On the low-temperature diffusion of localized Frenkel excitons in linear molecular aggregates A. V. Malyshev,∗ V. A. Malyshev,† and F. Dom´ınguez-Adame Departamento de F´ısica de Materiales, Universidad Complutense, E-28040 Madrid, Spain (Dated: February 2, 2008) 3 We study theoretically diffusion of one-dimensional Frenkel excitons in J-aggregates at temper- 0 atures that are smaller or of the order of the J-band width. We consider an aggregate as an open 0 linear chain with uncorrelated on-site (diagonal) disorder that localizes the exciton at chain seg- 2 mentsofsizesmallerthanthefullchainlength. Theexcitondiffusionoverthelocalization segments n is considered as incoherent hopping. The diffusion is probed by the exciton fluorescence quenching a which is due to the presence of point traps in the aggregate. The rate equation for populations of J the localized exciton states is used to describe the exciton diffusion and trapping. We show that 6 thereexist two regimes of theexciton diffusion at low temperatures. The first, slower one, involves 1 only the states of the very tail of the density of states, while the second, much faster one, also involves the higher states that are close to the bottom of the exciton band. The activation energy ] for thefirst regime of diffusion is of theorderof onefifth of theJ-bandwidth, while forthesecond n one it is of the order of the full J-band width. We discuss also the experimental data on the fast n low-temperature exciton-exciton annihilation reported recently by I. G. Scheblykin et al, J. Phys. - s Chem. B 104, 10949 (2000). i d PACSnumbers: 71.35.Aa;78.30.Ly;78.66.Qn;78.67.-nmaterialsandstructures . t a m I. INTRODUCTION this localization is the appearance of states below the - bottom of the bare exciton band. These states form the d tail of the density of states (DOS) and carry almost the n Since the seminal works by Jelley [1] and Scheibe [2], o the concept of Frenkel excitons [3, 4, 5] has been used whole oscillator strength of the aggregate. For this rea- c for explaining of the remarkable optical properties of son the one-exciton absorption in J-aggregates is spec- [ trally located at the tail of the DOS (see, for instance, molecular J-aggregates: (i) the appearance of a nar- Refs. 9, 10) and the width of the absorption band is of 1 row and intense line in the red-wing of the absorption v spectra (so called J-band), the full width of which is the order of the width of the DOS tail. 0 of the order of several tens of wavenumbers at cryo- 0 The exciton diffusion in a disordered aggregate is es- genic temperatures and (ii) the increase of the oscil- 3 sentially the transition from one localized eigenstate to lator strength of the optical transition by almost two 1 another. The transitionprobabilitydepends particularly 0 orders of magnitude [6, 7, 8, 9, 10]. During the on the temperature, the energy spacing between the in- 3 nineties, a considerable progressin understanding of lin- volved states and the overlap of these states. The lower 0 ear and nonlinear optical dynamics of J-aggregates was states,beinglocalizedatdifferent N∗-moleculesegments / made (for details see the reviews 11, 12, 13 and ref- t of the aggregate, overlap very weakly [19]. Contrary to a erences therein). In spite of the fact that monomers m that, the higher exciton states, that are localized at seg- which form the aggregateshave complex chemical struc- ments larger than N∗ molecules, overlap strongly with - ture, both linear and nonlinear optical dynamics in J- d severallowertailstates. Althoughhigherstatesarether- aggregates have been successfully described on the basis n mally less favorable, the hops from the lower to higher ofthesimplestone-dimensional(1D)tight-bindingmodel o statescanbefasterthanbetweenthelowerstatesbecause c with diagonal and/or off-diagonal disorder, both uncor- of the higher overlap. In this paper, we show that this : related [10, 14, 15] and correlated [14, 16, 17, 18]. v competitionbetween the overlapsand thermalfavorabil- i The eigenstates of a homogeneous (non-disordered) ity result in a complex scenario of the exciton transport X J-aggregate extend over the whole (N monomers) ag- atlow temperatures. At zerotemperature anexcitonre- r gregate. Disorder localizes the lowest in energy exci- a sides in one of the lower states at the tail of the DOS. ton states at segments of about N∗ molecules; N∗ de- As temperature rises, first, the exciton starts to diffuse pends on the disorder magnitude and is typically much over the weakly overlapped states of the DOS tail. The smaller than the total number of molecules in the chain: activationenergy for this regime is of the order of 1/4of N∗ N. One of the most important consequences of ≪ the DOS tail width (that is of the order of the J-band width). The diffusion in this regime is very slow. As thetemperatureincreasesfurther,thehigherstatescome into play. As these states overlap much better with the ∗Onleave from Ioffe Physiko-Technical Institute, 26Politechnich- lower states and each other and also are more extended, eskayastr.,194021 Saint-Petersburg,Russia †On leave from “S.I. Vavilov State Optical Institute”, Saint- the diffusion rate increases by several orders of magni- Petersburg,Russia. tude. The activation energy for this faster regime of the 2 excitondiffusionisofthe orderoftheDOStailwidthor, andnegative(J >0),whichcorrespondstothecaseofJ- in other words, of the order the J-band width. aggregates(see,e.g.,Ref.[6]). Inthiscasethestatescou- To the best of our knowledge, these aspects of the 1D pled to the light are those close to the bottom of the ex- diffusion problem have not been discussed in the litera- citon band. In what follows, moderate disorder (∆<J) ture yet. The same tight-binding Hamiltonian was used is considered. This implies that the exciton eigenstates to describe transport properties of electrons in doped ϕ (ν = 1,2,...,N), found from the Schr¨oedinger equa- ν semiconductors [20] as well as those of optical excita- tion tionsinactivatedglasses[21,22]. Itistobestressedthat N despite the seemingsimilarity ofthese problemsthe out- H ϕ =ε ϕ , H = nH m , (2) lined scenario of the low temperature 1D diffusion over nm νm ν νn nm h | | i m=1 the localization segments is more complex than the dif- X fusion over one-level point impurity centers in semicon- are extended over relatively large segments of the chain. ductors or glasses. The major complication comes from However, the typical size of these localization segments, the fact that the exciton can hop sideways to a differ- N∗, is small compared to the full chain length N (units ent segment not only directly (a ”horizontal” hop) but of the lattice constant are used throughout the paper). also indirectly via higher states, so that ”vertical” hops Having been excited into an eigenstate ν, an exciton up in energy become extremely important. In fact, it is cannot hop to other eigenstates if coupling to vibrations theindirecthopsthatprovidethedominantcontribution is not taken into account. We assume that this coupling to the diffusion rate at temperatures of the order of the is weak and do not consider polaron effects. This limit J-band width. is applicable to a number of J-aggregates as the Stokes We use the quenching of the exciton fluorescence by shift of the luminescence spectra with respect to the ab- point traps to probe the exciton diffusion. The temper- sorption spectra is usually small [7, 9]. The exciton- ature range lower or of the order of magnitude of the vibration interaction causes the incoherent hopping of J-bandwidth is of our primary interest; higher tempera- excitons from one eigenstate to another. We take the tures are beyond the scope of the present work. hopping rate from the state ν to the state µ in the fol- The outline of the paper is as follows. In Sec. II, we lowing form (see, e.g., Ref. 24) present the microscopic model of exciton trapping. Sec- tion III is focused on the qualitative discussion of the N 2 2 channels of the exciton diffusion over the localization Wµν =W0 S(|εν −εµ|) ϕνnϕnµ segments. The results of our numerical simulations of n=1 X the exciton fluorescence quenching, obtained on the ba- n(ε ε ), ε >ε µ ν µ ν sis of the rate equation approach, are the contents of − . (3) Sec. IV. In Sec. V we conclude the paper and discuss ×(1+n(εν −εµ), εµ <εν the results ofthe recentexperiments onthe fastexciton- Here, the constant W0 characterizes the amplitude of exciton annihilation in the aggregates of the triethylth- hopping and n(ε) = [exp(ε/T) 1]−1 is the occupa- iacarbocyanine salt of 3,3’-bis(sulfopropyl)-5,5’-dichloro- − tion number of the vibration mode with the energy 9-ethylthiacarbocyanine (THIATS) [23]. ε (the Boltzmann constant is set to unity). Due to the presence of the n(ε) and 1+n(ε) factors, the rate W meets the principle of detailed balance: W = II. MICROSCOPIC MODEL OF THE EXCITON µν µν W exp[(ε ε )/T]. Thus, in the absence of decay FLUORESCENCE QUENCHING νµ ν µ − channels, the eventual exciton distribution is the Boltz- mannequilibriumdistribution. Thesumoversitesin(3) We model a J-aggregate by N (N 1) optically ac- ≫ representstheoverlapintegralofexcitonprobabilitiesfor tive two-level molecules forming a regular in space 1D the states µ and ν. The spectral factor S(ε ε ) de- openchain. The correspondingFrenkelexcitonHamilto- | ν − µ| pends on the details of the exciton-phonon coupling as nianreads[4](forthesakeofsimplicityonlythenearest- well as on the DOS of the medium into which the ag- neighbor interaction is considered) gregate is embedded. For example, within the Debye model for the density of phonon states, this factor takes N N−1 the form S(E E ) = (E E /J)3 [25]. However, H = E n n J (n+1 n + n n+1) . (1) ν µ ν µ n − | − | | ih |− | ih | | ih | this model is applicable to glassy media (the media we n=1 n=1 X X assume as the host) only in a narrow frequency interval Here E is the excitation energy of the n-th molecule, of the order of several wavenumbers (see, for instance, n n denotesthe statevectorofthe n-thexcitedmolecule. Ref. 26, 27). Therefore, as in Refs. [28, 29], we re- | i The energies E are assumed to be Gaussian uncorre- strict ourselves to a linear approximation to this factor, n lated (for different sites) stochastic variables distributed S(E E ) = E E /J. This accounts for reduc- ν µ ν µ − | − | around the mean value ω0 (which is set to zero without tionoftheexciton-vibrationinteractioninthe long-wave loss of generality) with the standard deviation ∆. The acousticlimit [4, 5]. Also, it eliminates the divergenceof hopping integral, J, is considered to be non-random W at small values of E E . νµ ν µ − | − | 3 The diffusion of Frenkel excitons can be probed by where quenching of the exciton fluorescence by traps. Con- sider an aggregate with point traps, namely monomers N R = γ +Γ + W δ W . (9) at which anexciton decaysnon-radiativelyand veryfast νµ ν ν µν µν νµ comparedtothetypicalspontaneousemissionrateofthe µX=1 ! − aggregate. Then those excitons that reach the traps de- Aftersubstitutionof(8)intoEq.(6)andintegrationover cay non-radiatively and contribute to the fluorescence time, τ can be expressed in terms of the Rˆ-matrix: quenching. If an exciton is created far from the trap it has to diffuse to the trap to be quenched, the faster it N diffuses the more effective is the fluorescence quenching. τ = Rˆ−1P (0) . (10) Thus, the quenching rate depends on the diffusion rate νµ µ and can be used as a probe of the latter. νX,µ=1D E We define the quenchingrateofthe excitonstateν as: Calculation of the quenching rate W requires the cal- q culation of the inverse matrix Rˆ−1 for each realiza- Nq tion of disorder rather than the fluorescence kinetics. 2 Γν =Γ ϕνi , (4) The inverse matrix is to be found twice: for an ag- | | i=1 gregate with and without traps. Note that the decay X where Γ is the amplitude of exciton quenching and the time τ0 also depends on temperature (see, for example, sum runs over positions of the N traps. Thus, we take Refs. [25, 29, 30]). q the quenching rate to be proportional to the probability to find the exciton at trap sites. We describe the process of the exciton trapping by III. QUALITATIVE ARGUMENTS means of the rate equation: At low temperatures excitons reside in the tail of the N P˙ = (γ +Γ )P + (W P W P ) , (5) DOS,thatis,belowthebottomofthebareexcitonband, ν ν ν ν νµ µ µν ν − − E = 2J. As we show below, higher states that are µX=1 close t−o the bottom of the bare band contribute to the where Pν is the population of the νth exciton eigenstate exciton diffusion as well. Therefore, these two parts of and the dot denotes the time derivative, γν =γfν is the the exciton energy spectrum are of primary importance spontaneousemissionrateofthe νthexcitonstate,while for the low-temperature exciton transport. γ is that of a monomer, f = ( N ϕ )2 being the ν n=1 νn oscillator strength of the state ν. P Thetemperaturedependence ofthe excitonquenching A. Analyzing the low energy structure is calculated as follows. We admit the definition of the exciton fluorescence decay time as the integrated total Here, we recall briefly the concept of the local (hid- population [25]: den)energystructureoflocalized1Dexcitons[17,31,32], ∞ N whichwasprovedtoexistinthevicinityofthebandbot- τ = dt P (t) , (6) tom [19, 34]. According to this concept, the low-energy ν Z0 ν=1h i one-exciton eigenfunctions obtained for a fixed realiza- X tion of disorder are localized at segments of typical size where angle brackets denote the average over disorder N∗ (localization length). Some of these localized states realizations and traps positions. The decay time has to (about 30%) can be grouped into local manifolds of two be calculated for aggregates with traps (denoted as τ) (orsometimesmore)statesthatarelocalizedatthesame and without traps (denoted as τ0). The quenching rate N∗-molecule segment (see the states filled with black is then defined as color and joined by ellipses in Fig. 1). It turns out that 1 1 1 1 the structure of the exciton states in eachlocalmanifold W = . (7) q τ −(cid:18)τ(cid:19)Nq=0 ≡ τ − τ0 ihsomveorgyensiemouilsar(ntoont-dhiesosrtdruercetdu)relinoefatrhcehlaoiwneorfsletnagtetshoNf∗a. Thisquantitycarriesinformationaboutthediffusionrate In particular, the lowest state in a manifold has a wave and is the object of our analysis. function without nodes within its localization segment. Thedefinitionofthe decayrateastheintegratedtotal Such a state can be interpreted as the local groundstate population allows for considerable simplification of the of the segment (italic is used to distinguish this state calculation procedure. Write the solution of Eq. (5) in from the true ground state, that is, the state with the the formal matrix form lowest energy in each realization). A local ground state N carrieslargeoscillatorstrength,approximatelyN∗ times ˆ P (t)= e−Rt P (0) , (8) largerthan that of a monomer, so that the typical spon- ν µ µX=1(cid:16) (cid:17)νµ taneous emission rate is γ∗ = γN∗. The scaling law of 4 7 450 2 6 ∆ = 0.3J εf 1 ε = 0.1J DOS s o 5 σ 12= 0.14J nit 300 11 y, u 4 g er 3 n on e 2 150 P(E ) cit N* 1 Ex 1 0 0 0 50 100 150 200 250 300 -2.3 -2.2 -2.1 -2 -1.9 -1.8 -1.7 Site number FIG. 1: The energy structure of the exciton states in the 600 ∆ = 0.2J vicinity of the bottom of the exciton band. The states are ε = 0.06J obtainedbydiagonalizationoftheHamiltonian(1)foralinear DOS chainof300moleculesandthedisordermagnitude∆=0.1J. σ 12= 0.08J 11 Thebaselineofeachstaterepresentsitsenergyinunitsofthe 400 spacing in the local energy structure ε12. The origin of the exciton energy is set to the lowest energy for the realization. The wave functions are in arbitrary units. It is clearly seen that some lower states can be grouped into local manifolds 200 (themanifolds arejoined byellipses). Thestateswithineach P(E ) manifold are localized at the same segment of typical length 1 N∗ (thislengthisgivenbythebarinthelowerrightcorner), they overlap well with each other and overlap much weaker 0 withthestateslocalizedatothersegments. Thehigherstates -2.2 -2.1 -2 -1.9 -1.8 (filledwithgraycolor)aremoreextendedthanthelowerones. Typically,theyoverlapwellwithseverallowerstatesandwith each other. 900 ∆ = 0.1J ε = 0.02J DOS the localization length is [34] σ 12= 0.03J 11 600 −0.67 ∆ N∗ =8.7 . (11) J (cid:18) (cid:19) 300 The energy distribution of the local ground states, cal- culated as described in Ref. [34], and the total DOS are P(E ) 1 presentedinFig2. Thisfigureshowsthatalmostalllocal groundstates belong to the tail of the DOS, as has been 0 mentioned in the Introduction. -2.1 -2.05 -2 -1.95 -1.9 The second state in a manifold has a node within the localizationsegment(seethestatesfilledwithblackcolor Energy, units of J andjoinedbyellipsesinFig.1)andlookslikethefirstlo- calexcitedstate ofthe segment. Its oscillatorstrengthis FIG. 2: The total DOS (solid line) and the DOS of the local typically an order of magnitude smaller than that of the ground states P(E1) (dashed line) for different magnitudes localgroundstate. Itisimportanttorecallherethat,con- of disorder ∆. The DOS is normalized to N, P(E1) is nor- trarytotheeigenstatesfromthesamemanifold,thelower malized toN/N∗. Thevertical lines indicatepositions ofthe states localizedatdifferentsegments overlapweakly(see curves’ maxima. For all considered magnitudes of disorder all states filled with black color in Fig. 1). The energies themaximum of the local DOSis shifted with respect tothe ofthelocalgroundstatesaredistributedwithintheinter- maximumofthetotalDOSbyaboutthemeanspacinginthe val √2σ11 (σ11 being the average spacing between local local energy structure,ε12. ground states). This interval is larger than the typical energy spacing ε12 between the levels in a local mani- fold [34]. For this reason,the localenergystructure can- this structure determines the nonlinear optical response notbeseeneitherintheDOS(seeFig.2)orinthelinear of the aggregate [35, 36, 37]. absorptionspectra(see,forinstance,Ref.[10]). However, Higher states are more extended than the local states 5 as the localization length increases with energy (see the B. Hopping at zero temperature states filled with gray color in Fig. 1). Therefore, the higherstatescannotbeincludedintoanyparticularlocal At zero temperature an exciton can hop only down to manifold: theirwavefunctionscoversmorethanoneN∗- lower states. Let us assume that it is in the local excited molecule segments. Nevertheless, as all these states are state2. Thenitcaneitherhoptothelocalgroundstate1 closetothemaximumoftheDOS(seeFig.2),thetypical ofthesamesegmentortoalowerstateν′ localizedatan energyspacingbetweenthehigherstatesandthecovered adjacent segment (see Fig. 3, T =0). Because the intra- local states is of the order of ε12. Thus, the energy ε12 segment hopping is faster than the inter-segment one, is expected to be the characteristicenergyof the exciton first, the exciton hops down to the local ground state diffusion. 1 with the typical energy loss ε12 (ε12 being the mean It is clear from the above arguments that at temper- energy spacing in the local energy structure, see Fig. 3, atures T < ε12, it is the states from the local mani- T = 0). From the local ground state, the exciton can folds that determine the exciton diffusion. Two types hop only to a state ν′ of an adjacent segment, provided of hopping over these states can be distinguished: intra- segment hopping and inter-segment one, involving the 2 3 states of the same local manifold and those of different manifolds,respectively. Asthestatesfromdifferentlocal w manifoldsoverlapweakly,onlyinter-segmenthopstoad- w1 2 3 2 w ν ' 3 jacent segments are of importance. The disorder scaling 2 of the overlap integrals Iµν = nϕ2µnϕ2νn for the local 1 wν ' 2 wν ' 2 states of the same and adjacentPsegments was obtained w w ν' in Ref. 19: 1 2 2 1 w 0.70 γ1 ν ' 1 ν' 1 ∆ I12 =0.14 (12a) J (cid:18) (cid:19) T = 0 γ T > 0 1 FIG. 3: Schematic view of exciton hopping at zero and non- 0.75 zerotemperatures. Theindices1and2labelthelocalground ∆ Iν′1 Iν′2 =0.0025 . (12b) andthefirstlocalexcitedstatesof thesamesegment,respec- ≈ J tively. The ν′ state is localized at an adjacent segment. The (cid:18) (cid:19) index 3 label a higher state, which extends over two adja- centsegments. Hopsareshownbystraightarrows; thearrow Hereafter, the indices 1 and 2 label the local states of thickness represents magnitude of the corespondent hopping the samesegmentwhile those with primes labelthe local rate. Thin wavy arrows show spontaneous emission. Its rate is the slowest among all, which corresponds to the limit of states of an adjacent segment. As follows from Eq. (12), fast diffusion. the intra-segmentoverlapintegralis typically two orders of magnitude larger than the inter-segment one. Note, that both overlap integrals scale approximately propor- εν′ < ε1 and the spontaneous emission rate of the local tionaltotheinvertedN∗(compareEqs.(12a)-(12b)with ground state γ1 is small compared to the intra-segment Eq.(11)). Thisproportionalityholdsfortwoexponential hopping rate Wν′1 T=0. Hereafter, such a relationship | functions extended over the length N∗ and separated by between these rates is referred to as the limit of fast dif- the distance of the same order of magnitude, N∗. fusion; only this limit is considered in this work. The typicalenergylossduringsuchsidewayshopisofthe or- The intra-segment hops do not result in the spatial der of the average spacing between local ground states, displacement of excitons. Only the inter-segment hop- σ11 (σ11 is of the order of the J-band width). Thus, al- ping gives rise to the spatial motion. Nevertheless, we ready after one such sideways hop the exciton resides in show below that both types of hops are important for astateinthetailoftheDOS(seeFig. 2). Therefore,the understanding the features of the low-temperature exci- number of states with even lower energies decreasesdra- ton transport. matically,whichresultsinastrongincreaseofthetypical The overlapintegralsbetweenthe local states ofa seg- distancetothosestatesanddecreaseoftheprobabilityto ment and the higher states which are extended over this hopfurthersideways. Thentheexcitoneitherrelaxestoa segment and a few adjacent ones (see the states filled lowerstateofthesamesegment(ifthereisone)ordecays with gray color in Fig. 1) are of the order of I12. This spontaneously, i.e. this type of the spatio-energetic dif- fact implies that even at T < ε12, the indirect hops via fusion (towards lower energies) stops very quickly. Note these higher states can be more efficient than the direct thatthisdiffusionwouldmanifestitselfintheredshiftof inter-segmenthopsoverthestatesoflocalmanifolds(see the exciton emission spectrum relative to the absorption below). Our calculations support this assumption. spectrum. The experimental data shows that such red 6 shift is either absent [6, 7] or is smaller than the J-band A. Numerical results width [23, 38]. These experimental facts indicate unam- biguously that at low temperatures, T σ11, excitons As it was already mentioned in Sec. III, in the limit ≪ makefew hopsbeforetheydecaydue tothe spontaneous of fast diffusion the inter-segment down-hopping rate is emission,aswasarguedinRefs.39,40,41. Consequently, large compared to the typical spontaneous emission rate thezero-temperatureexcitonquenchingisexpectedtobe of a local ground state: weakprovidedthe concentrationofquenchersis low,the case we are interested in. Wν′1 T=0 W0 σ11 Iν′1 γ N∗ . (13) | ∼ J ≫ If a quencher is located within the localization segment C. Hopping at non-zero temperatures of a local state then the typical quenching rate for this state is Γ∗ = Γ/N∗ [see Eq. (4)]. As we are interested At non-zero but low temperatures (0 < T < ε12), an in the limit of fast quenching, this rate should be taken exciton can also hop up in energy. Consider ∼an exciton largerthanthetypicalintra-segmentdown-hoppingrate: in one of the lower states in the tail of the DOS, e.g. in the local ground state 1 (see Fig. 2, T > 0). For the reasonsdiscussedabove,first,the excitonhops uptothe Γ/N∗ W12 T=0 W0 ε12 I12 . (14) ≫ | ∼ J first local excited state 2 of the same segment, provided thehoppingratefortheconsideredtemperatureislarger This ensures that once an exciton hops to a local state thanthe spontaneousemissionrateγ1 oftheinitialstate of the segment with the trap, it is quenched almost in- 1. During this process the exciton typically gains the stantly. energy ε12. As ε12 is of the order of σ11 [34], already The inequalities (13) and (14) yield the relationship after the first hop up the exciton leaves the tail of the betweenthe rateequationparametersinthe limit offast DOS (see Fig. 2) and, hence, it is likely to have a lower diffusion and quenching: state ν′ localized at an adjacent segment. A hop down to this state with loss in energy is favorable and results σ11 I1′2 W0 γ (15a) in the spatial displacement of the exciton, i.e., in the J N∗ ≫ exciton diffusion. We stress that although only sideways hops result in the spatial displacement of the exciton, it is the initial hop up from the local ground state 1 to the W0 ε12 I12 N∗ Γ . (15b) J ≪ local excited state 2 that triggers the diffusion. Another way for the exciton to hop sideways to the The scaling laws of the values of σ11 and ε12 were ob- state ν′ is via the higher state 3 that overlaps well with tained in Ref. [34] and are given by both states 2 and ν′ (see Fig. 3, T > 0). As it has been mentioned, such hops compete with the sideways hops ∆ 1.33 over the local states; although the hop up to the state σ11 =0.7J J , (16a) 3 is thermally unfavorably, the overlap integral for this (cid:18) (cid:19) hop, I31, is large compared to that for an inter-segment hop,Iν′1. Weshowlaterthatthischannelofthediffusion ∆ 1.36 becomes dominant even at relatively low temperature. ε12 =0.4J . (16b) J (cid:18) (cid:19) Onthebasisofthescalinglaws(12)and(16)theparam- IV. TEMPERATURE DEPENDENCE OF THE etersetwaschosensothatforeachmagnitudeofthedis- QUENCHING RATE order ∆ the following equalities hold: Γ∗ = 10W12 T=0 and W1′1 T=0 = 10γ∗. Calculations were performed| for | In this section, we discuss the results of numericalcal- N = 1000 and averaged over 100 realizations of the dis- culationofthequenchingrateW . Weconsidertheinitial order. q conditionwheretheleftmostlocalgroundstateisexcited Figure 4 shows the temperature dependence of the whileasingletrapislocatedinthecenterofthelocaliza- quenching rate W for different magnitudes of the disor- q tion segment of the rightmost local ground state. In this der ∆. In each plot, the quenching rate is given in units case,the excitonquenchingis mostaffectedby the diffu- ofthetypicalexcitonradiativerateγ∗ =γN∗. Thetem- sion, as the createdexciton has to traveloveralmostthe perature is given in units of the mean energy spacing in whole chain to be quenched. Thus, the exciton quench- the local energy structure ε12. Note that both N∗ and ing at low concentration of traps can be studied. The ε12 depend on ∆ as described by (11) and (16b). Figure quenching rate was calculated as described in section II 4demonstratesveryclearlythatforallconsideredvalues for the parameter set corresponding to the limit of fast of ∆ at temperatures lower than ε12 the quenching rate diffusion and fast quenching (the latter limit is defined is vanishing. This indicates that the diffusion at these below). temperatures is not fast enough for the exciton to reach 7 B. Discussion 2.5 In order to understand which states contribute most 2 into the quenching process it is useful to estimate the effective sideways hopping rate W, which is required to 1.5 ∆ reach the quenching level Wq γ∗. To do this, con- = 0.3 J ∼ siderthesequenceoflocalizationsegmentsasaneffective 1 chain of ”sites”, the typical number of which is equal to the number of segments, N = N/N∗; the mean spac- s 0.5 ing between these ”sites” is N∗. The exciton diffusion coefficient is then estimated as D WN∗2 (the lattice * ∼ N 0 constant is set to unity). For the quenching to be as ef- γ fective as the spontaneous decay, the exciton has to be at the position of the trap (located on the opposite side of 4 of the chain) during the lifetime γ∗−1, i.e., it has to dif- s t fuse over the distance N during this time. Equating the i un 3 diffusionlength D/γ∗ toN,weobtaintheestimatefor ∆ = 0.2 J the required diffusion rate W: e, p at 2 W γ∗(N/N∗)2 . (17) r ∼ g n The localization length N∗ is equal to 38, 25 and 18 for i 1 h ∆=0.1,0.2and0.3,respectively. Thus,thecorrespond- c n ingdiffusionratesW areestimatedas625γ∗, 1600γ∗and e u 100 2500γ∗ (forN=1000). Thesevaluesareabouttwoorders Q ofmagnitude largerthanthe rates of sidewayshops over 8 thelocalstates,takentobe10γ∗ inallcalculations. This indicates that when the quenching rate becomes compa- rable to the spontaneous emission rate, the exciton does 6 ∆ nothop directly betweenthe local states ofadjacentseg- = 0.1 J ments(withthetypicalrateWν′1 10γ∗). Itratherhops 4 viathehigherstatesthatextendo∼vermorethanoneN∗- molecule segments (see the discussion in Sec. III). The 2 hopping rate via such states for T ∼2ε12 is of the order of W12 which is about two orders of magnitude larger than Wν′1 (see Eq. (3) and the scaling laws (12)). 0 In order to prove the above finding we performed cal- 0 0.5 1 1.5 2 2.5 3 culations of the quenching rate W varying the number q of states considered in Eq. (10). More specifically, we Temperature, units of ε 12 considered all states up to some (variable) cut-off state. Figure 5 shows the results of such study performed for ∆ = 0.1J. As it can be seen from the figure, W de- FIG. 4: Temperature dependence of the quenching rate Wq q calculatedforalinearchainofthelengthN =1000anddiffer- pends drastically on the number of states participating entmagnitudesofthedisorder∆. Theaveragingisperformed in the quenching process. In the region where Wq > γ∗, over100disorderrealizations. Foreachrealization ofthedis- approximately 6N/N∗ states are required to reach the order,theleftmostlocalgroundstateisexcited,whiletheonly true value of the quenching rate that is calculatedfor all trapislocatedinthecenterofthelocalization segmentofthe states (compare dashed and solid lines). Thus, at tem- rightmost local ground state. peraturesT >ε12thehigherstatesprovidethedominant contribution∼into the exciton quenching process. Figure 6 shows the regions of the DOS that corre- spond to different numbers of states that were used for the quencher during its (spontaneous) lifetime: it emits calculation of the data presented in Fig. 5. The higher aphotonbeforeitis trapped. Onthe contrary,justafter states lie just after the local ones, close to the maximum the temperature exceeds approximately ε12 the quench- of the DOS (see Fig. 6). Therefore, the typical energy ingbecomesnoticeable: theexcitonpartlydiffusestothe spacing between the local and higher states is about ε12. trap where it decays mostly due to quenching. Specifi- As the higher states extend over several, but not very cally, temperature of the order of 2ε12 are required for many,N∗-moleculesegments(seeFig.1),the overlapin- the quenching to become as effective as the spontaneous tegralbetweenthesestatesandthecoveredlocalstatesis emission: W γ∗ =γN∗. large. These two factors ensure high hopping rate from q ∼ 8 101 stateshavesmalloscillatorstrength,soaslongasanexci- ton remains in these states it does not decay radiatively. 0 10 The above qualitative arguments explain the dominant ∗ Ν contributionofthehigherstatesintotheexcitondiffusion γ of 10--21 and quenching within the temperature range T >∼ε12. s 10 t i n u 10-3 , q ∗ 100 100 W -4 Ν 10 γ 10-5 s of 10-2 10-2 0 1 2 3 nit Temperature, units of ε12 W, uq10-4 10-4 FIG. 5: Temperature dependence of the quenching rate Wq calculated for ∆ = 0.1J (N∗ ≈ 40) and different numbers -6 -6 of states considered in Eq. (10): solid line — all N = 1000 10 10 0.25 0.5 0.75 1 states, dashed line — 6N/N∗ = 150 states, dashed-dotted Temperature, units of ε line — 4N/N∗ = 100 states, dotted line — 3N/N∗ = 75 12 states,dashed-dashed-dottedline—2N/N∗ =50states. The averaging is performed over 100 disorder realizations. For FIG. 7: Temperature dependence of the quenching rate Wq eachrealizationofthedisorder,theleftmostlocalgroundstate calculated for∆=0.2J (N∗ ≈25). Solidline—allN =250 is excited, while the only trap is located in the center of the states, dashed line — 2N/N∗ =20 states. localization segment of therightmost local ground state. It is also seen from Fig. 5 that the difference between the true value of the quenching rate W and that calcu- q * 1.3 N/N * lated for a restrictednumber of states decreasesat lower 2 N/N 800 DOS temperatures. Figure 7 demonstrates the temperature 3 N/N* dependence of Wq obtained for ∆=0.2J (N∗ 25)and ≈ temperaturesT <ε12. Thesolidlinepresentsthedepen- 600 * 4 N/N dence calculatedfor all N =250states consideredin the * rate equation, the dashed line — that for 2N/N∗ = 20 6 N/N 400 states. These 2N/N∗ states include all the states of the local manifolds (1.3N/N∗) and 0.7N/N∗ higher ones. The parameters of the rate equation was set as fol- 200 lows: Γ∗ = 10W12 T=0 and W1′1 T=0 = 100γ∗. The | | chainlengthN =250is chosenso thatthe effective hop- 0 ping rate W γ∗(N/N∗)2 estimated as discussed above -2.1 -2 -1.9 -1.8 -1.7 ∼ (see Eq. (17)) is equal to the rate of the direct hopping Energy, units of J to an adjacent segment, W1′1 T=0. This yields the equa- tion(N/N∗)2 =100for the ch|ainlength. This condition FIG. 6: The total DOS calculated for N = 1000, ∆ = 0.1J ensures that the diffusion over the lower states only can (N∗ ≈40). The DOS is normalized to N. The vertical lines show the maximum energies corresponding to different num- pWrovid0e.t5hγe∗qautenthchisintgemrapteeraWtuqr∼e fγo∗r 2aNt T/N∼∗εs1t2a.teIsn.dTeehde, bers of states considered in the rate equation that was used q ≈ mostimportantpoint demonstratedby Fig.7 is thatbe- to calculate the dependencies plotted in Fig. 5 (in the sense that all states lower than the specified energy are consid- lowthetemperatureT1 0.25ε12 thetwocurvesdeviate ≈ ered). Note that the tail of the DOS is formed by 1.3N/N∗ slightly,which means that the contributionof the higher states, namely, by the states of the local manifolds (N/N∗ states into the diffusion becomes negligible: the exciton local ground states plus 0.3N/N∗ of the local excited states; hops mostly over the DOS tail states. On the contrary, recallthatabout30%ofthelocalgroundstatesformthedou- above the temperature T1 the higher states provide the blets). dominant contribution to the diffusion and quenching. Note also, that the value of the quenching rate at the criticaltemperatureT1 is typically verysmall,so the ex- the lower local to the higher states. Another important perimentalobservationofthis ”regimechange”is achal- point is that higher states are well overlapped and more lenging task. extended, so hops between them are typically faster and The criticaltemperatureT1 atwhich the higher states longerthanthosebetweenthelocalones. Also,thehigher come into play can be estimated by equating the typical 9 rateofthedirectsidewayshoppingfromalocalstate2to authors assumed that an exciton travels over about 104 an adjacent local state 1′ to the ”vertical” hopping rate dye molecules during its lifetime to meet another exci- fromthe local state 2 to a higher state 3 (see Fig.3, T > ton and annihilate. They found also that the activation 0): W1′2 = W32 W12 T=0 exp(ε12/T1). This equation energy of the exciton diffusion was 15K (10.5cm−1) and ≈ | yields the temperature T1: interpreted this energy as the typical energy spacing be- tween the states of adjacent localization segments. 1 T1 = ε12 0.25ε12 . (18) Despite the fact that the exciton-exciton annihilation ln(I12/I1′2) ≈ shouldtobe treateddifferently fromthe excitonquench- ing,the modelwearedealingwith caneasilybe adapted Westressthatthenumericalfactor1/ln(I12/I1′2) 0.25 ≈ for qualitative analysis of the exciton-exciton annihila- is almost independent of the disorder, as the disorder tion: oneofthetwoexcitonscanbeconsideredasanim- scalingsofthe overlapintegralsarealmostthe same (see mobile trap for the other, while the other diffuses twice Eqs. (12a)-(12b)). So, the estimate T1 0.25ε12 is uni- ≈ as fast. As reported in Ref. [23], the fluorescence spec- versal for a wide range of the disorder degree. trum of THIATS J-aggregates is narrowed by approxi- mately 26cm−1 and red-shifted by 23cm−1 as compared to the absorption spectrum. These results indicate un- V. SUMMARY AND CONCLUDING REMARKS ambiguously that excitons make sideways hops during their lifetime, i.e., the rate of sideways hops over local Inthispaper,westudytheoreticallythepeculiaritiesof states is larger than the exciton spontaneous emission the low-temperaturediffusion ofthe 1D Frenkelexcitons rate. Thus, the conditions for the exciton diffusion in localized by a moderate diagonal disorder. The exciton THIATS J-aggregatesare similar to those studied in the motion over localized states is considered as incoherent present paper (the limit of fast diffusion). hopping. The diffusion is probed by the exciton quench- Discussingtheabovementionedexperimentaldataand ing at a point trap. We consider a single trap located its interpretation presented in Ref. [23], the following at one end of the aggregate while the exciton is created points can be made. First, the typical energy spacing initially at the other end. In this case the exciton has between the states of the adjacent segments is of the or- to travel over almost the whole chain to be quenched. der of the J-band width [19], that is 82cm−1 and not Under this conditions, the quenching rate carries direct 10.5cm−1. The latter value is closer to 0.25 82cm−1, information about the diffusion length that the exciton × i.e. this temperature could be related to the tempera- travelsoverduring its lifetime. The excitonquenching is ture T1, the activation energy of the faster exciton diffu- described by the rate equation with the quenching rate sion regime. Above this temperature an exciton diffuses being proportionalto the probability of finding the exci- mostly over the higher states and not over the DOS tail ton at the trap site. states, as it was suggested in Ref. [23]. Another point, Both our qualitative arguments and numerical simu- and a more important one, is of the quantitative nature: lations show that there exist two regimes of the exci- The typical size of the localization segment in THIATS ton diffusion. At lower temperatures, those smaller than J-aggregatesis N∗ =30 [42]. In the model we are using, T1 0.25 J-band-width, the exciton diffuses mostly ≈ × this corresponds to the disorder magnitude ∆ 0.2J. over weakly overlapped DOS tail states which determine ≈ Our numerical data obtained for a chain of N = 1000 the optical response and form the J-band. This regime molecules demonstrates that for this value of the disor- of diffusion is very slow; the exciton cannot diffuse over der the exciton quenching is vanishingly small for the large distance during its lifetime at these temperatures. temperatures T (10.5/82) ε12 (we remind that ε12 is At higher temperatures, the higher states come into ∼ × of the order of the J-band width). In other words, the play. Thediffusionbeginstobuiltupduetothetwo-step exciton createdin the leftmost local groundstate cannot hopsviahigherstates. Thisacceleratestheexcitondiffu- diffuse over the whole chain of 1000 monomers during sion drastically, so that an exciton can diffuse over large itslifetime. However,itcandosoatthe temperaturesof distances during its lifetime. The higher states begin to theorderofT ε12 82cm−1. Thus,understandingthe contribute dominantly to the diffusion at temperatures ∼ ∼ fast low-temperature diffusion in the aggregates of THI- higher than about T1. However, the diffusion becomes ATS dye molecules, observed in Ref. [23], still remains reallyfast (in the sense thatthe quenching rate becomes an open question. comparable to the spontaneous emission rate of the ag- gregate) only at the temperatures of the order of the J-band width. InRef.[23],theanomalouslyfastlow-temperaturedif- Acknowledgments fusion of Frenkel excitons in J-aggregates of THIATS was reported. The authors of Ref. 23 studied experi- This work was supported by the DGI-MCyT (Project mentallytheexciton-excitonannihilationandfoundthat MAT2000-0734). A. V. M. and F. D. A. acknowledge this effect is pronounced even at T = 5 K (3.5cm−1), support from CAM (Project 07N/0075/2001). V. A. M. while the width of J-band of THIATS J-aggregates is acknowledges support from MECyD (Project SAB2000- 82cm−1. In order to explain the experimental data, the 0103) and through a NATO Fellowship. 10 [1] E. E. Jelley, Nature(London) 38, 1009 (1936). [23] I. G. Scheblykin, O. Yu. Sliusarenko, L. S. Lepnev, A. [2] G. Scheibe, Angew. Chem. 49, 563 (1936). G. Vitukhnovsky, and M. Van der Auweraer, J. Phys. [3] J. Frenkel,Phys. Rev. 17, 17 (1931). Chem. B 104, 10949 (2000); 105, 4636 (2001). [4] A. S. Davydov, Theory of Molecular Excitons (Plenum, [24] J.A.Leegwater,J.R.Durrant,andD.R.Klug,J.Phys. NewYork, 1971). Chem. B 101, 7205 (1997). [5] V. M. Agranovich and M. D. Galanin, in Electronic Ex- [25] M. Bednarz, V.A. Malyshev, and J. Knoester, J. Chem. citationEnergy TransferinCondensedMatter,editedby Phys. 117, 6200 (2002). V.M.AgranovichandA.A.Maradudin(North-Holland, [26] P. Dean, Rev.Mod. Phys 44, 127 (1972) Amsterdam 1982). [27] V. V. Ovsyankin, in Spectroscopy of Solids Containing [6] S. de Boer and D. A. Wiersma, Chem. Phys. Lett. 165, Rare-Earth Ions,eds.A.A.KaplyanskiiandR.M.Mac- 45 (1990). Farlane, (Amsterdam: North-Holland, 1987), p. 343. [7] H.Fidder,J.Knoester,andD.A.Wiersma,Chem.Phys. [28] M. Shimizu, S. Suto, and T. Goto, J. Chem. Phys. 114, Lett.171, 529 (1990). 2775 (2001). [8] H. Fidder, J. Terpstra, and D. A. Wiersma, J. Chem. [29] M. Bednarz, V. A. Malyshev, J. P. Lemaistre, and J. Phys.94, 6895 (1991). Knoester, J. Lumin. 94-95, 271 (2001). [9] H.Fidder, PhD Thesis, University of Groningen, 1993. [30] F. C. Spano, J. R. Kuklinsky, and S. Mukamel, Phys. [10] H. Fidder, J. Knoester, and D. A. Wiersma, J. Chem. Rev. Lett. 65, 211 (1990); J. Chem. Phys. 94, 7534 Phys.95, 7880 (1991). (1991). [11] F. C. Spano and J. Knoester, in Advances in Magnetic [31] V. A. Malyshev, Opt. Spektr. 71, 873 (1991) [Opt. and Optical Resonance, Vol.18,ed.W.S.Warren(Aca- Spectr. 71, 505 (1991)]; J. Lumin., 55, 225 (1993). demic, New York,1994), p. 117. [32] V. Malyshev and P. Moreno, Phys. Rev. B 51 14587 [12] Contributions to Adv.Mater. 7 (1995). (1995). [13] J-aggregates, ed. T. Kobayashi (World Scientific, Singa- [33] M. Shimizu, S. Suto, T. Goto, A. Watanabe, and M. pore, 1996). Matsuda, Phys. Rev.B 58, 5032 (1998). [14] E. W. Knapp,Chem. Phys. 85, 73 (1984). [34] A. V. Malyshev and V. A. Malyshev, Phys. Rev. B 63, [15] J. Knoester, J. Chem. Phys. 99, 8466 (1993); J. Lumin. 195111 (2001). 58, 107 (1994). [35] K. Minoshima, M. Taiji, K. Misawa, T. Kobayashi, [16] J. R. Durrant, J. Knoester, and D. A. Wiersma, Chem. Chem. Phys.Lett. 218, 67 (1994). Phys.Lett. 95, 7880 (1991). [36] J. Knoester and F. C. Spano,in Ref. 13, p.111. [17] V. A. Malyshev, A. Rodr´ıguez, and F. Dom´ınguez- [37] L.D.Bakalis andJ.Knoester,J.Phys.Chem.103, 6620 Adame,Phys. Rev.B 60, 14140 (1999). (1999); J. Lumin. 87-89, 66 (2000). [18] V.A.MalyshevandF.Dom´ınguez-Adame,Chem.Phys. [38] V. F. Kamalov, I. A. Struganova, and K. Ioshihara, J. Lett.313, 255 (1999). Phys. Chem. 100, 8640 (1996). [19] A. V. Malyshev and V. A. Malyshev, J. Lumin. 94-95, [39] V.A.Malyshev,H.GlaeskeandK.-H.FellerChem.Phys. 369 (2001). Lett. 305, 117 (1999). [20] B.I.ShklovskiiandA.L.Efros, Electronic Properties of [40] V. A. Malyshev, G. G. Kozlov, H. Glaeske and K.-H. Doped Semiconductors, Springer-Verlag, Berlin, 1984. Feller. Chem. Phys. 254, 31 (2000). [21] T. T. Basiev, V. A. Malyshev and A. K. Przhevuskii, in [41] I. V. Ryzhov, G. G. Kozlov, V. A. Malyshev, and J. Spectroscopy of Solids Containing Rare-Earth Ions, eds. Knoester, J. Chem. Phys. 114, 5322 (2001). A. A. Kaplyanskii and R. M. MacFarlane, (Amsterdam: [42] I. G. Scheblykin, M. M. Bataiev, M. Van der Auwer- North-Holland,1987), p. 275. aer,andA.G.Vitukhnovsky,Chem.Phys.Lett.316,37 [22] V. A. Malyshev, Fiz. Tv. Tela 34, 1109 (1992)[Engl. (2000). Transl.: Phys. Solid State 34, 590 (1992)].