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Comptes Rendus Physique
Volume 17, n° 3-4
pages 332-356 (mars 2016)
Doi : 10.1016/j.crhy.2015.11.008
The Peierls instability and charge density wave in one-dimensional electronic conductors
Instabilité de Peierls et onde de densité de charge dans les conducteurs électroniques à une dimension
 

Fig. 1




Fig. 1 : 

1D Electron dispersion together with an electron–hole ħω k (q ) excitation process. The Peierls gap 2Δ that develops below T P is expressed as a function of the creation ( ) and annihilation (c k ) electron operators involved in the q =2k F excitation process.


Fig. 2




Fig. 2 : 

π/2 phase shift between the BOW u (x ) and the CDW δρ (x ). Note that the internal site deformation d (x i ) behaves spatially as the CDW.


Fig. 3




Fig. 3 : 

Diffraction (in reciprocal space – left side) by the 2k F PLD, represented in direct space on the right side, in the 1D fluctuation regime (below  ), in the inter-chain short range coupling regime (between T CO and T P ) and in the 3D Peierls ground state (below T P ).


Fig. 4




Fig. 4 : 

X-ray patterns from K0.3 MoO3 (left) and NbSe3 (right) at RT showing different sets of 2k F diffuse scattering (adapted respectively from Refs. [26] and [27]). Note that the 2k F diffuse segments (black arrows) of K0.3 MoO3 perpendicular to the chain direction b are located in broad diffuse lines (white arrows) perpendicular to the a+2c direction and whose origin is commented in the text. In NbSe3 black-and-white arrows show two sets of diffuse lines perpendicular to the chain direction b , which respectively correspond to 2k F fluctuations on type-I and type-III chains defined in Fig. 14b.


Fig. 5




Fig. 5 : 

Thermal dependence of the reduced inverse intra-chain correlation length of 1D incommensurate 2k F PLD fluctuations measured in several organic charge-transfer salts. The continuous line gives the numerical solution of the fluctuating Peierls chain functional (9). The thermal dependence of ξ −1 for amplitude and phase fluctuations, respectively relevant at high and low temperatures, is separately shown (adapted from Ref. [32]).


Fig. 6




Fig. 6 : 

Thermal dependence of the magnetic susceptibility of TTF–[Ni(dmit)2 ]2 (adapted from Ref. [41]). The partial decrease of spin susceptibility corresponds to the complete development of a pseudo-gap in the LUMO conduction band structure of the Ni(dmit)2 stack. The continuous line reports the LRA calculation [39].


Fig. 7




Fig. 7 : 

Nature of the ground state in function of the mean-field SP gap,  , and of the critical phonon frequency Ω SP for the SP Heisenberg chain [44], together with the location of typical SP compounds (adapted from Ref. [45]).


Fig. 8




Fig. 8 : 

Thermal dependence of the spin susceptibility of (BCPTTF)2 AsF6 (left) [48] and of the reduced ESR susceptibility of MEM–(TCNQ)2 (right) [49]. The continuous line gives the “exact” thermal dependence of the spin susceptibility of S =1/2 chain with a near-neighbor AF exchange J that is indicated in each figure. T SP is the 30 K/16.5 K 3D SP transition temperature below which a singlet–triple gap of 133 K/–40 K opens in (BCPTTF)2 AsF6 /MEM–(TCNQ)2 . Note that in (BCPTTF)2 AsF6 the pseudo-gap opens at the onset, T1D , of 1D SP structural fluctuations.


Fig. 9




Fig. 9 : 

Left part (a): electron–phonon coupled Peierls chain: the modulation of atomic position (in red), by u q , leads via the electron–phonon coupling potential V ep (q ) to a modulation ρ (x ) of the electronic density. χ (q ,T ) is the static electron–hole response function of the 1D electron gas. Figures (b) and (c) in the right part give a schematic illustration of the effective potential experienced by the atom (in red) undergoing a PLD with respect to its surrounding (in blue); (b) corresponds to the displacive (adiabatic) limit and (c) to the order–disorder (anti-adiabatic) limit. The characteristic frequencies introduced in the text are also indicated. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)


Fig. 10




Fig. 10 : 

Mean-field thermal dependence of the 2k F critical phonon frequency,  , as a function of the dimensionless quantity ω A τ eh (see Appendix A).


Fig. 11




Fig. 11 : 

Imaginary part of the retarded 2k F phonon propagator ImD (2k F ,ω ,T ) defined by expression (A.9) of the Appendix for   (left) and 0.05 (right) as a function of the reduced frequency   [51]. This response is represented between   and   by steps of  .


Fig. 12




Fig. 12 : 

Coulomb couplings between (a) linear and (b) dipolar CDWs.


Fig. 13




Fig. 13 : 

(a) Structure of TMA–TCNQ-I projected along the stack direction b and transverse phase shift between CDWs located on zigzag-type TCNQ stacks. This figure gives the location of the dominant Coulomb interaction W d between CDWs (b) 3D Peierls transition temperature T P in function of W d calculated in Ref. [60] for the whole Y-TCNQ-I series. The size of the Y+ cation controls the diagonal coupling W d through the variation of the lattice parameter c . The continuous line gives T p calculated in the fluctuation regime of the amplitude of the Peierls order parameter (see expression (30) and Fig. 15).


Fig. 14




Fig. 14 : 

Structures of K0.3 MoO3 (a) and NbSe3 (b) projected along the stack direction b . Lateral phasing between the dipolar CDWs in (a) K0.3 MoO3 (red arrows) and (b) NbSe3 (red and blue arrows represent dipoles located on pairs of type-I and type-III chains, respectively). The anisotropy of the inter-chain couplings W given in the figure is obtained from the square of the anisotropy of the transverse correlation lengths ξ . (Fig. 14b is adapted from Ref. [61].) (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)


Fig. 15




Fig. 15 : 

Variation of T P scaled to   as a function of the inter-chain coupling W for the amplitude and phase (n =2)/domain wall (n =1) fluctuation regimes relevant for complex (n =2)/real Ising-type (n =1) order parameters. The variation of T P with W in the amplitude fluctuation regime is the same for complex- and real-order parameters.


Fig. 16




Fig. 16 : 

Two types of deformation of CDWs: (a) compression (dilatation) involves the longitudinal strain   and (b) shear involves the strain   (adapted from Ref. [66]). The vertical and curved continuous lines in (a) and (b) respectively are surfaces of constant argument Φ .


Fig. 17




Fig. 17 : 

(a) Elastic longitudinal compression ( )/dilatation ( ) of the phase φ (x ) of the CDW submitted to an electric field created by a difference of potential between the electrodes placed on the left and right sides of the sample (x is the chain direction). The continuous lines in the top panel are surfaces of constant argument Φ . The drift velocity, V D , of sliding CDWs for E >E T is indicated (adapted from Ref. [66]). (b) Difference of 2k F (x ) wave vector under the application of opposite ±I continuous (dc) and pulse (pc) currents larger than I T in NbSe3 . Note that 2k F (x ) varies linearly in the central part of the sample where 0.5 mm<x <2 mm. Below x 0.5 mm, near the current contact located in x =0, it is observed that 2k F decays exponentially with x (this effect is due to the ±I free-carrier charge conversion into the condensed moving CDW by adding or removing phase wave-fronts). (Adapted from Ref. [71].)


Fig. 18




Fig. 18 : 

Shear deformation of a sliding CDW pinned on a surface containing the chain direction. The interrupted curved lines are surfaces of constant argument Φ . The local wave vector q CDW (y ), perpendicular to the bent CDW surface fronts, induces a tilt of 2k F along the transverse direction y . (Adapted from Ref. [67].)


Fig. 19




Fig. 19 : 

Regular PLD modulation (left-hand side) and PLD phase dislocation (right-hand side) in K0.3 MoO3 . The top of the figure gives a representation of the PLD in direct space and the bottom of the figure shows the corresponding diffracted speckle spectra. Because of the tilting of the CDW wave vector q CDW with respect to chain direction, b , the dislocation mixes edge and screw wave front deformations (from Ref. [77]).


Fig. 20




Fig. 20 : 

CDW patterns for strong impurity pinning (a), weak impurity pinning (b) and weak pinning between strongly pinned CDW domains (c). In (a), the arrows show local shears of the CDW lattice due to individual phase adjustments on impurities. In (a) and (b), the dots represent local impurities and in (c) the boxes materialize strongly pinned domains. Parts (a) and (b) are from Ref. [82] and part (c) is from Ref. [84].


Fig. 21




Fig. 21 : 

Schematic representation of the so-called “white line” effect in diffraction and its analogy with holography in optics. (From Ref. [82].)


Fig. 22




Fig. 22 : 

Local variations of the phase Δφ (a) and of the amplitude u i (2k F ) (b) of the CDW around impurity centers (dots). Part (b) represents more specifically the induced 1D FO's of amplitude, u i (2k F ), in response to the impurity potential V imp .


Fig. 23




Fig. 23 : 

Right part: FOs with its phase shift of −π around a V impurity located at n =0 and its matching with the regular PLD/CDW via the formation of two solitons, each achieving a phase shift of π/2 at the modulation. The top panel gives the spatial dependence of the phase φ (n ) of the resulting modulation with respect to the regular PLD/CDW (interrupted lines in the bottom panel). Left part: experimental X-ray diffraction spectrum of K0.3 (Mo0.972 V0.028 )O3 at 15 K compared to the Fourier transform (continuous line) of the PLD/CDW modulation shown in the right-hand part. (Adapted from Ref. [88].)


Fig. 24




Fig. 24 : 

Graphical solution   of the implicit equation (A.10).

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