This last component gives rise to the 1D 4kF BOW pretransitional fluctuations shown Fig. 8 [30]. The (TMTTF)2X series exhibits a CO transition whose TCO varies between 230 K and 65 K with the anion X. As the TMTTF stack is already dimerized at high T, the CO removes the inversion symmetry leading to electronic ferroelectricity [37]. In addition to NMR, infrared studies [38] and a recent synchrotron radiation structural refinement (see the lecture of H. Sawa) reveal the presence of differently charged molecules in the PF6 salt below TCO. The CO transition is achieved by weak structural modifications [39], accompanied by lattice parameter anomalies [40] which have been associated to an anion shift at TCO. The recent structural refinement of d-(EDT-TTF-CONMe2)2Br in its CO ground state [41] shows that the charge disproportion is stabilized by a synchronous displacement of the Br anion towards the positively charged molecule. Finally, let us mention the recent finding of a 4kF localization in the non dimerized quarter filled band series (o-DMTTF)2X (X¼Cl, Br) [3]. Although the structural refinement has not been performed in this salt, symmetry arguments imply, as for the (DI-DCNQI)2Ag, that the charge localization is due to a mixture of CO and DM; this last component being achieved by the divergence of the experimentally observed 1D 4kF BOW pretransitional fluctuations.

9. Role of the anions The finding of a TCO strongly dependant on the anion X in the (TMTTF)2X’s [42] as well as the determination of an anion shift in the CO state of d-(EDT-TTF-CONMe2)2Br [41] mean that there is a coupling of the anions with the organic stacks. For example in the (TMTTF)2X’s this coupling should depend upon the size and polarizability of the anion, its interaction with the organic stack via the S–X contact distance (with the result to modulate the p electron density) or via the H-bonding network inside the methyl group cavity in which the anion is located (the polarization of the H bond leading to a negative charge shift of the s electrons which in turn induces a positive p charge modulation). These interactions appear to be quite general among the D2X organic salts such as the quarter filled (BEDTTTF)2X salts (see the lecture of E. Canadell) where they could help to stabilize a lattice deformation at a critical wave vector corresponding to maxima of the p electron-hole response function [43]. In this framework the anion shift appears to be an essential ingredient to stabilize the CO pattern [44]. It provides an efficient electron–phonon coupling mechanism modulating the site energy, an essential feature to stabilize the 4kF CDW. In addition, by their location in between the stacks the anions provide also a very efficient interchain coupling mechanism between the CDW’s. In the (TMTTF)2X’s the shift of the anions from the inversion centers stabilizes an excess of hole on the molecule towards which the anion moves and an excess of electron on the opposite molecule. This shift fixes also the magnitude of the ferroelectric polarization. In this context Fig. 9 shows that the Hartree potential due to the anion shift (or of its ordering if X is non centrosymmetric) can modify the convexity of the intrastack Coulomb potentials. With respect to the situation depicted Fig. 2: – the 4kF anion site potential e increases the Coulomb potential convexity, which favours the 4kF CDW/CO ground state, – the 2kF anion site potential d decreases the Coulomb potential convexity, which helps to stabilize the 2kF CDW ground state. In this framework a strong coupling of the organic stack with the anion sublattice could induce a neutral (N)–ionic (I) order

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Fig. 9. Effect of the 4kF (left) and 2kF (right) anion potentials on the ground states of Fig. 2.

with a charge disproportion of nearly half an electron forming neutral D0 and positively ionized D þ . The 4kF 1010 CO or ININ order is nearly achieved in d-(EDT-TTF-CONMe2)2Br (together with a weak 4kF BOW stack dimerization) [41], while the 2kF 0110 CDW or NIIN order is achieved in (EDO-TTF)2PF6 (together with a strong 2kF BOW stack tetramerization) [45].

10. The spin-Peierls transition Many A2Y and D2X salts undergo after the DM or CO transition an additional phase transition towards a SP ground state where the spins 1/2 are paired into singlets. This SP transition is achieved in MEM(TCNQ)2, (TMTTF)2X and (BCPTTF)2X with X¼ PF6 and AsF6. However, in spite of its resemblance with the Peierls transition, the mechanism of the SP transition is more subtle than it is usually believed. It relies on the answer at the simple question: is it energetically favorable to dimerize an AF chain? The SP ground state corresponds to a gain of magnetic energy if the zero point (quantum) fluctuations cannot populate the excited states or in other words if there is an energy gain to open a gap in the degenerate states of the AF chain. These degenerate states are present in the XY and Heisenberg S¼1/2 AF chains. However, there is no energy gain to dimerize the AF Ising chain, which already presents a gap in its excitation spectrum, or the AF chain of classical spins which, being ordered at 0 K, does not exhibit zero point quantum fluctuations. The opening of a gap in the magnetic excitations is a quantum process where the SP instability picks out the fluctuations of the AF chain to forms a condensate of local singlets; the latter ones being ordered by a concomitant lattice dimerization. This process presents a certain analogy with the Peierls transition which appears to be more transparent when spin operators are transformed into fermionic operators by the Wigner–Jordan transformation. By this operation the XY AF chain is transformed into an half filled tight binding band of spinless fermions. For such pseudo-fermions coupled to the phonon field, the 2kF Peierls instability stabilizes a chain dimerization. This corresponds for the SP chain to the pairing of S¼1/2 into magnetic singlets. The resulting gap opening in the pseudofermion dispersion occurs for the SP transition in the spin excitation spectrum. The case of the Heisenberg AF chain is more subtle [46] because the Wigner–Jordan transformation leads to interacting pseudo-fermions with an attractive g1. These pseudo-fermions behave as a LEL which dominant instability is towards the formation of a 2kF CDW (see Fig. 3). The SP transition of the AF Heisenberg chain is thus equivalent to the Peierls transition in a LEL. The attractive interactions lead to substantial differences with the SP transition of the XY chain. In particular the mean field gap of the SP Heisenberg chain (without logarithmic corrections),

Fig. 10. (DMF, O) phase diagram of the SP transition together with the location of typical SP compounds (adapted from Ref. [47b]).

given by the expression [47a]:

DMF  2:47kB T MF  1:3g 2 =_O,

ð3Þ

is larger than the gap given by the BCS relationship (DMF E 1.76kBTMF); the latter relationship being only valid for the SP XY chain. The confusion between these two relationships is frequently done in the literature. Another important difference with the classical Peierls transition relies on the presence of spinless fermions. While the classical Peierls transition exists whatever the strength of the electron-phonon coupling in a half filled band, the Peierls transition of spinless fermions exists only if the coupling with the phonon field (spin–phonon coupling g in (3)) is strong enough (g 4gC) [48]. This means that quantum fluctuations control the SP ground state. This leads to a quite subtle phase diagram [47b] shown Fig. 10. Its boundaries are controlled by the relative values of DMF, given by (3), and of the bare phonon frequency O: – if :O o DMF/2 the SP transition is in the classic (adiabatic) regime. The zero point fluctuations weakly reduce the amplitude of the SP gap with respect to its mean field value DMF, – if :O 4 DMF/2 and if g 4gC the SP transition is in the quantum (antiadiabatic) regime of Kosterlitz–Thouless type. The amplitude of the SP gap, which thus decreases exponentially with the spin–phonon coupling g, is strongly reduced, – if g ogC  0.7:O [49], the zero point fluctuations kill the lattice dimerization and the system remains in its spin liquid phase. For comparison with the theory, Fig. 10 includes various compounds undergoing a SP transition. For each of them DMF has been obtained from the relationship (3) with the mean field SP temperature TMF taken as the onset temperature of the 1D SP fluctuations precursor to the 3D SP transition (TSP), a procedure detailed in ref. [50]. For the organic compounds the bare phonon frequency O of the critical SP phonon mode is taken as the TA phonon frequency measured in similar materials [50]: TA phonons bear the Kohn anomaly in TTF-TCNQ [9] or are responsible of the transverse shift of TCNQ dimers in the SP ground state of MEM(TCNQ)2 [51]. Fig. 10 shows that (BCPTTF)2X’s are well located in the adiabatic regime where the pretransitional SP fluctuations are slow. They lead to the formation below TMF of a pseudo-gap in the spin susceptibility [25]. Indeed, Fig. 11(a) shows that the thermal dependence of the spin susceptibility of (BCPTTF)2AsF6 progressively deviates below TMF from the Bonner and Fisher (BF) behavior of the spin susceptibility of the 1D AF S¼1/2 Heisenberg chain governed by the hamiltonian: Hspin ¼ JSi Si USi þ 1

ð4Þ

MEM(TCNQ)2, TTF-CuBDT and Per2-Pt(mnt)2 belong to the antiadiabatic regime. Here, the pretransitional SP fluctuations as so rapid that a pseudo-gap cannot form. In consequence the spin

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Fig. 11. Thermal dependence of the spin susceptibility of (a) (BCPTTF)2AsF6 [25], (b) MEM(TCNQ)2 (C. Coulon, unpublished results) and (c) (TMTTF)2X for X indicated in the inset [42]. In (a) and (b) the continuous line is the fit of the high T dependence by the BF behavior. The exchange interaction defined by (4) is also indicated.

susceptibility follows the BF thermal dependence until the 3D TSP phase transition. Fig. 11(b) illustrates this situation for MEM(TCNQ)2 [34]. As the gapless boundary is approached the antiadiabatic corrections becomes more and more important. These corrections renormalize the 1st neighbor exchange interaction J and introduce next near neighbor exchange interactions J2, etcy in (4) [49,52]. The introduction of J2 leads to a deviation at the BF thermal dependence which is observed in CuGeO3 [53]. This last compound is located in Fig. 10 on the gapful–gapless transition line. The (TMTTF)2X’s which are located in the vicinity of the adiabatic-antiadiabatic crossover, present quite small pseudogap effects (Fig. 11(c)).

11. Coupling between the SP and CO transitions Fig. 12 shows that the SP transition is destabilized by the CO. This linear relationship between TCO and TSP is also followed by the experimental data of pressurized (TMTTF)2AsF6 [54]. Fig. 12 shows that when the CO vanishes a SP transition with a TSP larger than 25 K

Fig. 12. CO transition temperature (TCO) and mean field SP temperature (TMF - triangles) in function of the 3D SP transition temperature (TSP) of D2X salts. The adiabatic gapful, antiadiabatic gapful and gapless regimes are indicated.

occurs, as observed in the (BCPTTF)2X’s. In the opposite situation no SP transition is expected for TCO larger than 160 K. This is the case of (TMTTF)2SbF6 and SCN which present an AF ground state.

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Numerical simulations show that a decrease of the SP tetramerization follows the increase of the CO amplitude [55]. This result can be understood simply. In the localized limit the spin exchange J of a spin 1/2 located on the site i is mostly due to the coupling with its 1st neighbor i þ1. However, as the CO enhances the occupancy of next neighbors iþ2, a decrease of J is expected. The decrease of J leads to a decrease the spin–phonon coupling g: g  @J=@R ¼ mJ, for J  Jo expmR, which in turn leads to decrease of TMF, given by (3). Fig. 12 also reports the variations of TMF experimentally obtained from the onset temperature of the 1D SP pretransitional fluctuations. The consequences of the decrease of TMF are a destabilization of: – the SP instability and of the 3D SP critical temperature TSP, – the adiabatic nature of the SP instability. From the variations of TMF and DMF, related by the expression (3), Fig. 12 shows also that: – by changing PF6 into AsF6 in the H12 salts one passes from the adiabatic SP phase to the antiadiabatic SP phase; the PF6 (D12) salts being located on the crossover line, – the SbF6 and SCN salts located in the spin liquid gapless phase are any more subject to the SP instability.

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