14 May 1999

Ž .Chemical Physics Letters 305 1999 117–122

Exciton–exciton annihilation in linear molecular aggregates atlow temperature

V.A. Malyshev a,1, H. Glaeske a, K.-H. Feller a,)

a Fachhochschule Jena, Fachbereich MedizintechnikrPhysikalische Technik, Tatzendpromenade 1b, D-07745 Jena, Germany

Received 4 June 1998; in final form 19 February 1999

Abstract

Exciton–exciton annihilation is observable in linear molecular aggregates at low temperature as the pump intensity risesin spite of the fact that, due to weak exciton–phonon coupling, the exciton mobility at low temperatures is expected to beslow. Under the circumstances outlined, the applicability of the bimolecular theory, which implies the approach of twoexcitons before they annihilate, seems to be questionable. We propose an alternative channel of excitonic annihilationexpected to be an appropriate candidate in treating this effect and in estimating the corresponding rate. It appears to beinversely proportional to the cube of the localization length and reasonably fits the experimental data. q 1999 ElsevierScience B.V. All rights reserved.

1. Introduction

A great deal of interest has been shown during thelast two decades to one-dimensional structures suchas quasi-one-dimensional J-aggregates of polyme-thine dyes and conjugated polymers due to the ex-

w xtraordinary features of their photo-response 1–5Ž w x .see also reviews 6,7 and references therein mainlycaused by the collective character of the electronicexcitations presented by one-dimensional Frenkel ex-citons. The most important peculiarity of these ob-jects is the giant values of high-order susceptibilitiesscaled, moreover, with the size of an aggregate. This

) Corresponding author. E-mail: feller@mt.fh-jena.de; fax: q493641 205601

1 Permanent address: All-Russian Research Center, VavilovState Optical Institute, Birzhevaya Liniya 12, 199034 Saint Peters-burg, Russia.

makes them promising candidates in developing de-vices of optical logic.

w xAs first shown in Refs. 8,9 , the effect of exci-ton–exciton annihilation becomes of great impor-tance in forming the optical response of the aggre-gate subsystem as the pump intensity rises. Concern-ing data on luminescence, it results in noticeablereduction of the lifetime as well as in lowering the

w xquantum yield of luminescence 8–12 .Another manifestation of the exciton–exciton

w xannihilation was demonstrated in Ref. 13 with re-gard to the signal of four-wave mixing. Under theconditions of saturation of the exciton resonance, thiseffect is responsible for an enhancement of the four-wave mixing efficiency.

A definitely positive role of excitonic annihilationappears with respect to the possibility of obtaining a

w xbistable optical response of an aggregate 14–18being demonstrated firstly in the frame of the mean-

0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved.Ž .PII: S0009-2614 99 00369-3

( )V.A. MalysheÕ et al.rChemical Physics Letters 305 1999 117–122118

w xfield approach. Further studies 19,20 have shownthat, in reality, the bistable behaviour can hardly berealized since it is drastically disturbed by the spatialinhomogeneity of population resulting from the pres-ence of aggregate ends. The excitonic annihilation,the rate of which rises with the population, resists toa preferable creation of population at a certain regionof the aggregate and stabilizes it in such a way,providing conditions for bistability to be revealed.

In the present Letter we try to obtain an insightinto the possible channels of excitonic annihilation atlow temperatures, when one-dimensional excitons

Žbecome localized due to a weak static disorder e.g.,.of diagonal or off-diagonal nature . The term ‘locali-

zation’ will hereafter mean that a single-exciton stateŽis spread over part of the aggregate localization

.segment rather than over the whole. Different statescan be localized on different segments of the aggre-gate and, what is important, the localization seg-ments of lowest single-exciton states normally do not

w xoverlap 21–23 . Since two excitons take part in theprocess of excitonic annihilation, one should taketwo-exciton states into consideration. As is well

Ž w x.known see, e.g., Ref. 6 , one-dimensional Frenkelexcitons are weakly interacting fermions so that theeigenfunctions of states with two excitons can becomposed as Slater determinants of single-excitoneigenfunctions. Under the condition of localization,two different types of two-exciton states then appear:Ž .i with two excitons belonging to the same localiza-

Ž .tion segment, and ii with those localized on differ-ent localization segments. Notice that excitonic anni-hilation, in the former case, does not require any

Žapproach of two excitons just due to the fact that.they occupy the same aggregate segment while this

is not the case when excitons are created on differentlocalization segments. Here, excitons should ap-proach each other before they annihilate. Descriptionof such a movement is out of the scope of thepresent Letter as it requires treatment of theexciton–phonon coupling and, thus, obviously repre-sents a rather complicated problem. Nevertheless,there are indications that, at low temperatures, local-ized excitons are in fact immobile quasiparticles.Indeed, after excitation, the only way for a weaklylocalized exciton to move is to jump from thatlocalization segment where it has been created to

Žanother segment. At low temperatures we mean

.close to zero , it can do so if an adjacent localizationsegment is characterized by a lower energy. Thus,the movement of excitons in space will be accompa-nied by the spectral diffusion towards lowering theenergy, which in turn will manifest itself in a Stokesshift of the exciton luminescence spectra with re-spect to their absorption spectra. Experimental data

Žnormally show either no such shift see, e.g., Refs.w x. w x1,2,26 or only a small shift 25 related to the

Žpolaron effect for an analogous effect in systems ofw x.interacting impurities, see Ref. 24 . Thus, one can

expect a low mobility of the localized excitons atlow temperature. Therefore, we will mainly focusour attention on the annihilition of two excitonscreated on the same localization segment, calling itthe intra-segment exciton–exciton annihilation.

The remainder of the Letter is organized as fol-lows. In Section 2, we discuss the problem of possi-ble mechanisms of the exciton–exciton annihilationfor a system comprised of two molecules, selectingone in order to face the problem outlined above.Section 3 deals with the problem of excitonic anni-hilation for a regular aggregate of many molecules,which mimics intra-segment annihilation. We calcu-late the rate of annihilation for two excitons, assum-ing them to be in the lowest two-exciton state. InSection 4, we make an estimate of the intra-segmentannihilation rate, based on the low-temperature ex-

w xperimental data of Ref. 11 , and show that ourtheoretical result reasonably fits the experiment. Fi-nally, Section 5 summarizes the Letter.

2. Annihilation model

Before we continue to treat the general problemof excitonic annihilation in an aggregate of manymolecules, we will address the discussion of thepossible annihilation mechanism, making use of thesimplest model system consisting of only two identi-cal molecules coupled to each other by the dipole–dipole interaction.

Let initially both molecules be in their first ex-cited states. We are interested in the disappearenceof one or both excitations by any process exceptspontaneous emission of each molecule. There areseveral possibilities how this can occur. To begin, let

Žus assume that the twofold excited state both.molecules are in their first excited states is on

( )V.A. MalysheÕ et al.rChemical Physics Letters 305 1999 117–122 119

resonance with a high-lying electronic-vibrationalstate of an isolated molecule, which is able to un-

Ždergo multiphonon-assisted relaxation in recent lit-w xerature Refs. 27,28 , there is the hint that this is the

case for pseudoisocyanine, one of the most investi-.gated aggregate-forming dyes . Then, two channels

of the annihilation can be considered as competitive.First, the energy of two excitations can be resonantlytransferred to this electronic-vibrational state whichthen rapidly relaxes, emitting phonons either to the

Ž .ground state both excitations disappear or to theŽfirst excited state one excitation disappears; see Fig.

.1 .The second possible channel for degradation of

the twofold excited state is a direct transformation ofits energy into the energy of vibrations. Concerningthe case of J-aggregates, we consider that such aprocess seems to be rather less probable compared tothe process just considered since the exciton–phononcoupling in J-aggregates is rather weak, as men-tioned in Section 1.

One more process has, in principle, to be taken inaccount that looks like annihilation, namely coopera-tive luminescence, when both excitations disappear

Fig. 1. Scheme of the exciton–exciton annihilation process throughŽ .a high-lying molecular level the resonance case, 2v s v .21 31

The first step is that molecule a goes to the ground state whilemolecule b passes to the high-lying molecular level. The secondstep is the radiative or radiationless relaxation from the third to

Ž .second level one excitation disappears and from the third toŽ .ground level both excitations disappear with the rates G and1

G , respectively.2

w xemitting a photon of double frequency 29 . Its effi-ciency is governed by the following matrix element² < < : ² < < : Ža1b1 H a3b1 a3b1 V a2b2 r E y Ea – ph ab 3a 2 a

.yE , where the term H stands for the interac-2 b a – ph

tion of the molecule a with the quantized electro-magnetic field and E , E , E are energies of the3a 2 a 2 b

corresponding molecular states. Note that here theenergy of a high-lying level 3 has not necessarily to

< :be in resonance with the doubly excited state 2 a2b ,i.e., E /E qE . If the molecule has a well-de-3a 2 a 2 b

fined parity and the interaction V is of dipole–di-ab

pole nature, the transition a3b1™1a1b is allowedeither as a quadrupole or magneto-dipole transitions.This determines a low efficiency expected for a suchtype of annihilation channel. Nevertheless, in thecase of no resonance of the doubly excited state withany high-lying molecular state as well as when parityis a bad quantum number, this channel may come toits own.

Hereafter, we choose for further study the anni-hilation channel involving multiphonon degradationof the high-lying molecular electronic-vibrationalstate. The Hamiltonian describing this model reads

HsH qH qU qV , 1aŽ .a b ab ab

2 2q qH s E a a , H s E b b , 1bŽ .Ý Ýa i i i b i i i

is1 is1

U syU a bqqaqb , 1cŽ .Ž .ab 1 1 1 1

q q � 4V sVa b b qa q h.a. , 1dŽ .Ž .ab 1 1 2 2

where the E and E are the energies of the first1 2

excited and high-lying molecular states, respectivelyŽ .we choose the ground state energy equal to zero ;

q Ž . q Ž . Ž .a a and b b are the creation annihilationi i i i

operators of the ith molecular state of molecules aŽand b the state with both molecules in their ground

< :.states serves as the vacuum state 0 ; U is theab

Hamiltonian of the resonant dipole–dipole interac-tion between molecules responsible for the excitonic

² < < : Žeffect, Us a1 U b1 for definiteness, we as-abw x.sumed U)0, as in the case of J-aggregates 1,3–7 ;

V is the Hamiltonian of annihilation with Vsab² < < : ² < < :b2 V a1,b1 s a2 V a1,b1 .ab ab

In order to calculate the rate of the exciton–exci-< : < : < :ton annihilation process a1,b1 ™ b2 or a1,b1

< :™ a2 we will make use of the perturbation theoryassuming that the rate of vibrational relaxation of thehigh-lying level G exceeds the intermolecular cou-

( )V.A. MalysheÕ et al.rChemical Physics Letters 305 1999 117–122120

pling V. Then, according to the ‘golden rule’, oneobtains for the exciton–exciton annihilation rate

2p 4pV 22<² < < : <w s a 2 V a1,b1 s . 2Ž .Ýa ab

G Gasa ,b

Ž . Ž .In Eq. 2 , the density of final states r e is substi-f

tuted by the inverse relaxation constant Gy1. Be-sides, starting from here we choose the Planck con-stant "s1.

3. N-molecule problem

In this section, we address the more general prob-lem of an elementary event of the exciton–exciton

Žannihilation, considering a homogeneous chain with.no disorder of a finite number N)2 of molecules

and assuming a nearest-neighbour interaction be-tween them. The latter approximation makes one-di-mensional excitons being non-interacting fermionsŽ w x.see, e.g., Refs. 30–34 , such that the energy of theexciton system can take the values

N

Es E n , 3aŽ .Ý k kks1

pkE sy2U cos . 3bŽ .k Nq1

Here n s0,1 is the occupation number of the k thk

excitonic mode, the E are the eigenenergies of thek

one-exciton problem. Then the eigenfunctions of ann-exciton state can be composed as Slater determi-nants of the eigenfunctions of the one-exciton prob-lem

N

< : < :k s w n1 , 4aŽ .Ý k nns1

1r22 pknw s sin , 4bŽ .k n ž /Nq1 Nq1

< :where n1 is the state vector of the nth moleculebeing in its first excited state. In particular, thetwo-exciton states are completely represented by theset of eigenfunctions

N

< : < :k k s c n1,m1 , 5aŽ .Ý1 2 k n ;k m1 2n-m

c sw w yw w . 5bŽ .k n ;k m k n k m k m k n1 2 1 2 1 2

Our goal now is to calculate the rate of the annihila-< : < :tion event k1,k2 ™ n2 making use of the pertur-

bation theoryN2p

2<² < < : <w s n2 V k ,k , 6Ž .Ýa annih 1 2G ns1

where the annihilation operator V , written in theannih

site representation basis, now has the formNy1

q q � 4V sV b b b qb q h.a. .Ž .Ýannih 1n 1,nq1 2 n 2,nq1ns1

7Ž .ŽHere, we have assumed that two excitations in the

.local basis picture disappear when they are nearestneighbours. The object of our interest is the annihila-tion of the lowest state of the two-exciton manifoldŽ Ž ..with k s1 and k s2 in Eq. 6 since precisely1 2

this state has the dominating probability of excitationŽ w x.by the external field see, e.g., Ref. 6 . Substituting

Ž . Ž .Eq. 5a into Eq. 6 at k s1, k s2 and making1 2

the necessary summations, one obtains2 N2pV 2w s c qcŽ .Ýa 1n ;2 ,nq1 1,ny1;2 n

G ns1

8pV 2 2p p2 2s sin qsin . 8Ž .ž /G Nq1 Nq1 Nq1Ž .

Assuming the number of molecules N to be large,Ž .then one finds from Eq. 8

40p3V 2

w s . 9Ž .a 3G N

Comparison of this equation with the analogous onefor the two-molecule problem shows the great differ-ence between them: the annihilation rate for a chainof size N scales with the inverse cube of the chainlength.

4. Analysis of the experimental data and discus-sion

ŽReal one-dimensional objects as mentioned, e.g.,.in Section 1 generally cannot be considered as

Žhomogeneous due to the presence of disorder mainly.of the static nature at low temperature . As was

already mentioned in Section 1, the disorder resultsin localization of the exciton states. According to the

( )V.A. MalysheÕ et al.rChemical Physics Letters 305 1999 117–122 121

wconcept of hidden structure proposed in Refs. 21–x23 , one may classify the single excitonic states,

close to the bottom of the exciton band, into severalŽ .groups of few states two or sometimes three such

that the states of each group are localized within aŽ . )particular coherent segment of average size L s

) Ž )N a N is the so-called number of coherentlybound molecules, and a the ‘lattice constant’, i.e. the

.chain length per monomer and do not overlap withthe others. It is remarkable that they can be consid-ered as the lowest excitonic states of a regular

Ž . )aggregate in the absence of disorder of size N a.Ž .From this it becomes clear that one can use Eq. 9

for estimation of the rate of the intra-segment anni-hilation w intra, simply substituting N by N ) in Eq.aŽ .9 which gives us

40p3V 2

intraw ; . 10Ž .a) 3G N

In order to make an estimate of the magnitude ofw intra we need the experimental data concerning thea

parameters U, V, N ) and G . Since we have noinformation with regard to the magnitude of interac-tion in the annihilation channel, assume it as being ofdipole–dipole nature and consider V of the sameorder of magnitude as the dipole–dipole interactionU responsible for the formation of the exciton band.The value U;600 cmy1 is widely admitted for

w xJ-aggregates of pseudo-isocyanine 1,2,6 . Substitut-y1 ) Žing V by 600 cm , N by 20 the data of Ref.

w x. 13 y111 and the relaxation constant G by 10 sw x intra Ž .11,10 we obtain for w , according to Eq. 10 ,a

intra 12 y1 Ž .y1w f5=10 s s 200 fs . Unfortunately,aw xthe authors of Ref. 11 did not provide the whole set

of experimental data relative to the kinetics of theexciton–exciton annihilation except that at leastthree exponential functions with the characteristic

( y 1) ( y 1)time constants of 200 fs G , 1.5 ps G , and1 2( y 1)20 ps G were needed, besides an instanta-3

neously responding component and a long-life com-( )ponent 4100 ps , it should moreover be noted,

with no indications of their weights. Our estimate isin rather good agreement with the fastest component.We refer it to the excitonic annihilation within the

Ž ) .typical localization segment of size N s20 . AŽ .possible channel for the slower component 1.5 ps is

the annihilation of two excitons created within sepa-

rated localization segments as was suggested in Ref.w x11 . Probably, this channel starts to play a role at 20

ŽK the temperature, at which the measurements inw x .Ref. 11 were done . An estimate of the correspond-

ing annihilation rate is out of scope of this Letter andwill be considered in a forthcoming paper.

Recall that the model of intra-segment annihila-tion implies two excitons created within the same

w xlocalization segment. The authors of Ref. 11 madean estimate that 100 molecules per aggregate were

Žproduced by the highest excitation power 0.982 . w xGWrcm . Based on the data of Ref. 8 that an

aggregate normally comprises of ;104 molecules,they concluded that less than one exciton was cre-

Žated per segment of localization containing typically.20 molecules . Assuming a Poisson distribution for

the probability of having an integer number of exci-tation per segment, we found that for the data of Ref.w x Ž 411 100 excitations per an aggregate of 10

. )molecules , 10 segments of size N s20 are ex-pected in average to be more than once excited,while for the once excited segments this number is80. This suggests for the weight of the 200 fscomponent a value of the order of 1r10. Making amore correct estimate of this value requires a moredetailed study, based on the numerical solution of theeigenvalue problem, and will be given elsewhere.

A number of experiments was devoted to theexciton–exciton annihilation in linear molecular ag-

w xgregate at room temperature 8–10,12 . We cannotcompare directly our theoretical estimate with the

w x Ž .data obtained in Refs. 8–10,12 . Eq. 10 includestwo quantities that could be affected as the tempera-

Ž .ture rises: 1 the relaxation rate of the high-lyingŽ .molecular level G , and 2 the number of coherently

bound molecules N ). The former will go up, givingrise to decrease of the annihilation rate. On thecontrary, the latter will go down, resulting in growthof the annihilation rate. The interplay of these twofactors will determine the exciton annihilation effi-ciency at higher temperatures. Thus, fitting the ex-

w xperimental data of Refs. 8–10,12 requires mesure-ments of the temperature behaviours of G and N ).Besides, one should keep in mind the fact that, athigher temperatures, the annihilation of two excitons

Žcreated on separated segments of localization not.considered in the present Letter may also come to

its own.

( )V.A. MalysheÕ et al.rChemical Physics Letters 305 1999 117–122122

5. Summary

The problem of excitonic annihilation in one-di-mensional molecular aggregates is considered underthe conditions of localization of the exciton states.The perturbation theory is used in order to estimatethe rates of the intra-segment exciton–exciton anni-hilation under the assumption of an annihilationmechanism involving multiphonon degradation of ahigh-lying molecular electronic-vibrational state. It isshown that the corresponding rate w intra scales witha

the inverse cube of the localization length. Ourestimates of the annihilation rate are in a goodqualitative agreement with the fastest component ofthe experimental low-temperature annihilation data.

Acknowledgements

This work was supported by the Federal Ministryof Education, Science, Research, and Technologyunder the project ‘Bistable optical switching ele-ments on the basis of organic macromolecular mate-rials’. VAM thanks the Deutsche Forschungsgemein-schaft for a grant.

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