Charge orderings and lattice distortions in complex TCNQ salts

Physica 121B (1983) 134-152 North-Holland Publishing Company

CHARGE

ORDERINGS

S. ROBASZKIEWICZ

AND LATTICE

DISTORTIONS

IN COMPLEX

TCNQ SALTS

and T. KOSTYRKO

Instituteof Physics, A. Mickiewicz

University,

Poznan’, Poland

Received 12 July 1982 Revised 28 September 1982

A model for the description of charge orderings and distortions in complex TCNQ salts involving intra- and interchain electron interactions and electron-lattice couplings is investigated. Its thermodynamical properties are analysed in the limit of infinitely strong intramolecular Coulomb repulsion, treating intermolecular interactions within the two-sublattice H-F approximation. Stability conditions for various types of orderings are determined. Site charge ordering and dimerization accompanied by bond charge ordering are found to coexist only in a limited range of parameters diminished by the hopping and the electron-phonon coupling via Coulomb forces. Depending on the value of the latter, interaction phase transitions in the system can be continuous or not. Intermolecular Coulomb interaction stabilizes site charge ordering but it can also essentially enhance transition temperatures and energy gap in dimerized phases. For experimentally reliable values of parameters the energy gaps predicted by the model are in good agreement with those observed in D+(TCNQ)I.

1. Introduction Quasi-one-dimensional (q-1D) complex TCNQ salts with stoichiometry D+(TCNQ); (D’ denotes a diamagnetic donor with a closed-shell electron configuration) constitute an interesting class of compounds with various electronic and crystal structures and a wide variety of electrical and magnetic properties [l-lo]. The complexes range from insulators to highly conducting materials. Among them are the systems forming regular TCNQ chains (e.g. D = Qn) as well as the ones exhibiting dimerization (e.g. D = MEM) or tetramerization (e.g. D = TEA) [4]. In this group of salts electron charge orderings characterized by a periodic modulation of charge density at sites or bonds along TCNQ chain are frequently found. The site charge ordering can be alternating (i.e., . . . ABAB. . . type -different electron densities on the alternate sites, e.g. D = DECA [6]) or of the “island” type (i.e., . . . ABBA . . . type - different electron densities on the alternate pairs, e.g. [l, 4-Di-N-pyridinium methyl benzene]‘+ TCNQf [6]) whereas a characteristic example of the bond charge ordering is the modulation of electron density on alternate bonds in a dimerized (“Peierls”) state. Charge orderings can be accompanied by inter0378-4363/83/0000-0000/$03.00

@ 1983 North-Holland

molecular distortion (e.g. MEM [ll]), or can occur independently, i.e. in a regular chain (e.g. D = (CH$X-I&Fe [12]). Some of these salts exhibit also phase tranusually discontinuous, related with a sitions, change in the crystal structure or in a type of charge ordering, (e.g. D = MEM,HEM [ 111, DECA [13], Py [14]) as well as various anomalies of magnetic properties and conductivity (e.g. Me&P, Qn [41). Such a variety of physical properties cannot be explained in the framework of the usual band theory, taking into account only the hopping term t and its modulation by phonons (PeierlsFrohlich model, e.g. [3,4]). Moreover, there are unambiguous evidences from optical and magnetic investigations [l-17], thermoelectric power measurements [18] as well as theoretical estimations [19], that the electron interactions (both intra- and intermolecular) are important in the salts considered. Thus, we expect that a number of electron orderings and distortions can be explained if one includes into the model the electron-electron interactions and, consequently, their modulations by lattice vibrations. Evidently, there is a possibility of alternative explanations of the distortions and orderingsby treating them as induced by the chemical struc-

S. Robasrkiewicz,

T. Kostyrko

13s

/ Charge orderings and lattice distortions

ture of donors (pseudosoric chains [7,8]). In fact, the structure of donor chains in several TCNQ salts may have essential influence on the molecular and electron structure of the TCNQ chains as well as on the phase transitions. However, in this work we will take into account the donor chain effects, in an indirect way only, as a factor modifying (renormalizing) the parameters of the TCNQ subsystem (see also section 6). We will consider a q-1D tight-binding model as being an extension of the Hubbard model and take into account the direct electron interactions (intra- and intermolecular) as well as the couplings of electrons to lattice vibrations. Our main purpose is to find on the basis of this model the stability conditions for various types of charge orderings and to determine the influence of the considered interactions on thermodynamical properties of the system. In some aspects the present work is a conof our recent investigation [20] tinuation concerning the effects of interatomic Coulomb interactions and electron-phonon couplings via Coulomb forces in strictly 1D systems. We have found that these latter interactions can influence the electron and lattice subsystems in a qualitatively different way than the electron-phonon couplings usually considered. However, in the previous work we omitted the finite bandwidth effects, i.e. the hopping term and its modulation by intermolecular vibrationsresponsible for Peierls distortion as well as neglected interchain couplingsresponsible for long-range orderings (LROs) at T > 0 and phase transitions (only the effect of 3D phonons on the short-range intrachain interactions was taken into account). The model considered here is free from the above restrictions, though in turn due to its complexity approximate methods are needed for analysis. Our paper is also an extension of another work of one of us (SR) which analysed in the Hartree and Hartree-Fock approximations a 1D spinless fermion model with nearest-neighbours electron-electron interaction and electron-lattice couplings [21]. This model did not include the interchain Coulomb interactions and the electron-lattice couplings via Coulomb forces. In suitable limits our results reduce to those found in the aforementioned work.

The plan of the paper is as follows: In section 2 we define a Hamiltonian of the system and mention the approximations used. In section 3 we derive expressions for the free energy of the model and the self-consistent equations determining electron thermal averages and lattice distortion. Section 4 includes the calculations of the phase diagrams, the order parameters and the energy gaps. The ground state analysis of the model (section 4.1) as well as the results for T > 0 (section 4.2) are presented. At the end of this section we also point out the effects of the coupling vibrations electron-intramolecular (section 4.3). In section 5 we comment on the validity of the approximations and relate our results with other approaches. Then we discuss the results in connection with the experimental situation in complex TCNQ salts. A short comment on the effects omitted in our analysis follows in section 6.

2. Hamiltonian

of the model and approximations

We consider a system of linear chains built of N identical molecules each of them possessing n = NJN = l/2 electrons on the highest, partially filled orbital of n-type. The model will be described by the following Hamiltonian: H=He+H,_e+Hp,

(2.1)

where

Aiu

Aia n,4iaAi+1rt + WI

+ W” C AiUU

C

nAunA’id

?

(AA’)~OU (2.2)

Hc+t

=

2

(XAi

+

He=

c

(r’

-

X*i+l)[f~t(~~bCAi+lo

~W~Ai&Ai+l,’

+

h-c.)

1

(2.3)

cf&b;b,.

(2.4)

k

Here,

He is the Hamiltonian

of the 3D lattice,

H,

136

S. Robaszkiewicz,

T. Kostyrko

/ Charge orderings and lattice distortions

is the extended Hubbard model Hamiltonian and H,c represents the electron-intermolecular vibrations coupling terms (for clarity we omit here the coupling of electrons to intramolecular phonons which will be discussed separately in section 4.3). c and b refer to the electron and + phonon operators, respectively. hi, = CAiuCAim where A is the chain index, i numbers the sites in the chain, and u denotes spin. t is the intrachain hopping integral, CT is the intramolecular electron interaction. WI and W’ are the intermolecular Coulomb interactions from the same chain and from neighbouring chains, respectively. For simplicity, the intermolecular vibrations are taken to consist of a single acoustical branch Q only- the generalization on the other possible low-frequency vibrations such as torsional or libration modes is straightforward. The electron-acoustic phonon coupling is taken to occur via the modulation of the hopping integral (the E, term) [22,23,3] and the nearest-neighbour Coulomb interaction (the eW term) in intermolecular vibrations [20]:

da (RI

E =dR a

I

)

Ly =

t,

=

7

which excludes even virtual double T n*i J occupancies of sites by electrons (for n < 1) and leads to a complete decoupling of the translational and spin degrees of freedom. Recently, it has been shown rigorous by Bernasconi et al. [25] and Brandt [26] that in the lJ -+m limit the thermodynamical properties of the 1D Hubbard model as well as of its extension’ including additionally the intersite Coulomb interactions are equivalent to those of a composite system consisting of (1 - n)N spinless fermions and nN independent spins. Since the considerations of [25,26] are valid for an arbitrary form of the nearest-neighbour (nn) intrachain hopping integral and short-range Coulomb interactions (both intra- and interchain), they can be as well applied for our model. We obtain that for the limit U+ CC the Hamiltonian (2.1) is equivalent to the following one: 7

H=~+~+fi,_~t-H~,

(2.7)

where

w”.

WA.-&+I)

qi is the intermolecular displacement vector of the {Ai}th molecule. Transformation of XAito the phonon operators bk is given by

XZi

zAi nAi

(&)‘I*

e*R*lek6k,

(2.5)

where M is the effective mass of vibrational mode, ek the polarization vector, and & = bk + blk. The electron number condition, determining the chemical potential p reads

+ WL

c,

(AA’ i

nAinA’i

(2.8)

,

eiqR~,[~Ab(C:iCAi+l

+

ArnAiflAi+l]

+

h.c.) (2.9)

3

cE) denotes the creation (annihilation) operators for a spinless fermion at the site hi, nAL= c:icAi, and l/2 ek$(W){l

-

exp[ik(RAi

-

(2.6) We will investigate the presented model for n = l/2 under the assumption that U + CQ, i.e. that on-site Coulomb repulsion is much greater than any other interaction in the system (see discussion in section 5.3) [24-261. In such a limit we only have to discuss the Hamiltonian (2.1) within the subspace of minima1 eigenvalue of

RAi+,)])

.

(2.10) In eq. (2.9) we have taken into account the transformation (2.5). The number condition (2.6) transforms into (2.11)

We will analyse

the above

Hamiltonian

using

S. Robuszkiewicz,

T. Kostyrko

the error introduced by the approximation the those conclusions, in particular qualitative concerning the influence of the considered interactions on a mutual stability of the charge orderings will remain unchanged with respect to the exact solution (see discussion in section 5.1). In the following we restrict our considerations to an analysis of the two-sublattice orderings only. Such restrictions may be justified by the structure of the Hamiltonian (nn interactions, half-filled spinless fermion band). Comments on the more complicated orderings will be given in section 5.2.

the (broken symmetry) Hartree-Fock approximation (HFA) for the intersite Coulomb interactions and the mean-field approximation (MFA) for the electron-phonon coupling terms. MFA seems to be well substantiated by a threedimensionality of the lattice. Similarly, HFA may be justified by the 3D character of the Coulomb interactions. The most important advantage of the HFA treatment is its connection with a variational principle allowing the calculation of the approximate free energy which is an @per bound of the exact energy with a relatively small error. Applying HFA we expect that in spite of 3. Derivation

of an approximate

free energy

In order to obtain the free energy of the Hamiltonian use Bogoliubov’s variational principle [27]. Restricting assume a variational Hamiltonian Ho of the form

HO=

C

137

I Charge orderings and laffice distortions

(2.7) in the Hartree-Fock approximation the analysis to two-sublattice orderings

we we

~?nj(cLncrm+j + h.c.),

AmjfjPO)

or in k-space: k=2Q HO

=

c k=O

(4

k=k*a,

+

fk

fkbk

+

=

-fk+o,

(gk

+

ihk)ci+Qck

hk

=

(3.1)

,

-

hk+a

gk

=

gk+Q

,

where 29 = a* + b* + c* and a*, -b*, c* are elementary vectors in a chain direction, nkj, 4, fk, gk,hk are variational calculated by minimization of F,, given by

VeCtOr

of a reciprocal lattice, a is an elementary parameters (real numbers) which are

L=(H-Hoh,+Fo, F. = - 8 In tr exp[-(Ho Ho conserves

h,)

=

( c:,cA,+r

an electron-hole II +

+Aexp(iQR,)

- &?)/tY], symmetry

,

+ h.c.) = - /3 + 5(-l)“.

8 = kBT. and leads to an alternation

(3.2) of the relevant

averages: (3.3) (3.4)

For the strictly 1D model, Ho given by (3.1) is the most general one-particle variational Hamiltonian of the above properties. One can see that averaging with Ho (3.1) corresponds to the Hartree approximation (HA) for the interchain interaction W’ terms and the variational free energy F,, becomes equivalent to a free energy of a system of NL independent 1D chains in the effective site-staggered field - W’A; hence in the following the chain indices can be dropped. A and 6 are the charge order

138

S. Robaszkiewicz,

T. Kostyrko

/ Charge orderings and lattice distortions

parameters describing a modulation of the electron density at alternate sites and bonds, respectively, II is the average number of electrons per site and /3 is the average transfer correlation function. As we are interested in macroscopic distortions only we may treat phonon amplitudes classically according to large quantum number correspondence principle [28]. Thus in the calculations of (H)e we replace phonon operators by c-numbers:

and find their values 9 = Q = nJa only.

by a minimization

A ge = 4,oAocpo + %oA~o where A&Q, Aorpoare effects of Aopo # 0 are will not consider them In order to obtain transformation v and

of FVaT.Due to the assumed

form of Ho, A,cp,f

0 for 9 = 0 and

(3.6)

9

the measures of dimerization and dilatation of the lattice, respectively. As tht mainly quantitative (a temperature-dependent renormalization of t and WI’) we in further analysis and will assume Aoqo = 0. relevant thermodynamic averages we diagonalize Ho with the help of unitary obtain

(3.7) where

ek=p-++dfi+h2kfg2k,

7=

+,k>Q, -,ksQ.

I

(3.8)

Thus F. is given by

FO = - 8 c, After

finding

ln[2ch(+/28)]

the elements

A, 5; p on the variational A

+ i C,

(3.9)

ck.

of matrix Q we can calculate parameters fk, gk, hk, 4:

an explicit

dependence

of the parameters

2 =~~k(&k+Q)=-+

c,Bkgk> k
(3.10)

,$ =

-

$

c,

(c;ck+o)sin

k = - & c k B&k, k
where

Bk =;

th(a/28),

(3.11)

S. Robaszkiewicz,

here Jo = 4 fulfills the number

Using

the Wick theorem F,,/N

+&

c

139

/ Charge orderings and lattice distortions

condition:

we derive

= - i@+(W”+

T. Kostyrko

(H) and F,, becomes

zW1/2)n2 - i( W” + zW1/2)A2

Ek th(EJ20)-$

- i W’p*-

c ln[2 ch(&8)1+

where z is the number of neighbouring chains. After parameters giving minima F,, are found to be

a W”t’+

iA&&

+ ~&J’rpo&+

f&+2,.

tedious

though

(3.12)

simple

calculations

the variational

(3.13)

/_&=Cp= w”+zwL/2, Agcp&) cos k ,

fk = (t + i%‘lp-

(3.14)

gk=-(W”+2W1/2)A,

(3.15)

hk = (Ab+ A,Wcp& - W’@sin k >

(3.16)

CPQ= - &4b

(3.17)

+ Ao”P)l.n,.

Inserting eqs (3.13-3.17) into (3.8) (3.10) and (3.12) one obtains one-particle energy spectrum lk, the free energy F and the internal

the following energy E:

expressions

for the

lK = T{( W” + 2 W1/2)*A2 + e*[ WI’+ (Ab + Ag/3)2/f2Q]2 sin* k

+ [t + W”p + A$(Ab

+ A@)~2/LtQ]2 cos* k}“*,

(3.18)

F/N=-~(W”+.W1/2)+~(W”+zW1/2)A2+~W”/3*+~U/(’~* +&A%Ab+A,WP)Pt*+&j-

0

(Ah + Ao”p)‘p + &Fo,

(3.19)

E/N=(H)/N=:(W”+zW’/2)+I-$(W”+zWL/2)A2 - : W”p* - : W”c$’- +(Ab + Ao”p)2~2/f2Q, whereas the self-consistent state can be written as

A = ;(W’l+

equations

defining

the values

(3.20) of thermal

averages,

A, /3,.$ in equilibrium

.zW1/2)A 1 dk&, 0

p

=

$2

+

wlrp

+

Ag(Ab

+ A,wp)~‘lf2Q] j- dk Bk cos* k

,

0 ml2 5

=

f[

W” + (A& + A&)*/f&]5

1 dk Bk sin2 k 0

.

(3.21)

140

S. Robaszkiewicz,

The sums appearing

T. Kosiyrko

in (3.10) were replaced

/ Charge orderings and lattice distortions

in eqs. (3.21) by integrals

?r/z

Provided

that A # 0 and (or) t# 0, the one-particle

E, = 2{(W”+ zW’/~)~A~+

energy

spectrum

has a gap Eg: (3.22)

[ W"+ (A& + A~~)*/~,]*~*}‘“.

As the number of states per site is equal to l/2 in each subband the considered system is nonmetallic Eg # 0. In a dimerized phase (pof 0) the hopping and electron interaction parameters alternate along chain: Hz = C ~t[l+(-l)i~,](c;c*i+1+

h.C.) + W’[l+(-l)‘~w]n*;n*i+l+

C

the

(3.23)

W’nAinA,i.

(A\r’)i

Ai

The alternation

for

parameters

@, = Ab~o’r,

@, and aw are related

@w = Ag&

with Q, by (3.24)

W” .

The system of self-consistent equations (3.21) (3.17) has four different types of METAL: A metallic solution for which Eg = A = 5 = 40~ = 0. Cl: A solution describing site-charge-ordered phase (A # 0) without dimerization 5 = 0). P: A solution describing a phase with charge ordering on bonds (t# 0, A = 0) and the lattice (po# 0), i.e. Peierls (P) distortion. MIXED: A solution describing coexisting site- and bond-charge orderings (A # 0, tion of the lattice. The thermodynamically stable state can be found by comparing the free energy of

4. Thermodynamics

of the system

solutions: of the lattice

(cpo =

with dimerization

of

g# 0) with dimerizaall these solutions.

where

4.1. Ground state properties equations in the Analysis of the self-consistent because the inground state (GS) is simplified to the tegrals in these equations are reduced elliptic ones:

C, = ( W” + z W1/2)A ,

(4.2a)

C, = [WI + (Ab + A’&I)*/n&,

(4.2b)

C, = t + W”p + AF(Ab

(4.2~)

d*= C,+ Dq = (K, -

(4.1)

C’,,

+ A,Wp)/.n,,

q = (Cl - C;)/d2,

E,)lq .

Kq and Eq are the the first and second branch of solutions be reduced to one quantity q. In limiting cases

(4.3) complete elliptic integrals of kind, respectively. For every this system of equations can equation with an unknown some

analytical

expressions

S. Robaszkiewicz, T. Kostyrko I Charge orderings and lattice distom’ons

for the ordering parameters be derived:

and energy gaps can

141

form:

A. Cl phase (A # O,t = 0) (4.8)

E;’ = 2( W” + z W1/2)A , A ~1,

t 4 WI+ rWL/2:

p so;

(4.4a)

t % w” + .zWL/2:

A = % yZ exp(-$J)

,

zwI’+ 7rt

(4.4b)

YZr2(Wl+zWL/2);

Inequalities (4.8) show explicitly that in a weak coupling limit MIXED phase can develop in a very narrow range of parameters only (cf. figs. 1 and 2). As we checked by comparison with rigorous numerical solutions of eqs (4.1) the above approximate formulas describe dependences of Eg, A and 5 on the model parameters quite accurately, e.g. eq. (4.4b) gives the values of A and

B.P phase (f# 0, A = 0) E: = 2( WI’+ (Ab + Ao"#/&)

,

5 s p s l/4 .

t 4 W”3 (Ab)*/& t S W”, (Ab)*/&

(4Sa)

:

7r(t + p

w”)

Y~_2[W11+(Ab+Aowp)2/~o]’ 8-$

(4’5b) 0

C. MIXED phase (A, E# 0)

0.5

10

1.5

21)

2.5

:I

w4t (tG’ - z WL/2)t Pmix = (

~92

+

G’ = (Ab)*Ir&,

z IV”

t-t”

+

ztG’ W1/2 ’

,

(4.6a)

Fig. 1. The ground several fixed values denote the values is complete (/@,I =

state phase diagram of the model (2.7) for of b = A&/Ah and W’ = 0. Dashed lines of G’ = (Ab)2/f20t for which dimerization 1).

t s= wl’ + 2 WL/2:

(4.6b) (4.6c)

The solutions (4.6) exist if

(4.7) For WL = 0 inequalities

(4.7) take the following

Fig. 2. The ground state phase diagram of the model (2.7) for several fixed values of W”/t(W’ = 0). Above the dotted line (at b -0.52) the MIXED phase is unstable and the phase boundary between the ordered phases P and Cl is of the first order.

142

S. Robaszkiewicz,

T. Kostyrko

I Charge orderings and lattice distortions

Eg with an error less than 7% in comparison to numerical solution even for WI %- r. We performed complete numerical analysis of the self-consistent equations (4.1) and the internal energy (3.20) for W’ = 0, t, AL > 0 (the case t, Ah < 0 is connected with the previous one by Cc,+)= (- l)“Cc,+); canonical transformation: p, 5 -+ -p, -5; W’ can be easily incorporated but its effects are evident from analytical expressions). Ground state phase diagrams for several values of W”/t and b = Ag/Ab are shown in figs 1,2. The P state is stabilized by both types of electron-acoustic phonon couplings A& A$, whereas Coulomb interactions stabilize the Cl state. In a relatively narrow part of the phase diagrams, situated between the stability regions of the pure phases, the MIXED phase is stable. Its range of stability diminishes with increasing b and t. Moreover for b b 0.52 it becomes unstable with respect to the P and Cl phases at any values of other parameters. Transitions in the GS between the MIXED phase and the Cl and P ones are of the second order. This allows to use a condition /&-I = Pmix (t-0) for calculation of a phase boundary between Cl and MIXED phases. We then get G;=;

K,-D4

_ dqD;[

Gt

~

- D,)]”

(Ab)* a($

where

1+ ;d;b(K,

’

q is a solution

of the equation

2 lrt 2D,(W”+ zW1/2) 1

(4.10)

GS defines (for b ~0.52) a threshold value of the electron-phonon interactions necessary to stabilize the distortion in the GS. For ( W’l+ 2 W1/2) 4 t this quantity is well approximated by * 1 . As it is seen

from

this expression

(4.11)

both intra- and interchain Coulomb interactions destabilize the dimerized phase with respect to the Cl one. GE is quite low in the wide-band limit but it increases rather quickly with decreasing bandwidth. For b 3 0.52 only the pure phases Cl and P can be stable in the system and a phase boundary between them (Ec, = EP) is of the first order. In this case Gi is slightly different than the quantity given by (4.9) but its dependence on the parameters of the model is similar (figs. 1,2). The value b -0.52 (weakly dependent on WI/t) separates two types of the GS behaviour of the model. This is exemplified in figs. 3,4 and 5, for WI/t = 1.5. For 6 ~0.52 the order parameters 5, d (fig. 3) and the energy gap Eg (fig. 4) change continuously with increasing G’ = (A;2)2/&t, whereas for b > 0.52 we observe an abrupt change of all these quantities at the phase boundary. Eg increases at the transition. In each case the electron-phonon coupling via Couplomb forces A,W enhances the distortion cpo (fig. 5) and the energy gap (fig. 4) of the P and MIXED phases. Let us stress that in the approximation used A’ and AW influence only the P and MIXED phases (A’b” does not enter into the self-consistent equations determining the Cl state nor the corresponding free energy formula). Similarly the P phase is not affected by WI. On the other hand, all phases are affected by intrachain Coulomb interaction. In particular, the increase in W” reduces the internal energy of the Cl, MIXED and P phases with respect to that of the METAL phase (though the reduction is the greatest for the Cl phase) and enhances Eg in all these phases. It also raises the value of distortion in the P state and reduces this value in the case of MIXED state. As vibrations of the lattice cannot change the signs of parameters t and W”, the conditions ]@,1]< 1, j&] 6 1 provide us with an upper limit for the electron-lattice coupling parameters. The limiting distortion I@,,1= 1 is obtained for G& = 2/(1+

b/2) ;

(4.12)

in this case: and figs. 1,2

p = 5 = l/2 )

.

S. Robaszkiewicz,

T. Kostyrko

143

/ Charge orderings and lattice distortions

Fig. 4.

Fig. 3. 1.0 1 0.8-

Fig. 5. Figs. S5. The charge order parameters A, .f (fig. 3), the energy gap E8 (fig. 4) and the alternation parameter ground state as functions of G’ for U/ii/t = 1.5 and several fixed values of b = Ag/Ab (W’ = 0).

It means that only sically feasible.

G’ <2/(1+

b/2) may be phy-

4.2. Results for T > 0 K At T > 0 the system of equations (3.17), (3.21) cannot be reduced to one equation, as for T = 0, and numerical difficulties increase considerably. Nevertheless, in a weak and strong coupling limits several analytical results can be derived for some particular cases. In general, as the temperature increases there is either a phase transition (PT) P-METAL or Cl *METAL or two successive PTs in the system depending on the values of the interaction parameters. For b = 0 (i.e. A,W = 0), numerical calculations indicate that two successive transitions occur only for such values of the parameters for which the MIXED state is stable in the ground state. In that case there exists a

@, = Ah cpojt (fig. 5) in the

range of parameters with a sequence of transitions: MIXED t* Pt, METAL as well as a range with a sequence MIXEDttCl++METAL (see fig. 6). All phase transitions for b = 0 are of the second order. We have not performed numerical analysis of solutions for b# 0, but analytical results (see below) and conclusions of section. 4.1 suggest

I

I

I

I

-G’Fig. 6. A schematic plot of the kgT/t-G’ fixed u/l/it, W’lt (b = 0).

phase diagram for

144

S. Robaszkiewicz,

T. Kostyrko

/ Charge orderings and lattice distortions

that for b 60.52 the situation is qualitatively the same as for b = 0 (though some of the transitions may become of the first order), whereas for b > 0.52 besides the transitions Cl++METAL (2nd order) and P-METAL (1st or 2nd order), the discontinuous transitions between P and Cl phases are possible in a narrow range of parameters. As the transition Cl c, METAL is continuous for any values of the parameters the critical temperature Tcl is given by the equations:

n/2

2 P, = ;

I

t + w’lp, W” + (Ah + Ao”&)*/~, 42

= 2 1 dk th(e!i!/2kaT,) rr E$ = (t +

t + W”, (A$‘/&

I, 2

w1I+*w:/2=-rr

;

(4.16a)

:

kBTp = 0.834(t + W@,) exp(- 7,)

742

w’lp

t +

(4.15)

cases we obtain

kBTp = $( W” + (Ab)‘/f&)

t-=3 W’:

cos k,

tg k sin k,

W@,)cos k

and in the limiting n/z PC = $ I dk th(e’ki2kaTJ

cos k ,

dk th(eQ/2k,T,)

dk th(E;/2kaT&os

k

,

(4.16b)

= 0.57B;(0)/2, where

E; =

(t +

W’&) cos k .

(4.13)

Using approximate computations the limiting cases one can get

of integrals

in

t 6 w” + zWL/2: kBTC-;(

W”+ zW1/2) -&F;‘(O)

;

(4.14a)

t B W’ + 2 WlJ2: keTC = 2.28(t + Wap,) exp(-y,) (4.14b)

= O.S7B;‘(0)/2, where

pc= 2/F

t+p,w”

Yc=2(wI’+zwL,2)7-YP.

As we can see the ratio of gap to critical temperature is similar to that of BCS theory but it slightly depends on the bandwidth (similar conclusion concerns Tp- see below). Tel is increased by the Coulomb interactions W”, WI, diminished by the hopping whereas the electronphonon couplings A& A,W do not influence it. The P t, METAL transition can be continuous or not, depending on the b value. In the former case T, is determined by

p, = 2h,

Yp =

t +

2[

p,wfl

Wb+ (Ab + A:fl,ylL&]

Z- = ‘“, ’

As we see Tp is enhanced not only by both types of the electron-acoustic phonon coupling but also by the intrachain Coulomb interaction. Thus we conclude that WI increases both Tel and Tp, though an increase in T,-, by WI is relatively larger. In order to find whether the P-METAL transition could be discontinuous we calculated the fourth derivative of free energy of P phase with respect to dimerization parameter (a0 @). In a weak coupling limit the following simplified expression is found: 1 d4 -3t4 N a@; F T=Tp= (2G’)z(l+ b/3,)

1 dk ch2(&2kgTp)(# (w”/t +

G’(1+ bP,))

1

sin2 k+ (2,t)bEQ cos k 2

G’(1 + b&J

x [(&W-) -

sh(&kgT)l

+ (4/t)b2ef sh(ef/2kBT) cos2 k

p,=$

I

, (4.17)

S. Robaszkiewicz,

T. Kostyrko

/ Charge orderings and lattice distortions

For b = 0 the integrand is negative for every G’, so the phase transition is continuous (in agreement with numerical findings for this case), but b > 0 could change this conclusion. The possibility of the discontinuous Pt, METAL transition for b# 0 is also indicated by the results from the opposite limit: t+O. For t -0, Ab+O the self-consistent equations (3.21), for A = 0, are reduced to the following ones:

14.5

vibrations to the free energy vanishes in the considered case of two-sublattice orderings (see Appendix B). In the U + m limit the electron-intramolecular vibrations coupling term can be written as follows: H = He-imv + Hi,

(4.22)

where /3 = ;

W”p(l+

GWt2) 1 dk $ th(e,J28)

cos2 k ,

0 d2 tJ =

s

W@l

+ GWp2) 1 dk $ th(ek/20) sin2 k , 0

(4.18) where flqu are the optical phonon energy branches arisvibrations (imv), GqV= ing from intramolecular

Ek = W1[p”(l + GWe2)2 cos* k + e2(1 + GWp2)2 sin2 k]'" , GW = (A,W)*/fl& We get a solution X = $ th[X(l

(4.19)

.

bqv + b+qv,

(4.20)

p = 5 = X, where

+ GWX2)Wfl/2kgT]

.

(4.21)

Analysing eq. (4.21) one finds that for any GW > 4/3 the transition to the non-ordered phase (X = 0) is of the first order. 4.3. The effects of electron-phonon modulation of the site energy

coupling

via

For the sake of clarity of derivations we have not included into the Hamiltonian analysed (1.1) the electron-phonon coupling via modulation of the site energy E (i.e. molecular orbital energy plus the effect of the crystal potential). Within the MFA applied for the electron-phonon interactions in this work (section. 3.1) it appears that only the modulation of site energy by intramolecular vibrations influences the thermodynamics of the system, whereas the contribution of the electron-phonon coupling due to the modulation of site energy by intermolecular

are the parameters of the electron-imv coupling, M, is the effective mass of vth imv mode. To the site energy E in obtain Heh, we expanded terms of the intramolecular displacements Xin kept the linear terms only and transformed Xi, to the phonon operators bqy [20,23,29]. In the framework of the approach presented in section 3, the effect of the above term can be easily incorporated. Consequently, one finds that basic equations for thermal averages A, /I, 5 and free energy have the same form as eqs. (3.21), (3.18), and (3.19) if in the latter;zW’ is replaced by $W’ + G& where Gz = Z, (A&ylf20V, whereas the amplitudes of internal distortions induced by this coupling are given by (pov = - A&AI&.

(4.23)

(po,, describe static periodic deformations of the molecules along the chain (e.g. alternate expansions and contractions). As we can see the effect of A& couplings on the electron subsystem is

146

S. Robaszkiewicz,

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/ Charge orderings and latrice distortions

formally identical with that of WI, i.e. it enhances instability towards Cl ordering and raises the threshold value of electron-acoustic phonon coupling G: required for the occurrence of Peierls distortion. Following eq. (4.23) the intramolecular distortions (ID) are proportional to A, therefore they vanish simultaneously with Cl ordering and the intermolecular Coulomb interactions WI, WI stabilize them (see also ref.

WI). Recently the model with both types of electron-lattice coupling (A:” and A:) has been considered by Madhukar [23] and Rice et al. [29] with somewhat contradictory conclusions. While we find that the internal distortion and the dimerization of the lattice are strongly incompatible and their coexistence is possible only in a rather restricted range of interaction parameters, Madhukar [23] reported existence of the mixed P-ID phase for any tG’/GE, > 1, whereas Rice et al. [29] concluded that Ai and A;,, cooperate and the mixed phase is stable for any nonzero value of both couplings. However, the author of the former paper did not perform a numerical analysis of the phase stability and his incorrect conclusion relies upon the necessary (but insufficient) condition for the existence of the solutions rpo# 0, (po,, # 0. On the other hand, the finding of the other authors follows from a specific treatment of the electron-acoustic phonon coupling Ai- the first term in eqs. (2.3) and (2.9). Namely, the authors [29] approximate this term, which in k-space reads

where g

k,q

=

-

2i(2M0g)-“2e,

. e,

x [sin k + sin (4 - k)] , by

Our

results

indicate

that

the

neglect

of k-

conclusions comdependence Of gk,, changes pletely. (Taking into consideration the electronacoustic phonon coupling A: does not influence this statement because A$ = 0 - see Appendix B.) One should stress that our conclusions indicating strong competition between A& and Ai and the importance of k-dependence of glr,q are derived for the half-filled band case. On the contrary, when dealing with incommensurate deformations and partially filled bands the kstructure of gk,, may be roughly neglected, particularly in the limit of a nearly empty (or nearly full) band. In that case the inter- and intramolecular distortions become roughly proportional and the treatment of Rice et al. is justifiable.

5. Discussion 5.1. Comments proximations

on the applicability of the ap-

Our (broken symmetry) HFA treatment of the intersite electron interactions is undoubtedly best substantiated if the interactions have 3D character, i.e. if WI and IV’ are of the same order: WI b IV’ b 0.1 WI’. Of course one should keep in mind that mean-field approaches overestimate a tendency towards long-range orderings and the correct critical temperatures as well as semiconducting energy gaps can be (even in such case) essentially lower than those predicted by the approximation. For W’ 6 WNthe range of temperatures within which the applied approximation can give reasonable results diminishes. Some indications regarding the applicability range of the approximation for (qualitatively proper) description of the Cl phase in the case of small W’ can be obtained from the rigorous results for the zerobandwidth limit. In that case the model is equivalent to q-1D antiferromagnetic Ising model for which one has an exact solution in 2D and very accurate estimations T, in 3D [30]. Taking the rigorous T, as a measure of the 3-dimensionality of the problem it appears that the model exhibits 3D character in a relatively wide range of temperatures (0 6 ksT d W”/4 6

S. Robaszkiewicz,

T. Kostyrko

/ Charge orderings and lattice distortions

kBTc) already for very small values of W’/w” 3 10e5. Unfortunately, it is difficult to say something definite about the case of opposite limit t> wI*,l. In the case W’ = 0 the range of applicability of the HFA is restricted, in principle, to the case T = 0 only, because the HFA yields LROs and phase transitions at T > 0 which do not occur in strictly 1D system with short-range interactions. The results of the HFA for T > 0 may nevertheless be useful since the correct mean-square fluctuations of the order parameter may behave similar to the square of the order parameter in the HFA [31]. At T = 0 in the strictly 1D case the approximation used gives the excellent agreement with the known exact results concerning E0 and the value of short-range order both for the uniform case [32,33] as well as for the case of strong dimerizations [34] (among others one obtains the exact results for the isolated dimers limit and in the uniform case, for the t + 0 and WI+ 0 limits). It also correctly predicts the existence of the EB and LRO Cl for WI’> t in the case of a regular chain [33]. Disagreement in comparison with the exact solution concerns Eg for W’I < t. In the uniform case the HFA results predict E,>O for any WI > 0 whereas the exact solution indicates that for WI < t, Eg = 0 [33]. 5.2. Comparison

with other works

For Ag # 0 the effective Hamiltonian (3.23) can be reduced by the Jordan-Wigner transformation [35] to a rather general case of anisotropic compressible Heisenberg model (n = I/2):

H, = r C [I+(-I)i@t](SZSii+l+

SKiSKi+*)

Ai +

[P+(-l)‘b@,]S$S~i+*

+ W’

C

S&Si’i + const .

(5.1)

(AA%

The charge orderings in the electron model correspond to spin orderings in this Heisenberg model: Cl + Antiferromagnetic (AF); P + SpinPeierls (SP), MIXED + SP-AF ordered phase.

147

For WI = 0, p = b = 1 (p = WI/t), our ground state results are consistent with those obtained by Lepine [36], who investigated the possibilities of coexistence of AF and SP phases in the 1D isotropic Heisenberg model and found them incompatible. Lepine applied the approximation developed by Soos [37] which is effectively equivalent to the method described by us in section 3 of this work. For W’ = 0, p = b = 1, the model (5.1) was also studied in refs. [3w] with the help of the restricted HFA introduced by Bulaevskii [41], which however cannot describe magnetically ordered phases: AF, and SP-AF (for our comment to the above treatment see Appendix). Similar approach to the problem of dimerization was used by Kondo [42] who investigated the spinless fermion model identical with our effective Hamiltonian (2.7) except that he put W’ = 0. Kondo used HFA corresponding to that applied here with Ho given by (3.1) for gk = 0. Thus his results are identical with ours for “pure” Peierls phase but differ in an area of stability of MIXED and Cl phases in our solution, where we admit gkf 0. One of more important results of Kondo’s work was an assertion that electron-electron interaction (WI) can appreciably enhance the stability of P phase in finite temperatures. According to Kondo TP would depend mainly on the WI value. Kondo’s conclusion that electron interaction stabilizes the distortion of the chain is inconsistent with our results for G essentially lower than W”/t, (G’ < GS, eq. (4.9)) -where according to our solution the distorted phase is unstable with respect to the site-charge-ordered one. Our results are supported by rigorous solution of the l-, 0 (“Ising”) limit [43], which indicates that in such case dimerization is unstable for any WI, GW (w’(x) > 0). Kondo’s theory cannot describe this limit properly due to neglect of the solutions with A # 0 (gk # 0). The limit of very strong correlations (A;, t + 0) was investigated in our last work [20] where the Hamiltonian studied included also the nextnearest-neighbour interactions ( W!) term: WY C Ai

nAin*i+Z

.

(5.2)

148

S. Robaszkiewicz,

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/ Charge orderings and lattice distortions

This particular limit gives a possibility of rigorous treatment of the electron interactions (for W’ = 0) as well as the electron-phonon terms of coupling Ar -if we neglect long-range the latter interaction. The calculations made for t, Ai+ 0 complete in a substantial way the present results. As we found [20], for 2 W, < W1 (W, is WI renormalized by electron-phonon couplings) the Cl ordering is a ground state of the model. For 2W2 > W1 the ground state is a state ordered charge tetramerized (C2): . . . ABBA .. . and the AW term supports this phase. The tetramerized state is particularly interesting from the experimental point of view and poses a question about the coexistence of the ordered phases: Cl, C2, P, in a finite bandwidth model of the real q-1D crystal. This problem can be investigated theoretically by an extension of the present approach through the inclusion of four-sublattice solutions. However, the number of possible phases increases considerably causing great analytical and numerical difficulties. These problems will be treated in our future work on this subject. 5.3. Connection ments

between

the theory

and experi-

The most general structure of the D’(TCNQ); salts can be described as a four-sublattice ordering, in which each identity period along the stack contains one, two or four TCNQ molecules which can be unequally occupied by electrons. Due to the limitations of our model we can interpret within the presented theory only the (Cl, P, MIXED, orderings two-sublattice METAL)which are, however, very common among the complex TCNQs salts. The Cl state, with disproportion of electron density at alternating sites is realized in D+ D+ = (&H&r+ with (TCNQ); L4-41, (CH3CJ-I&Fe’ [12], NMeQn’ [45] and probably in Py+(TCNQ); [14]. Examples of the P state, with the dimerization of a chain and the bondcharge-ordering are D’ (TCNQ); with D+ = TPP+, EE,TCC, TEamine+, NEtPh+, NMePy+ [2,4,46,47]. The MIXED state, involving both

types of orderings (the site- and bond-chargeordering) and with dimerization of the lattice is rather rare in D’(TCNQ); salts. This is consistent with our theory which predicts a very narrow range of stability for the MIXED state. Till now, the only known examples are: DECA+(TCNQ): [13] and MEM+(TCNQ); (below T = 335 K) [ 111. Discussion of the results of the presented model in relation to experiments involves evaluation of the empirically reasonable values of the model’s parameters. From the analysis of the low-energy charge transfer excitations observable in infrared absorption spectrum, Mazumdar and Soos [17] concluded that in q_lD ion-radical WI\ - 3t/2 and t - 0.2 eV. salts: U - 3 w’], Theoretical estimations of Hubbard [19] give also rather high values of U, and Wl(U = 2.4 eV, Wj -L 0.9 eV, W! = 0.35 eV). These results supported also by other optical magnetic and TEP measurements [15, 16, 181 indicate that short-range electron interactions and in particular intrasite Coulomb interaction U are much greater than hopping, justifying the assumption of U + 00 in the analysis of charge orderings in the q-1D systems. In D’(TCNQ); salts, where D and A molecules crystallize in segregated stacks, the quantities U, t, WI’ depend probably only weakly on the donor’s type. This assertion applies also to some extent to the GE, value for which typical estimations give GE, - 0.02-0.1 eV [29]. Other parameters, i.e. G’, GW, W’ as well as the parameters omitted in the model, i.e. interchain hopping tL and disorder in donor’s subsystem, depend much stronger on particular donor and this very fact seems to explain a great variety of electron and crystal properties observed in D+(TCNQ); salts. For the interaction W’ every value between 0 and WI can be reliable in a particular case [48]. There are several, often mutually inconsistent estimations of G’ [29,42,48]. Anyway, the considerations of Rice et al. [29], Kondo [42] and Woynarowich [48] strongly suggest that in TCNQ salts G&/t cannot exceed 0.2-0.4 (the upper limit for the electron-phonon coupling derived in section 4.1, eq. (4.12) gives G’ < 2).

S. Robaszkiewicz,

T. Kostyrko

/ Charge orderings and lattice distortions

So far there is no evaluation of GW in the literature. For tentative estimations of b = assume reasonable to Ag/Ab it is phenomenological expressions for the dependence of WI and t on the separation x between TCNQ molecules: t(x) = to exp(W’(x) = T

ZJ~),

exp(-KX)

,

149

ported by the fact that Eg (inferred from conductivity measurements [l, 7,501) is slightly changed for various distorted D+(TCNQ); compounds. The systems which strongly differ from each other in values of distortion and in mode of overlap often have Eg o,fthe same order 8.x - O.lz A, Eg = 0.65 eV (e.g. D+ = MEM’: [ll], D+ = &PCH;: 6x - 0.35 A, E, = 0.3 eV [4]).

(5.3) 6. Final remarks

where v, n are the screening these expressions one finds twl’

bZ!L-!Lc 6 &=(l+K)/l’,

’ eQ

(

parameters.

Using

) >

(5.4)

and e.g., for v = 8 [49] and K = 0.6 (on the basis of the Hubbard’s estimations [19]) we get E = 0.2. This estimation leads to the conclusion that irrespective of the values of Wilt the area of stability of the MIXED state is very narrow and vanishes for W/l large enough. From this evaluation it follows that the MIXED phases observed in real systems should rather be explained by the effects disregarded in the model. Analysis of our results on the basis of the above estimations of parameters supports the hypothesis about the importance of Coulomb interactions in complex TCNQs salts. In particular, it suggest that the interaction WI can determine the value of Eg, for the salts exhibiting the Cl ordering as well as in distorted phases (P, MIXED). For the above-cited values of parameters WI and t both the approximate expressions (4.4), (4.5), (4.14) (4.16) and the numerical calculations (see fig. 4) predict very high values of TcI-METAL and TP-~~ti as well as of Eg (T = 0 K) (E, = 0.4-0.9 eV) which could explain why in real ordered systems phase transitions to a metallic phase are hardly observed. On the contrary, for W( = 0 even the largest, experimentally reliable value G’ = 0.2-0.4 is much too low to explain typical values of E,(O) (0.2-0.5 eV). Our conclusions on the importance of Wit are also sup-

Under the assumption U + 00, applied in our analysis of model (2.1) the electronic and spin degrees of freedom become completely decoupled which implies the Curie low for susceptibility [24-261. It is obvious that for a description of magnetic properties at low temperatures such an assumption is inadequate as it overlooks the effects of the intersite magnetic correlations (e.g. kinetic exchange - t2/ U). As follows from [ll, 521, these very effects cause a dimerized system with n = l/2 to have, with respect to the magnetic properties, the character of an 1D spin-l/2-Heisenberg model (with J 0~ 2t? Interdimer/Uee). Thus, at low temperatures the system can be unstable towards a spin-Peierls distortion which in such a case corresponds to a dimerization of the dimers, i.e. a tetramerization. An example of such tetramerization seems to be MEM(TCNQ), at T < 19 K [ll]. As the tetramerized phase develops on the basis of the dimerized one (P phase in our notation) one can expect that this phase is strongly incompatible with Cl ordering, similarly to the P phase. Moreover, because in the systems considered WI, tG’ S JGJ one can predict that energy gaps in the P and SP phases will be of the same order of magnitude (as in both cases the contribution to Eg from WI and G is dominating). Conductivity measurements in MEM(TCNQ), and HEM(TCNQ), confirm this suggestion. In our model the effects of donors are taken into account only in an indirect way, as a factor renormalizing the parameters of the acceptor chains. As the first step in the analysis of systems with complete charge transfer this approach

150

S. Robaszkiewicz,

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/ Charge orderings and lattice distortions

seems to be quite reasonable though, undoubtedly, some effects of the donors cannot be properly described in this way. In particular, in concrete compounds detailed properties of given donor molecules, i.e. sterical factors, distribution of charge, etc., can in a crucial way influence the crystal and the electronic structure of the acceptor subsystem -e.g. by imposing a definite type of intermolecular distortion or site charge ordering (in the case of strongly asymmetric donors with ordered dipole moments [S]) or by introducing significant disorder (in the case of random donor arrangements [3,8]). The latter effect, together with strong one-dimensionality, is probably the reason for the absence of longrange orderings in the highly conducting complex TCNQ salts (D = Qn, Ad, Acd) [2-4,8]. In addition, some phase transitions (e.g. for D+ = Me&P+, NPQ,‘, [4,51]) occuring with no change in periodicity of TCNQ chains may be attributed to internal reconstructions of the donor subsystems.

e&k sin k = hk cos k and HE takes the form:

Wi = xf&k

Appendix

for

A

In the case W’ = 0, b = 1 the model (5.1) was investigated in terms of constrained HFA developed by Bulaevskii [41] and applied for the problem of a spin-Peierls distortion [3W+O]. From the point of view of the variational principle a difference between our treatment and the constrained HFA can be put as follows: one can obtain Bulaevskii’s results by omitting in Ho (3.1) a term describing magnetic ordering (gk = 0) and by assuming that Ho commutes with the XY part of the Hamiltonian (5.1):

(A-1)

V-G,Hz,1 = 0 . (A.l) yields the following relation ational parameters fk, hk:

between

vari-

nk+O

(A.3)

Applying variational method (3.2) with the above trial Hamiltonian for isotropic dimerized Heisenberg model (p = b = 1, W’ = 0) and looking for min F,,, we obtain (in t units)

d7

+932

w; n-

I dq 5

th&/20)]

cos k

,

0

(A.4)

fk‘d,&osk ,

wk = ~cos*

The authors wish to thank Dr. R. Micnas many helpful discussions during this work.

-

+i@, tg k(cick+o + h.c.)] .

6k = Acknowledgement

(A4

k + CD:sin* k ,

(A.5) c4.6)

thus rederiving Bulaevskii’s results [41]. Constraining by (A.2) a number of independent variational parameters we get a form of the energy spectrum (AS) which is different from (3.18), and a free-energy whose value is higher than that obtained in the unrestricted treatment in the whole range of parameters (besides the @ = GW = 1 limit, where both methods give the exact value of energy). An important advantage of the restricted approach is that, in the case of a uniform, strictly 1D system, it predicts correctly the absence of an energy gap for p < 1 in contradiction to the result obtained with Ho given by (3.1). However, this constrained I-WA does not allow us to describe long-range ordered phases AF and dimerized-AF (in the spin model) or Cl and MIXED (in the electronic one). Moreover, it does not give the energy gap for p > 1, (@, = aW = 0) in contradiction to the exact result for T = 0 K [33]. For our main purpose in this work, i.e. investigations of LRO orderings in D’(TCNQ);

S. Robaszkiewicz,

T. Kostyrko

salts (where WI1> t and interchain Coulomb teractions seem to be important), we applied unrestricted HFA in our calculations.

/ Charge orderings and lattice distortions

inthe

151

Obviously, if more complicated orderings are considered, e.g. orderings with periodicity q = 2kr # Q in non-half-filled bands (i.e. for n # l/2 in the U--SW limit), the contribution of the above term can be important.

Appendix B In the considered Hamiltonian (2.1) we omit the electron-acoustic phonon coupling due to a change of the site energy E in intermolecular vibrations [S-55]. In the U+m limit the corresponding term of the Hamiltonian, site diagonal in electron operators, can be written as

HL?=

2

EE(X*i+m

-

X,4-m)&

7

(B.1)

Ai. m

where I$! are gradients of the degenerate threecenter integrals [%I, which stand for the rate of change of E with distance between Ai and Ai+, molecules. Other notations are the same as in section 2. Transforming x*i to the phonon operators (eq. (2.5)) and taking the momentum representation for the fermion operators one obtains:

=i+

Af&t-,G,,

U3.2)

where A; = 2i(2ML$,)-“*

c e& m

sin(qum)

.

Neglecting expansion of the lattice, from eq. (3.6) we get A$ = &A& but here Afj = 0, i.e. the term proportional to A - Q~ in the free energy vanishes. Thus we find that in the case of two-sublattice orderings the considered interaction does not influence thermodynamical properties of the system. It means, among others, that for the half-filled band case the Peierls instability in a discrete lattice cannot arise from the site diagonal terms of the electron-acoustic phonon couplingwhich is in agreement with the remarks of Pincus [56] and Madhukar [57].

References

[ll

W. Siemons, P. Bierstedt and R.G. Kepler, J. Chem. Phys. 39 (1963) 3523; R.G. Kepler, J. Chem. Phys. (1963) 3528. 121Z.G. Soos, Ann. Rev. Phys. Chem. 25 (1974) 121; Z.G. Soos and D.J. Klein, in Molecular Association, Vol. 1, R. Foster, ed. (Academic Press, London , 1975) p. 1. [31 L.N. Bulaevskii, Usp. Fiz. Nauk 115 (1975) 263. A. Bieber and F. Gautier, Ann. Phys. 1 [41 I.J. Andret, (1976) 145. and Semiconductors, L. Pal, G. PI Organic Conductors Griiner, A. Janossy and J. .%lyom, eds. (Akademiai Kiado, Budapest, and Springer-Verlag. Berlin, 1977). @I S. Flandrois and D. Chasseau, Acta Cryst. B33 (1977) 2744. I71 J. Ashwell, Phys. Stat. Sol. (b) 86 (1978) 705. PI K. Holczer, G. Mihaly, A. Janossy, G. Griiner and M. Kerttsz, J. Phys. Cl1 (1978) 4707. Conductors, Vols. I, II, S. [91 Quasi One-Dimensional BariSiC et al., eds. (Springer-Verlag, Berlin, 1979). Metals, W.E. Hatfield, ed. (Plenum Press, [lOI Molecular New York and London, 1979). Thesis, University of Groningen (1980) [111 S. Huizinga, unpublished; and in ref. [9], Vol. II, p. 34. [121 S.R. Wilson et al., in ref [lo], p. 407. [I31 S. Flandrois, P. Delhaes, J.C. Guintini and P. Dupuis, Phys. Lett. 45A (1973) 339; and in ref. [S], p. 499. [I41 K. Holczer, G. Mihaly, K. Pinter, A. Janossy, G. Griiner and W.G. Clark, in ref. [5], p. 507. [I51 J. Tanaka, M. Tanaka, T. Kawai, T. Takube and 0. Maki, Bull. Chem. Sot. Japan 49 (1976) 2358. Sol. Stat. [161 B. Torrance, B.A. Scott and F.B. Kaufmann, Commun. 17 (1975) 1369; J.B. Torrance, B.A. Scott, B. Welber, F.B. Kaufmann and P.E. Seiden, Phys. Rev. B19 (1979) 730.

[I71 S. Mazumdar iI81 [I91 PO1 WI m i231 [241 WI

and Z.G. Soos, Phys. Rev. B23 (1981) 2810. J.F. Kwak and G. Beni, Phys. Rev. B13 (1976) 652. J. Hubbard, Phys. Rev. B17 (1978) 494. S. Robaszkiewicz and T. Kostyrko, Physica B112 (1982) 389. S. Robaszkiewicz, Acta Phys. Polon. A56 (1979) 75. S. BariSiE, Phys. Rev. B5 (1972) 932,941. A. Madhukar, Chem. Phys. Lett. 27 (1974) 606. A. Gvchinnikov, Zh. Eksp. Teor. Fiz. 64 (1973) 342; D.J. Klein, Phys. Rev. B8 (1973) 3452. J. Bernasconi, M.J. Rice, W.R. Schneider and S.S.

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/ Charge orderings and lattice distortions

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