PEIERLS INSTABILITIES II. GINZBURG-LANDAU MODEL

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PEIERLS INSTABILITIES II. GINZBURG-LANDAU

MODEL

S. Bari Ič, A. Bjeli

To cite this version:

JOURNAL DE PHYSIQUE Colloque C7, supplkment au no 12, Tome 38, dkcemhre 1977, page C7-254

PEIERLS INSTABILITIES 11. GINZBURG-LANDAU MODEL

S.

BARISIC and A. BJELIS

Institute of Physics of the University, Zagreb, Croatia, Yugoslavia

Rhumb. -- On donne une revue de I'approche Ginzburg-Landau des instabilites structurelles, qui sont observies dans les conducteurs unidimensionnels. La statique des fluctuations d'une chaine G o l k est dtcrite. Le traitement par Brazovskii et Dzyaloshinskii de la dynamique correspondante est aussi brikvement mentionnt, en relation avec la supraconductivitt de Frohlich. Finalement nous dCcrivons les effets de couplages interchaines sur I'ordre transversal des systkmes a chaines. Le modkle est utilisC pour interprkter les instabilitks de TTF-TCNQ a 2 k,.

Abstract.

-

The Ginzburg-Landau approach to the structural instabilities observed in quasi- one-dimensional conductors is reviewed. The statics of fluctuations in a single chain is described. The treatment of the corresponding dynamics recently proposed by Brazovskii and Dzyaloshinskii is further briefly mentioned, together with its relation to the Frohlich superconductivity. Finally, we describe the effects of various interchain couplings on the transverse ordering of the chain systems. This model is used for the interpretation of the 2 k,-instabilities in TTF-TCNQ.

1. Introduction. - The many-body theory for one-

dimensional electrons interacting via phonons and Coulomb forces has been intensively studied over the last few years [l]. In particular, the important role of the retardation effects in the phonon mediated and in the dynamically screened Coulomb interaction was reviewed in the companion paper I (Ref. [ l ] ) . The phonon retardation is particularly important at tempe- ratures higher than the Debye frequency (2 nT > m,) :

Within the logarithmic approximation then, the purely Coulomb CDW correlation function plays the role of the phonon self-energy. This scheme can be further simplified provided that the phonon mediated electron- electron adiabatic backward scattering is stronger than the Coulomb matrix elements (V/I U

I

2 1 ) . Close to the mean-field temperature the phonon self-energy 1s then given by a simple perturbation expansion in powers of U . The first and to some extent the second of those conditions suits well the interaction picture for some organic conductors and Pt-salts.

In the present text we start by elaborating upon the perturbative limit { 2 xT

>

m,, V

>

I

U

I

) using the Ginzburg-Landau (GL) expansion. The G L model for a single chain shows the strong dimensionality effects. We discuss here some of them, namely the static fluctuations [2-41, the commensurate pinning [5, 61, the dynamics of fluctuations and the related Frohlich conductivity [7, 81. Furthermore we extend the sta- tic GL model to the problem of coupled chains. We apply this model to the most studied among chain conductors, the organic salt TTF-TCNQ :

The observed three-dimensional ordering in TTF- TCNQ [9-1 l] shows the interesting feature of the tem- perature activated sliding in the relative phase of

CDWs on the neighboring chains. At a low tempera- ture the sliding is arrested by the commensurate pinning. We shall interpret this behavior within the GL model for the crystal lattice with two different sets of chains.

2. Ginzburg-Landau expansion for a single chain. -

The first few terms in the

1

U J / V < 1 expansion of phonon self-energy in the vicinity of the mean-field transition temperature T,, are given by

Here the full line represents the electron propagator, the dots the electron-phonon coupling constant I ( V =

I

I (2/Mmi), while the dashed lines are Coulomb

interactions. It is understood that the condition

1

U

1

< V is valid for both Coulomb potentials

U ( q l l w 2

k,)

and U ( q l , w 0) appearing respectively in the second and the third diagram of the expan- sion (1). We note that all terms in Eq. (1) are bilinear in the linear electron-phonon coupling I, i.e. that

Eq. ( I ) amounts to the harmonic approximation in the calculation of the effective deformation energy.

Let us first consider the intrachain deformation energy. The leading harmonic contribution coming from band electrons is given by the first term in Eq. (1). This term determines both the position of the most unstable mode as q l l 2z 2 k , and the longitudinal correlation length

5,

2z u,/T, as the coefficient in the expansion of the harmonic deformation energy in powers of q l l

-

2 k,. In order to rewrite these terms

PEIERLS INSTABILITIES JI. GINZBURG-LANDAU MODEL (37-255

in the GL form it is convenient to introduce the order parameter $( j ) through the relation

The slowly varying quantity $(j) is real for 2

kF

= 0, d!/2, and complex otherwise. It comprises Fourier components $, lying within the Kohn anomaly (1 q ( < l). uj in Eq. (2) is therefore defined non- locally, as the sum over displacements in a spatial range of the order of

to,

except for the site dependence associated with the factors exp(+ i 2

k,

d,, j).

The first term in Eq. (l), combined with the elastic energy and expressed in terms of +(j) gives rise to the usual GL form (with X = d,, j)

The anharmonic terms can arise from different sources, and in particular from the linear electron- phonon coupling itself. In order to fix ideas we can start from the anharmonic contributions expressed in terms of uj, and write

f'4'

= C

2

b(81,82,83) X

f 61.2,3

U j X Ujfdl.Ujf62.Uj*63

.

(4)

The coefficients b(S1, 62, 63) are expected to fall off rapidly as the neighbor indices Si increase. Written in terms of +(X) =

I

$(X)

1.

exp(iO(x)), Eq. (4) reduces to

f

(4' = bl

I

$(X)

l4

+

b2

1

$(X)

l4

COS [4 O(x)

+

8

kF

X]

+

Here we neglect the dependence of $ ( j

+

hi)

on Si since $ ( j ) is slowly varying in space and Si dll

<

to

for all relevant 6,'s. These terms were perturbatively taken into account in Ref. [8].

3. Static d = 1 fluctuations. - The harmonic ener- gy (3) and the first anharmonic term in Eq. (5) repre- sent the basic GL problem. The b, and b, terms of Eq. (5) oscillate strongly in space and thus average out to zero, except when 8

kF

or 4

kF

are very close to d f . This particular situation will be briefly discussed at the end of this section.

The problem defined above can be mapped on the GL problem with the more convenient coefficients by the simple rescaling [l21 of the variable X and the field $, .X' = .x/s and ~ ( x ' ) = or$(x). The corres- pondence obtained by an appropriate choice of s

and cc is given by

Here,

5,

= ( - c / u ) ' ! ~ and T , = a 2 (,/h,. We see that the associated problem is characterized by the single energy T,. In the purely electron-phonon model 6 , is such that the zero temperature electron gap A

-

I

I

$,

I

15 ol' the order 01' 7,. 1-he tcmprlalura I h IS t h z n

also of the order of A .

We note that the coefficients of the amplitude terms ( cp

1'

and

I

cp

l4

in the associated problem become large at temperatures T

<

T,. The amplitude fluc- tuations around its mean-field value are thus very costly. In contrast, the phase fluctuations are asso- ciated with the finite coupling constant (equal to one). In the low temperature limit the associated problem reduces thus to the noncritical continuous Heiesenberg model. It should be kept in mind that the described procedure is rendered possible by the absence of the ultraviolet cut-off, which is unnecessary for low (d 2) dimensions.

Te correlation function ( 9(xr) cp(0)

>

of the conti- nuous Heisenberg model cannot develop a singularity at short distances, because at X = 0 it has to match the

finite value of the auto-correlation function. The latter is equal to the squared length of the spin (- A 2).

In terms of the usual definitions for the critical expo- nents, the absence of the short distance singularity means = 1. Obviously, the matching argument is inapplicable in the discrete case, i.e. when the finite ultra-violet cut-off is required.

Cl-256 S. BARISIC AND A. BJELlS correlation length diverges according to the simple

power low T - ' (i.e. the critical index v equals one). The critical behavior of other thermodynamic func- tions can be derived in a similar way.

As already mentioned, the b, and b, terms become important if 8 k, and 4 k, are close to df

.

The situation in TTF-TCNQ suggests the consideration of the case 8 k , = dif. It is then convenient to rewrite the two remaining anharmonic terms as

where

+,

and

+,

denote the Cartesian components of the complex field

JI.

The coefficient b, can now be interpreted as introducing the cubic anisotropy into the symmetric d = 1, n = 2 problem. It is well known that such problem has the Ising critical behavior [l31 : The phase fluctuations are pinned down by the b, term (note that for b , = 3 b, the problem splits in two uncoupled Ising problems [14]).

Of course, for this effect to occur 8

k,

need not be just equal to d f . This question was examined in Ref. [ 6 ] . There, the b, term is treated approximately, neglecting the variation of the amplitude brought about by its activation. According to this work the b , term is unimportant if

The detailed account of the pinning effects in the opposite limit is beyond the present short review, and can be found in the mentioned paper [6]. We shall only state here that the exponential critical behavior of the d = 1 Ising model is not observed in TTF-TCNQ. However, similar terms might be of some importance for some Pt-salts. Also, as we shall see later, the analogous phase dependent terms are important in the transverse ordering problem of TTF-TCNQ.

4. Dynamics of the d = 1 fluctuations. - The main reason for the extensive dynamical studies of the electron-phonon model is that the dynamical proper- ties of vibrations determine the Frohlich conductivity. It was pointed out some time ago [l51 that the characteristic frequency m, of the Peierls phonon self- energy (1) is of the order of T,/(n, V)'''. It was then argued that o, is larger than wD in the chain conductors known up to date. We note that this assumption is consistent with the requirement Tp > W,, for the validity of the simple Peierls theory. o, > o, means simply that the time of the electron redistribution m;' is short with respect to the phonon time, i.e. that the Born-Oppenheimer approximation is valid.

On generalizing the low temperature mean-field treatment of Ref. [7], it was recently claimed [8] that this result remains essentially unchanged in the fluctuative low temperature limit. The starting point of this calculation is the previously mentioned fact that the amplitude of the order parameter weakly

fluctuates around its mean-field value

1

3/,

1

-

All at T

<<

T,

X A. To the zeroth-order and at low

temperatures the electrons then exhibit the gap A in their spectrum.

The fluctuations of the phase are instanteneously followed by electrons in the next order of this calcula- tion. Since there is no dephasing between phasons and electrons the former are dispersive. They obey the simple wave equation

i.e. they propagate with the velocity u

-

U, w,/o,.

The small amplitude fluctuations are also dispersive and occur at frequency o, A / o , z (n, V)'I2 U,. The

spectrum of phasons and amplitons is shown in figure I.

FIG. I . -- Dispersion curves for phasons ( 0 ) = u(q - 2 k,)) and amplitons (w = (n, V)'!' W,,) in the harmonic approximation.

According to Ref. [g] the phasons are weakly

attenuated by the higher order couplings such as ( ~ O I ~ X ) ~ , neglected in Eq. (5). The phason is optically active and its correlation function ( €)(X, t ) 8(0,0) ) determines the conductivity. The conductivity is then singular in the small o limit, due to the essentially dispersive nature of phasons. The surprising result is reached in this way, namely that the system is Frohlich superconducting at a finite temperature, in absence of the long range phase coherence.

Concerning now the electrons, their coupling to phasons and arnplitons destroys the band edges and introduces the exponential tails of the density of states into the gap region

[a].

For T

>

(n, V)'I2 oD the phasons and amplitons act on electrons as an entirely stochastic static disorder potential [8]. The change in the density of electron states is small, and the starting point with A-gap in the electron spectrum seems justified. However, in the presence of static disorder the nature of the states is changed from the extended to the localized type [16, 171. It is not clear that with this latter type the electron response to phasons remains adiabatic.

PEIEKLS INSTABILITIES 11. GINZBUKG-LANDAU M O D E L (-7-257

(phason) [7] this question to some extent is academic. Such a mode is obviously very sensitive to commen- surability, impurities or other phase pinning mecha- nisms, which are thus very harmful to the Frohlich superconductivity.

Let us consider again, briefly, the commensurability pinning with 8

k,

= d r . Neglecting after Ref. [6] the

amplitude changes, the phase obeys the wave equa- tion [18, 191

a''

- = - p I a'8

+

p,

sin 4 B ,

a t b 2 ax2

instead of Eq. (8). This is the well known sine-Gordon equation with /l2

-

b,. The solution of this equation is such that the phase lies in one of the four extrema over the large part of the chain, switching to another equivalent extremum in the narrow domain wall. The appearance of domain walls is not surprising since we have seen that the corresponding static problem contains the cubic anisotropy.

When the described configuration moves with the uniform velocity it is called soliton. The solitons produce a sharp peak in the infrared spectrum with a DC value which decreases exponentially with decreasing temperature [20]. Rather than pre- senting the detailed discussion of this effect, the authors of Ref. [83 consider the homogeneous phase motion around the extremum of cos 4 B. The lowest phason frequency is then shifted to the finite value

cc),

x

2(2 which is further attenuated by the 84 term. The corresponding conductivity decreases expo- nentially at low frequencies, after going through the maximum at o = 3 m,.

The effect of impurities on the conductivity was also considered starting from Eq. (8). It has been shown [21] that at T = 0 the low frequency conductivity is exponentially small, this result being independent of the impurity concentration.

5. Interchain coupling. - According to the discus- sion of Section 2 we describe each chain by an energy density comprising Eq. (3) and the first term of Eq. (5). Further we introduce the bilinear interchain terms

The Coulomb contribution to Eq. (10) comes from the second diagram of Eq. (1). The corresponding coupling constant is equal to the potential [22]

where d, is the distance between n-th and (n

+

p)-th

chain. The complete coupling constants in,,+, in Eq. (10) may contain, beside the contribution (11),

also the contributions of other origins. Indeed, the data on transverse elastic constants [23] indicate that

in the case of TTF-TCNQ the elastic contributions may also be important. In contrast to the Coulomb term (I I) which is limited to the narrow range of the Kohn anomaly, this term is present in the large range of the wave numbers q,,. Other contributions may come from the direct interchain overlap, the splitting of CDWs across the molecules, etc. The molecular polarization model with the particular emphasis on the coupling between CN group of the TCNQ mole- cule and the sulphur atom of the TTF molecule was discussed in Ref. [24].

Beside the bilinear coupling (IO), it will appear important to assume (at least) a weak fourth order interchain interaction. Before introducing these terms we shall determine the effects of the bilinear interaction terms (10) on the three-dimensional ordering.

6. Approximations. - In the GL theory for the single chain set it is necessary to distinguish between the cases of strong and weak interchain coupling [25]. For

CO,

> d,, i.e. l.,,,,, > a', some simple transfor- mations of variables transform the problem into the isotropic three-dimensional form even though

CO,

.< ~ O l l . However, if

C,,

< d , (Rn,,+, < a') the one-dimensional fluctuations can develop m the high temperature regime,, above the so called crossover temperature. In this case one has to treat the large one- dimensional fluctuations exactly. Performing again the operations which led to Eq. (7), we find the addi- tional correspondences which concern the interchain couplings

It is clear from Eq. (12) that the crossover tempera- ture in the case of bilinear coupling is proportional to T,(l/a1)"'. The same dependence is obeyed by the ordering temperature 141. This exact result can be also obtained from the mean-field treatment [26-28, 31, which shall be thus used below for double chain systems.

We continue to distinguish between these two extreme limits for the interchain coupling in the case of two sets of chains. In the case of the TTF-TCNQ chain lattice (figure 2) we assume that both transverse direc- tions, although not equivalent, correspond to the same type of conditions on

c,,.

Thus we d o not consider here the intermediate situation in which the coupling is strong in one direction and weak in the other.

S . BARISIC AND A. B J E L I S

FIG. 2. -- Chain lattice of TTF-TCNQ. In the GL pointcharge model the effective unit cell is two times smaller than the actual

unit cell.

FIG. 3. - CDW ordering in 'ITF-TCNQ at (a) q., = a*/2 and ( b ) g. = 0.

The following extremely simple reasoning leads to the law for the change of q, with temperature : There are two bilinear, q,-dependent, energy terms. The first renormalized out [29]. The meaning of this expression _{one comes from the interaction between TCNQ} differs from that of Eqs. (3), (4)9 (l0). Its _{and T T F CDWs on neighboring ,-hains. It vanishes by} variables are the quantities $.(X) averaged over the _{symmetry when q, = a*/2 (figun 3a). ~h~ energy} intrachain fluctuations. As was already stated in _{gain produced by the finite}

_I

@,

I

on T T F chains is Section 3, the crossover to the three-dimensional _{given by}

behavior occurs in the regime with only phase ffuc- tuations present. The coefficients of

(

(

*,

)

l2

and

I

<

$, )

l4

are then easily found to be

respectively. Here ( $,

I2

= a'lb. Finally, since

<

$, ) is the intrachain average over the strong phase fluc- tuations, the only interchain terms contributing to the effective expansion are those which depend on the phases. This includes of course the bilinear contri- bution (IO), but only some among the fourth order terms.

The steps which follow after the above intrachain averaging are the same in both limits and consist in the straightforward minimization over the remaining variables in the free energy. Here we shall thus discuss the physical ideas behind the full mean-field limit, keeping in mind that after the renormalization of the coefficients a,, 6, (Eq. (13)) this discussion also applies to the weak coupling case.

7. Phase transitions at 54 K and 49 K in 'ITF-

TCNQ. - The three-dimensional ordering of ITF- TCNQ with the longitudinal wave number equal to 0.295 h* [9, IO] (interpreted as 2

k,)

is then explained as follows [22, 30, 311 : At 54 K TCNQ chains get deformed and order with q, = a*/2. The T T F chains are inactive (figure 3a) until the temperature reaches 49 K. The CDW developed on the TTF chain below 49 K increases as the temperature decreases. If only the bilinear coupling were retained this CDW would at very low temperatures tend to order in opposition of phase with the CDW on the neighboring TCNQ chain. As shown in figure 36 this would correspond to g,

=

0.

for small a*/2

-

q,. Here OQ - 0, is the relative phase between TCNQ and T T F CDWs. Its equilibrium value is determined by

cos (OQ

-

8,) =

-

sign ;l,,,

.

(14a) so that

fd2,'

is always negative, as is seen from the last expression in Eq. (14). The second energy term comes from the coupling between next neighbor TCNQ chains. It is minimal when the TCNQ CDWs are opposed in phase (ga = a*/2), assuming that R, is positive. The small departure of q, from a*/2 leads to the energy loss

The terms (14) and (15) are the only q,dependent contributions to the deformation energy arising from the bilinear coupling. The minimization of the total energy with respect to g, yields

PEIEKLS INSTABILITIES 11. GINZBURG-LANDAU MODEL ('7-259

ordering are then the solutions of the respective equations [31]

aQ(T) =

+

l

).c,2

l

7 (1 7a)

and

For TTF-TCNQ the temperature (17a) (54 K) is larger than the temperature (17b) (49 K). In the opposite case in which the temperature (1 76) is larger than the temperature (17a). both chains order simul- taneously at an intermediate temperature and with the finite value of a*/2

-

q, [3 l].

The complete analysis of the interchain problem confirms therefore the original statement [22] concern- ing the phase sliding from the edge of Brillouin zone (g, = a*/2), i.e. that it occurs

if

the 'ITF chains order separately. This is the case which was considered by Bak and Emery and developed later by Bak 1301. The discussion of all possible types of ordering, with the bilinear terms (14) and (15) taken in the non- expanded form, is carried out in Ref. [29]. It should be however pointed out that the question of the a-axis ordering is not settled. The alternative explanations can be found in references [24, 32, 331.

The long range order in the c-direction occurs together with that in the a-direction. The neighboring chains will have either the same or the opposite phases depending on whether A,/, is negative or positive. In both cases g, = 0 since the TTF-TCNQ lattice constant in figure 2 contains two unequivalent pairs of chains. The interchain coupling of TTF and TCNQ chains in the diagonal a-c directions might lead to the temperature variation of qc [34]. However this coupl- ing is probably weak since no departure of 9, from zero was observed.

8. Commensurate ordering at 38 K. - Upon decreasing the temperature below 49 K the sliding of g,

progresses towards the interior of the Brillouin zone until the position 9, = a*/4 is reached. This position corresponds to the commensurability of the order four in the transverse direction, and requijes a particular treatment due to the appearance of Umklapp terms in the GL energy. The problem differs from the usual commensurability treatments since one (2 k,) of the three components of the wave vector is incommen- surate at all temperatures. This particularity leads to the discussion presented in this Section.

For simplicity we work at first with the order parameter of the only one set of chains (TCNQ) [35].

The most unstable deformation at q, f, a*/4 is u+,(n,j) =

I

$ + Q

l

cos (O+,

+

2 kFjb

+

%an)

( 18a) or

~ - ~ ( n , j ) =

I

$-Q

I

cos ( K Q

-

2 k,jb

+

q,an), (186)

written in terms of the Fourier components

$Q(qa3

+

2 k p q, = 0) =

1

$ * Q

l

exp(ifl*~) (figure 4)

FIG. 4. -- The star of wave-vectors in a-b plane for the ordering with qo < a*/2.

The moduli of the components $,, are normally involved in two fourth-order invariants of the transla- tional group, entering the mean-field energy

This energy does not depend on the phases

.

O

,

,

When qa = a*/4 an additional Umklapp invariant can be constructed, namely

We notice that $+, and I/-, are combined so as to

eliminate the incommensurate wave-Yector compo- nent 2 k,, taking advantage of the Umklapp in the transverse direction. This adds the phase dependent energy to the contribution (19)

Within the mean-field treatment of interchain coupl- ing the commensurability term (20) acts only when

qa = a*/4, while the discussion concerning Eq. (5)

suggests that it might be important within the width (qa

-

a*/4) <

u(n, j ) . The fourth order terms are positive i.e. the GL theory stable, if

B ,

> O ,

B 2 + . 2 B 1 - 2 1 B 3 1 > 0 .

In the particular case of local anharmonic energy one has B , = B , = B,/4 > 0.

The transverse sliding starts out continuously from the point q, = a*/2 only if a single leg of the wave- vector star in figure 4, $ + Q or $ -Q, is activated for q,

5

a*/2. This is not surprising since for qa = a*/2,

$ + Q = $-Q, i.e. the star reduces to only one leg. For q, < a*/2 the one leg solution u + ~ (Eq. (18a)) o r u - ~ (Eq. (18b)), is more stable than the solution

U + Q

+

U - Q (with

I

$+,

I

=

1

$-Q

l),

which corres- ponds to the simultaneous activation of both legs in figure 4, if the anharmonic intrachain coupling dominates over the anharmonic interchain coupling. This is indeed a very natural assumption. However, the free energy of the one-leg solution is continuous upon reaching the point a*/4, whereas the two-leg solution reduces suddenly its energy by the activation of the commensurability term (20). This term can always be made negative by a suitable choice of the phase 8 + Q

+

If the anharmonic coupling is purely intrachain, the two solutions become energeti- cally degenerate. The small attractive interchain coupling stabilizes the two-leg solution with respect to the one-leg solution. Of course, the repulsive coupf ing acts in the opposite direction. The possible explanation of the first order transition at 38 K and q, = a*/4 is thus that there the deformation suddenly changes from u + ~ (or u - ~ ) to U+.Q

+

u - ~ , due to the attractive anharmonic interchain ~nteraction.

The presence of T T F chains complicates somewhat the above picture. This problem is fully discussed in Ref. [29], while here we consider only the simplest case. Beside Eq. (20) one also has to take into account Umklapp terms with the T T F chains. The Umklapp contribution in which only T T F chains are involved is given by

analogously to Eq. (20). It contains the additional phase 4 qa(a/2) which arises from the position of T T F chain at a12 with respect to theTCNQchain.

In the limit of entirely intrachain anharmonic coupling the terms (20) and (21) are the only Umklapp terms. Then the one-leg solution remains more stable than the two-leg solution even at g, = a*/4. The reason is that one cannot determine all the relative

[ l ] BARISIC, S., J . Physique Coiloq., to be published in Proceedings

of the Congrh de la Societe Franqaisc de Physique.

[2] BALIAN, R. and TOULOUSE, G., Ann. Phys. 83 (1974) 28.

phases in such a way as to have both terms (20) and (21) negative. Note in this respect that the bili- near Q-F coupling (Eq. (14)) imposes the values of both relative phases 8 + Q - O+, and

-

8-, as equal to x or 0, depending on whether

A,,,

is negative or positive (see Eq. (14~)). In both cases one has

Consequently the relative phase which favors the two- leg TCNQ deformation implies simultaneously an energetically inconvenient phase relation between the TTF deformations. In consequence the two-leg and the one-leg solution are not exactly degenerate at qa = a*/4. This leads to the additional condition for the stability of the two-leg solution : The energy gain coming from the interchain anharmonic terms (e.g. BdF

@G

$: and/or B&

I

14)

has to overcome the intrachain T T F loss (-- BF I I),

14).

This require- ment does not seem too restrictive. In particular it is expected from Eq. (16) that the ratio (

l/(

$,

I

is large and may thus compensate for the possible smallness of the ratio B,/BF.

9. Conclusion. - The GL model is presented here in its simplest form and can serve only as a crude description of the physical situation in the real chain systems. Nevertheless, already in the case of the simple chain it gives some conceptually interesting and pro- bably observable fluctuative effects like Frohlich conductivity and phase pinning followed by the non- linear soliton behavior. An important question which arises in this context concerns the electron localization caused by one-dimensional low frequency CDW fluc- tuations. The answer to this question will eventually lead to the picture in which soft phonons play the role of the intrinsic intrachain disorder.

Other improvements making the GL model suit better the situation in real systems are also necessary. Thus, although the GL model explains satisfactorily the rather complicated 2 kF-ordering in TTF-TCNQ, very probably it does not contain the interpretation for the 4 kF-scattering. Two improvements seem to be promising. The first one is the treatment beyond the logarithmic approximation of the situation in which the Coulomb interaction is of the same order of magnitude as the phonon-mediated coupling. The second concerns taking into account the internal structure of (planar) molecules which form the chains. The effects of the intramolecular charge redistribution and of the intra- and interchain bonding then may strongly influence, not only the overall cohesion of the crystal, but also the details of the CDW ordering. Actually each of the above mechanisms were already tentatively used [36, 37, 241 in the interpretation of the 4 kF-ordering.

[3] YEFETOV, K. B. and LARKIN. A. I., Zh. Eksp. Teor. Fiz. 66

(1974) 2290.

PEIERLS INSTABILITIES 11. GINZBURG-LANDAU MODEL C7-261

[5] BKALVVSK~I, S. A. and D~YALVSHINSKII, 1. E., Low Tempera- ture Physics, LT-14 4 (1975) 337.

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