Festk6rperprobleme XIV (1974)

Experimental Studies of the Electron-Phonon Interaction in One-Dimensional Conducting Systems

Wolfgang G I~serInstitut fiir Angewandte Kernphysik, Kernforschungszentrum Karlsruhe, 75 Karlsruhe, Germany

Summary: Recent experiments on pseudo-one-dimensional conducting systems are reviewed.Diffuse X-ray and neutron-scattering experiments on the metal-organic compoundK2Pt(CN)4Bro. 3 • 3H20 are emphasized. Theseexperiments support the idea that the metal-insulator transition is caused by a Koim-Peierls instabilitydue to the strongelectron-phononcouplingin one dimension.The experimental evidence for the transition to 3-dimensionalorder at low temperatures is discussed, and limitations of current theoretical models atepointed out. Experiments on the organiccharge transfer salt TTF-TCNQ are discussed whichsupport the general picture of the physics of one-dimensional conductors emergingfrom thescattering studies. In relation to the experimental results, the present view of the Fr6hlichcollective mode conductivity is summarized.

1. Introduction

In recent years, there has been agrowing activity both experimentally and theoreti-cally in studying the properties of one-dimensional or pseudo-one-dimensional solidstate systems. In particular, the questions of the existence of a l-dimensional metallicstate and of the possibility and nature of phase transitions in these systems are ofinterest. This is mainly because during the last few years some materials have beendiscovered which, in certain respects, show such properties.

There are several rasons why the study of these compounds is challenging:

a) Concepts in solid state physics are very often developed for 1-dimensional models.In the 1-dimensional case, the chances are highest to solve more realistic modelsfor the solid in a good approximation or even exactly. A direct experimentalcheck of such modelswould be of great help.

b) On the other hand, 1-dimensional solids should, in several respects, differ in acharacteristic way from 2- and 3-dimensional solids. There are a number oftheorems, predicting, e. g., that there is no phase transition at finite temperaturesin ideal 1dimensional solids with short range forces [ 1], that arbitrary smalldisorder in one dimension should lead t o localized eigenstates of the electrons[2], and that in the one-electron approximation a 1-dimensional metal can notbe stable [3]. It is still not clear, at present, how far thesetheorems apply t oreal systems.

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c) And finally there are speculations,going back to Little's idea [4] o f a nonphononmechanism o f superconductivity in a 1-dimensionalconductor, that it should bepossible to synthesize superconductors with high transition temperatures on thebasis o f l-dimensional metals, ff they exist.

There are several classes o f materials containing groups o f electrons which seem tobe restricted to move in one direction. One o f them is the class o f intermetallic com-pounds o f the r-tungsten (A-15) structure.Among these are the best superconductorsknown today. The 1-dimensionalbehavior o f the electrons in these compounds isthe basis o f the Labbd-Friedel model [5] which explains many o f their properties.In this paper, we do not cover these materials on which a recent review has beengiven by Neger and Goldberg [6].Presently two other classes o f compounds are known, which show high anisotropiesin their electric conductivities essentially due to the 1-dimensional behavior o f theelectrons. These are some organic charge transfer salts based on the tetracyanoqui-nodimethane molecule (TCNQ) and some metal-organic compounds based on thesquare planar platinumtetracyano complex. The conductivities,magnetic proper-ties, optic properties, etc., o f these compounds have been studied extensively. Atroom temperature these materials behave like metals, whereas at sufficient lowtemperatures they are insulators. There are excellent reviews o f this work pr iorto 1973 [7, 8]. Severalmodels have been proposed to explain the experimentalresults. Among theseare the Mott-Hubbard transition due to electron correlation[9], phononassisted hopping due to disorder [ 10], and the interrupted strandmodel [11 ].

More recently, it became likely that most o f the properties o f the quasi-one-dimen-sional systems observed experimentally can be understood on the basis o f theelectron-phonon interaction in a linear chain leading to an enhanced Kohn effect[12] and eventually to aPeierls instability [3].

In this paper, we will discuss mainly the experiments conclusive in this direction.These are investigations o f the lattice dynamics and o f structural changes b y diffuseX-ray and neutron-scattering techniques. Extensive scattering experiments havebeen performed up to now only on one compound o f the class o f the "mixed-valence" platinum chain complexes, namely on K2 Pt(CN)4 Br0.3 • 3H20. Suchexperiments have not been reported up to now on the charge transfer salts o fTCNQ probably because o f the lack o f sufficient large single crystals o f thesecompounds which are a prerequisite for low intensity scattering experiments.Therefore, we will concentrate on the results o f the scattering experiments onK2Pt(CN)4Bro.3 • 3H20 and their interpretation. The discussion of the TCNQcompounds will be restricted to experimental results which indirectly support thepicture emerging from these scattering experiments. Finally a remark on the Fr6h-lich conductivity mechanism will be made.

206

2. K2Pt(CN)4Bro. 3 • 3 H 2 0

The structure o f K2 Pt(CN)4 Bro.3 " 3H20 (in short: KCP) at room temperaturehas been investigated by Krogmann and Hausen [ 13] with X-rays.Figure 1 sumrn-adzes their results.The unit cell o f KCP is tetragonal and contains two platinumatoms. The planar Pt(CN)4 complexes are stacked in such a way that the Pt atomsform linear chains with a Pt-Pt distance o f 2.89 A. The spacing between chains(9.87 .~) is much larger. According to the X-ray analysis,only 0.6 of the availablebromine sites and 0.5 o f the available potassium sites are occupied, and the occu-pation should be random.

bl

~ .:3" '"'- -.'": . ; . .

1

• : @ ~. 9 7~

o

~ : P t ~ Q : K o c:C o c~:N ~ ~:::H20

Fig. 1Structure ofK2Pt (CN)4Br0. 3 •3H20according to [131

The short interchain Pt-Pt distance (2.77 A in metallic platinum) led to thesuggestion that KCP may exhibit 1-dimensional metallic properties. It is believedthat the partial oxidation o f the platinum ions by bromine leads to a 5dz2 con-duction band having only 5/6 o f the electronic states occupied [ 14].The temperature dependence o f the de conductivities Oll and a t parallel and per-pendicular to the platinum chains observed by ZeUer and Beck [15] is illustratedin Fig. 2. At room temperature, all is about 5 orders o f magnitude higher than o1and has a value o f several hundred (~crn)-l . With decreasing temperature, theconductivity decreases. At 35 K KCP seems to be an insulator, and the anisotropya H/Ol has decreased to about 103. The conductivity data can not be explained ina satisfying way either by Mott's variable range hopping mechanism[ 10, 16] o rwith the interrupted strand model [ 11, 17]. The other tempting possibility is to

207

(Qcm)'

I '°

. . . . . . . . . i . . . . . . . . . I . . . . . . . . . I . . . . . . . . . i

" ~ Longitudinal (andtransversal\ conductivity of~ K z Pt (CN)= Br.a 3H=O

1

10"

io-~

I fit!

'~ , , i . . . . . . , . . . . . . . . . i , . . . . . . . . r . . . . . . . . . i

0 .010 .020 .030 .0t.0= T -1

Fig 2

Temperature dependence o f thelongitudinal and transversal dcconductivity ofK2Pt(CN)4Bro.3 • 3H20 measuredby Zeller and Beck [151

assume that the temperature dependence of the conductivity is due t o a tempe-rature-dependent gap caused by a Peierls distortion.

According t o Peierls [3], in the one-electron approximation the electrons in a parti-ally filled l-dimensional band can always lower their kinetic energy by a symmetryreduction of the 1-dimensional lattice. A spontaneous periodic distortion of thelinear chain with a period rr/kt~ - where kF is the Fermi wave number of theelectrons - would lead to a splitting of the band. This is illustrated in Fig. 3 fora linear chain with lattice constant e having a half-filled conduction band. If inthis case, the ions order pairwise; the electrons feel a potential with a period 2c;this leads t o a splitting of the conduction band at k = + kF = -+7r/2c. The filledand empty band are separated by a gap 2A.

Because the splitting lowers the electron and the electron-phonon contribution tothe energy by an amount proportional to In A and the increase in elastic energy dueto the distortion goes proportional to A2, for small distortions the Peierls insulatorwould be energetiely more favorable than the metallic state. However, it is not clearwhether this picture is still valid if the electron-electron interaction is taken into

208

l , ), I i , I l, - @ - , -C>- , - @ - ,, - C ~ - , - C ~ - , - O -

METAL !

k F N c N (E)~

+ - o - o +

INSULATOR ! 2A, ,.. ] . . . . . . .

N(el2c

Fig. 3Schematic picture of the Peierlstransition fora linear chain witha half-filled conduction band

account [ 18]. Ovchinnikov [ 19] has shown that the 1-dimensionalHubbard modelwith a noninteger number o f electrons per unit cell can undergo a metal-insulatortransition for certain values o f the electron interaction constants.

2.1. Diffuse X-ray scattering

In the case o f KCP, the first indication o f a lattice instability was given by a diffuseX-ray scattering experiment by Com~s e t al. [20]. Investigating a room temperaturesample o f KCP with the "monochromatic Laue technique," theseauthors foundsuperlattice structure planes in reciprocal space. Intensity planes in reciprocal spacecorrespond to a 1-dimensional periodic structure in real space. In the case o f KCP,the satellite planes can be explained with an uncorrelated sinusoidaldistortion o fthe platinum chains having a period o f about 6 Pt-Pt distances. Because the necessarylattice distortion for splitting the electronic band at the Fermi level corresponds toa reciprocal lattice vector 2 kF, the diffuse X-ray scattering result on KCP wouldbe compatible with the assumption o f a Peierls distortions which splits the 5/6filled band in 5 filled and one empty band.However, from the X-ray result, it is not possible to decide whether the observedsuperlattice structure is static or dynamic. It could be caused also by a latticevibration with a wave length o f 6 Pt-Pt distances. If the frequency o f this parti-cular phonon is small compared to the phonon frequencies for neighboringwavevectors, the diffuse X-ray intensity should be especially large because the cross-section for the scatteringo f X-rays on a lattice vibration o f frequency u~ goes like1/co2. Such a low phonon frequency can, in principle, be caused by a strong ele-ctron-phonon interaction.

209

3. Electron-Phonon Interaction and Kohn-Effect

The coupled electron-phonon system for a 1-dimensional solid is usually describedby the Fr6hlich Hamiltonian,

H = ~ ekC~Ck+ ~ h I 2 q b q b q + N"x/2 ~ gq+r4Ck+q+r(b-q+bq),(1)k q k,q,7"

+where c~, Ck, bq, bq are electron and phonon creation and annihilation operators,12q isthe unrenormalized phonon frequency, gq isthe electron-phonon couplingconstant, and r isa reciprocal-lattice vector. In Eq. (I),the electron-electron inter-action is neglected, respectively itis assumed that its effects are includedin theparameters ek, ~q, gq. Often the strength of the electron-phonon coupling isexpressed by a dimensionless parameter ~.:

= N(eF)),=g~h ~ 2 q ' (2)

where N(eF) isthe electronic density of states at the Fermi surface.Ifthe electrons are treated as free, g is assumed to be q-independent and umklappprocesses are neglected, then in linear responsethe solution of Eq. (1) for the re-normalized phonon frequency can be written,

6o2(q) = ~2~(1 - ), x(q)), (3)

withh2k~ i°° f(ek)(I - f(ek+q))

x ( q ) = m dk (4)ek + q -- e k

×(q) is the static limit of the free electron gas susceptibility in the random phaseapproximation (RPA), and f(ek) is the Fermi distribution function for freeelectron states. For T = 0, the integration of Eq. (4) yields:

x(q)=kqFln 2 k F + q lq J" (5)

For comparison in Fig. 4a, x(q) asgiven by Eq. (5) is plotted together with theq-dependent susceptibilities of the 2- and 3-dimensional electron gas also calcu-lated in RPA. The susceptibilities have singularities of different character atq = 2kF. Whereas in the 2- and 3-dimensional case, only the derivatives of X(c0are singular, the l-dimensional x(q) itself has a logarithmic divergence.However, independent of the dimensionality, whenever the phonon wave vectorcorresponds to a diameter of the Fermi surface, the screening of the ions by the

210

2.0

~ 41-5

O~

"' //

........... iii ,, [ .......".-:. _..._.:

0.5 1. 1.5 2.0q/2K~

. . . .

2N ~

Fig. 4aStatic susceptibilites x(q) of the 1-, 2- and 3-dimensional electron gas in RPA approximation

Fi 8 . 4 bEffectof the singularities x(q) atq = 2kF onthe phonon dispersion

electrons changes suddenly, leading to an image o f the Fermi surface in the phonondispersion. This effect was first predicted by Kohn [ 12]. In general the Kohn effectis a rather weak effect, but for a 1-dimensionalmetal it should be considerablyenhanced [21]. According to Eq. (3), the logarithmic singularity in x(q) can, inprinciple, also lead to an instability in the phonon dispersion,a lattice instabilityas illustrated in Fig. 4b.

The picture described is a first approximation and probably gives only a qualitativeaccount o f the effect. In fact, FrOhlich [22] already considered the coupling o fnoninteracting electrons to the phonons as a model for the Peierls transition. Thecoupling o f phonons to a 1-dimensional electron gas and its effect on the phonondispersion around 2kF and on the stability o f the lattice has been treated moreextensively by Horovitz et al. [23, 24]. The influence o f the long-range repulsiveelectron-electron interaction on the electron-phonon coupling in the tight-bindingapproximation has been discussed by Bari~i6 [25] in a linearized self-consistenttheory. Recently Chui et al. [26] have shown in a more rigorous treatment thatthe electron-electron interaction leads to a modification o f the logarithmic diver-gence o f the susceptibility o f the 1-dimensional electron gas to a power law diver-gence.

211

4. Inelast ic Neut ron Scat te r ing Studies

The existence o f an enhanced Kohn anomaly in KCP a t room temperature has beendemonstrated by Renker et al. [27]. They measured the longitudinal acoustic (LA)phonon dispersion in the direction o f the platinum chainsby means o f inelasticneutron scattering. Figure 5 shows the result o f this experiment. The pronouncedanomaly in the LA phonon branch o f KCP is at the same q-value where the 1-dimensional superlattice structure has been observed in diffuse X-ray scattering[20]. The anomaly has been analyzed in terms o f a very simple model, treatingthe electrons as a i-dimensional free electron gas, describing the electron-ion inter-action with the Ashcroft [28] pseudopotential model and including umklappprocesses.In this model, the LA phonon branch can be written:

~=(q)= a~ + f ~ {~, IF(q+r) -l -11cos=IRc(q+r)[T

I-e(r)-'-1 Icos=lRcrl }, (6)r~0

where fZp is the ion plasma frequency o f the bare platinum ions, and Re the cut-offradius o f the Ashcroft potential. The q-dependent dielectric function e (q ) wasassumed to be given by,

kTF 2e(q ) = 1 + q0

10.0

3

~.~ 5.0-

WZW

×(q, r), (7)

0.5 1.0K z P t (CN}~ Br3 -3H20

oo Exper imentFit T=3OO*K

O o

o

• ' * , • t 0

0 0.5 1.0WAVENUMBER q [~r]2cl

10.0

5.0

Fig. 5Longitudinal acoustic phononbranch in the chain direction ofKzPt(CN)4Bro. 3 • 3H20. Thesolid curve is a fit of the modeldiscussed in the text to the data

212

where KTF is a reciprocal Thomas-Fermi screening length and x(q, T ) is given byEq. (4). At finite temperature, the logarithmic singularity of x(q, T = 0) is roundedto a rmitevalue. The full curve shown in Fig. 5 represents a fit of the unknownparameters of Eq. (6) to the experimental data. The resulting values: hI2q = 29.4 meV,Re = 0.54 A and kTF = 0.78 A-~ seem to be quite reasonable for a metal. The dipat q = 0.32,8, -t reflects the remainder of the singularity of x(q, T) at room tempe-rature. It may be worthwhile to mention that the q-position of the dip correspondst o a wave length of 6.66 Pt-Pt distances which is not commensurable with thelattice. From the expedmental data, a value between 0.2 and 0.3 can be esti-mated for the electron-phonon coupling constant X.

These data have been measured on KCP crystals containing ordinary crystal water.The incoherent neutron scattering of hydrogen makes phonon measurements andeven more so quasi-elastic studies rather difficult. The availability of deuteratedsingle crystals considerably improved the experimental conditions [29].

Figure 6 shows a selection of phonons measured up to now in deuterated KCP atroom temperature. Part of these data have been collected at the High Flux Reactorin Grenoble by Renker and ComEs [30]. Besides the LA phonons in the chain

direction (0, 0, ~), LA phonons at the zone boundary a' a ' of the Brillouin

. 2.0-~

1.0- / 1.5-

,, / / / ° 1o.s- Z 1 " 1 |

4f- 0

(~'~..2 = 2

'~/,~,~ .s .i .2 .3 .i .s1 1

Fig. 6. Acousticphonon modes measured in K2Pt(CN)4Br0. 3 •3D20 at room temperature

213

{lr 7r 2k \zone are presented. The phonon anomaly appears also at ~ a ' a ' F/ and was

further measured at several other q-points in the 2kF-plane o f the reciprocal lattice,which proved in a rather direct way that this effect is due to a 1-dimensional pro-perty of the crystal.

That the anomaly at ( ~ , ~ , 2 k F ) i s somewhat deeper than in the chain direction

was explained in a qualitative way byBariJid and ~aub [31] in the frame of theself-consistent tight-binding theory o f the electron-phonon interaction. Theseauthors argued that because the long-range Coulomb interaction o f the electronsdepends on all three components o f the wave vector ( , among all phonons withqz = 2kF the softest mode should be the one with the shortestwavelength and

[_~ rr 2kthis isthe mode at the point \a' a' F]- Figure 6 also shows some transverse

acoustic (TA) phonons in the chain directions, which obviouslyare not influencedby the electrons, and further some T A phonons in the (f,~', 0)-direction. Thediagram of the phonon spectrum shown in Fig. 6 demonstrates rather convincinglythat itis only at larger q-values in the vicinity of qz = 2kF where the electronsimpose l-dimensional features on the phonon dispersion. The phonon spectrumisotherwisequite 3 dimensional. In the region of the anomaly, the phonons seemnot to be well defined. For example, Rietschel [32] calculated a resonance structureo f the phonon line width due to electron-phonon interaction with a maximum linewidth o f about 30 % o f the phonon frequency.

5. Metal-Insulator Transit ion

The experimentally determined phonon dispersion o f KCP together with the knowntemperature dependence o f the susceptibility ×(q, T) of the 1-dimensional electrongas suggest to interpret the phonons in the range o f the anomaly as "soft modes"as precursors o f a structural phase transition. Because x(2kF, T) goes to inf'mitywith T ~ 0, W2kF should decrease with decreasing temperature. If W2kF wereto reach zero, the corresponding atomic motions would freeze, and the dynamicdistortions o f the lattice would proceed to a permanent superlattice distortion,a Peierls instability.

A mean field theory o f this type o f transition o f a 1-dimensionalmetal for a nearlyfree-electron model was already given in 1954 by Fr6hlich [22] and recently hasbeen worked out in more detail, in the tight-binding approximation, by Rice andStrdssler [33]. The latter authors started from the Fr6hlich Hamiltonian (Eq. (1)),treated the electrons in the tight-binding approximation, neglectedumklapp processes,and calculated the temperature Tp at which the screened phonon vanishes. Theirresults are the following:

214

a) Temperature dependence of the phonon frequency above Tp,

g2N(eF ) '~ k F = XS2~kv t n ( T f f p ) X = - - ; (8)

h~"~2 kF

b) Transition temperature Te of the Peierls transition,

kBTp = 2.28 eF e -t/~ eF = Fermi energy; (9)

c) Electronic gap below Tp at T = 0,

A(0) = 4eF e"Ux. (10)

The gap A has been calculated by assuming that in the Peierls insulator the phononstates q = -+ 2kF are occupied macroscopieally. Formal analogies of Eqs. (9) and(10) with the BCS theory are obvious, the Debyeenergy has been replaced by theFermi energy. The change of the logarithmic singularity of the electron suscepti-bility to a power law divergence due t o electron-electron interaction leads also t oa modification of the Peierls transition temperature [26]. Equation (9) has t o bereplaced by,

kBTp = 2.28 eF e o~ (11)

where a is a measure of the electron-electron coupling. Because for Coulomb inter-action a should be negative, the transition temperature should be enhanced relativeto the result of Eq. (9). This holds in the mean field approximation.

The molecular field approach has an essential weakness. The influence of thefluctuations of the order parameter, which are known to be important in l-dimen-sional systems, has been taken into account only in linear response. The criticaltemperature region in which the interactions between fluctuations or, in otherwords, anharmonic effects have to be included is of the order of Tp itself.

In the case of KCP, these would be the fluctuations of the distortions of the platinumchains. Lee et al. [34] have tried to estimate the influence of the fluctuations on thetransition temperature Tp in a better approximation. They calculated for a 1-dimen-sional model in a generalized Landau theory the temperature dependence of a corre-lation length ~ for the order parameter. This correlation length increases with decreas-ing temperature but diverges only at T = 0. As expected for an ideal 1-dimensionalsystem, there is no phase transition at f'mite temperatures. But at sufficient lowtemperatures the correlation lengthgrows rapidly enough so that a weak 3-dimen-sional coupling - always present in a real system - can cause a 3-dimensional phasetransition. The estimate of Lee et al. for this 3-dimensional transition temperatureof KCP is about one-quarter of the mean field value TM'F'. Rice and Striissler [35]

215

Z i

i .I t

I I / T - O1\o.3 i ,,I \ I "I \ l ' ,

0 1 2 3E / ~ T p

Fig. 7Effect of the fluctuations of the orderparameter on the electronic density ofstates for severaltemperatures

performed a calculation along similar lines but included interchain coupling. Forweak interchain coupling, they arrived essentially at the same result as Lee et al.

The scattering of the electrons on the fluctuations leads also to a change o f thespectrum of the electrons, t o fluctuations of the electronic density of states atthe Fermi surface. In the 1-dimensional model [34], a sharp energy gap in theelectronic spectrum exists only at T = 0. With increasing temperature, this gapwill be filled and changed to a "pseudo-gap" as illustrated schematically in Fig. 7.Including increasing-interchain coupling will decrease the influence of fluctuationsand reduce the pseudo-gap. As a consequence of the change of the density of statesat the Fermi level, one expects also a change of the Kohn effect.That this picture is at least qualitatively correct has been shown in recent Knightshift measurements byNiedoba et al. [36]. The Knight shift at the platinum atomsin KCP increases with increasing temperature between 120 K and room temperature.The data can be described with the susceptibility calculated by Lee et al. [34] forthe 1-dimensional model including the interaction between fluctuations. A fit ofthe model to the data yields a mean field-transition temperature of about 600 K.

6. Three-Dimensional Phase Transition

If a structural phase transition occurs in KCP, it should be detectable, in principle,by scattering experiments. Studying the temperature dependence of the diffuseX-ray scattering, Comts et al. [37] found that the superlattice structure in KCPobserved first at room temperature loses gradually its 1-dimensional character,and that at 77 K a 3-dimensional superlattice structure characterized by the super-

lattice point a ' a ' 2kF remains. According to a theoretical suggestion o f Bari~'d

216

CO

Z

o

800

600

t.O0

200

room temperature

600[ ~ Od : o

'°°I / \

[o.o,4.~;1 [ : os?

o p

0 1 2 3 /. 5 6 7 meV

Fig. 8Experimentat evidence for quasi-elasticcoherent scattering inK2Pt(CN)4Bro. 3 • 3D20 at q = 2kF(/i = cq/Tr). The inelastic ridge in theconstant Q scan is due to the narrowanomaly in the phonon dispersion

[25], this phase transition was interpreted as beingdue to the condensation o f thesoftest phonons o f the enhanced Kohn anomaly, corresponding to an antiparallelcoupling of the Kohn-Peierls waves in neighboringchains.

Renker et al. [38] tried to supplement the X-ray results by neutron-scattering experi-ments in order to get additional information on this transition. Figure 8 shows atypical room temperature neutron constant Q scan at the (0, 0, 2kF) point in thereduced zone o f the reciprocal lattice o f KCP where the Kohn anomaly previouslyhas been observed. The insert shows the result o f a constant energy scan with energytransfer zero, along the direction (0, O, ~').

Besides the phonon ridge which corresponds to the extremely sharp anomaly in thephonon dispersionextra intensity appears around 60 = 0 at qz = 2kF- This "centralpeak" is observed also in all off-symmetry directions studied, at -~ vectors having acomponent 2kF parallel to the platinum chains and therefore proving, in additionto the Kohn anomaly, the existence o f l-dimensional periodic distortions in thelattice.

If the extra scattering would be truly elastic, one would have to assume uncorrelatedstatic distortions o f the platinum chainsas was pointed out byRietschel [39]. How-ever, it also could be critical scattering due to the fluctuations into the distortedstate. Because o f the limited experimental ~o- and q-resolution, no direct decisionwas possible up to now.

It has been shown by Strdssleret al. [40] that in a l-dimensional model the scatter-ing o f the soft phonon modes with the fluctuations into the distorted state shouldlead to a large increase o f the damping o f thesemodes well above Tp and thereforeto a central peak in the dynamic structure factor. Central peaks previously have

15 Festk6rperproblemeXIV 217

7000-

6 0 0 0 -

¢ J

5000

t, 000U .

,,5t~o 3000eio

2000-

N i rZ

, ° ° ° I

00 4'0 80 120 160

T (K)

tl.

300-300-d

200-[O'O'~kFl ,

loo-

. n

200 2io 280 320

Fig. 9. Temperature dependence of the extra-quasi-elastic scattering in K2Pt(CN)4Bro.3 • 3D20at two selected poin ts of the 2kF-plane in reciprocal space

been observed in structural phase transitions o f other systems [41, 42]. However,the microscopic explanation o f these centralpeaks is still a matter o f controversy[431.The temperature dependence o f the measured intensity o f the central peak in KCPat two specialpoints o f the 2kF-plane in reciprocal space namely at (0, 0, 2kF) and

E r r 2k ~a ' a ' F] is shown in Fig. 9. In the temperature range 200 K to 320 K, the

intensity at both points is practically the same, as expected for 1-dimensional

distortions. Below 160 K, the intensity at the superlattice point ' a ' 2kF

increases rather rapidly and saturates at about 40 K. In the same temperaturerange, the intensity at the point (0, 0, 2kF) decreases but does not go to zero aswould be expected for a long range ordered 3-dimensional superlattice structure.From the neutron data, it is clear that the 3-dimensionalordering observed in theX-ray measurements around 80 K essentially is due to the "central peak."It may be also o f interest to note that the temperature dependence o f the gap whichZeller [8] derived from the conductMty data seems to be directly correlated to the

218

6 0 -

I,o_ 3 0 -

~ 20-

10-

Kz Pt(CN)¢ Br.3-3D20

I

I &O 80 120 160

fl- 3

200 TEI<]

Fig. 10Temperature dependence of theline width r' of quasi-elasticscattering at the point

~-, 2kF) and of the deducedcorrelation length ~ l perpendi-cular to the platinum chains inK2Pt(CN)4Bro.3 • 3D20

temperature dependence o f the "3-dimensional structure factor" observed in neu-tron scattering.

The absence o f long-range order is also evident from a finite line width o f the cen-

tral peak at (~, ~, 2kF)perpendicular to the chain direction in reciprocal space.

This line width has been interpreted as inverse correlation length ~1 for the orderperpendicular to the platinum chains [38]. The temperature dependence o f ~1 isshown in Fig. 10. Below 100 K, ~l reaches about three interchain distances andremains constant at lower temperatures. This absence o f long-range order perpen-dicular to the chains below the transition at I00 K has up to now not been ex-plained in a satisfying way. According to the X-ray structure analysis [13], thebromine sites are occupied in a random way at least at room temperature. In anideal 1-dimensional system, a random potential should lead to electron localization.Probably these effects are overcome in KCP by the strong electron-phonon coupling.On the other hand, one can also imagine a pinning o f the distorted lattice waves toimpurities and lattice defects. In an ideal system o f parallel chains, the charge densitywaves connected with the distortions would order in phasedue to the long-rangeCoulomb interaction. This already should take place for small correlation lengthso f the chain distortions because the energies involvedare small. But if the chargedensity waves are pinned to the defects in and outside the chains, much largerenergies are required for 3-dimensionalordering.

219

Up to now no experimental determination o f the correlation length ~II o f thedistortions in the platinum chainshas been possible. The present data [38] giveonly a lower limit o f 20 Pt-Pt distances which is considerably greater than theestimated value o f Lee et al. [34]. The discrepancy may be explainable by anenhancement o f ~II due to a weak interchain coupling.

The simple molecular field theory picture suggested a study o f the temperaturedependence of the inelastic neutron scattering in the q-range of the anomal in orderto get information on the temperature dependence o f the soft modes. Measurementsbelow the transition at 100 K showed that the 2kF anomaly in the phonort disper-sion completely disappeared [44]. But in the wide transition region between 200 Kand 100 K, the definition o f a lowest phonon frequency encounters difficulties.This is partly because the very narrow anomaly cannot be tackled with the resolu-tion presently available. But, on the other hand, it is known that several strongdamping mechanism make it questionable whether very close to qz = 2kF definedphonons exist at all. Some theoretical studies o f the retarded phonon self-energy[24, 45] have also shown that the 1-dimensional behavior o f the electrons may giverise to a peculiarbehavior o f the phonon spectrum in the vicinity o f q = 2kF. Butin these studies,the influence o f fluctuations has not been considered.What isneeded is a careful experimental study o f the dynamic structure factor S(q, co)at q = 2kF with an extreme high q-resolution.

Although some detailed problems have still to be solved, from the present availableexperimental data - mainly from the scattering experiments - the following pictureo f the properties o f KCP emerges.

At high temperatures, the platinum chains show uncorrelated dynamic distortions -an enhanced Kohn effect - due to a strong electron-phonon interaction. The conden-sation o f these soft modes in a static Peierls distortion is prevented in a considerabletemperature range by large fluctuations of the order parameter. With decreasingtemperature, the correlation length o f the intrachain distortions increases, and theinteractions between chains - probably due to long-range Coulomb forces -- be-come more and more important. This 3-dimensional chain interaction finally leadsto a gradual disorder-order transition, namely an antiparallel ordering o f the Kohn-Peierls waves.

The temperature dependence o f the conductivity can be explained in this modelb y assuming that the contribution o f interchain scattering to the resistivity increaseswith decreasing temperature. Essentially all experiments performed so far on KCPare consistent with this picture. Besides this dominant mechanism, defects anddisorder o fcourse play an important role in determining some o f the propertieso f such a pseudo- 1-dimensionalsystem.

220

7. Experimental Studies o n Tet racyanoquinod imethane Compounds

Among all the organic charge transfer saltsbased on TCNQ, the compound tetra-thiofulvalinium tetracyanoquinodimethane (TTF-TCNQ) [46] has attracted themost attention. In the solid state, the molecules are arranged in alternating parallellinear TTF- and TCNQ-chains. It is believed that both the acceptor (TCNQ) anddonor (TTF) chainsare conducting. TTF-TCNQ shows a room temperature dcconductivity oil parallel to the chains as high as 400 (~2cm)-1 . The temperature-dependent conductivity measurements [47, 48] revealed a maximum o f all atabout 60 K being about a factor 10 to 20 higher than at room temperature. Belowthe maximum, the conductivity drops sharply. In a few samples, extraordinarymaxima - about two to three orders of magnitude higher - had been observedand were suggested to be due to superconducting fluctuations [47]. But furtherstudies did not confirm the extra maxima and seriously questioned the presenceo f superconducting fluctuations [48, 49]. If we exclude these spurious effectsfrom further discussion, it remains that the studied properties o f the "normal"samples clearly indicate that 1-dimensionalbehavior o f the electrons plays animportant role.Measurements o f the thermoelectric power o f TTF-TCNQ by Chaikin et al. [50]showed a linear temperature dependence of the negativeSeebeck coefficient above140 K which is characteristic for metallic conduction independent o f the dimen-sionaLity. Near the transition temperature, the thermopower decreases rapidly andchanges the sign which is interpreted as being due to a metal-insulator transition.

It is likely that the picture o f the Kohn-Peierls transition derived from the experi-ments on KCP applies also on TrF-TCNQ, althoughup to now there is no directsupport by scattering experiments.On the basis of the 1-dimensionalmodel for the electron-phonon interaction,Patton and Sham [51] derived a simple approximation for the dc conductivity:

2o0o(T) = (12)

1 + e 6/kBT

This formula should hold below and above the Peierls transition, ao is the metallicconductivity i n the absence of the phase transition and 6 a fluctuation gap. Themeasured temperature dependence of the dc conductivity of the normal TTF-TCNQ samples can be described by Eq. (12).Measurements of the ac conductivity [52], optical reflectivity [53], and magneticsusceptibility [54] are aU consistent with an interpretation by a Kohn-Peierls tran-si t ion although they do not rule out unambiguously other possible explanationsof the metal-insulator transition in TTF-TCNQ around 60 K . In measurementsof the specific heat of TTF-TCNQ [54a] a weak steplike change in the specificheat at about 55 K was observed and interpreted as evidence for a Peierls tran-sition.

221

Recently, the pressure dependence o f the conductivity o f TTF-TCNQ has beeninvestigated by Chu et al. [55]. This work showed an increase in the transitiontemperature (position of the maximum o f Oil) with increasing pressure which onedoes not expect, e. g., for a Mort-Hubbard transition due to electron correlationeffects. In the Mott-Hubbard model, one would expect the electron transfer inte-gral (hopping probability o f the electrons) to increase,and the effective Coulombinteraction o f the electrons (responsible for the localization) to decrease with in-creasing pressure.This would mean a stabilization o f the metallic state to lowertemperatures with pressure,whereas with the Kohn-Peierls concept the observedeffect can be explained even in the simple mean field model via Eq. (9). eF and)~ are expected to increasewith pressure at least in the tight-binding approximation[51]. The same arguments hold also for the observed increaseo f ou at roomtemperature with pressure. But the observed much weaker pressure dependenceo f Oll near the transition temperature is not so obvious in the simple l-dimen-sional model. Chu et al. concluded that their results on TTF-TCNQ are consistentwith the idea o f a Kohn-Peierls instability, but that interchain coupling may beimportant near the transition and that there interchain scattering may contributesignificantly to the resistivity.

The importance o f interchain coupling in the transition range is essentially whathas been observed in quasi-elastic neutron scattering on KCP [38]. For a directsupport o f the Kohn-Peierls interpretation o f the TTF-TCNQ properties, diffuseX-ray and neutron-scattering experiments would be highly desirable.

8. Fr6hlich Conduct iv i ty

So far, the view has been adopted that the Kohn-Peierls transition leads to a semi-conducting or insulating state. In fact, with the Peierls distortion o f a linear chainhaving a partially filled band, a chargedensity wave (CDW) along the chain isconnected. This is illustrated in Fig. 11 for the half-filled band case in which twodistortions relative to the undistorted chain are possible. Where the atomic distanceis smaller than c, extra negative chargebuilds up, whereas in the regions o f largerdistances there is a lack o f negative charge relative to the equilibrium charge densityin undistorted chains. In KCP below the transition, this induced CDW (having awavelength o f 6.66 Pt-Pt distances) is equal in amplitude but opposite in phase inadjacent platinum chains. If the CDW is fixed relative to the host lattice, t h e systemshould be an insulator.

The discovery o f pseudo-l-dimensional conductors renewed the interest in a sugges-tion o fFr6hlich [22] made before the event o f the BCS theory for superconductivity.Fr6hlich argued that a kind o f superconductivity in l-dimensional systems mightresult if the periodic chain distortion and the CDW coupled to it are not fixed

222

t l

I e @ e e eb c - - - ~

Fig. 11 . Schematic view of the charge density waves (CDW) connected to the Peierls distortionsof linear chains with half-filled bands

2c x . ~ 2c

F~. 12Shift o f the gap in the electron dispersionof a half-f'dled band in an electric field i fthe gap caused by the Peierls distortion iscoupled to the Fermi surface and canmove withit

relative t o the rest lattice but are able t o move with the electrons due to the strongelectron-phonon coupling. In this model, no pairingof the electrons is needed. Asimple interpretation of the Fr6hlich model has been given by Bardeen [56]. Inthis picture, the energy gap in the electron band caused by the lattice distortiondue to electron-phonon coupling can move with the Fermi surface if the electronsare displaced in momentum space to give a net current as shown schematically inFig. 12. At low temperatures, only the electron states below the gap are occupied.If in an electric field there is a current flow with velocity vs, the electrons shouldnot be scattered as long as ttkvs is smaller than the gap; a supercurrent can persist.hkvs is the difference of the electron energy in the host lattice and the movingframe. Resistance appears when hkvs increases above A because electrons earl bescattered back in the next higher band. This view has been worked out in more

223

II

r.rp

r<rpI to

2~

Fig. 13Schematic illustration of the splitting o f the coupledcollective electrorbphonon modes suggested byLee et al. 1581

detail recently by Allender et al. [57] in a phenomenological approach treatingthe electrons in the tight-binding approximation instead of the nearly free-electronapproximation used by Fr6hlich. Although these authors conclude that in theirmodel below the Peierls transition temperature superconductivity is possible inprinciple, they also conclude that attractive interaction betweenchains is necessaryto achieve a sufficient high momentum o f the system and therefore a reasonablelifetime for the high conductingstate.

The essential point o f these models is the existence o f a traveling coupled electron-phonon mode, called Fr6hlich mode. A microscopic although still mean fieldapproach to understand the macroscopic-occupied traveling lattice wavehas beensuggested by Lee et al. [58]. These authors have shown that, below the Kohn-Peierlstransition o f a 1-dimensionalsystem, the coupled collective electron-phonon modessplit in two distinct modes, as illustrated schematically in Fig. 13. Below a distortivephase transition in a normal insulator, both modes would rise to a f'mite frequency.However, the mode frequencies calculated by Lee et al. are,

with

6o+2= ;~I2~kv +4m3m* v~, I q - 2kFI 2, (13)

_ m 2~ _ ~ - ~ g - v F 12kF - q l 2, (14)

m 4A2= 1 + ~I22k~ ' 1 ~ 2 (15)m *

224

The effective mass ratio m*/m (m is the band mass o f the electrons) can be ratherlarge because the response o f the phonon system is much slower than that o f theelectron system. The ~+-mode can be visualized as being due to the amplitudefluctuations and the ~o_-mode as describing the phase fluctuations o f the orderparameter. If the wave length o f the CDW is commensurable with the lattice, themodes degenerate (standingwave). It is the phase mode which under ideal con-ditions - if restoring forces are absent - has zero frequency and can, therefore,carry current. In the frequency dependent conductivity a(~o), a 5-function peakwould appear in the electronic gap (Fig. 14). The weight o f this peak relative tothe single particle contribution is givenby the ratio m/m*.

/- A

due toFr~hllch mode

0 A w

Fig. 14Zero frequency component of the acconductivity due to the traveling phasemode in the Peierls distorted phase ofa 1-dimensional conductor

However, any mechanism which can supply a restoring force would lead to a finitefrequency o f the phase mode and, therefore, to a loss o f de conductivity. Restoringforces can be caused by 3-dimensional interactions and also through pinning o f theCDW by impurities and defects. It may be conceivable that already in the transitionregion o f the Kohn-Peierls transition, a CDW exists if the correlation length o f thedistortion is large enough. In this case,the t'mite lifetime o f the CDW would leadonly to a f'mite de conductivity.However, it is not clear how the picture changesif interactions between the fluctuations o f the order parameter are taken intoaccount in a more complete theory.Recently, indirect evidence for a current-carrying mode at low frequencies has beendeduced from optical reflectivity data on KCP [59] and TTF-TCNQ [60] in thelow-temperature phase. However, up to now no direct experimental proof o f theexistence of a special current carrying electron-phonon collective mode in pseudo-1-dimensional systems has been given.

225

9. Conclusions

Extensive experimental and theoretical studies o f the present known pseudo-l-dimensional conductingsystems,especially KCP and TTF-TCNQ, have been per-formed during the last years. But only the recent scattering experiments usingdiffuse X-ray and neutron-scattering techniques revealed the basic physical mech-anism responsible for the observed properties o f these materials. Particularly neutron-scattering experiments give a deeper insight in the peculiar dynamical properties andstructural changes o f pseudo-l-dimensional conductors and supply direct informationon the electron-phonon interaction in these systems. Further systematic and moredetailed investigations o f this interaction and o f the damping mechanismin therange o f strong eleetron-phonon coupling may be of principal interest also fo r abetter understanding of distortive phasetransitions in general.

In the case o f KCP, the experiments performed so far strongly support the assump-tion that the transition from the 1-dimensional metallic state to the Peierls distortedstate coincides with the gradual development o f a 3-dimensional superlattice stru-cture. The metallic state is characterized by an enhanced Kohn effect. Althoughthere is no phase transition in the uneorrelated chainsdue to the fluctuations o fthe order parameter, these fluctuations seem to die away with decreasing tempe-rature because o f the 3-dimensional coupling between chains.

Probably the metal-insulator transitions observed in other pseudo-1-dimensionalconducting systems are governed by the same mechanism, but this still has t o beverified by direct scattering experiments.Besides this dominant mechanism, impurities, defects, and disorder certainly havea great influence on the properties o f pseudo-l-dimensional systems. Systematicstudies o f this influence on the electron-phonon properties are necessary. Theymay also help to elucidate how far the presently emerging picture o f the currentcarrying collective Fr6hlich mode is an appropriate concept for pseudo-l-dimen-sional conducting systems.

Acknowledgements: The author would like to point out that a large part of the experimentalwork reported in this paper is the result ofa collaborative effort of groups at Brown Boveri(Baden, Switzerland), Orsay (France), Institut Laue-Langevin (Grenoble) and Kernforschungs-zentrum (Karlsmhe). He would also like to acknowledge clarifying discussions with severalcollegues working on 1-dimensional systems and thank them formaking results available priorto publication.

226

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