Coupled charge density waves in nearly one dimensional systems

Solid State Communications,Vol. 19, pp. 1189—1192, 1976.

Pergamon Press.

Printed in Great Britain

COUPLED CHARGE DENSITY WAVES IN NEARLY ONE DIMENSIONAL SYSTEMS F. Woynarovich, L. Mthály and G. GrUner Central Research Institute for Physics, H-1525 Budapest, P.O.B. 49., Hungary (Received 19 March 1976 by P.G. de Gennes) A mean field theory of coupling between charge density waves CDW in linear chain systems is described. It is shown that interchain coupling can stabilize CDW’s and lead to a semiconducting behavior. The model is applied to semiconducting TCNQ salts. LATTICE distortions observed in charge transfer salts of the organic acceptor TCNQ are usually discussed in terms of electron—phonon interaction which leads to the well known Peierls ground state. This behavior is well documented in TTF—TCNQ’ and it is believed2 that a gap of the order of 0.1 eV observed in semiconducting salts has the same origin. A simple estimation of the relevant energies however suggests that this may not be the case. In the conventional Peierls argument the energy gain due to the opening up the gap at the Fermi surface is balanced by the energy loss arising from the distortion of the lattice. The former is estimated to be Ee i~.(i~s/W)where ~ is the energy gap, W is the bandwidth. The distortion energy is given byEi~~ 2u~mwhere u is the lattice distortion, m the mass of ~ ,j0ion or molecule, co the 0 the bare phonon frequency at 3 with 2kF. 0.15 Taking typical eV,TEA(TCNQ)2 W 1.0 eV, mas a 200 mH,example ~ 0.035 A and w~ 6meV4 we obtain Ee 0.025 eV and Eia~ 0.001 eV. The order of magnitude difference between the two characteristic energies5 implies that the gap ~ -~

-~

has a different origin and that6the lattice distortion has Both on-chain and interonly aCoulomb secondaryforces importance. chain may be responsible for this gap. When on-chain Coulomb effects are important, strong ld fluctuations well above the phase transition are expected, while for dominating interchain coupling typical 3d effects are expected. No clear-cut distinction exists between these two cases, although it is generally accepted that the former situation holds for the well conducting TCNQ salts. We believe however that a large interchain coupling regime may apply for some organic salts. As a prime example we mention NMeQn(TCNQ) 7 a sharp, non reversible phase transition from well conducting 2. In thisa material modification (with a gap ~ 0.12 eV obtained from the conductivity), to a phase with intermediate conductivity and with ~ 0.6 eV occurs. This transition is associated with the evaporation of the solvent molecules between the chains. We believe that the plausible explanation for

this behavior is that the highly polar solvent molecules destroy the interchain interactions. Solvent evaporation increases the interchain coupling and leads to coupled Id chains. We suggest that the Coulomb coupling between the charge density waves (CDW) on the neighbouring TCNQ chains is the main source of the gap, and that the lattice distortion if it exists arises from a coupling between the CDW’s and the lattice. We present a mean field calculation of this coupling mechanism, and discuss shortly the iattice distortion under such circumstances. The Hamiltonian we investigate is given by —

H

=

~

—

V(q)/N

ekakiakl +-~ ~

i,k

1~a~+qjakja~’_qjak’j, (1)

qkk’ iJ

where a’s are the electron operators. The first term describes the kinetic energy of the electrons propagating along the chains, V(q) is the Fourier transform of the Coulomb interaction between electrons on chain i and /, N 11 is the number of sites in the chains and the spin variables of the operators are not denoted. Thus, we neglect interchain and Coulomb interaction within the chains. hopping, In the light ofthe renormalization calcuiation of coupled one dimensional chains8 one would expect that a weak on-chain interaction does not change the physics of the problem drastically. We assume that due to the mutual stabiisation CDW’s develop on each chain, and calculate first the ground state energy in terms of the charge density wave (Pq)

=

~

A

(a,~_qak).

Decoupling into separate chains (Hq) = 2 ~ i,k =

1189

2

6k (na)

+1 2

ii&J

4V(q) ((Pq )i(P Nil

~I ~ ~knk)1 +(~2V(q) ~)() A

\i#j

Nii

)~+ c.c.)

+

1190

CDW IN NEARLY ONE DIMENSIONAL SYSTEMS Vol. 19, No. 12 2kF. 1 ~ 4V(q)~><> + c.c.), (2) where q =Hamiltonian can be diagonalised by introThis 2 j~j N 11 ducing new fermion operators that are linear combiwhere is the expection value of the number of nations of ak and ak We obtain electrons in the state k and the factor 2 comes from the summing up the spin components. = ~ E~(n~ — ,42)) (7) Self-consistency requires that a CDW developing on A chain corresponds to a field described by the Fourier where (if Ek is symmetric with respect to EF) components of the neighbouring chains 2z V(q) 0 they are opposite in phase, i.e. the terms are real]. (p)1 = ±(p)~= (p). Therefore the second term in The gap is determined either by minimalising E~0~ equation (2) is given by or by the seifconsistency condition 2)/N 2N1(2z I V(q) II (Pq ) 1 11, (4) 112(J and the total energy U1 = 2zV(q)/N~1 ~A (42) _4i))/(~~ + ~ (i 0) ~ = Eknk + 2zIV(q)lI(Po)12/NII) = 2zV(q)/N 2U 1~ ~ (n~—n~)/(e~ + ~)~ 1. +~.

I

(5) =

2NiNii

~ ~

+

~2 ~ 2zIV(q)l)’

This gives the gap equation 2zIVI/N11

where i~(or p) is determined on a self-consistent way. This equation describes the energy of the system only for the selfconsistent CDW. One can show that E~1as a function of p (or ~) is extremal at this value of ~ (ori~). This mutual stabiiisation of the CDW’s holds for arbitrary wave vector q. However, the energy gain due to coupled CDW formation is maximal when q = 2kF, i.e. when the gap opens at the Fermi surface. In this case the energy gain is proportional to ~2 in ~, the second term in equation (5) givesthe a positive contribution proportional to z~2thus CDW formation is always energetically favourable. For the sake of simplicity we consider coupled chains with half filled bands. Performing the average over chains j(r’ i) we obtain 117

ekahIak,

=

k

+

~ (~(+))+ A

2V(q) N 11

\k,’j

eka,~a,~ + ~ Ujak+qjak,,

=

A

A

(nt?~—n~~)/Ek = 1

~

(ii)

k

where the sum is over the reduced Brilouin zone. Considering linear dispersion relation we obtain IUI = ~(T—0) = Wexp~—W/(2zIVI)} (12) and kB r~= W/4 exp {— W/(2z I VI)}. (13) The gap equation and thus equation (12) holds only in the small coupling limit. In the opposite, strong9cor~ is relation given bylimit, in analogy to the Hubbard model = (J~ffUe°ff4t where t is the correlation-narrowed bandwidth, Ue°ffand U~ffare the Coulomb energies in the ground state, and in the excited state with one electron removed from site a and placed to a distant site 13 on the same chain. It is apparent that the self-consistent mean field theory of the coupled CDW’s closely parallels with the mean field theory of the Peierls instability.’0 The main reason for this is that in this approximation the selfconsistent wave function is the same type as the wave function of the electrons in the distorted lattice. In our case the periodical potential, thus the gap in the one

Vol. 19, No. 12

CDW IN NEARLY ONE DIMENSIONAL SYSTEMS

particle energy spectrum is due to the charge density waves, and there is a term in the total energy quadratic in the CDW amplitude. This close analogy between two calculations most probably exists only in the mean field approximation. Because in our model the CDW is due to a three dimensional coupling one expects that fluctuations will not have such a drastic effect like in the ld Peierls case. As the CDW’s are coupled not only to each other but to the phonons too, as in other cases,11 CDW formation can cause periodical lattice distortion (PLD). To be more precise CDW formation and PLD can enhance each other. This can be proved by showing that a small PLD with an appropriate phase further lowers the energy of the system with CDW’s. The full Hamiltonian of the coupled chains with phonons 7~ Oq,IP_q,i + c.c. (14) H = H°+ jjPh + ~ ig/~/J where ~0 is the Hamiltonian of equation (1); HPh the bare phonon Hamiltonian and the last two terms describe the on chain electron—phonon coupling. Distorting the lattice by cb~small compared with V(q)(p 0 )/ 0q,i = sign V(q)Øqj, the energy of the (~/VR~) and system is given by

+

1191 (—

~

I

V(q)I/Nii)

(18)

~ a (Pq) + 2z I V(q) (p ~q )INii] 5Pg~+ c.c. The second term vanishes because (Pq) minimalizes the energy if ~ = 0. The first term can be estimated assuming that Ek depends on (p) only through its modulus I(p)I. Then =

2N

1I

R (‘~~—~_‘). (19) e ~ I V ~, The expression in the first bracket is positive, thus we can choose the phase of 0 in such a way that the whole expression is less than zero i.e. lattice distortion leads to further reduction of the ground state energy. For a full discussion of the effect of CDW’s on phonons see reference 12. The basic difference between the conventional

aE~

I(~))

a I(p)I

Peierls distortion and our proposal is that this model leads to a large band gap (stabilized by electron—electron interaction), and small lattice distortion i.e. Ee ~ E~. The rather small (compared with Ee) lattice distortion energy was discussed before in the case of TEA(TCNQ) 2. 2 /(r r The gap can be13estimated using equation (12) with V = e 1e1), 1 7 A being the interchain distance and e~ 133, With the dielectric z = 3 we constant obtain a of gapthe ~ interchain 0.4 eV in medium. good agreement with the measured value. Similar estimates for other semiconducting TCNQ salts leads to similar good agreement between the calculated and measured energy gaps. We believe that the model describes well the ‘-~

E

=

~

(~Mco.~ I O,~12 +

~

Ek (ak)

+ 2z1 V(q)I I(pq)12/Nji)

(15)

where M is the ionic mass and wq is the phonon frequency. Now the gap in Ek is approximately =

Iig~~/V~~ 2z I V(q) IPq/N~jI —

(16)

properties of charge transfer salts with large semiconducting energy gap. While it is firmly believed that other one dimensional systems like KPC are prime examples of

(17)

the Peieris distortion, Coulomb effects discussed above may have an important role in this class of materials too.

and Pq can be split into two terms Pg

=

(P 0) +

Spq(cbq)

) being the self-consistent amplitude for Ø,.~ = 0. As çb is small, E can be expanded around 0 = 0. Keeping only the linear terms in ~, the change in energy due to the PLD is given by (Pq

=

~

f~ I.

A

nk

aEk

—i

igcb~/\/N~

a(~~

Acknowledgements We wish to thank J. Sólyom, S. Barisic, K. Holczer and G. Miháiy for many helpful discussions and comments. —

REFERENCES 1.

DENOYER F., COMES R., GARITO A.F. & HEEGER A.J.,Phys. Rev. Lett. 35, 445 (1975).

2. 3.

BERNASCONI J., RICE M.J., SCHNEIDER W.R. & STRASSLER S., Phys. Rev. B12, 1090 (1975). FARGES J.P. & BRAU A.,Phys. Status. Solidib 61,669(1974).

4.

MOOK H.A. & WATSON Ch.R. (to be published). The value quoted was measured in TTF(TCNQ) but it is expected to be a good representative figure for charge transfer salts of TCNQ base.

1192

CDW IN NEARLY ONE DIMENSIONAL SYSTEMS

Vol. 19, No. 12

5. 6.

More accurate estimations based on the mean field theory give essentially the same results. Using the results of phonon measurements [LYNNJ.W. eta!., Phys. Rev. B12, 1154 (1975)] one gets similar values for KCP: Ee 0.04 eV,Eia~ 0.001 eV.

7.

MIHALY G. eta!. (to be published).

8.

MIHALY L. & SOLYOM J. (to be published).

9.

MOTT N.F.,Metal Insulator Transitions. Taylor and Francis (1974).

10.

RICE M.J. & STRASSLER S., Solid State Commun. 13, 123 (1973).

ii.

BYCHKOV Yu.A., GORKOV L.P. & DZYALOSHINSKY I.E., Soy. Phys. JETP 23,489(1966).

12.

BJELIS A., SAUB K. & BARISIC S.,Nuovo Cimento 23B, 102 (1974).

13.

This gives an order of magnitude estimate of the Coulomb correlation energy. Using an anisotropic dielectric constant, and summing up all the Coulomb interactions along the chains one gets V= e

L ru \e± where Eli and e~the perpendicular and parallel dielectric constants, K0 the Bessel function. Using Cj cii 1 [see BUSH L.R., Phys. Rev. B12, 5698 (1975)] one obtains ~ = 0.29 eV. See also SAUB K., BARISIC S. & FRIEDEL J., Phys. Lett. (to be published). 1r1

‘-~

r11

3 and