Journal of Luminescence 81 (1999) 127—134

Free-induction fluorescence from Frenkel excitons inone-dimensional systems with substitutional traps:

Effects of long-range interactions

V.A. Malyshev!, A. Rodrıguez", F. Domınguez-Adame#,*! All-Russian Research Center, ‘‘Vavilov State Optical Institute++, Birzhevaya Liniya 12, 199034 Saint-Petersburg, Russian Federation

" GISC, Departamento de Materma& tica Aplicada y Estadı&stica, Universidad Polite&cnica, E-20840 Madrid, Spain# GISC, Departamento de Fı&sica de Materiales, Universidad Complutense, E-20840 Madrid, Spain

Received 26 May 1998; received in revised form 1 October 1998; accepted 1 October 1998

Abstract

We study the effect of long-range interactions on the free-induction fluorescence from Frenkel excitons following afterexcitation by an ultrashort pulse of a small area. A non-perturbative contribution of far neighbors to the free-inductiondecay is found. The effect results from the non-perturbative renormalization of that part of the exciton energy spectrum(in the vicinity of the bottom of the excitonic band), which mainly contribute to the free-induction signal. We also analyzethe applicability of the segment model to the problem of interest when including the coupling to far neighbors. ( 1999Elsevier Science B.V. All rights reserved.

PACS: 71.35.Aa; 36.20.Kd; 78.30.Ly

Keywords: Frenkel excitons; Exciton trapping; Free induction fluorescence

1. Introduction

The model of Frenkel excitons in one-dimen-sional disordered lattices has been proven to bevery useful in order to explain the low-temperaturetransport properties and photophysics of molecularaggregates [1] (see also the reviews [2,3] and refer-ences therein), polymer chains [4—9] as well asseveral antiferromagnets [10—12]. The latter sys-tems present strong anisotropy in the hopping inte-grals for three crystallographic directions. One of

*Corresponding author. Tel.: 34 91 3944488; fax: 34 913944547; e-mail: adame@valbuena.fis.ucm.es.

the integrals is typically several orders of magni-tude larger than the others and, consequently, thesystem can be treated within the quasi-one-dimen-sional approach.

Dynamics of one-dimensional Frenkel excitonsis strongly affected by the disorder of differenttypes. Fluctuations of both hopping integrals andsite energies result in the localization of the excitonstates, which is also reflected in the exciton opticaldynamics. A great deal of work has been devoted tosuch problems during the last two decades (see thereviews [2,3]). In this paper we mainly focus ourattention on the disorder produced by the presenceof ions substituting host atoms. Such ions are ran-domly distributed in the host chain and can act in

0022-2313/99/$ — see front matter ( 1999 Elsevier Science B.V. All rights reserved.PII: S 0 0 2 2 - 2 3 1 3 ( 9 8 ) 0 0 1 5 4 - 9

a twofold way, first, localizing the excitons, andsecond, trapping them. Both effects have differentmanifestation in the exciton optical dynamics[8—11,13,14]. The authors of all just referred papersused the nearest-neighbor (NN) approximation inorder to describe the exciton optical dynamics, as-suming that coupling to far neighbors is negligible.However, previous works have shown that theinclusion of the long-range interactions stronglyaffects Frenkel exciton states in one-dimensionalsystems with continuous diagonal or off-diagonaldisorder [15,16]. The main goal of the presentpapers is to show that the contribution of far neigh-bors to the linear optical response of one-dimen-sional Frenkel excitons cannot be perturbative. Thereason of this effect becomes clear from the fact thatthe major contribution to the linear optical re-sponse is determined by the states of the bottom ofthe exciton band. The energies of these statesundergo non-perturbative changes with inclusionof the coupling to far neighbors [17].

The remainder of the paper is organized as fol-lows. In Section 2 we describe our model Hamil-tonian and the way to calculate the free-inductionfluorescence from Frenkel excitons under the con-dition of a short pulse excitation. In Section 3 wepresent the results of the nearest-neighbor approxi-mation as applied to the case of low temperatures.Section 4 deals with the analytical treatment of theeffect caused by the coupling to far neighbors. Sec-tion 5 is devoted to results of numerical simula-tions and discussions. As a main point, we willshow that the long-range dipole—dipole interactionleads to a much faster fluorescence decay as com-pared to the nearest-neighbor interaction. Sec-tion 6 concludes the paper with a brief summary ofresults and some general remarks on their physicalimplications.

2. Model and basic relationships

We assume that a quencher substitutes a hostatom, in such a way that the whole chain consists ofa number of segments separated by quenchers (theso-called model of disruptive quenchers, accordingto the terminology proposed in Refs. [4]). Theprobability p(N) to find a segment of N host atoms

is p (N)"c (1!c)N~1. At large N, which is of inter-est in most experimental situations, this quantityreduces to the Poissonian distribution p (N)"c exp (!cN).

In the model with only nearest-neighbor interac-tions between the host atoms, the problem of theoptical response of the whole chain can be reducedto that for a segment with traps at either end. Theresult must then be averaged over the segmentlength distribution p(N). On the one hand, inclu-sion of the coupling between far neighbors makesthe segments dependent from each other and, onthe other hand, renormalizes the exciton spectrumof each segment. The tight-binding Hamiltonian ofthe whole chain with all interactions reads

H" +n,m(mEn)

ºmn

asman!+

n

mn[º

n~1,n(as

nan~1

#asn~1

an)#º

n,n`1(as

nan`1

#asn`1

an)

#iC (asn~1

an~1

#asn`1

an`1

)], (1)

where asn

(an) creates (annihilates) an exciton at site

n and ºmn"!º/Dm!nD3 (º'0) is the dipole—

dipole hopping integral. The stochastic variablemntakes the values 1 and 0 with probability c and

1!c, respectively. Here C is the quenching con-stant. We will only account for quenching in thosehost atoms which are nearest to traps. All the siteenergies are set to zero since we do not take intoaccount any on-site disorder.

As it was established in Refs. [13,18], the opticalresponse of the excitonic system to the action of anultrashort pulse of small area consists of two com-ponents of emission, namely incoherent and coher-ent. The incoherent part describes the decay of theexciton population due to trapping. It is deter-mined by the expectation value of the exciton num-ber operator nL "+

jasjaj, which is proportional to

the Q-function introduced by Huber and Ching[13]. This part of emission will not be considered inthe present work.

The coherent component, which we will be inter-ested in, is associated with the expectation value ofthe dipole operator of the system with N sites

DM (t)"ihN

S0DDe~*HtDD0T, (2)

128 V.A. Malyshev et al. / Journal of Luminescence 81 (1999) 127—134

where

D"

N

+n/1

(asn#a

n) (3)

is the dipole operator of the whole chain (for thesake of simplicity, hereafter we substitute all thetransition dipole matrix elements by unity and as-sume that the whole chain has a length less than theemission wavelength) and D0T denotes the groundstate of the system. The evolution in time of themean dipole is responsible for the free inductionfluorescence emitted from the system. Its intensityis proportional to the square of the second deriva-tive of DM (t) averaged over an oscillation periodand is linked with the P-function introduced inRef. [13]

P(t)"1

N2DS0DDe~*HtDD0TD2. (4)

This equation is the basis of our analysis of theeffects of long-range interactions on the decay ofthe excitonic free-induction fluorescence.

3. Nearest-neighbor approximation

To make clear the gist of the effects we are goingto treat, let us turn first to the NN approximation.As it was mentioned above, one may then considerthe segments to be independent of each other, ow-ing to the presence of traps. It is obvious that underthese conditions, both the Hamiltonian H and thedipole operator D split into a sum of independentblocks

H"+MNN

HN, (5a)

HN"!º

N+n/1

(asnan`1

#asn`1

an)

!iC(as1a1#as

NaN), (5b)

D"+MNN

DN, (5c)

DN"

N+n/1

(asn#a

n), (5d)

where the symbol +MNN means the summation overa stochastic realization of segments. The operatorsH

Nand D

Nare, respectively, the Hamiltonian and

the dipole operator of a segment of size N, spannedon the corresponding subspace of eigenfunctions.Taking into consideration such a representationand assuming that the number of segments is largeenough to treat P(t) as a self-averaging quantity,one can rewrite Eq. (4) in the form

P(t)"K1

N+MNN

DMN(t)K

2"Kc+

N

p (N)DMN(t)K

2, (6a)

DMN(t)"S0DD

Ne~*HNtD

ND0T, (6b)

where p (N)"c (1!c)N~1 has the meaning definedabove.

Note that Eq. (6a) differs from the analogous oneused in previous studies [4,5,10], P(t)"+

Np(N)

DDMN(t)D2. The latter implicitly assumed that there is

no relative phase coherence in the radiation emit-ted by the individual segments so that the inten-sities of the radiation fields from the segments areadded rather than the amplitudes as it takes placein Eq. (6a). In our opinion, such an assumption canbe only justified at high temperatures when therelative phase coherence of segments can be de-stroyed due to the exciton—phonon interaction be-fore the emission process starts. On the contrary,Eq. (6a) seems to be applicable at low temper-atures. We will compare both coherent and inco-herent approaches later.

There are two reasons for P(t) to decay withtime. The first one is naturally the presence oftraps, while the second exists even in the absence ofthe trapping process. Indeed, using the complete-ness of the excitonic eigenstates DkT (related toa segment of length N), one can rewrite Eq. (6b) inthe form

DMN(t)"

N+k/1

DS0DDNDkTD2 e~*Ek(N)t, (7)

where Ek(N) are the excitonic eigenenergies cal-

culated for the segment of length N

Ek(N)"!2º cos

pk

N#1, k"1, 2,2, N, (8)

V.A. Malyshev et al. / Journal of Luminescence 81 (1999) 127—134 129

and DS0DDNDkTD2 is given by the expression

DS0DDNDkTD2"

1

N#1[1!(!1)k] cot2

pk

2(N#1). (9)

From Eq. (7) it follows that the magnitude DMN(t)

is a finite linear combination of periodic functionsof non-multiple frequencies. Therefore, the evolu-tion in time of DM

N(t) represents beatings, with no

decay. Nevertheless, as the frequencies of compo-nents fluctuate, the averaged contribution of eachone and, as a result, P(t) will decrease in time, owingto the destructive interference of oscillations withcontinuously distributed frequencies. Our maingoal now is to discuss the latter mechanism ofdecay of the free induction fluorescence, neglectingthe traps in the Hamiltonian (5b) for a moment.

Let us assume further that c;1, which meansthat typically one has N<1. In this case,DS0DD

NDkTD2+8N/(pk)2, with k"1, 3, 5,2 . One

can also use in Eq. (7) an approximate expressionfor the exciton energies, E

k(N)"!2º#º(pk)2/

N2, and substitute p (N) in Eq. (6a) by the Poissonlaw. Due to the steep decrease of DS0DD

NDkTD2 with k,

the main contribution to P(t) comes obviously fromthe lowest excitonic state (with k"1). This allowsus to limit the summation over k in Eq. (7) by onlythe first term. Thus, finally we will get

P(t)"A8

p2B2

Kc2+N

e~cNN expA!ip2ºt

N2 BK2. (10)

Passing in Eq. (10) from the summation over N tothe integration, Eq. (6a) will read

P(t)"A8

p2B2

KP=

c

dm e~mm expA!ip2c2ºt

m2 BK2. (11)

The factor exp (!m) approximately restricts theregion of integration to m)1. On the other hand,the interval m(pc (ºt)1@2 does not contribute tothe integral in Eq. (11) due to fast oscillations of thetemporal exponent. As the size of this intervalgrows with time, the magnitude of integral willdecrease and approach zero at p2c2ºt'1. It isa matter of simple calculations to show fromEq. (11) that the decay is quadratic for very shorttimes (p2ºt;1): P(t)"(8/p2)2 [1!(1

2) (p2cºt)2],

whereas the decay is linear for larger times(c~2'p2ºt<1): P(t)"(8/p2)2 [1!(1

2) p3c2ºt].

In the latter case, one can replace the lower limit ofintegration by zero.

4. All interactions

There are at least two reasons for P(t) to berenormalized with including all interaction as com-pared to the NN approach. First, the representa-tion of the whole chain as a number of independentsegments here is no longer valid since neighboringsegments are coupled to each other by thedipole—dipole interactions of the next neighbors.Second, even if we neglect this coupling, non-per-turbative effect arises from renormalization of theeigenenergies of each segment, as we will see below.

4.1. Effect of segment coupling

In order to gain insight into the problem ofsegment coupling, let us turn to the simplest caseassuming that (i) the chain has only one trap, whichdivides it into two segments of sizes N<1 andM<1, (ii) the exciton problem of each isolatedsegment is treated in the NN approximation, (iii)the segments are coupled by the dipole—dipole in-teraction of sites being nearest to the trap, (iv) thetrapping itself is neglected. The Hamiltonian ofsuch a problem reads

H"HN#H

M#», (12)

where the Hamiltonians of the isolated segmentsare

HN"!º

N+n/1

(asnan`1

#asn`1

an), (13a)

HM"!º

m+

m/1

(bsmbm`1

#bsm`1

bm), (13b)

where the operators an,as

nand b

m,bs

mare related to

the segments with N and M sites, respectively, andthe interaction Hamiltonian reads

»"!

º

8(as

Nb1#bs

1aN). (14)

Under these conditions, we aim to elucidatewhether the interaction Hamiltonian » can be con-sidered as a perturbation. Making use of the

130 V.A. Malyshev et al. / Journal of Luminescence 81 (1999) 127—134

excitonic transformations of the site operatorsanand b

m

an"A

2

N#1B1@2 N

+k/1

aksin

pkn

N#1, (15a)

bm"A

2

M#1B1@2 m

+k/1

bksin

pkm

M#1, (15b)

the different terms of the Hamiltonian (12) become

HN"

N+k/1

Ek(N) as

kak, E

k(N)"!2º cos

pk

N#1,

(16a)

HM"

m+k/1

Ek(M)bs

kbk, E

k(M)"!2º cos

pk

M#1,

(16b)

»"

N+k/1

m+

k{/1

»kk{

(askbk{#bs

k{ak), (16c)

where

»kk{

"

(!1)kº

4 (N#1)1@2(M#1)1@2sin

pk

N#1sin

pk@M#1

.

(17)

As the interaction Hamiltonian couples the ex-citon states of one segment, Dk,NT, to those of theother one, Dk@,MT, we should compare the energydifferences *E

kk{"2ºDcos[pk/(N#1)]!cos[pk@/

(M#1)]D with the magnitudes of »kk{

. At the regionof interest (k;N#1, k@;M#1), one has

K»

kk{*E

kk{K"

(N#1)1@2(M#1)1@2

4DN!MD(N#M#2)kk@. (18)

Accounting for the fact that only values ofk"k@"1 actually contribute to the exciton freeinduction fluorescence, from Eq. (18) it follows thatD»

kk{D(D*

kk{D, excluding the low probable event

N"M. Thus, the segment coupling through the farneighbors can be neglected in average and, sub-sequently, the segments themselves can be con-sidered as independent one from another.

This result can appear somewhat counterintui-tive. In principle, one would expect that the largerthe segments, the closer their energies, which, inturn, would imply stronger interaction, in contrastto what is found in Eq. (18). However, a more

careful consideration of this puzzling result allowsto unconver its explanation: The key fact is that theprobability amplitude of finding an exciton on thenearest site to the trap is given by [2/(N#1)]1@2sin[pk/(N#1)], which is of the order of(N#1)~3@2 for k"1 as the segment size N in-creases. This effectively decouples the segments,thus counteracting the opposite tendency (indeed,as we have seen, prevailing over it) due to theabove-mentioned energetic approaching.

4.2. Effect of renormalization of the energy spectrum

Now, let us turn to the effect of the eigenenergyrenormalization on the exciton free induction sig-nal. We will exploit the fact that the characteristictime of the free induction decay is determined bythe energy of the lowest excitonic levels. In consist-ence with the results obtained in Ref. [17], just overthis region the excitonic spectrum is noticeablyrenormalized with including all interactions, takingthe form (N<1)

Ek(N)"!2ºm(3)#ºK2 (3

2!lnK),

K"

pk

N, k"1, 2,2, k;N, (19)

where m(3)"+nn~3"1.202. On the other hand, as

it was established in Ref. [17], the exciton eigen-functions are not changed, in fact, with including alldipolar interactions. Thus, we do not expect anyinfluence of this type of renormalization on thesignal of the free induction fluorescence. Conse-quently, we can approximate the matrix elementDS0DD

NDk"1TD2 by (8/p2)N, as we have already

done to arrive at Eqs. (10) and (11).Accounting for the facts just mentioned, we

finally obtain

P(t)"A8

p2B2

KP=

c

dm e~mm

]expC!ip2c2ºt

m2 A3

2!ln

pc

m BDK2, (20)

instead of the expression (11). Now, the ex-citon free induction fluorescence will disappearwhen p2c2ºt[3/2!ln (pc)]'1. This inequalitydiffers from the analogous one obtained in the

V.A. Malyshev et al. / Journal of Luminescence 81 (1999) 127—134 131

NN-approximation (see Section 3) by the factor inparenthesis. It can noticeably deviate of unity atlow concentration of traps, giving rise, respectively,to a faster decay of the free induction signal.

5. Numerical results and discussions

In this section we present results of numericalsimulations demonstrating the correctness of ourdifferent approximations and particularly the ef-fects of the long-range dipolar coupling. In whatfollows, we will evaluate the free induction fluores-cence intensity as a function of the dimensionlesstime p2c2ºt. Fig. 1 shows the free induction flu-orescence signal in the NN approximation for dif-ferent values of c according to Eq. (10) and theresults from Eq. (11), as a function of the dimen-sionless time p2c2ºt. We notice that P(t) convergestowards a universal curve on decreasing c and thiscurve is given by Eq. (11). In fact, for lower valuesof c (not shown in the figure), the results fromEqs. (10) and (11) are superposed.

Up to now, we have neglected the effects ofexcitonic states with k greater than unity. The nextstate with an appreciable contribution to the freeinduction signal is that with k"3. Its oscillator

Fig. 1. Plots of the free induction fluorescence decay calculatedusing Eq. (10) for c"0.05 (dashed line) and c"0.01 (dottedline). Solid line shows the result obtained from Eq. (11). Allcurves are normalized to unity at t"0.

strength is reduced by a factor 19as compared to the

oscillator strength of the ground exciton state,whereas its energy is E

3(N)"!2º#9p2º/N2.

The free induction fluorescence in the limit N<1then becomes

P(t)"A8

p2B2

KP=

c

dm e~mmCexpA!ip2c2ºt

m2 B#

1

9expA!i

9p2c2ºt

m2 BDK2, (21)

in the NN approximation. From Eq. (21) we obtainthat P(t)"(8/p2)2[1!(9

2)(p2cºt)2] for p2ºt;1

and P(t)"(80/9p2)2 [1!(9/10)p3c2ºt] for c~2'

p2ºt<1, so that the initial decay is faster than thatobtained for k"1. However, at longer times thecontribution of k"3 can be confidently neglected,as seen in Fig. 2, at least compared to far neighborcoupling effects (see below). Higher values of k pro-duce even smaller corrections. Thus, we concludethat in the coherent approximation the decay of thefree induction signal mainly arises from the destruc-tive interference between the lowest excitonicmodes (k"1) belonging to different segments. Itshould be noticed that in the model with no phase

Fig. 2. Plots of the free induction fluorescence decay calculatedusing Eq. (11) (dashed line) and Eq. (21) (solid line). All curvesare normalized to unity at t"0. The inset compares the resultsfor the coherent (solid line) and incoherent (dashed line) approx-imations for the two lowest exciton modes (k"1,3).

132 V.A. Malyshev et al. / Journal of Luminescence 81 (1999) 127—134

coherence between segments [4,5,10] one obtainsno decay when higher modes (k'1) are neglected.Therefore, the next state with an appreciable oscil-lator strength (k"3) should be taken into account.In such a case, the free induction fluorescencecoming from each segment is proportional to8281#(2

9) cos [(E

3!E

1)t]. The average of this

quantity results in a decaying signal as time elapses.Since the weight of k"3 mode in the whole signalis 1

9, one should expect a drop of the same order in

the emission intensity from the initial value. Theinset of Fig. 2 compares the results for the coherent[P(t)&D+

Np(N)DM

N(t)D2] and incoherent [P(t)&

+N

p(N)DDMN(t)D2] approximations for two lowest

exciton modes (k"1, 3), showing a dramatical dif-ference between these two approximations andconfirming our qualitative considerations.

According to Ref. [10], finite values of thequenching constant modifies the eigenenergyE1(N). In the limit C;º, which is of interest in

most experiments, the perturbed eigenenergy isgiven by EI

1(N)"E

1(N)!4ip2C/N3 when N<1.

As the correction is of order N~1 as compared tothe unperturbed eigenenergy, we cannot expecta significant contribution of finite values of thequenching constant for small c. In fact, this is thecase for parameters of interest. Replacing E

1(N) by

EI1(N) in Eq. (6a) and passing to integration we

obtain

P(t)"A8

p2B2

KP=

c

dm e~mm

]expC!ip2c2ºt

m2 A1!i4cC

ºmBDK2. (22)

Notice that the correction depends on cC/º,which is usually rather small in most physical sys-tems since C;º and c;1. Fig. 3 shows that thedecay curve obtained from Eq. (22) for small valuesof the parameter cC/º is essentially the same asthat obtained from Eq. (10). We observe deviationsonly for higher (and rather unphysical) values ofthat parameter. Thus we are led to the conclusionthat the limit CP0 provides a fairly good descrip-tion of the free induction fluorescence as far as theparameter cC/º remains small.

Finally, let us now turn to the main aim of thepaper, namely the effects of the dipolar coupling

Fig. 3. Plots of the free induction fluorescence decay calculatedusing Eq. (22) with cC/º"0.01 (solid line), 0.10 (dotted line)and 1.00 (dashed line). All curves are normalized to unity att"0.

Fig. 4. Plots of the free induction fluorescence decay calculatedusing Eq. (20) for c"0.05 (dotted line) and c"0.01 (solid line).Dashed line shows the result obtained from Eq. (11). All curvesare normalized to unity at t"0.

between far neighbors. From Eq. (20) it is clear thatthe free induction fluorescence does not convergetowards a universal curve as a function of thedimensionless time on decreasing c, as it is seen inFig. 4. We notice that there exists an appreci-able deviation on the fluorescence decay curveswhen dipolar coupling is taken into account, as

V.A. Malyshev et al. / Journal of Luminescence 81 (1999) 127—134 133

compared to the NN approximation. This effect ishigher on decreasing c, namely when the segmentlengths are larger.

6. Conclusions

We have shown that the coupling to far neigh-bors substantially affects the decay rate of the freeinduction fluorescence from one-dimensional Fren-kel excitons created by an ultrashort pulse ofa small area in a chain with substitutional traps.Due to the non-perturbative nature of the effect, itmust be accounted for at the fitting of the experi-mental data. Exciton trapping almost does not in-fluence the free induction decay, excluding the caseof very high concentration of traps. The free induc-tion signal vanishes in time mainly due to fluctu-ations of the exciton energies of segments in whichthe whole chain is divided by the traps.

Acknowledgements

The authors gratefully thank D.L. Huber, R.Brito, A. Sanchez for numerous helpful conversa-tions. Work at Russia has been supported by theRussian Foundation for Basic Research under Pro-ject No. 96-02-18292. Work at Madrid has been

supported by CICYT (Spain) under Project No.MAT95-0325.

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134 V.A. Malyshev et al. / Journal of Luminescence 81 (1999) 127—134