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Physica B 362 (2005) 187–192

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Small-polaron resistivity of the narrow band molecular chain:The influence of phonon hardening

Zoran Ivic�, Slobodan Zekovic, Dalibor Cevizovic, Dragan Kostic,Gradimir Vujicic

Theoretical Physics Department, The ‘‘Vinca’’ Institute of Nuclear Sciences, 020, 11001 Belgrade, P.O. Box 522, Serbia and Montenegro

Received 24 December 2004; received in revised form 7 February 2005; accepted 7 February 2005

Abstract

We study the effect of the small-polaron induced changes of phonon spectra on charge transfer in molecular solids.

Noticeable discrepancy with the predictions of the traditional small-polaron theories is observed: accounting of phonon

hardening yields significantly lower estimates for small-polaron resistivity than that obtained on the basis of

conventional approaches. This especially concerns the low temperature limit in which transport processes are

dominated by the band like motion of charge carriers. This difference gradually disappears with the increasing of

coupling constant and temperature.

r 2005 Elsevier B.V. All rights reserved.

PACS: 63.20.Dj; 71.38.�k

Keywords: Polaron resistivity; Phonon hardening; Molecular chain

Strong coupling of charge carriers to localstructural deformations of underlying crystallattice plays an important role in determiningboth electronic and vibrational properties of manyof the substances currently of interest in condensedmatter physics. Thus, for example, it is believedthat small-polaron effect could be crucial for theunderstanding of various physical properties of the

e front matter r 2005 Elsevier B.V. All rights reserve

ysb.2005.02.010

ng author. Tel.: +38111 2441126;

40100.

ss: zivic@rt270.vin.bg.ac.yu (Z. Ivic).

wide class of very important substances such ashigh-temperature superconductors [1], colossalmagnetoresistive manganites [2,3], highly conduct-ing polymers and molecular chains like acetanilide(ACN) and biological macromolecules, a-helix andDNA [4–6], for example.

The subject of this paper is the examination ofthe possible consequences of the polaron inducedmodifications of phonon spectra on charge trans-port in the strongly coupled electron–phononsystems. The importance of the accounting ofsuch changes has been pointed out recently by Lee

d.

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Z. Ivic et al. / Physica B 362 (2005) 187–192188

and Min and Yu and Min [3] who examined theimpact of phonon frequency hardening on thesmall-polaron transport in colossal magnetoresis-tive manganites. However, in these studies themodification of phonon spectra was consideredwithin the model with the linear electron–phononinteraction. This is not justified since thesmall-polaron effect causes quite different, highlynonlinear in phonon operators, polaron–phononinteraction which would produce the specificchanges of the phonon spectra. The character ofthese changes has been examined in Refs. [7,8] bymeans of the diagrammatic perturbation theory. Itwas found that the small-polaron effect leads to asoftening of phonon modes which displays quitedifferent behavior in the low and high temperatureregimes. The magnitude of the phonon frequencycorrection is estimated to be proportional to thesquare of the small-polaron narrowing factor. Sopredicted behavior of phonon frequencies is quitedifferent than that which should be expected onthe basis of the variational treatments of thesmall-polaron problem by means of the modifiedLang–Firsov method [9] combined with thesqueezed state ansatz [10,11], where theso-called squeezing parameter characterizes thedegree of the phonon spectra renormalization.Namely, as shown in our recent papers [12],calculations of the renormalized phononfrequencies within the pseudo phonon approxima-tion or by means of Green function andsqueezed state approaches imply that the small-polaron effect would lead to a phonon hardeningrather than softening. The magnitude ofthe so estimated changes of phonon spectra is ofthe first order in the small-polaron narrowingfactor and fairly exceeds the contribution esti-mated in Refs. [7,8] where the phonon self-energyhas been calculated by means of the diagrammaticperturbation theory. However, only the terms ofthe second order in polaron–phonon interactionwere accounted and consequently, the mostdominant contribution, linear in small-polaronnarrowing factor, has been neglected. For thatreason we base our further analysis upon theresults of papers [12] where we, employing thenonadiabatic strong coupling approximation, de-rived the following expression for the renormalized

phonon frequency

~o2q ¼ o2

q 1 þ8jF qj

2

_2o2q

Xk;g

Jeff ðgÞ

_oq

sinq � g

2nk cos k � g

!.

(1)

In terms of the below listed physical parameters ofsystem condition for the applicability of the aboveequation reads 2J5_o05EB: Throughout thispaper we have used the same labels as in Ref. [12]:Jeff ¼ Je�SðTÞ denotes the effective intersite trans-fer integral reduced on account of phonon dres-sing. It is determined by the value of the couplingconstant

SgðTÞ ¼2

N

Xq

j F qj2

ð_oqÞ2sin2 q � g

2ð2~nq þ 1Þ

in which nq ¼ ðe_oq=kBT � 1Þ�1 is the phononaverage number, g is the vector connecting thenearest neighbors, while F q ¼ F�

�q denotes elec-tron–phonon coupling parameter whose explicitdependence on phonon quasi-momentum (q) isdetermined by the type of the electron–phononinteraction and phonon dispersion law oq ¼ oðqÞ:Finally, nk ¼ ðeðEðkÞ�Ef Þ=kBT þ 1Þ�1 denotes thesmall-polaron average number in which Ef andEðkÞ ¼ �

Pg Jeff ðjgjÞe

ik�g represent Fermi level andthe small-polaron band energy measured from thesmall-polaron binding energy (EB ¼ ð1=NÞ

Pq

ðjFqj2=_oqÞ), respectively.

This result implies that the small-polarons mayinduce the noticeable enhancement of phononfrequencies which, in turn, causes the significantmoderation of the rise of the small-polaroneffective mass as a function of coupling constantand temperature [12]. Such changes of phononspectra may have certain consequences for trans-port processes which will be examined below. Forthat purpose we have calculated the small-polaronresistivity taking into account the renormalizationof phonon frequencies as specified by Eq. (1). Ouranalysis is based upon the conventional theories ofthe small-polaron transport [9,13–15]. For theparticular calculations we have used the wellknown results for band and hopping conductivityobtained within the one-dimensional Holstein’smolecular crystal model which is specified by thefollowing choice of electron–phonon coupling

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Z. Ivic et al. / Physica B 362 (2005) 187–192 189

parameter [13] Fq ¼ wð_=2MoqÞ1=2 and phonon

frequency oq ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffio2

0 þ o21 cos qR0

q: Such an

approach represents rather crude idealization ofa real solid but still captures the essentialphenomena involved in the small-polaron trans-port. Moreover, present results may be useful forthe understanding of the transport processes insome realistic substances such as quasi one-dimensional conductors [4,5], biological macro-molecules, DNA and a-helix [6], and titaniumoxide compounds [16] (LaTiO3:41 for example),which may be successfully described within theone-dimensional models.

By means of the above introduced simplifica-tions, Eq. (1) becomes

~oq ¼ oq

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ 4SBe�SðTÞ

ð1 � cos qR0Þ ~n

ð1 þ ðo1=o0Þ2 cos qR0Þ

2

s.

(2)

The meaning of system parameters is as in Ref.[12]: o0 denotes the frequency of the intramole-cular oscillations, o1 characterizes the vibrationalenergy transfer between neighboring sites. M ; w;and R0 denote mass of the molecules constitutingthe solid, the strength of the electron–phononinteraction and lattice constant, respectively.Finally, ~n ¼ ð2=NÞ

Pk nk cos kR0 is the effective

electron (small-polaron) concentration, factor 2comes from the summation over the two possibleprojections of electron spin, S SðT ¼ 0Þ ¼EB=_o0 denotes the zero temperature couplingconstant while B ¼ 2J=_o0 is so-called adiabaticparameter.

We shall use the above result in order to explorehow the small-polaron induced renormalization ofphonon spectra affects the charge transfer in thesesystems. For that purpose we shall calculateresistivity substituting the above result into theknown expressions for band and hopping con-ductivity obtained within the conventional small-polaron theories [9,13–15] which predict substan-tially different character of polaron motion in thelow and high temperature regimes. At lowtemperatures (1=N

Pq 2jF qj

2=ð_oqÞ2ð1 � cos qÞ

cschðboq=2Þ51) polarons behave as heavy quasi-particles in a band with renormalized width Jeff

and the metallic-like conductivity is expected.Upon the increasing of temperature, band picturegradually breaks down and competition betweenthe coherent and incoherent, hopping, motion ofsmall-polaron arises.

In the low temperature regime conductivity maybe expressed in terms of the above introducedparameters as follows [15]:

sband

s0¼ 4

S4

B2

_o0

kBT

Doo0

exp �2S þ_o0

kBT

� , (3)

where s0 ¼ nce2R2

0=_; nc is carrier concentration,while Do denotes the phonon bandwidth. In thehigh temperature regime we have calculatedresistivity of system by virtue of R ¼ 1=ðsband þ

shopÞ where sband and shop denote band andhopping conductivities specified by means offollowing expressions:

shop

s0¼

J2

kBT

2p

ð1=pÞR p0

4jF ~qj2 sin2

ð ~q=2Þ cschð_o ~q=2kBTÞd ~q

" #1=2

� exp �1

p

Z p

0

4jF ~qj2

_2o2~q

sin2 ~q

2tanh

_o ~q

4kBT

� d ~q

( ), ð4Þ

sband

s0¼

1

kBT

2p

ð1=pÞR p

04jF ~qj

2 sin2ð ~q=2Þ csch _o ~q=2kBT

� �d ~q

" #�1=2

� exp �1

p

Z p

0

4jF ~qj2

_2o2~q

sin2 ~q

2

(

�csch_o ~q

2kBT

� d ~q

); ~q ¼ qR0. ð5Þ

We stress that, in accordance with the approxima-tions involved in the derivation of these expres-sions [9,13,14], we need ð1=NÞ

Pq 2jFqj

2=ð_oqÞ2

ð1 � cos qÞ cschðboq=2Þb1 for Eqs. (4) and (5) tobe valid.

Substituting the explicit expression for ~oq andoq into the above formulas we have calculated theconductivity and resistivity of the system with andwithout taking into account the phonon renorma-lization. These calculations have been performedin accordance with the approximations introducedin Ref. [7]. Thus, for the small-polaron Fermienergy we assume that it is of the same order asthe characteristic phonon energy (EF ’ _o0) sothat nk ¼ ½expðEk � EF=kBTÞ þ 1��1 ’ 1=2ð1 þ

tanhðEF=2kBTÞÞ 1=2ð1 þ tanhð_o0=2kBTÞÞ:

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τ

ρ/ρ 0

700

600

500

400

300

200

100

0.50.0 1.0 1.5 2.0

S=4B=0.3

B=0.5

B=0.7

Fig. 2. Small-polaron resistivity versus normalized temperature

t ¼ kBT=_o0: full lines correspond to RðTÞ calculated taking

into account renormalization of phonon spectra.τ t

B=0.3

0.75

0.70

0.65

0.60

0.55

0.50

Z. Ivic et al. / Physica B 362 (2005) 187–192190

Next, we substitute summation over the electronquasi-momenta by the integration adopting the

rule ð1=NÞP

� � � ! ðR0=pÞR kf

�kf� � � dk: kf denotes

the Fermi quasi-momentum which at half fillingbecomes kf ¼ p=2R0: In such a way for theeffective carrier concentration we obtain ~n �

1=2ð1 þ tanhð_o0=2kBTÞÞ:Our results are illustrated in Figs. 1–3 where we

have plotted the temperature dependence of thesmall-polaron resistivity in the low (Fig. 1) andhigh (Fig. 2) temperature regime. In Fig. 3transition temperature (T t) from the coherent tothe incoherent small-polaron motion versus cou-pling constant, has been plotted. Full linescorrespond to the results which are obtainedtaking into account the renormalization of thephonon spectra. In both cases, i.e. with andwithout accounting of phonon hardening, weobserve practically the same qualitative behaviorof the resistance as a function of temperature. Thatis, at low temperatures (Fig. 1), where bandconductivity dominates transport processes, RðTÞ

increases with the rise of temperature. Thistendency maintains in the intermediate tempera-ture region (Fig. 2) where the competition betweenband like and hopping conduction takes place.When temperature exceeds come characteristic

0

1

2

3

4

5

6

τ

ρ/ρ 0

1.00.80.60.40.20.0

B=0.3

B=0.5

B=0.3B=0.5

S=4

Fig. 1. Low temperature small-polaron resistivity versus

normalized temperature t ¼ kBT=_o0 for S ¼ 4 and two fixed

value of adiabatic parameter (B ¼ 0:3 and 0:5). Full lines

correspond to RðTÞ calculated taking into account renormaliza-

tion of phonon spectra.

2 4 6 8 10

S

B=0.5

B=0.70.45

0.40

Fig. 3. Transition temperature from the coherent (band like) to

the incoherent (hopping) small-polaron motion versus coupling

constant. As above, full lines refer to renormalized phonon

spectrum, while dotted, dashed and dash–dotted lines corre-

spond to nonrenormalized phonons for B ¼ 0:3; 0:5 and 0.7,

respectively.

value, determined by the particular values ofcoupling constant and adiabatic parameter, hop-ping conductivity prevails and we observe, in acertain narrow temperature interval, the suddendrop of RðTÞ: Further temperature rise yields theincrease of resistivity. The influence of the phononhardening is manifested through the noticeablereduction of the small-polaron resistivity, i.e.

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Z. Ivic et al. / Physica B 362 (2005) 187–192 191

enhancement of conductance, in the low andintermediate temperature regime and through theshift of the position of the peak of RðTÞ towardsthe higher temperatures. Latter indicates that thetransition from the coherent, band like small-polaron motion, to the incoherent one achievedthrough random site jumps, is moved towards thehigher temperatures. It is demonstrated in Fig. 3where we have plotted the transition temperaturefrom the coherent to the incoherent regime as afunction of the coupling constant. Increasing ofcoupling constant and temperature tends to cancelthe renormalization of phonon spectra so thatRð ~oqÞ ! RðoqÞ and T tð ~oqÞ ! T tðoqÞ when S andT rise.

All these effects are the consequence of theexponential dependence of the effective hoppingintegral (Jeff ) on coupling constant which behavesas S 1=o2

q and decreases when oq rises.Concluding this paper, we emphasize that the

accounting of the phonon hardening yields quali-tatively the same behavior of the small-polaronresistivity (conductivity) as a function of thesystem parameters and temperature as it waspredicted within the conventional small-polarontheories. Quantitative predictions, however, maybe quite different. This especially concerns the lowtemperature regime Fig. 1 in which we predict thehuge difference between the resistivity estimatedwith and without the phonon renormalization.This difference persists in the intermediate tem-perature regime and disappears in the hightemperature limit. The degree of the deviationfrom the previously obtained results is determinedby the temperature and the values of systemparameters and disappears with the increasing oftemperature and coupling constant. In such a wayour results point out that in the low andintermediate temperature regimes and for not toolarge values of coupling constant, traditionaltheories of small-polaron transport should berevised in order to account for the changes of thephonon spectra. So revised theory indicates thatthe validity of the small-polaron band theory isextended towards the higher values of couplingconstant and higher temperatures. This fact shouldbe taken into account in the interpretation of theexperiments in which the presence of polaronic

excitations was reported. We must underline that,due to limited validity of Eq. (1) which is the basisof present analysis and which applies in thenonadiabatic strong coupling case, our resultscould be relevant only for the narrow bandsystems. Extension towards the wide band systemsis possible by means of the mentioned [10,11]variational methods which encompasses Lang–-Firsov method combined with the squeezedcoherent state ansatz. Further limitation on theapplicability of the present analysis to realisticsystems comes from the disregarding of the direct(Coulomb) and indirect phonon mediated polar-on–polaron interaction which may change thecharacter and the degree of the modification ofphonon spectra. The influence of polaron–polaroncorrelations will be examined subsequently.

Let us finally note that although we explicitlyconsidered a molecular chain, above results maybe generalized to two and three dimensions underthe restriction to the narrow-band tight-bindingmodel.

We would like to acknowledge useful conversa-tions with Dr. F. Vukajlovic, Dr. D. Kapor andDr. L.J. Hadzijevski. This work was supported bythe Serbian Ministry of Science and Technology:under the contract number 1961—Nonequlibriumand cooperative phenomena and processes incomplex systems of condensed matter physics.

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