PHYSICAL REVIEW B VOLUME 26, NUMBER 6 15 SEPTEMBER 1982

Fluctuations of polarization energy and commensurate-incommensurateconduction in quasi-one-dimensional systems

M. KavehDepartment ofPhysics, Bar-Ilan University, Israel

M. WegerDepartment of Physics, The Hebrew Uni versity, Jerusalem, Israel

L. FriedmanGeneral Telephone and Electronics Laboratories, 8'altham, Massachusetts 02254

(Received 5 April 1982)

The recent proposal of scattering by fluctuations of polarization energy is used to calculate thedifference in conductivity between the commensurate and the incommensurate phases. It is

shown that this difference increases when higher pressures are required to achieve commensura-bility. Another effect of polarization energy is to increase the magnitude of second-orderelectron-phonon scattering relative to first-order scattering at ambient pressure. However, when

the pressure is increased, the contribution to the resistivity from second-order electron-phononscattering decreases with pressure much more rapidly than the contribution to the resistivityfrom first-order electron-phonon scattering until, at sufficiently high pressures, the first-orderelectron-phonon scattering is dominant.

The electrical resistivity p of quasi-one-dimensionalsystems above the Peierls transition temperature hasrecently attracted a great deal of interest. ' " We be-lieve that the temperature"' and pressure" "dependence of p are now well understood. ' ' Therecently observed' increase of p with frequency inthe far infrared range yeilds a large, negative dielectricconstant which serves as experimental confirmationof the phonon-drag theory proposed' "' for quasi-one-dimensional conductors. The phonon-dragtheory also explains ' why the resistivity in the com-mensurate phase is larger than the resistivity in theincommensurate phase. The main point is that in theincommensurate phase, first-order electron-phononscattering is not resistive. This explains the absenceof a linear term in the measured p(T) in the incom-mensurate phase. According to this picture, the elec-trical resistivity at low temperatures T & 150 K oc-curs only via second-order electron-phonon scatteringp2. '2' At high temperatures, the phonon-phononinteraction redistributes the momentum that the 2kFphonons receive from the electrons and phonon dragis quenched. However, since pi &( p2 at low tem-peratures (T ( 150 K), the linear term is maskednear ambient Tby the T dependence of p2 evennear room temperature.

In the commensurate phase, we expect ' that first-order electron-phonon scattering is resistive and thelinear term pi shows up experimentally. ' We may

summarize the above ideas as follows:

Pine —P2 ~

Pcom = Pi + P2 ~

= l + (pt/p2), (3)Pine

where p„and p;„, are the resistivities in the com-mensurate and the incommensurate phase, respec-tively. We see from (3) that the ratio p~/p2 plays acrucial role in determining the difference between theresistivities of the two phases. Moreover, since thecommensurate phase is achieved by applying pres-sure, it is important to know the pressure dependenceof pi/p2

Very recently, Friedman' suggested that fluctua-tions of polariziation energy are important and pro-vide a strong scattering mechanism for electrons inquasi-one-dimensional organic systems. We here ap-ply the fluctuations-of-polarization-energy model tothe analysis of the pressure dependence of pq/p2. Weshow that including polarization-energy fluctuationsaccounts for the observed difference between theresistivity in the commensurate and the incommensu-rate phases. In addition, it also accounts for thechange of the temperature dependence of p above150 K under pressure in the incommensurate phase.

The contribution of a molecule (at a distance rfrom a single excess electron on another molecule) to

26 3456 1982 The American Physical Society

BRIEF REPORTS 3457

the polarization energy is

noe 2

Vp=-2r' (4)

bp2 cc

CO

Employing a Lennard-Jones potential for the b

dependence of ~, we found"

Q lnoJ1p

8 lnb

Therefore from Eqs. (5) and (6), respectively,

(6)

~ lnpi 9 inc)= —2 —10=+21—10=+11, (8)9 lnb g lnb

~ lnp2 9 inc@= —4 —12 =+42 —12 =+ 30 . (9)9 lnb 9 lnb

Thus the polarization effect increases the relative

dependence of pt/p2 on pressure. For a given changeof pressure, without the polarization effectSp2/p2= 25p~/p~. However, including the 1/rpolarization-energy effect yields Sp2/p2 =—35p &/pt.

Moreover, the linear term is due mainly to longitu-dinal phonons, which have a small Gruneisen con-stant because the molecular dimension along thestacking axis is small (the molecules are flat). How-ever, transverse phonons and librons also contributeto the linear term, since the energy of an electron ona given chain is modulated by a transverse phonon orlibron on a neighboring chain. This interaction was

suggested by Weger and Friedel. Following thissuggestion, Megtert et al."looked for such phononsand librons in the diffuse reflections, and failed todetect any. This indicates that the coupling strengthof these phonons is weaker than that of the longitu-dinal phonons, and the phonons polarized in the c

where 0.0 is the molecular polarizability. The first-order electron-phonon coupling is therefore propor-tional to (8 Vp/dr )„y cc b 5 and p& is proportional to

b-10pi cK

QJ

where ~ is the phonon frequency. Friedman es-timated p~p due to fluctuation of polarization energyto be three times larger than the Bloch mechanism

pts [due to thermal modulation of the bandwidth

(S)] at ambient pressure. However, ptp/p~s ~ b '

and therefore increases rapidly with pressure. %emay therefore conclude that scattering by fluctuationsof polarization energy is the dominant scatteringmechanism in the commensurate phase, as given by(2).

We now analyze, pt/p2 as a function of pressure.The second-order electron-phonon coupling constant,derived from (4), is proportional to(d'Vp/'dr') „&~ b 6 and therefore

I.O

0.5

0.0 0l

20

P (k bar)

FICT. 1. p)/p2 as a function of pressure. The arrows indi-

cate the pressures at which the indicated compound becomescommensurate.

direction, which also have a longitudinal component,due to the tilt of the molecules. The contribution tothe quadratic term p2 is also due to transverse pho-nons and librons, which have a large Gruneisen con-stant because the molecular dimensions along the aand c axes are large. By analogy with naphthaleneand anthracene, we may expect a Griineisen contanttwice as large. Also, the compressibility measure-ments of Debray et al. indicates that the Griineisenconstant for transverse motion is more than twice aslarge as that for longitudinal motion. If taken intoaccount, this would further enhance the pressuredependence of the quadratic contribution to the resis-tivity relative to the linear contribution. In Fig. 1, wepresent our estimated pt/pq as a function of pressure.

We estimate that pt/p2= 0.2 at 80 K for ambientpressure. For I' = 20 kbar, p~ decreases by about afactor of 2 whereas p2 decreases by about an order ofmagnitude. This is illustrated in Fig. 2.

Figure 1 completely explains all the available resis-tivity data for the commensurate phase.TMTSF2(tetramethyltetraselenafulvalenium) -PF6 isalready commensurate at I' = D, and therefore p mustbe given by (4). Fitting (2) to the observed resistivi-ty yields"" pt/p2—-0.1. For TSF-TCNQ(tetraselenafulvalenium-tetracyanoquinodimethane) itwas found that the commensurate phase is achievedby applying a pressure of about 4—5 kbar. For thispressure, it follows from Fig. 1 that pt/p2= 0.2—0.3,accounting for the measured ratio p„ /p;„, . Themost dramatic change in resistivity in the commensu-rate phase relative to the incommensurate phase isobserved'2 for TTF(tetrathiafulvalenium)-TCNQ.This is also in agreement with Fig. 1, since the com-mensurate phase for TTF-TCNQ is not obtained until19 kbar, and for such a high pressure pt/p2= 1.Therefore, according to (3), p„ /p;„, ——2, in agree-ment with experiment. Since the commensuratephase is usually obtained by applying pressure, wemay conc1ude that the effect of polarization is to in-crease the resistivity in the commensurate phase.

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200

150

E

100

of the phonon frequencies. 9 " By increasing thepressure to about 30 kbar, we see from Fig. 2 that

p~ & p2 and m drops toward 1.The small value of p~/p2 at ambient pressure is also

in accord with the theory of polarization-energy fluc-tuations. These enhance p2 over p~ since

p2 (x (f) Vp/Br ) and p~ cc (8 Vp/8r )'. For Vp cc r

it V /Br ~ ~ (n + 1)nr "+2

00 IO

P (kbar)

20

FIG. 2. p2 and p~ as a function of pressure for p~ =0.2p2at P =0. This corresponds to TTF-TCNQ at T = 100 K.

Figure 2 indicates that p~ & p2 for P & 20 kbar.This may explain the change in temperature depen-dence of the resistivity upon increasing the pressure.Experimentally, p c T, where m depends on pres-sure, decreasing from m = 2.3 at ambient pressure tom = 1.3 at p = 30 kbar. ' For T & 150 K, there isalmost no experimental difference in the measuredresistivity between the commensurate and the incom-mensurate phases. The reason for this may be due tothe fact9 ""that phonon drag is quenched at hightemperature and therefore first-order scattering doescontribute to the resistivity even in the incommensu-rate phase. However, at ambient pressures, p& &( p2and p = p2 n 'r" (the fact that m is larger than 2 isaccounted for by the weak temperature dependence

and since n = 4 for the polarization eriergy, the factor(n + 1)n = 20 is large and enhances the quadraticterm relative to the linear one. This is an additionalreason why the quadratic contribution to the resitivityis large in organic metals. Other reasons were givenin Refs. 27 and 28. An explicit calculation of p2, us-ing the polarization energy in the same manner aswas done by Friedman, yields a value within a fac-tor of 3 of the experiment value. In view of thelarge uncertainties of such a calculation, such agree-ment may be considered to be satisfactory.

In summary, under pressure the dominant first-order electron-phonon contribution to the resistivityis due to fluctuations in the polarization energy. Thiscontribution is observed only under conditions whichlead to the suppression of phonon drag, namely, in

the commensurate phase or at high temperatures inthe incommensurate phase. Moreover, the fluctua-tions of polarization energy enhance the first-orderelectron-phonon contribution to the resistivity rela-tive to the second-order contribution at high pres-sures. This explains the observed resistivity in thecommensurate phase for TMTSF2-PF6 (P = 0).TSF-TCNQ (P = 4 kbar) and TTF-TCNQ (P = 19kbar). Finally, the weaker temperature dependenceof the resistivity (smaller m) under increasing pres-sures is also accounted for.

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