First-order CIC transition induced by Coulomb forces

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First-order CIC transition induced by Coulomb forces

S. Barišić, I. Barišić

To cite this version:

S. Barišić, I. Barišić. First-order CIC transition induced by Coulomb forces. Journal de Physique Let-

tres, Edp sciences, 1985, 46 (17), pp.819-824. �10.1051/jphyslet:019850046017081900�. �jpa-00232904�

L-819

First-order CIC transition induced by Coulomb forces

S. Bari0161i0107

Laboratoire de Physique ^des Solides, ^F-91405 Orsay Cedex, ^France

and Department ^of Physics, Faculty ^of Science, ^P.O.B. 162, ⁴¹⁰⁰¹ Zagreb, Croatia, Yugoslavia (*) and I. Batisti0107

Institute of Physics ^{of the} University, ^P.O.B. 304,41001 Zagreb, Croatia, Yugoslavia (Re~u le 17 avril 1985, accepte ^{le 4} juillet 1985)

Résumé.

Nous montrons que l’interaction Coulombienne entre les solitons chargés ^est attractive.

La transition de l’état commensurable à l’état incommensurable se produit alors pour ^une densité de solitons finie. Une telle transition est du premier ordre. Nous discutons brièvement l’importance

de ces résultats _pour les matériaux réels.

Abstract

It is shown that the long range Coulomb interaction between charged solitons is attrac- tive. The transition from the commensurate to the incommensurate state occurs therefore at finite soliton density. ^Such transition is of the first order. The relevance of those results to real materials is

briefly ^discussed.

J. Physique ^{Lett. 46} (1985) L-819 - L-824 ler SEPTEMBRE 1985, ¹

Classification

Physics ^Abstracts

64.70-71.45-71.SO

Much of the recent theoretical interest in the field of quasi one-dimensional (ld) conductors is related to the concept ^of charged ^solitons [ 1, 2]. However, ^{until now there is no} conclusive experi-

mental evidence that such solitons exist [3, 4]. ^The purpose of this Letter is to discuss briefly ^{one of}

the possible ^reasons for the absence of such evidence, namely ^the long-range ( 1 - r) ^Coulomb

interaction between the soliton charges. ^For simplicity ^{we shall be} dealing ^{here with} phase ^soli- tons, although similar results apply ^{to the} amplitude ^{solitons too.}

Our starting point is the usual tight-binding (TB) Hamiltonian [5]

HSSH denotes the much studied Su-Schrieffer-Heeger [ 1, 2] part ^{of the TB} Hamiltonian and

(*) ^{Permanent address.}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyslet:019850046017081900

L-820 JOURNAL DE PHYSIQUE2013LETTRES

is the 1

^r Coulomb interaction between the excess [5] charges in the unit cells n and m ^at dis- tance Rnm. The other electrostatic terms, ^which appear in the TB analysis [5] ^and depend ^{on the}

details of the electron distribution within the unit cell, ^{are of} the shorter range than H~ ^of equa- tion (2). They ^are therefore omitted here.

The long-range ^{nature of} H~ manifests itself in phenomena which involve large characteristic

lengths. ^{Indeed earlier} investigations [5] ^{of the} ordinary ^{1 - r} phonons ^have revealed that longi-

tudinal phonons, unlike the transverse ones, activate H~. ^{It will be seen here that} similar conclu- sions hold for longitudinal ^and transverse phonons and for solitons.

Close to the commensurability ^{of the order M # 2 it is customary to} introduce the phase ^and

the amplitude of the order parameter t/J = I t/J ei~ by singling ^{out the fast} component of the dis-

placement ^{with the wave vector 2} 7r/M~ ^along the chain axis x [6]. Assuming ^{that the} tempera-

ture is sufficiently ^{low to} suppress the amplitude variations, ^the charge redistribution is associated with a phase gradient according ^to

where 5p ^{denotes the extra} charge ^{due to electrons} accumulated within the unit cell of longitudinal

dimension 2 TrAf/6’

AMn at ^{the position x. The bar in equation (3)} denotes the spatial averaging along ^the chain and the overall charge of the chain is conserved due to the second term in equa- tion (3). Although ^this usually ^{small term} is omitted [1, 2, 6, 7] ^{it will} prove important ^{in the} pre-

sence of 1

r Coulomb forces.

The Coulomb _energy F~ corresponding ^to expression (3) is obtained by inserting expression (3)

in equation (2) ^and by changing ^the summation in the latter equation ^{from the} original dll ^{to the}

new Af~j. cells. This shows that the 1

r nature of F~ ^stems directly ^{from the 1}

^{r nature of} H~

in the TB Hamiltonian ( 1 ). On the other hand the short-range redistribution of electrons within the Md~~ unit cell is entirely ignored ⁱⁿ equation (3) and therefore in F~ ^{too. The role} of the short- range interactions has already been discussed at some length [8, 9], whereas the terms similar to

F~ ^were used earlier _e.g. in reference [10] ^{and more or less} implicitly ^{in reference} [11].

F~ should be added to the usual free energy Fph ^{of free phasons and to the} sine-Gordon com-

mensurability ^term FM ^[12], arising ^from HssH ^of equation (1). ^{It was} recently argued ^{that this} sepa- ration which is obvious in the Peierls limit when the electron-phonon coupling constant ~, is larger

than the Coulomb coupling constant v ⁼ np e2/d~~ ( 1 > ~, > v), ^is also valid in the opposite ^case

1 > v > ~ [6~ 13]. The effect of large ^{v can be absorbed in the value} [6, 13] of the low temperature

gap L1 (i.e. in the value of the amplitude ~ ~ (), the form of the phase Hamiltonian remaining ^the

same as in the Peierls case. The equation of motion for the phase ^{is thus}

irrespectively of the ratio of £ and v, with

,

2

1~2 d ^{M 2} 2 ~ y§

Here [11] ~

~ ^{M =} ~(1 ⁺ 4J~~,)-~ ~ - j~T ’ ^{and ~}

^{1 +} ~201320132013 ^is

Here [ 11 ] Ço = uF/d, ^{u =} uFC 1 ⁺ 4 d /~,cv2kF) , ~ ~ E ^{B~/ and Eo - 1 +} 3 v ³ d 2 d2 II ^{~ II is}

the dielectric constant due to the single particle electron-hole excitations across the semiconduct-

ing gap J. Only ^the parallel component JE~ ^{of the external fields couples to the phase.}

L-821 COULOMB FIRST-ORDER CIC TRANSITION

The ground ^state configuration follows from the integro-differential equation ^{obtained on}

setting ^{0 in equation (4). Let us also assume} _Elixt ^{= 0. In} solving ^the resulting equation ^we

start with the observation that the commensurate § = ~o ^{and the} single harmonic § = xx confi-

gurations ^are the solutions of equations (4) ^and (5). Indeed in both cases 5p ^of equation (3) ^and

hence Ell ^{and Fe vanish} identically. These solutions thus correspond ^{to an extremum of F} provid- ed that 0 = 27r/Mand/

2 kF - GIM respectively, just ^as in the absence of the Coulomb term.

Let us now turn to the soliton lattices, ^{which are the} remaining solutions of the Coulomb-free, sine-Gordon problem. ^{It can be} expected ^on topological grounds ^that they ^{continue to exist} although ^{in a} modified form in the _presence of F~ (i.e. Ell). ^{In order to} determine the nature of this modification let us first assume that v 1, and consider the commensurate-incommensurate

(CIC) transition. For v small it can be reasonably expected that the dilute soliton lattice remains a

reasonable concept, provided ^{that the} modification by Fe ^{of the} large ^{distance behaviour is allow-}

ed for. This implies ^that primarily the inter soliton distance d is altered whereas the soliton width I and the soliton form are less affected. Restricting ourselves for qualitative purposes ^to the single

chain solitons [1, 2] ^{we thus assume}

where the form

is taken over from the Id sine-Gordon theory [14]. Inserting the trial functions (6) ^and (7) ⁱⁿ Fe

we get

where

is the Coulomb self _energy of the soliton and

is the interaction energy of the solitons. The logarithmic singularity ⁱⁿ equations (8) ^and (9) ^is

related to the logarithmic behaviour of the Id Fourier transform of the long-range ^Coulomb

interaction [6]. Equation (9) shows that the long-range interaction of solitons is attractive. The effective attraction arises from the compensating p g charge - 1 g 00 2 ^of equations (3)

7C ax AfM q ( )

and (6), which resides between the solitons. Negative Vs ^{can also} be understood as describing ^the tendency towards the cancellation of Coulomb forces in the single-harmonic solution 4> ⁼ Xx.

The attraction between solitons favours the multi-soliton configurations. Indeed, ^{the mini-}

mization of the total free _{energy F} with respect ^{to the two} parameters d and 1 of equations (6) ^to (9)

leads to a transition from the commensurate phase ^{to a} soliton lattice of finite density. This transi-

tion is of the first order. It occurs for the band filling X ⁼ 2 kF - G/M equal ^to

L-822 JOURNAL DE PHYSIQUE - LETTRES The corresponding soliton width is

and the equilibrium distance, which results from the competition ^{of the} long-range ^Coulomb

attraction and the exponential, sine-Gordon repulsion ^is

Throughout ^the equations (10) the index zero denotes the familiar sine-Gordon values [14]

Xo ⁼ 4 ¹⁰¹ =4 /b-/~0- ^{As Xc} > Xo the region ^{of the} commensurate phase is extended and the

’It n

soliton width I increased with respect ^to 10. ^The inter soliton distance d is modified (decreased)

in an essential _{way. In} other words the Coulomb forces are smearing ^{out the soliton lattice,} making

the transition first order.

This is particularly ^{clear on} leaving ^{the extreme weak} coupling ^{limit v} log y~2013-7- _{~/o ~jj} ^{1. Let us}

then consider the two parameter ^solution of the soliton lattice type [14]

which for p ^{small is close to the} single ^{harmonic state. The Coulomb} energy corresponding ^to equation (11) ^is

The minimization of the total F with respect ^{to a} and ~ gives

where flo

~/~~ Z~ ₀ is the sine-Gordon value and v ⁼ 7E2 ^{X log} , ^The CIC transition j ^{v x} ~Z~)j ^II

*

value Xc ^{must be} sought ^{under the} assumption ^v > 1, ^{because for v 1 it is} given by ^equa-

tion (lOa). ^{In this} way ^we find

i.e. for large vxc tends ^to the value 1.1 Xo at ^{which the} commensurate phase exchanges ^its stability

with the single ^harmonic solution. The values of a, /3 ^and Fc at Xc ^are easily ^found from equa- tions (14), (13) ^and (12) : /! ^is considerably ^{reduced with} respect ^to flo, ^{and the} soliton lattice is smeared out again.

Equations (11) ^to (13) ^are meaningful ^not only ^at v > 1 of ( 14) but also for ^v 1 if this value of v is reached on assuming ^that x > Xc in either of the regimes (lOc) or (14). ^{However this corres-} ponds ^{to the less} interesting situation in which the soliton lattice (and thus the Coulomb effect)

is eliminated by ^the departure ^{from the} commensurability rather than by ^{the Coulomb} force itself.

L-823 COULOMB FIRST-ORDER CIC TRANSITION

In summary, the logarithmic ^Coulomb singularity, ^which appears in the expressions ^for xa

(I Oc) ^{for v 1 and} (14) ^for v > 1, makes the CIC transition first order. The result differs quali- tatively ^{from that} [8, 9] ^{for the} short-range forces which only broaden the solitons but conserve

the smooth CIC transition.

The logarithms appear due to the assumption that the soliton lattice is isolated on one chain.

In multi-chain materials charged soliton walls should be considered [15] instead. The 3d Coulomb

singularities ^are stronger than the Id logarithm ^{one and} the first order nature of the CIC transi- tion is then enhanced The appearence of the charged soliton walls is costly ^{in Coulomb} energy but not entirely prohibitive ^{Yp due to} the existence of compensating ^{~ g g} charges - ! _7C ^2013. _7~C

Let us finally briefly consider the relevance of the present ^{results to} the actual materials such as

TTF-TCNQ, conducting trichalcogenides and blue bronzes. As already ^mentioned above, ^v is entirely ^{absent from the} ground ^state energy of the C phase and, provided that it is not too

small, appears only asymptotically ^{in the IC} phase. ^v should be therefore determined from the excitations of those phases (full ^Eq. (4)) rather than from the ground ^state properties. ^{In the} single

harmonic limit of the IC phase ^the phason spectrum ^{is known} [11, 13] ^to contain in particular ^the

hard _{(9,, :--~} o~~/1.5 ~ longitudinal, beside the soft, transverse Frohlich mode c,~T : 0 (there

are other modes in between [ 13] ). a~L (and ^thus v) ^{can be} determined as the zero of the dielectric constant in the infrared _range [11]. Alternatively v ^{can be} determined from the plasmon ultraviolet

zero of the dielectric constant [6]. ^{It turns out} [6] ^{that v is the} largest ⁱⁿ TTF-TCNQ (u ~ 1) ^and

not very small (v ~ 0.1 ) in other materials. As concerns 6 it is the largest ⁱⁿ TTF-TCNQ (M ⁼ 3)

and perhaps ⁱⁿ o-TaS3 (M ^{= 1)} [16]. ^Indeed TTF-TCNQ ^{or rather} TSF-TCNQ ^{is the} only

material where the commensurability pinning ^{due to} FM ^has undoubtedly [ 17] ^{been observed} (in TSF-TCNQ ⁴ ~p ~ ² kF - G ^is absent). ^The transition from the undistorted phase ^{to the IC} phase is 2nd order and to the M ⁼ 3 C phase ^{1 st order. However} the first order nature of the CIC transition has not been observed either due to the experimental difficulties or to the electron and hole nature of the two types ^of chains which constitute those materials. Indeed if the structural transition concerns only [6] the electron chains the charges ^{of the} corresponding soliton lattice

are metallically ^screened by the hole chains. If on the other hand the transition involves both types ^{of chains on} equal footing ^{it can lead to} the alternation oppositely charged ^{solitons on} adjacent electron and hole chains. These possibilities, together with intermediate cases are

currently ^under investigation following ^{the mode} analysis of references [18] ^and [6]. ^Similar

remarks apply ^to o-TaS3, ^when it is described by ^{the M = 1 model of} weakly coupled inequiva-

lent chains [16]. ^In blue bronzes and in NbSe3 (M ⁼ 4) ^{6 is} very small [6]. ^Indeed no commensu-

rability pinning ^was observed in those materials despite ^the very small values of _X which attain 10-3 2 kF in blue bronzes [20]. ^{Even such} X corresponds ^{to the} x > Xc regime ^of equations (11)

to (13).

Acknowledgments.

Useful discussions with J. Friedel and L. P. Gor’kov are gratefully acknowledged

This work has been partially supported by ^YU-US collaboration contract DOE 438.

L-824 JOURNAL DE PHYSIQUE - ^LETTRES

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