University of Groningen

Dynamics of Frenkel excitons in J-aggregatesBednarz, Mariusz

IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.

Document VersionPublisher's PDF, also known as Version of record

Publication date:2003

Link to publication in University of Groningen/UMCG research database

Citation for published version (APA):Bednarz, M. (2003). Dynamics of Frenkel excitons in J-aggregates. s.n.

CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).

Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.

Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.

Download date: 09-06-2021

https://research.rug.nl/en/publications/dynamics-of-frenkel-excitons-in-jaggregates(a9ad3e94-012c-429f-9a5f-1b693327e7d3).html

Dynamics of Frenkel excitonsin J-aggregates

Mariusz Bednarz

Printed by Ponsen & Looijen bv, Wageningen, The Netherlands

Copyright © 2003 by Mariusz Bednarz

ISBN 90-367-1920-8

MSC Ph.D.-thesis series 2003-10ISSN 1570-1530

Rijksuniversiteit Groningen

Dynamics of Frenkel excitons in J–aggregates

Proefschrift

ter verkrijging van het doctoraat in deWiskunde en Natuurwetenschappenaan de Rijksuniversiteit Groningen

op gezag van deRector Magnificus, dr. F. Zwarts,in het openbaar te verdedigen op

maandag 3 november 2003om 14.15 uur

door

Mariusz Piotr Bednarz

geboren op 29 december 1969te Szczebrzeszyn, Polen

Promotor:

Prof. dr. J. Knoester

Beoordelingscommissie:

Prof. dr. L.D.A. SiebbelesProf. dr. F.C. SpanoProf. dr. D.A. Wiersma

v

Contents

1 Introduction 11.1 Thesis overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Intraband relaxation and fluorescence decay in homogeneous aggregates 72.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Model Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 The Pauli master equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4 Intra-band exciton scattering rates . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4.1 Glassy host . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4.2 Crystalline host . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.5 Defining the fluorescence decay time . . . . . . . . . . . . . . . . . . . . . . . . 152.6 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.6.1 Glassy host . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.6.2 Crystalline host . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.7 Summary and concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . 28

3 Dynamics of Frenkel excitons in disordered linear chains 333.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2 Disordered Frenkel exciton model . . . . . . . . . . . . . . . . . . . . . . . . . 353.3 Intraband relaxation model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.4 Steady-state fluorescence spectrum . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.4.1 Steady-state fluorescence spectra at zero temperature . . . . . . . . . . . 433.4.2 Temperature dependence of the Stokes shift . . . . . . . . . . . . . . . . 45

3.5 Fluorescence decay time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.5.1 Broadband resonance excitation . . . . . . . . . . . . . . . . . . . . . . 473.5.2 Off-resonance blue-tail excitation . . . . . . . . . . . . . . . . . . . . . 493.5.3 Dependence on the detection energy . . . . . . . . . . . . . . . . . . . . 51

3.6 Summary and concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . 53

4 Comparison to experiment 594.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.3 Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

vi CONTENTS

5 Absorption and pump-probe spectra of cylindrical aggregates 675.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.2 Model and linear absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.3 Pump-probe spectrum: general theory . . . . . . . . . . . . . . . . . . . . . . . 715.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.5 Interpretation of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.5.1 Intra-ring and inter-ring two-exciton states . . . . . . . . . . . . . . . . 775.5.2 Transition dipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.5.3 Explanation of the pump-probe spectrum forJ1 = 0 . . . . . . . . . . . . 805.5.4 Explanation of the pump-probe spectrum forJ1 6= 0 . . . . . . . . . . . . 82

5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

Samenvatting 89

List of publications 91

Acknowledgements 93

Chapter 1

Introduction

The main subject of this thesis is the theoretical study of exciton dynamics in Frenkel excitonsystems. A wide class of Frenkel exciton systems exists as aggregates of dye molecules [1].The term aggregate is used to characterize a, usually self-assembled, collection of molecules.The molecules are kept together by electrostatic forces in assemblies of reduced dimensionalityoften with a chain-like configuration. The unique optical properties of dye aggregates were firstobserved independently by Jelley and Scheibe in the mid-thirties for the dye pseudo-isocyanine(PIC) [2, 3]. Following a spontaneous self-aggregation process which occurs in solution at highdye concentrations or in thin solid films, these molecules form rod-like arrangements consistingof a large number of molecules [2–4]. The most spectacular indication for the formation ofaggregates is the dramatic change of the absorption spectrum. In addition to a broad monomericband, one observes the rise of a new absorption band, which is significantly shifted and stronglynarrowed compared to the monomer absorption band. It is confirmed that the excited states insuch materials are of excitonic nature with the electrons tightly bound to their molecular centers(Frenkel excitons) [5]. The exciton states (collective eigenstates) are delocalized electronic statescaused by the strong long-range dipole-dipole interactions between the monomers within thechain. The coupling of the optical transitions of the molecules by these interactions results in theformation of a band of Frenkel exciton states. The width of the exciton band is proportional tothe coupling energy. In the simplest case of one molecule per unit cell and a negative couplingenergy, only transitions to states at the bottom of the exciton band are optically allowed and formthe so called J-band. The total oscillator strength of the coupled molecules is thus swept togetherin a few eigenstates only and the majority of the exciton states is not accessible in absorption. Asa consequence, the radiative rates of the bottom states are strongly enhanced as compared to thesingle molecule, a phenomenon usually referred to as exciton superradiance [6].

Despite the intensive investigations over the past several decades directed toward better un-derstanding the basic photophysical properties of molecular aggregates, there still remain funda-mental questions regarding the dynamics of excitons in such materials. One of the most chal-lenging theoretical problems is to understand the emission properties of these systems in theframework of a microscopic theory. The emission results from the complicated interplay be-

2 Introduction

tween radiative and nonradiative processes and is particularly sensitive to the details of the exci-tons’s spatial structure (delocalization) and its dynamics (motion). The motion may be coherentor incoherent. In general, the degree of delocalization and the type of motion of the excitondepend crucially on the interaction between the exciton system with other degrees of freedomin its environment. The exciton dynamics may be coherent when the intermolecular transfer ofexcitation energy is much faster than any relaxation time scale introduced by the fluctuating en-vironment. In the opposite case the exciton motion will be incoherent, leading to a hopping pro-cess between different sites. Coherent transfer requires fixed phase relations of the exciton wavefunction on different molecules, while incoherent transfer takes place when relaxation processesintroduce fast intermolecular dephasing of the wave function [7]. The degree of delocalizationand dephasing are fundamental quantities which can be determined experimentally using linearand nonlinear spectroscopy [8]. During the last decade, J aggregates have been intensively in-vestigated using various laser-spectroscopical techniques, such as spectral hole-burning [9, 10],photon-echo [10], pump-probe [11], fluorescence lifetime measurements [6], etc. Using suchmeasurements the exciton properties have been probed and it has been confirmed that the actualcoherence length, the number of monomers involved in the delocalized excitons, is much smallerthan the total aggregate size [11]. Additionally at very low temperatures (below 1.5K) dephasingis not present and only gradually starts to play a role with increasing temperature [10].

At low temperatures, usually in the order of the J-band width, the absorption spectra of J-aggregates do not change position and profile [12–14]. It has been shown that the low-temperatureabsorption lineshape of dye aggregates is inhomogeneously broadened and can be explainedquantitatively by taking into account disorder in the Frenkel chain. The inhomogeneous widthand profile of the absorption band was explained by two models of disorder. In the so-calledsegment model one assumes that defects are distributed at random over the chain and that theylocalize the excitons on mutually decoupled homogeneous segments. In this model, inhomoge-neous broadening arises from the distribution of segment lengths [9,15,16] and(or) the transitionenergy common to all molecules within a given segment [17]. In the second model, one assumesthat the molecular transitions energies and (or) interactions are randomly distributed as a resultof solvent shifts. Extensive numerical simulations of the linear optical spectra of aggregateshave been performed and applied successfully to a variety of aggreagtes, for instance to PICaggregate [18,19], to THIATS (3,3’-bis-[3-sulfopropyl]-5, 5’-dichloro-9-ethylthiacarbocyanine)aggregate [16], or mixed cyanine aggrgates [20]. Knapp in his fundamental work about motionalnarrowing [21] in dye aggregates also considered the effect of intersite correlations in the dis-order. In this frame both models of disorder can be interpreted as limiting cases. The segmentmodel corresponds to the completely correlated limit and continuous site energy disorder as thetotally uncorrelated one.

In the literature, a large number of theoretical investigations exists which focus on the dy-namics of excitons in molecular aggregates when the influence of the environment on the excitonmotion is taken into account. The dominant influence of the environment is the coupling to lat-tice vibrations. Most theoretical models on exciton-phonon coupling have restricted to idealized,homogeneous exciton systems [5, 22]. One of the simplest approaches models the effect of thephonons as giving rise to classical stochastic fluctuations in the molecular transition frequencies.The δ-correlated Haken-Strobl model is a well-known example of such an approach [23–25].A full quantum mechanical approach leads to interesting predictions, such as exciton-self trap-ping [26]. In one-dimensional systems, self-trapping occurs at arbitrarily weak exciton-phononcoupling, which is the reason that it has been considered by various authors as the source ofexciton dynamics and fluorescence Stokes shifts in molecular aggregates [27, 28]. On the other

1.1 Thesis overview 3

hand, in J-aggregates the exciton-phonon coupling generally seems to be rather weak, while en-ergetic disorder is ubiquitous. Hence, the localizing effect of energetic disorder may lead to astronger limitation of the extent of the exciton wave function than the effect of self-trapping. Insuch a case, it is more useful to start from the basis of exciton states localized by the disorder andconsider the effect of the weak exciton-phonon coupling through scattering of these localizedexciton states. This is the point of view that will be taken in this thesis.

The models in this thesis have been formulated mostly against the background of molecu-lar aggregates. In general, however, our study has strong relevance to other low-dimensionalexciton systems as well. Examples are conjugated polymers [29], such us theσ-conjugatedpolymer polysilane [9,30], or polydiacetylene (PDA) chains [31], and recently studied oligomer-aggregates [32].

1.1 Thesis overview

The contents of this thesis is such that at the beginning of each chapter an extensive introductionis given. Here, in brief only, the order of the thesis and the main results presented in each chapterwill be summarized. The sequence of chapters 2, 3, and 4 constitutes one extended theoreticalstudy, both analytical and numerical, of the dynamics and relaxation of Frenkel excitons in lin-ear chains. For a start, in chapter 2, we study the fluorescence decay time for a homogeneousone-dimensional aggregate placed in a host medium. The primary goal in this chapter is to gainbetter insight into intra-band relaxation in homogeneous aggregates. In particular, we modelthe scattering between different exciton states, arising from their coupling to acoustic vibrationsin the host medium. A Pauli master equation is used to describe the redistribution of excitonsover the band. The rates entering this equation are calculated within the framework of first-orderperturbation theory, assuming a weak linear on-site interaction between excitons and acousticphonons. Solving the master equation numerically for aggregates of up to 100 molecules, wecalculate the temperature dependence of the fluorescence kinetics in general and the decay timescale in particular. The proper definition of the fluorescence decay time is discussed in detail.We demonstrate that, even at a quantum yield of unity, the possibility to directly interpret flu-orescence experiments in terms of a simple radiative time scale depends crucially on the initialexcitation conditions in combination with the competition between spontaneous emission andintra-band phonon-assisted relaxation. This holds both for glassy and for crystalline hosts, whichwe find to differ widely in the nature of the intraband relaxation processes.

In chapter 3 we extend our study and an attempt is made to account in our kinetic model forthe following factors, relevant to common experimental conditions at low temperatures: weaklocalization of the exciton states by static disorder (essential for a proper description of the in-homogeneous broadening of the absorption spectra at low temperatures), and factors, alreadyconsidered in chapter 2, like coupling of the localized excitons to vibrations in the host medium,a possible non-equilibrium of the subsystem of localized Frenkel excitons on the time scale ofthe emission process, and different excitation conditions (resonant or non-resonant). A Paulimaster equation, with microscopically calculated transition rates, is used to describe the redis-tribution of the exciton population over the manifold of localized exciton states. We calculatethe temperature dependence of the fluorescence Stokes shift and the fluorescence decay time inlinear Frenkel exciton systems resulting from the thermal redistribution of exciton populationover the band states. We find a counterintuitive non-monotonic temperature dependence of theStokes shift and wavelength dependence of the exciton decay time within the fluorescence band.

4 Introduction

In addition, like in the case for homogeneous aggregates considered in the previous chapter, weshow that depending on the initial excitation condition, the observed fluorescence decay timemay be determined by vibration-induced intra-band relaxation, rather than radiative relaxation tothe ground state.

In chapter 4 a comparison of our theory to experimental data will be presented. We will theo-retically analyze the exciton dynamics for THIATS dye aggregate. In recent experiments, a wideset of temperature sensitive optical properties, such as absorption line broadening, fluorescenceline broadening, Stokes shift, fluorescence decay time, and the exciton-exciton annihilation ratewere reportered for this material [14]. Such en extensive experimental study using various spec-troscopic tools on one material over a large temperature range (from 5K to 130K) make thismaterial unique for theoretical study. We show that our microscopic model developed in theprevious chapters, with the effect of inhomogeneity included, is able to describe in a quantitativemanner the exciton dynamics in THIATS aggregates at low-temperatures (up to≈ 100K). Wesuccessfully fit the experimental data of absorption and fluorescence spectra, Stokes shift, andfluorescence decay time of THIATS aggregates with one set of microscopic parameters.

In biology, molecular aggregates are of general interest, since most photobiological pro-cesses, especially photosynthesis [33], rely on aggregates for energy or charge transport. Forexample, in biological light-harvesting systems ring-shaped molecular aggregates [33] act as an-tenna complexes and are responsible for capturing the sun light and transporting the associatedexcitation energy to the photosynthetic reaction center. A new class of Frenkel exciton systemswhich can mimic the energy transfer processes that occur in photosynthetic systems are den-drimers [34] with a highly branched structure. The interest in novel aggregates with cylindricalsymmetry has increased strongly since the recent discovery that the photosynthetic systems ofcertain bacteria contain aggregates of chlorophyl molecules with such symmetry [35]. At thesame time, self-assembled cylindrical aggregates can be formed synthetically, using cyanine dyemolecules substituted with amphiphilic side groups [36]. In chapter 5, we study the opticalresponse of Frenkel excitons in molecular J-aggregates with a simple cylindrical geometry (per-pendicular stack of rings), which can be considered as a first step toward better understandingthe linear and nonlinear spectroscopy of such novel aggregates. We show that the linear absorp-tion spectrum exhibits two lines with perpendicular polarization that are separated by a “ringenergy scale”, which is set by the circumference of the cylinder and the intermolecular transferinteraction in the circumferential direction. On the other hand, the pump-probe spectrum showsbleaching and induced absorption features that are separated by a much smaller energy scale,which is determined by an effective Pauli gap imposed by the length of the cylinder and thetransfer interaction in its longitudinal direction. We show that this can be well-understood fromthe approximate separation of the set of two-exciton states, into a class of inter-ring and intra-ringtwo-exciton states. Our calculations show that the experimental linear absorption spectrum maybe used to estimate the cylinder circumference, while the pump-probe spectrum yields informa-tion on the length of the cylinder or on the delocalization length of the excitons in its longitudinaldirection.

5

Bibliography

[1] J-aggregates, ed. T. Kobayashi (World Scientific, Singapore, 1996).

[2] G. Scheibe, Angew. Chem.49, 563 (1936); ibid.50, 51 (1937).

[3] E.E. Jelly, Nature138, 1009 (136); ibid.139, 631 (137).

[4] H. von Berlepsch and C. Böttcher, J. Phys. Chem. B106, 3146 (2002).

[5] A. S. Davydov,Theory of Molecular Excitons (Plenum, New York, 1971).

[6] S. de Boer and D.A. Wiersma, Chem. Phys. Lett.165, 45 (1990).

[7] V. May and O. Kühn,Charge and energy transfer dynamics in molecular systems (WILEY-VCH, Berlin, 2000).

[8] S. Mukamel,Principles of Nonlinear Optical Spectroscopy (Oxford University Press, NewYork, 1995).

[9] A. Tilgner, H.P Trommsdorff, J.M. Zeigler, and R.M. Hochstrasser, J. Chem. Phys.96, 781(1992).

[10] S. de Boer, K.J. Vink and D.A. Wiersma, Chem. Phys. Lett.137, 99 (1987).

[11] H. Fidder, J. Knoester, and D.A. Wiersma, J. Chem. Phys.98, 6564 (1993).

[12] H. Fidder, Ph.D Thesis, University of Groningen, 1993.

[13] V.F. Kamalov, I.A. Struganova, and K. Yoshihara, J. Phys. Chem.100, 8640 (1996).

[14] I.G. Scheblykin, O.Yu. Slisarenko, L.S. Lepnev, A.G. Vitukhnovsky, and M. Van der Auw-eraer, J. Phys. Chem. B105, 4636 (2001).

[15] S.N. Yaliraki and R.J. Silbey, J. Chem. Phys.104, 1245 (1995).

[16] P. Argyrakis, D.M. Basko, M.A. Drobizhev, A.N. Lobanov, A.V. Pimenov, O.P. Varnavsky,M. Van der Auweraer, A.G. Vitukhnovsky, Chem. Phys. Lett.268, 372 (372).

[17] F.C. Spano, J. Phys. Chem.96, 2843 (1992).

[18] H. Fidder, J. Knoester, and D.A. Wiersma, J. Chem. Phys.95, 7880 (1991).

[19] J. Knoester in “Organic Nanostructures: Science and Application” (IOS Press, Amsterdam,2002), pp. 149-186.

6 BIBLIOGRAPHY

[20] L.D. Bakalis and J. Knoester, J. Chem. Phys.117, 5393 (2002).

[21] E.W. Knapp, Chem. Phys.85, 73 (1984).

[22] V. M. Agranovich and M. D. Galanin, inElectronic Excitation Energy Transfer in Con-densed Matter, edited by V. M. Agranovich and A. A. Maradudin (North-Holland, Amster-dam 1982).

[23] H. Haken and G. Strobl, inThe Triplet State, edited by A. Zahlan (Cambridge UniversityPress, Cambridge, 1967).

[24] H. Haken and G. Strobl, Z. Phys.262, 135 (1973).

[25] V.M. Kenkre P. Reineker,Exciton Dynamics in Molecular Crystals and Aggregates,Springer-Verlag Berlin Heidelberg 1982.

[26] E.I. Rashba, inExcitons, edited by E.I. Rashba, M.D. Sturge (North-Holland, 1982).

[27] V. M. Agranovich and A. M. Kamchatnov, Chem. Phys.245, 175 (1999).

[28] M. Drobizhev, C. Sigel, A. Rebane, J. Lumin.86, 107 (2000).

[29] Relaxation in Polymers, ed. T. Kobayashi (World Scientific, Singapore, 1993).

[30] M. Shimizu, S. Suto, and T. Goto, J. Chem. Phys.114, 2775 (2001)

[31] R. Lécuiller, J. Berréhar, J.D. Ganière, C. Lapersonne-Meyer, P. Lavallard, and M. Schott,Phys. Rev. B66, 125205 (2002).

[32] F.C. Spano, J. Chem. Phys.118, 981 (2003).

[33] H. van Amerongen, L. Valkunas, R. van Grondelle,Photosynthetic excitons (World Scien-tific, Singapore, 2000).

[34] D. Jiang and T. Aida, Nature388, 454 (1997).

[35] V.I. Prokhorenko, D.B. Steensgaard, A.R. Holzwarth, Biophys. J.79, 2105 (2000).

[36] A. Pawlik, S. Kirstein, U. De Rossi, S. Daehne, J. Phys. Chem. B101, 5646 (1997).

Chapter 2

Intraband relaxation andfluorescence decay in homogeneousaggregates

2.1 Introduction

The concept of one-dimensional (1D) Frenkel excitons [1, 2] has proven to be very useful inexplaining the low-temperature optical properties of molecular aggregates and conjugated poly-mers (for reviews, see Refs. [3–5] and references therein). One of the remarkable features of 1DFrenkel exciton systems is that only a few states accumulate the entire oscillator strength. Aslong as the chain length is small compared to the emission wavelength, this leads to an enhance-ment of the corresponding spontaneous emission rates by approximately a factor ofN over theradiative rate of a single molecule. HereN denotes the number of molecules in the chain or, inthe case of a disordered chain, the number of molecules within a localization domain of the ex-citons [6–8]. For a perfectly ordered aggregate whose length exceeds the emission wavelength,the enhancement factor saturates at the number of molecules within this wavelength [9,10].

Experiments on various types of cyanine J-aggregates in (glassy) solution, in particular 1,1’-diethyl-2,2’-cyanine (PIC) [11–15], 5,5’,6,6’-tetrachloro-1,1’-diethyl-3,3’-di(4-sulfobutyl) ben-zimidazolocarbocyanine (TDBC) [16], 1,1’-diethyl-3,3’- bis (sulfopropyl)-5,5’,6,6’- tetrachloro-benzimidacarbocyanine (BIC) [17], and 3,3’-bis (sulfopropyl)-5,5’-dichloro-9- ethylthiacarbo-cyanine (THIATS) [18], have revealed that the exciton radiative lifetime grows with increasingtemperature. Typically, the temperature dependence consists of a plateau that extends to severaltens of Kelvin, followed by a power-like growth of the lifetime at higher temperatures. This slow-ing down of the aggregate’s radiative dynamics is usually attributed to the thermal population ofhigher exciton states, which in J-aggregates have oscillator strengths that are small compared tothose of the optically active states near the bottom of the exciton band [8,12,13].

8 Intraband relaxation and fluorescence decay in homogeneous aggregates

The first attempt to fit the experimental data on PIC reported in Ref. [11], was based on amicroscopic model of Frenkel excitons coupled to the vibrations of the aggregate itself [19]. Anintegro-differential equation of motion for the populations of the exciton states, derived by elim-inating the phonon variables through a factorization, was used to describe the exciton dynamics.Assuming this dynamics to be dominated by an optical phonon of suitable frequency, the ex-perimental data were fitted reasonably well over the entire temperature range. However, aftercorrection of the experimental data for the temperature dependence of the quantum yield [15],it turned out impossible to fit the experiments by the theory developed in Ref. [19], unless PICJ-aggregates were assumed to be two-dimensional [20]. The issue of structure and dimensional-ity of cyanine J-aggregates in solution is a difficult and intriguing one. While usually assumedto be 1D, the nearly linear temperature dependence of the exciton radiative lifetime measured inBIC aggregates has lead to the conclusion that these aggregates would be really two-dimensionalas well [17]. The rationale for this conclusion was the similarity of this (linear) dependenceto the behavior observed in quasi-two-dimensional semiconductor nanostructures [21]. On theother hand, the exciton radiative lifetime of THIATS aggregates can be understood in terms of aone-dimensional model, provided that the Davydov splitting [2] is correctly accounted for [18].Moreover, recent cryogenic transmission electron microscopy images have revealed that PIC ag-gregates in solution do in fact assume a one-dimensional structure, in which a few molecularchains bundle up to form one aggregate [22].

The state of affairs described above, calls for a renewed critical discussion of the tempera-ture dependent radiative lifetime of J-aggregates. The present chapter contributes several newelements to this discussion. In particular, we will point out the important role of the experi-mental excitation conditions in relation to the competition between spontaneous emission andvibration-assisted intraband exciton relaxation.

It is important to realize that in all the above quoted measurements of the exciton radiativelifetime [11–18], the system was excited in the blue tail of the absorption band, while the fluores-cence was observed either within the entire band or at a particular energy close to the absorptionmaximum. Thus, between the absorption and emission events an additional step existed: thevibration-assisted relaxation from the initially excited states to the radiating ones. From this itis immediately clear that the ratio between the rates of two processes, namely vibration-assistedintra-band exciton relaxation and exciton spontaneous emission, determines the kinetics of thefluorescence decay. Two limiting cases can be distinguished. If the intra-band relaxation is fasterthan the spontaneous emission, the population of the excited state is rapidly transferred to theradiating state, whereupon this state will slowly (on the scale of the intra-band relaxation) ra-diate. In this limit, it is thespontaneous emissionrate that determines the rate of the excitonfluorescence decay. Analyzing this decay properly (quantum yield correction, etc.), one may ex-tract from such measurements the actual exciton radiative decay time. In the opposite limit, it isthe slowintraband relaxationthat acts as bottleneck in the exciton fluorescence decay and thusgoverns the measured lifetime. It is then unlikely that one obtains accurate information aboutthe exciton radiative lifetime from such experiments. Rather, it seems that then the only way toproperly measure the exciton radiative lifetime is to resonantly excite the exciton fluorescence.This may be done using accumulated photon echo experiments. However, as we will show inthis chapter, even under resonant excitation, it is not always easy to extract information about theexciton radiative lifetime from spectroscopic data.

The goal of this chapter, motivated by the above observations, is a systematic analysis of thetemperature dependence of the fluorescence decay time of 1D Frenkel excitons under variousexcitation conditions, taking into account the effects of intraband exciton relaxation. As the

2.2 Model Hamiltonian 9

physics of the above noted effects does not depend strongly on whether the system is disorderedor not [23], we will restrict our study to the simplest case of an ordered aggregate, where theseeffects can be demonstrated in their purest form. A study of the additional effects of disorder willbe deferred to a later chapters. We will assume that the relaxation dynamics of the excitons inthe aggregate is governed by their coupling to vibrations in the host medium [24–27], rather thanto vibrations of the aggregate itself [19]. The exciton dynamics will be described at the level of aPauli master equation for the populations of the exciton states. The vibration-assisted populationtransfer rates governing this equation are obtained within a first-order perturbation expansion ina linear exciton-phonon interaction. As we mainly aim to study the low-temperature behavior ofthe exciton fluorescence, we focus on a coupling to acoustic phonons. The thus obtained masterequation is solved numerically to describe the fluorescence kinetics both as a function of timeand temperature. As the case of fast relaxation is rather uninteresting (because the excitons reachthermal equalibrium before emission), we will throughout this chapter mostly focus on the caseof slow relaxation. By this we mean that at least the zero-temperature intraband relaxation ratesare small compared to the superradiant emission rate.

This chapter is organized as follows. In Sec. 2.2, we present the model Hamiltonian ofFrenkel excitons interacting with host vibrations. The Pauli master equation for the exciton pop-ulations is introduced in Sec. 2.3. Section 2.4 deals with calculating the exciton scattering ratesthat enter this equation. In Sec. 2.5, we demonstrate that the proper definition of a fluorescencedecay time is a subtle problem and is in fact affected by the competition between radiative decayand intra-band relaxation. Our numerical results for the temperature dependence of the excitonfluorescence decay time in different limits of this competition and for different initial excitationconditions are given and discussed in Sec. 2.6. Finally, we conclude in Sec. 2.7.

2.2 Model Hamiltonian

We model an aggregate asN (N À 1) optically active two-level molecules forming a regular 1Dlattice with spacinga. If the aggregate is considered fixed in its equilibrium geometry, its opticalexcitations are described by the Frenkel exciton Hamiltonian: [2]

Hex =N

ån=1

e0n |n〉〈n|+N

ån,m

Jnm|n〉〈m| . (2.1)

Here, |n〉 is the state with moleculen of the aggregate excited and all other molecules in theirground state. This basis state has energye0n = e0 +U0n , with e0 the energy of the excited stateof an isolated molecule andU0n = å sU0ns the shift due to the interactionsUns of thenth excitedmolecule with all other aggregate and host molecules in their ground states (the superscript “0”indicates that these interactions are taken for the molecular equilibrium positions). As we will notconsider effects of electronic disorder, all energiesen are assumed identical and from now on willbe set to zero. Likewise, disorder in the excitation hopping integralsJnm will not be considered.These interactions are assumed to be of dipole-dipole origin:Jnm=−J/|n−m|3(Jnn≡ 0), where−J is the nearest-neighbor coupling. We will takeJ to be positive, as is appropriate for J-aggregates [11–13]. Then, the optically allowed states are those in the vicinity of the bottom ofthe exciton band.

Accounting for all dipole-dipole interactions, the Hamiltonian Eq. (2.1) can be diagonalized

10 Intraband relaxation and fluorescence decay in homogeneous aggregates

with a precision of the order ofN−1 (see Ref. [28]):

Hex = åK

Ek |k〉〈k| , (2.2)

with the new basis

|k〉 =(

2N

+1

)1/2 N

ån=1

sin

(

pknN+1

)

|n〉 , (2.3)

and

Ek = −2JN

ån=1

1n3

cos

(

pknN+1

)

. (2.4)

Here,k= 1,2, . . . ,N. For future use it is convenient to introduce the compact notation (wavenum-ber)K = pk/(N + 1). The statek = 1 lies at the bottom of the exciton band. Near the bottom(k¿ N or K ¿ 1) and in the limit of largeN [28], the exciton dispersion relation reads

Ek = −2.404J+J(

32− lnK

)

K2 . (2.5)

For comparison, within the nearest-neighbor approximation one obtainsEk = −2J+JK2.The oscillator strengths of the exciton states near the bottom of the band are given by

Fk =2

N+1

(

N

ån=1

sinKn

)2

=1− (−1)k

N+14

K2. (2.6)

Here, the oscillator strength of a single molecule is set to unity. According to Eq. (2.6), thebottom statek = 1 (with energyE1 =−2.404J) accumulates almost the entire oscillator strength,F1 = 0.81(N+1); it is referred to as the superradiant state. The oscillator strengths of the otherodd states(k = 3,5, ...) are much smaller,Fk = F1/k2, while the even states (k = 2,4, ...) carryno oscillator strength at all,Fk = 0. We note that the small corrections to the sine wave functionsin Eq. (2.3) due to the longe-range dipole-dipole interactions, lead to a small change in thesuperradiant prefactor, which for aggregates of 100 molecules reads 0.84(N + 1) in stead of0.81(N+1) [7].

Thermal motion of the surrounding molecules as well as the molecules of the aggregate itself,result in fluctuations of both the on-site energiesen (due to the fluctuations inUn = å sUns) and thedipole-dipole interactionsJmn. This causes scattering of the excitons from one state|k〉 to otherstates|k′〉. In this chapter, we only deal explicitly with the on-site part of the exciton-vibrationcoupling and neglect the intermolecular part. We start with the Hamiltonian in the form [2]

Hex−vib =N

ån=1

ås

δUsn|n〉〈n| , (2.7)

δUsn =(

¶Usn¶Rsn

)

0· (δRs−δRn) . (2.8)

Here,Usn is the interaction of the excited aggregate moleculen with a surrounding molecules(either a host molecule or another aggregate molecule - the latter being in its ground state; see

2.2 Model Hamiltonian 11

dRn

dRs

Usn

Figure 2.1:Schematic representation of an aggregate embedded in a host medium.Usn is the interaction of the excitedaggregate moleculen with a surrounding molecules. δR̄s andδR̄n describe the displacements of molecules from theirequilibrium positions.

Fig. 2.1). Furthermore,δUsn is the variation of this energy resulting from the displacements,δRsandδRn, of these molecules from their equilibrium positions. The subscript “0” denotes that thederivative must be taken at the equilibrium value ofRsn.

We express the displacement operators in terms of operators of the normal vibration modes,labeledq, in the standard way (h̄ = 1)

δRn = åq

(

12wq

)1/2

Qnqaq +h.c. (2.9)

Here,wq andQnq are, respectively, the eigenfrequencies and eigenvectors of the vibration Hamil-tonian of the entire system (host + aggregates), andaq (a†q) is the annihilation (creation) operatorof this mode. Within the normal mode representation, the Hamiltonian (2.7) takes the form

Hex−vib =N

ån=1

åq

Vnq|n〉〈n|aq +h.c. , (2.10)

Vnq = ås

(

12wq

)1/2( ¶Usn¶Rsn

)

0· (Qsq−Qnq) . (2.11)

We focus on the coupling of excitons to acoustic phonons, in which case theQnq are repre-sented by plane waves:

Qnq =(

1M

)1/2

uqeiq·Rn , (2.12)

where the mode indexq = (q,a ) specifies the wavenumber (q) and polarization (a = 1,2,3) ofthe acoustic phonons,M is the mass of the total system, anduq is the polarization vector of theacoustic modeq. Note that, in contrast to the case of optical phonons,uq does not depend on themolecular position. The dispersion relation in the long-wavelength limit is given bywq = va |q|,v1 = v2 = vt andv3 = vl being the speed of transverse and longitudinal sound, respectively.

12 Intraband relaxation and fluorescence decay in homogeneous aggregates

Substituting Eq. (2.12) into Eq. (2.11), we obtain

Vnq =

(

12Mwq

)1/2

eiq·Rn ås

(

¶Usn¶Rsn

uq

)

0

[

eiq·(Rs−Rn) −1]

. (2.13)

As Usn decreases fast with increasing distance|Rsn|, only the nearest surroundings contribute tothe sum in Eq. (2.13). As a result, we may expand the exponential inside this sum, keeping onlythe first two terms. Doing so and using the dispersion relationwq = va |q|, we get

Vnq = i

( |q|2Mva

)1/2

cnqeiq·Rn , (2.14)

where we introduced

cnq = ås

(

¶Usn¶Rsn

uq

)

0

q · (Rs−Rn)|q| . (2.15)

In a crystalline host medium,cnq does not depend on the position of the molecule in the aggre-gate, while it is a stochastic function of this position in the case of a disordered host medium. Itshould be noted thatcnq depends only on the orientation of the phonon wavevectorq. Moreover,due to the summation over many surrounding molecules, this dependence is expected to be rathersmooth. The∼

√

|q| scaling ofVnq expresses the fact that the coupling of excitons to acousticphonons diminishes in the long wavelength limit.

Within the exciton representation, the Hamiltonian Eq. (2.10) takes the form

Hex−vib =N

åk,k′=1

åq

Vqkk′ |k〉〈k′|aq +h.c. , (2.16)

where the exciton-phonon couplingVqkk′ is given by

Vqkk′ = i

( |q|2Mva

)1/2 2N+1

N

ån=1

cnqeiq·Rn sin(Kn)sin(K′n) . (2.17)

In this chapter, we distinguish two models for theRn-dependence ofcnq. In the first one,we assume no dependence:cnq = cq. This corresponds to the situation of an aggregate placedin a crystalline host. In the second model,cnq is regarded a stochastic function of the molecularposition, having the correlation properties:

〈cnq〉 = 0 , (2.18)

〈cnqcn′q〉 = c 2qδnn′ . (2.19)This may serve as a model to describe exciton-phonon coupling for an aggregate placed inside adisordered host. The Kronecker symbol in Eq. (2.19) implies that the surroundings of differentmolecules in the aggregate are not correlated.

Throughout this chapter, we assume that the exciton-phonon coupling is too weak to renor-malize the exciton band structure and wave functions (no polaron effects) and, thus, can be treatedas a perturbation. This allows us, first, to calculate the rates of the intra-band exciton scattering(k → k′) using first-order perturbation theory, and second, to exploit the Pauli master equationfor describing the kinetics of the intra-band exciton relaxation. As we will see, both models forcnq introduced above, lead to analytical expressions for the exciton scattering rates.

2.3 The Pauli master equation 13

2.3 The Pauli master equation

In order to describe the kinetics of intra-band exciton relaxation, we employ the Pauli masterequation for the populationsPk(t) of the exciton states:

Ṗk = −gkPk +åk′

(Wkk′Pk′ −Wk′kPk) . (2.20)

Here, the dot denotes the time derivative,gk = g0Fk is the spontaneous emission rate of thekthexciton state, which is enhanced relative to the single-molecule emission rateg0 by a factor ofFkgiven by Eq. (2.6), andWkk′ is the rate of phonon-assisted scattering of excitons from statek

′ tostatek. Within first-order perturbation theory, the latter reads

Wkk′ = 2p åq|Vqkk′ |

2[ nq δ(Ek−Ek′ −wq)+(1+nq) δ(Ek−Ek′ +wq) ] , (2.21)

whereVqkk′ is given by Eq. (2.17) andnq = [exp(wq/T)−1]−1 is the thermal occupation of theqth acoustic mode, which has energywq (h̄ = kB = 1). In the next section, we will use Eq. (2.21)to determine the scattering rates for crystalline and glassy hosts. At this moment, we restrictourselves to the general observation that Eq. (2.21) implies these rates to obey the principle ofdetailed balance

Wkk′ = Wk′k exp

(

Ek′ −EkT

)

, (2.22)

which guarantees that eventually the excitons will arrive at the proper equilibrium state, charac-terized the Boltzmann distribution over energy.

The initial conditions to Eq. (2.20) depend on the excitation conditions. In the experimentson J-aggregates reported in Refs. [11–18], the fluorescence was observed after exciting weaklyallowed excitons in the high-energy tail of the J-bands. In our numerical calculations, we willconsider in addition to such blue-tail excitation, also the case of resonant excitation, where onlythe superradiant bottom state (k = 1) is initially excited.

The presence of two types of rates in the Master equation (2.20), namely for spontaneousemission and intra-band relaxation, makes the competition between both types of processes ex-plicit. Throughout this chapter, we will define the limit of slow relaxation through the relationW12(T = 0) ¿ g1, which implies that at zero temperature, the phonon-induced transfer betweenthe two bottom states of the exciton band is small compared to the superradiant emission rate ofthe bottom state.

2.4 Intra-band exciton scattering rates

We now turn to evaluating the scattering ratesWkk′ given by Eq. (2.21). To this end, we firstreplace the summation over the acoustic modes by an integration according to the standard rule:

åq

=3

åa =1

åq→ V

(2p)33

åa =1

∫

dq , (2.23)

whereV is the quantization volume. Next, to simplify the algebra, we will restrict ourselves toan isotropic model for the acoustic phonons, implying equal speed for transverse and longitudinal

14 Intraband relaxation and fluorescence decay in homogeneous aggregates

sound waves:v1 = v2 = v3 = v. Thenwq = v|q|, so that Eq. (2.23) becomes

åq

= åq

3

åa =1

→ V(2pv)3

3

åa =1

∫

dWq∫

dwqw2q , (2.24)

where the first integration is over the orientations ofq. Due to theδ-functions in Eq. (2.21), theintegration overwq can be performed explicitly. This leads to the substitution|q| = |Ek−Ek′ |/vin any function that depends on|q|.

2.4.1 Glassy host

For an aggregate embedded in a glassy host,cnq is a stochastic function with correlation prop-erties given by Eq. (2.18) and Eq. (2.19). This allows us to find an analytical expression for theaverage ofWkk′ . Namely, using the equality

〈∣

∣

∣

∣

∣

N

ån=1

cnqeiq·Rn sin(Kn)sin(K′n)

∣

∣

∣

∣

∣

2〉

= |cq|2N

ån=1

sin2(Kn)sin2(K′n) =14|cq|2(N+1)

in Eq. (2.21) and performing the integrations overwq andWq, we obtain

Wkk′ =Wgl0

N+1

∣

∣

∣

∣

Ek−Ek′J

∣

∣

∣

∣

3

{Q(Ek−Ek′)n(Ek−Ek′)+Q(Ek′ −Ek)[1+n(Ek′ −Ek)]} , (2.25)

with

Wgl0 =J3V

8p2v5M

3

åa =1

∫

dWq|cq|2 . (2.26)

Here,Q(x) = 1 for x> 0 andQ(x) = 0 otherwise. As is seen, in the glassy-host modelWkk′ scalesinversely proportional toN + 1, which is similar to the scaling obtained within the stochasticfluctuation model of exciton-phonon coupling (see, e.g., Ref. [29]). The cubic dependence ofWkk′ in Eq. (2.26) on|Ek −Ek′ |, however, sharply contrasts with the Lorentzian dependenceobtained within the stochastic fluctuation model, and causes the hopping process to slow downwith decreasing energy mismatch, i.e., towards the bottom of the exciton band.

2.4.2 Crystalline host

For an aggregate embedded in a crystalline host, we havecnq = cq, which can be taken out ofthe summation in Eq. (2.17). The remaining geometrical series yields

Vqkk′ =1

N+1

( |q|2Mv

)1/2

cqsinQ[

1− (−1)k+k′eiQ(N+1)]

× sinK sinK′

[cosK′−cos(K +Q)][cosK′−cos(K−Q)] , (2.27)

whereQ = |q|acosq, q being the angle between the phonon wavevectorq and the aggregateaxis.

2.5 Defining the fluorescence decay time 15

Let us now estimate the value ofQ, which can be done using|q| = |Ek−Ek′ |/v (see above).Then, for the exciton states near the bottom of the band (the region of primary interest), whereK,K′ ¿ 1, we haveQ ∼ (aJ/v)|(3/2− lnK)K2− (3/2− lnK′)K′2|. Typical parameter valuesare:a= 10−7 cm,J = 600 cm−1 (2×1013 s−1) andv= 5×105 cm/s. Thus, the factoraJ/v≈ 4,i.e., of the order of unity. This fact and the quadratic scaling ofQ with bothK andK′ allows us toneglectQ as compared toK andK′ in the denominator of Eq. (2.27) as well as to substitute sinQby Q. We also recall thatcq is a smooth function ofq, and thus can be replaced by a constantwhen integrating over the orientations ofq in Eq. (2.21). With these simplifications, the latterintegration can be performed analytically, leading to

Wkk′ =Wcr0

(N+1)2sin2K sin2K′

|Ek−Ek′ |5J5(cosK−cosK′)4 f [X(|Ek−Ek′ |)]

×{Q(Ek−Ek′)n(Ek−Ek′)+Q(Ek′ −Ek)[1+n(Ek′ −Ek)]} , (2.28)

where we introduced

Wcr0 =a2J5V3pv7M

3

åa =1

|cq|2 , (2.29)

X(|Ek−Ek′ |) =|Ek−Ek′ |a

v(N+1) , (2.30)

f (X) = 1− (−1)k+k′ 3X

[(

1− 2X2

)

sinX +2X

cosX

]

. (2.31)

We first note that thesin-functions in Eq. (2.28) reflect a strong suppression of scatteringfor the exciton states near the bottom of the band as compared to those in the center of theband. Second, the scattering between energetically close exciton states is less probable thanbetween well separated ones (independent of their location within the band), because the factor|Ek −Ek′ |5/(cosK − cosK′)4 is roughly proportional to|Ek −Ek′ |. Finally, it turns out that inpractice the energy dependent factorf (X) is always of the order of unity and does not yield afurther suppression of the scattering rate. To see this, we first estimateX, which is obviouslysmallest for the two bottom states:k = 1 andk′ = 2. Using Eq. (2.5), one thus obtains a minimalvalue ofX given byXmin ≈ (9p2/2N)(aJ/v). We have estimated before thataJ/v≈ 4; moreover,in practice aggregate (coherence) lengths are limited to a few hundred molecules or less. We thusfind thatXmin ≥ 1, which from Eq. (2.31) is seen to give values forf that are indeed of the orderof unity.

2.5 Defining the fluorescence decay time

To characterize the fluorescence decay that follows a short-pulse excitation att = 0, it is mostconvenient to have a single decay time. The definition of such a time is straightforward onlyfor mono-exponential decay, which, as we will see in Sec. 2.6 does generally not take place. Inaddition, we will see that, even if a decay time seems straightforward to define, it may not alwaysrelate to a time scale of radiative emission. In this section, we address the two most obviousdefinitions of a fluorescence decay time and, by applying these definitions to the analyticallysolvable example of an exciton “band” consisting of two states only, we explain the nature of theproblems that may arise and how they are affected by the experimental conditions.

16 Intraband relaxation and fluorescence decay in homogeneous aggregates

The quantity observed in a fluorescence experiment is the radiative intensityI(t), which isthe number of emitted photons per unit time. Obviously, this equals the rate of loss of totalexciton population:I(t) = −Ṗ(t), with P(t) = å k Pk(t). As in multi-level systemsI(t) generallydoes not show a mono-exponential decay, it is mostly impossible to obtain a lifetime from asimple exponential fit. The simplest solution is to define a decay time,t e, as the time it takes theintensity to decay to 1/eof its peak valueI(tpeak):

I(tpeak+ t e) = I(tpeak)/e . (2.32)

We note that for blue-tail excitation generallytpeak 6= 0. Alternatively, and maybe mathematicallysomewhat better-founded, one may define a lifetime,t , as the expectation value of the photonemission time, given by

t =∫ ¥

0dt I(t) t =

∫ ¥

0dt P(t) . (2.33)

Clearly, for mono-exponential decay,P(t) = exp(−t/t ), Eq. (2.33) gives the appropriate decaytime t . However, also for nonexponential fluorescence kinetics, the thus defined decay timeseems to make sense. This indeed turns out to be correct, unless the total population kineticsconsists of a large-weight component that rapidly decays and a much smaller-weight very slowcomponent, comparable in integrated area to the fast component. Then, the tail contribution maymask the decay time of the fast component, which for all practical purposes should be consideredthe proper decay time. It appears that such a peculiar situation may easily occur in the case of J-aggregate fluorescence, in particular in the limit where the intra-band relaxation is slow comparedto the spontaneous emission from the superradiant bottom state.

To demonstrate this, we consider a model of two exciton levels, labeledk = 1 andk = 2.Level 1 is lowest in energy and has a radiative emission rateg1, while level 2 is dark. We willassume that level 1 is initially excited. The Pauli master equation (2.20) now reduces to

Ṗ1 = −(g1 +W21)P1 +W12P2 , (2.34)

Ṗ2 = −W12P2 +W21P1 , (2.35)with initial conditionsP1(0) = 1, P2(0) = 0. Solving Eq. (2.5) through Laplace transformation,yields for the total population,P(t) = P1(t)+P2(t), the result

P(t) =g1

l 1− l 2

[(

1+W12l 2

)

el 2t −(

1+W12l 1

)

el 1t]

, (2.36)

where

l 1,2 = −12

(g1 +W12+W21)±12

√

(g1 +W12+W21)2−4g1W12 . (2.37)

Substituting Eq. (2.36) into Eq. (2.33), we arrive at the exact result

t =1g1

(

1+W21W12

)

. (2.38)

From this it follows that at high temperatures (T À E2 −E1 and thusW21 = W12 ≡ W), onewill get t = 2g−11 , independently of the relation betweenW andg1. In the fast-relaxation limit

2.5 Defining the fluorescence decay time 17

(defined asW12(T = 0)À g1, implyingW À g1 as well), this result is not surprising, because thepopulation is then distributed uniformly over the two levels before the emission occurs. As onlythe lower level is radiating, this naturally leads to the effective division of the decay rate of level1 by a factor of two. However, in the limit of slow relaxation at the temperature considered (i.e.,not onlyW12(T = 0) ¿ g1, but alsoW ¿ g1), the above result seems physically counterintuitive,because in this limit only a small part of the population can be transferred to the upper (dark)level before the lower level radiates, so that the upper level remains almost unpopulated. Onethus expects to find a fluorescence decay timet = g−11 , which obviously contradicts the exactresult.

In order to discover the nature of the above contradiction, let us analyze in more detail thehigh-temperature kinetics of the total population in the slow-relaxation limit (W ¿ g1):

P(t) =

(

1− Wg1

)

e−g1t +Wg1

e−Wt . (2.39)

This kinetics contains a fast exponential (first term) and a much slower one (second term). Thefast component has the dominant weight, 1−W/g1 ≈ 1ÀW/g1; this reflects the fact that state1 carries nearly all population, which decays rapidly with the decay timeg−11 . The much slowersecond term of the kinetics, describes the decay of that (small) part of the total population that istransferred to level 2. Substituting Eq. (2.39) into Eq. (2.33) yieldst = 2g−11 , i.e., twice as largeas expected from the physical arguments. This originates from the long-time tail in the secondterm, which gives, despite its small weight, a contribution tot that is exactly equal to the onefrom the fast component.

In the opposite limit of fast relaxation,W À g1, we have

P(t) =g1

4We−2Wt +

(

1− g14W

)

e−g12 t . (2.40)

The first term, having a small weight, describes the fast (on the scale of 1/g1) equilibration of thepopulation over both levels. After that, the total population decays with the rateg1/2. We nowarrive att = 2g−11 , which meets our physical expectation. Thus, in the fast-relaxation limit thedefinition Eq. (2.33) as fluorescence decay time seems to work properly.

To end this section, we re-consider the slow- and fast-relaxation limits, but now usingt e forthe decay time. In the slow-relaxation limit,W ¿ g1, the intensity analog of Eq. (2.39) reads

I(t) = g1e−g1t +W2

g1e−Wt . (2.41)

Obviously, the second (slower) exponential has a negligible weight compared to the first (fast)one and we arrive att e = g−11 , which is the physically expected value and does not suffer fromthe long-time tail.

In the fast-relaxation limit the intensity analog to Eq. (2.40) is given by

I(t) =g12

(

e−2Wt +e−g12 t

)

. (2.42)

Here, the fast and slow components have equal weights, so that the intensity will decay rapidly(within t ≈W−1) to half of its initial value,I(0) = g1. This reflects the already encountered factthat due to the fast transfer of population to level 2, the effective radiative constant is reducedfrom g1 to g1/2. It is straightforward to generalize this to the situation wherel non-radiating

18 Intraband relaxation and fluorescence decay in homogeneous aggregates

levels are rapidly populated due to intra-band relaxation from the superradiant level. At timezero, only the lowest state is populated, and the intensity is given by its decay rateg1. However,the population of that state is very rapidly (within a time∼ 1/W ¿ 1/g1) redistributed over alllstates, which will cause the effective rate to drop by a factorl +1 and thus also give an intensitydrop fromg1 to g1/(l +1) over a time scale 1/W. This results in a value fort e in the order of theinverse relaxation rate (1/W), which has nothing to do with the actual radiative emission timescale in the system.

In conclusion, both most obvious definitions of fluorescence decay time,t andt e, may leadto counter-intuitive results when trying to interpret them as exciton radiative lifetimes. Thisis unavoidable, due to the role of intraband relaxation, and simply means that all such measuresshould be considered with care and in relation to the experimental conditions. In the next section,we will see the above peculiarities show up for actual aggregates as well.

2.6 Numerical results

We now turn to our numerical study of the temperature dependence of the exciton fluorescencedecay time in 1D molecular aggregates, described by the model presented in Sec. 2.2. In allcalculations, we have considered an aggregate of 100 molecules, which is a typical exciton co-herence length for cyanine aggregates at low temperature. The monomer radiative rate was setto g0 = 2× 10−5J, which yields for the radiative rate of the superradiantk = 1 exciton stateg1 = 0.84(N +1)g0 = 1.68×10−3J. In order to obtain actual time, frequency, and temperaturescales, we have used in all our graphs the valueJ = 600 cm−1 (1.8×1013 s−1), which is appro-priate for PIC J-aggregates. This translates tog0 = 3.6×108 s−1 andg1 = 3.0×1010 s−1, whichare indeed typical of monomer and aggregate radiative decay rates. Finally, it is useful to notethat for an aggregate of 100 molecules, the separation between the two lowest exciton states fromEq. (2.5) is found to beD≡ E2−E1 ≈ 0.01J.

To solve the Pauli master equation Eq. (2.20), we use the numerical procedure proposed inRef. [24], based on passing from the equation’s differential form to its integral version

Pk(t) = Pk(0)e−Wkt +å

k′Wkk′

∫ t

0dt′e−Wk(t−t

′)Pk′(t′) . (2.43)

Here,Wk = gk + å k′ Wk′k is the total rate of population loss from statek, both due to scatteringand due to spontaneous emission. The equivalence of Eq. (2.43) to Eq. (2.20) is proved bystraightforward differentiation of the former. Next, using Eq. (2.43), one relatesPk(t + dt) toPk(t) through

Pk(t +δt) = Pk(t)e−Wkδt +1

Wk

(

1−e−Wkδt)

åk′

Wkk′Pk′(t) . (2.44)

Finally, to avoid divergencies at smallWk in Eq. (2.44), we expand

1Wk

(

1−e−Wkδt)

= δt (1−Wkδt) . (2.45)

The iteration procedure defined by Eq. (2.44) in combination with Eq. (2.45) turned out to pro-vide a stable numerical algorithm to solve the master equation.

We will present results for four types of situations. First, we will consider a glassy host,where we distinguish between initial excitation of the superradiant bottom (k = 1) state and blue-tail initial excitation. Next, we reconsider both cases for a crystalline host.

2.6 Numerical results 19

0

1

2

3

4

5

6

0 40 80 120 160

T (K)

(b)

0

1

2

3

4

5

6 (a)

γ1τ

γ1τ

Figure 2.2:Temperature dependence of the fluorescence decay timet measured in units of 1/g1 after bottom excitationof an aggregate of lengthN = 100 withJ = 600 cm−1. The glassy host exciton scattering model was used with scatteringstrengthWgl0 /J = 10 (solid), 10

2 (dashed), and 104 (dotted). The thicker dots indicate the data points generated inour numerical simulations, while the curves provide a smooth guide to the eye. Data in (a) were calculated using thedefinition Eq. (2.33), while in (b), the long-time tail of the total population kinetics was neglected by truncating theintegral in Eq. (2.33) attmax for whichP(tmax) = 0.1.

2.6.1 Glassy host

First of all, let us estimate the value of the parameterWgl0 that distinguishes between the limits offast and slow relaxation. To this end, we equateW12(T = 0) to g1. SubstitutingE2−E1 ≈ 0.01Jinto Eq. (2.25), one obtainsWgl0c ≈ 105J. ForW

gl0 > W

gl0c (W

gl0 < W

gl0c), we are in the fast (slow)

relaxation limit. As argued in the Introduction, we will mostly be interested in the slow limit.

A. Bottom excitation

Figure 2.2 shows the T dependence of the fluorescence decay timet for three different valuesof the exciton scattering strength (Wgl0 = 10J,10

2J, and 104J), calculated after direct initial ex-citation of the superradiant statek = 1. The data presented in Fig. 2.2(a) were obtained usingthe definition Eq. (2.33), where the time-integration was carried out till the total population haddecayed to the valueP(tmax) = 0.005 (which for all practical purposes agrees with integrating tillt = ¥ ), while in Fig. 2.2(b) we used a relaxed definition, withP(tmax) = 0.1, thus ignoring anylong-time tails (cf. Sec. 2.5). For further discussion of this figure, it is convenient to also plot the

20 Intraband relaxation and fluorescence decay in homogeneous aggregates

time dependence of the total populationP(t) and the partial populationsPk(t) of the lowest fourexciton states (k = 1, . . . ,4), which is done in Figs. 2.3 and 2.4 for three different temperatures,in the caseWgl0 = 10J.

Analyzing Fig. 2.2, we first note that all curves yield aT = 0 decay time that equals thesuperradiant lifetime,g−11 , of statek = 1 (the small deviation fromg

−11 in Fig. 2.2(b) is due to the

truncation of the integral in Eq. (2.33)). This is the natural result, because at zero temperature,the exciton created in the lowest (superradiant) state can not be scattered to the higher (weaklyradiating) states, due to the absence of phonons. Being stuck in the bottom state, the excitonemits a photon on the average after a timeg−11 has passed, meaning that the fluorescence decaytime actually reflects the emission process itself. The fact that at low temperatures, hardly anyexcitation is transferred to the higher exciton states before emission takes place, is clearly visiblein Figs. 2.3(a) and 2.4(a).

Next, increasing the temperature fromT = 0, we observe a well-pronounced plateau in thevalue of t (if Wgl0 ¿ W

gl0c = 10

5J). The plateau is seen to widen with decreasingWgl0 . Thephysics of this phenomenon is clear:t will only increase if the presence of the higher lyingdark exciton states becomes noticeable, i.e., if those states actually become populated throughexciton-phonon scattering before the energy is spontaneously emitted from the bottom state.Estimating such effects to become visible if 10% of the excitons undergo such scattering beforeemission, we may calculate the extent,Tpl , of the plateau by requiring thatå kWk1(Tpl) = 0.1g1.The left-hand-side in this criterion is the total scattering rate out of the bottom statek = 1 atT = Tpl .

To obtain an analytical estimate ofTpl , we first replace the summation overk by an integrationaccording to the standard ruleå k → [(N + 1)/p]

∫ ¥0 dK. This step implies that many exciton

levels fall within the energy interval[0,Tpl ]. In view of the nearly quadratic dispersion near thebottom of the exciton band, the values ofK that mainly contribute to the integral, are of theorder ofKpl = (Tpl/J)1/2. As by assumptionKpl À K1 ≡ p/(N+1), we thus also typically haveK À K1, which allows us to approximate(Ek−E1)/|J| ≈ (3/2− lnK)K2. Finally, we substitutelnK by lnKpl , because this function is changing slowly in the interval close toKpl , where thedominant contribution to the integral comes from. Now, the integral may be performed, to yield

åk

Wk,1(Tpl) =15Wgl016√

pK7pl

(3/2− lnKpl)1/2. (2.46)

Thus, we arrive at a transcendental equation forKpl = (Tpl/J)1/2:

K7pl(3/2− lnKpl)1/2

=8√

p75

g1Wgl0

. (2.47)

If we apply this estimate toWgl0 = 102J andWgl0 = 10J, we find Tpl = 28 K andTpl = 52 K,

respectively. These values are in a good agreement with the data presented in Fig. 2.2(b) andapproximately twice as large as those presented in Fig. 2.2(a). We conclude that the long tailof the total population kinetics, which exists at elevated temperatures due to population of thehigher exciton states (cf. Figs. 2.3(b) and 2.4(b)) decreases the extent of the plateau by a factorof two. This is due to the overestimating effect which such tails have on the radiative lifetimedefined through Eq. (2.33) (see discussion in Sec. 2.5).

We note that the estimate Eq. (2.47) is not applicable to the caseWgl0 = 104J, as it yields

Tpl ≈ 7.5 K, which is smaller then the energy differenceDbetween the two lowest exciton states.

2.6 Numerical results 21

0

0.2

0.4

0.6

0.8

1

0 200 400 600

Pop

ulat

ion

Time (ps)

(c)

0

0.2

0.4

0.6

0.8

1

Pop

ulat

ion

(b)

0

0.2

0.4

0.6

0.8

1

Pop

ulat

ion

(a)

Figure 2.3:Kinetics of the total population (P(t), solid line) and the population of the bottom exciton state (P1(t), dot-ted line) following bottom excitation of the same aggregate as in Fig. 2.2 withWgl0 = 10J at three different temperatures,T = 17 K (a),T = 42 K (b), andT = 84 K (c). Clearly seen is the occurrence of a long-time tail inP(t) at T = 42 K,caused by transferring a small amount of the population from the statek = 1 to the statek = 2, and its disappearanceagain atT = 84K.

Thus, replacing thek summation by an integration is not allowed for this scattering strength. Thesmall plateau that can still be observed forWgl0 = 10

4J, originates from the discreteness of theexciton levels.

For temperatures beyondTpl , all curves in Fig. 2.2 show an increase oft , due to the popu-lation of dark states. As is observed, at higher temperatures all curves approach one asymptoticcurve that has an approximateT1/2 behavior. The latter reflects two facts: (i) - the excitons reachthermal equilibrium before the emission occurs (see Figs. 2.3(c) and 2.4(c), where the lowest fewexciton states are seen to quickly acquire equal populations) and (ii) - the number of states thatbecome populated is approximately proportional toT1/2 due to the approximate(E−E1)−1/2behavior of the density of exciton states near the bottom of the band. A small deviation from theT1/2-dependence is expected, because the exact spectrum, Eq. (2.5), differs logarithmically fromtheK2-dependence characteristic for the nearest-neighbor approximation.

We finally note that at higher temperatures and (or) higher scattering strength, the curves inFigs. 2.2(a) and 2.2(b) approach each other. This is due to the fact that a large part of the popu-lation is then transferred from the initially excited bottom state to the dark states before emissionoccurs. As a result, there is no special long-time tail in the kinetics of the total population any-

22 Intraband relaxation and fluorescence decay in homogeneous aggregates

0

0.2

0.4

0.6

0 500 1000 1500

Pop

ulat

ion

(x10

-1)

Time (ps)

(c)

0

0.4

0.8

1.2

Pop

ulat

ion

(x10

-2)

(b)0

0.2

0.4

0.6

Pop

ulat

ion

(x10

-3)

(a)

Figure 2.4:As Fig. 2.3, but now are shown the populations of the exciton statesk = 2 (solid),k = 3 (dashed), andk = 4 (dotted).

more. To illustrate this, consider Figs. 2.3(b) and (c). In the former, a separate long-time tailis still seen to follow the fast initial decay, while in the latter the overall kinetics time scale hasbecome longer.

To complete the discussion for bottom excitation, we briefly consider the alternative defini-tion t e for the fluorescence decay time. The corresponding results are plotted in the left panelof Fig. 2.5. As in Fig. 2.2, we observe a plateau, whose extent correlates well with that inFig. 2.2(b). This confirms our observation below Eq. (2.41) that the measuret e does not sufferfrom the slow-tail problem. However, beyond the plateau, the curves in the left panel of Fig. 2.5differ drastically from those in Fig. 2.2 and are counterintuitive, as they go down instead of up.This is due to a rapid decrease of the emission intensity, not because of radiation from the bottomstate, but because population is rapidly transferred from this state to the higher lying dark states(see discussion below Eq. (2.42)). This is illustrated clearly by the right panel of Fig. 2.5, whichshowsI(t) for four temperatures atWgl0 = 10

4J. With increasing temperature, the initial decayof the intensity becomes faster, while its tail (reached after equilibration) slows down. The latteris the intuitive effect on the radiative decay time, but this is not captured by the decay timet e.

2.6 Numerical results 23

0

0.2

0.4

0.6

0.8

1

0 40 80 120 160

T (K)

γ1τe

0.001

0.01

0.1

1

0 50 100 150

Inte

nsity

(a.

u.)

Time (ps)

Figure 2.5: The left panel represents the temperature dependence of the fluorescence decay timet e defined inEq. (2.32), calculated for the same system and conditions as in Fig. 2.2. Curves labeled as in Fig. 2.2. The rightpanel shows the fluorescence intensity as a function of time for the same system as in Fig. 2.2, using bottom excitationand a scattering strengthWgl0 = 10

4J. Curves correspond toT = 0 K (solid), 8 K (dashed), 17 K (dotted), and 84 K(dash-dotted).

B. Blue-tail excitation

We now turn to the case where the initial excitation takes place above the bottom of the excitonband. Then spontaneous emission can only occur after the exciton has relaxed to the bottomstate. Obviously, in the slow-relaxation limit this may cause a bottle-neck for the emission pro-cess. This reflects itself in a particular temperature dependence of the fluorescence decay time.In Fig. 2.6 we depictt (T), calculated using Eq. (2.33), for initial excitation of theki = 7 excitonstate forWgl0 = 10

4J. Clearly, the observed behavior differs drastically from the one found forbottom state excitation, in particular at low temperatures. First of all, atT = 0, t deviates fromg−11 , which for bottom excitation was always found to be the zero-temperature decay rate. Fur-thermore, the curve shows a region wheret decreases upon increasingT. The noted peculiaritiesoriginate from the interplay between intra-band relaxation and spontaneous emission. We willclarify this further by estimating the scattering ratesWk7 that feature in this interplay forT = 0.

Due to the cubic dependence ofWkki on |Ek−Eki | (Eq. (2.25)), the transitions from the initialexciton stateki to the ones with quantum numbersk = 1 and 2 are the most probable; moreover,the corresponding rates are almost equal to each other. This is becauseE7 −E1 andE7 −E2both are of the order of 0.1J, while they differ from each other byD≈ 0.01J. For estimates,we will useE7 = −2.3J, which for N = 100 leads to theW17 ≈ W27 ≈ 10−5Wgl0 . Thus, for ourexample case ofWgl0 = 10

4J, we arrive atW17 ≈ 0.1J ≈ 102g1. This means that the populationfrom the initially excited state is transferred rapidly (on the scale ofg−11 ) to the statesk = 1 andk = 2. Thek = 1 state decays radiatively with the rateg1, while thek = 2 state only decaysvia relaxation to thek = 1 state. The rate of the latter process is given byW12 ≈ 0.1g1. Thisrelationship between the dominant rates leads to a bi-exponential kinetics of the total population:a fast decay with a time constant∼ g−11 , followed by a slow decay with time constant∼ W−112 .This analysis is nicely confirmed by Fig. 2.7, in which the kinetics of the total populationP(t)as well as the partial populationsP1(t) andP2(t) states are shown. Both exponentials are seento have comparable weight, as was to be expected, because the relaxation rates from the excited

24 Intraband relaxation and fluorescence decay in homogeneous aggregates

2

4

6

8

0 40 80 120 160

T (K)

γ1τ

Figure 2.6:Temperature dependence of the fluorescence decay timet (Eq. (2.33)) for an aggregate of lengthN = 100with J = 600 cm−1 after initial excitation of thek = 7 state. The glassy-host exciton scattering model was used withscattering strengthWgl0 = 10

4J. The dots indicate the data points generated in our numerical simulations, while the curveprovides a smooth guide to the eye.

stateki = 7 into the statesk = 1 andk = 2 are roughly equal. As a consequence, the integratedcontribution of the slow component dominates over the one deriving from the fast component,which explains why the value oft (T = 0) in Fig. 2.6 is larger thang−11 . The same argumentsexplain why at low temperaturet goes down upon increasingT: this is simply due to the fact thatW12 increases with increasing the temperature, so that the contribution of the slower componentto t diminishes. If temperature increases further,W12 becomes larger thang1 and we approachthe limit of fast (on the time scale of emission) equilibration. Thent (T) should not stronglydepend on the initial condition anymore, which is why Fig. 2.6 at higher temperatures is verysimilar to Fig. 2.2.

At smallWgl0 , the decay scenario differs from the one described above. Let us, for instance,

takeWgl0 = 10J, so thatW17 ≈ 10−4J ≈ 0.1g1. Now, the intra-band relaxation is so slow thatit completely governs the total population decay. Again, a bi-exponential behavior is found inthe population kinetics, with the fast component now being limited by the rateW17, while theslow one is again characterized by the (now very slow) rateW12. This limit of extremely slowrelaxation is illustrated in the right panel of Fig. 2.7.

Finally, we address theT-dependence of the decay time defined byt e (Eq. (2.32)). Theresults are presented in the left panel of Fig. 2.8 for three scattering strengths and, for furtherclarification, the corresponding kinetics atT = 0 is shown in the right panel of Fig. 2.8. At shorttimes, before the excitation is scattered to the band bottom, the intensity is very small, as theemission rateg7 is small (Eq. (2.6)). The intensity then grows due to population of the bottomstate(s) through intra-band relaxation and, after reaching a maximum, it finally decays again witha rate that may be interpreted as the relaxed exciton’s radiative rate. This results in a temperaturedependence oft e that is again characterized by a low-temperature plateau. This plateau onlyoccurs att e = g−11 if the relaxation is sufficiently fast to bring all population down to the bottomstate before emission from it starts. For the small scattering strengthWgl0 = 10J, this is seennot to be the case. In contrast to the case of bottom excitation (Fig. 2.5), the plateau is generallyfollowed by an increasing decay time. The reason is that now the populations of all exciton levels

2.6 Numerical results 25

0

0.2

0.4

0.6

0.8

1

0 80 160 240 300

Pop

ulat

ion

Time (ps)

0

0.2

0.4

0.6

0.8

1

0 400 800 1200 1600

Pop

ulat

ion

Time (ps)

Figure 2.7:The left panel represents the zero-temperature kinetics of the total populationP(t) (solid line) and thepopulationsP1(t) (dashed) andP2(t) (dotted) of the two lowest exciton states for the same system and conditions asconsidered in Fig. 2.6. The figure clearly shows a bi-exponential behavior ofP(t). The right panel shows the samekinettics as the left panel, but now forWgl0 = 10J.

have indeed equilibrated by the time the spontaneous decay from the bottom state starts. This isnot the case forWgl0 = 10J, which explains the more complicatedT dependence observed in theleft panel of Fig. 2.8.

2.6.2 Crystalline host

Similar to the case of the glassy-host model of exciton-phonon scattering, we have carried outa series of numerical calculations for the crystalline-host model. In many respects, the essentialphysics within both models is the same and we will therefore discuss in detail only the newfeatures, dealing with the already discussed phenomena only in passing.

As before, we first estimate the value ofWcr0 that distinguishes between the limits of fast andslow relaxation. Recall that this implies equatingW12(T = 0) to g1. Using in Eq. (2.28) withk = 1, k′ = 2 the approximations sin[p/(N + 1)] ≈ p/(N + 1), sin[2p/(N + 1)] ≈ 2p/(N + 1),E2 −E1 ≈ 0.01|J|, and f (X) = 1, we arrive atWcr0c ≈ 105J, which happens to be equal to thecritical scattering strength for the glassy-host model.

A. Bottom excitation

In Fig. 2.9 we depictt (T), calculated after direct initial excitation of the bottom exciton state,for three different scattering strengths:Wcr0 = 10

2J,103J, and 104J. Figure 2.9(a) shows theresults using the definition Eq. (2.33) by integrating up to the timetmax where the total excitonpopulation has decayed toP(tmax) = 0.005 (i.e., uptot = ¥ for all practical purposes), while inFig. 2.9(b) the slow-tail contribution tot is discarded by integrating only till the time at whichP(tmax) = 0.1. As in the case of the glassy host (Fig. 2.2), all curves start from the same point(corresponding to the radiative life timeg−11 of the superradiant statek = 1), then show a plateau,and finally go up with increasing temperature.

As in Sec. 2.6.1, we estimate the extentTpl of the plateau by equatingå kWk1(Tpl) to 0.1g1.Substituting(Ek−E1)/(cosK −cosK1) ≈ 2J(3/2− lnK) (typically K À K1 ≡ p/(N +1)) and

26 Intraband relaxation and fluorescence decay in homogeneous aggregates

2

4

6

8

0 40 80 120 160

T (K)

γ1τe

1

10

100

0 60 120 180

Inte

nsity

(a.

u.)

Time (ps)

Figure 2.8:The left panel represents the temperature dependence of the fluorescence decay time as in Fig. 2.6, exceptthat nowt e (Eq. (2.32)) has been used as measure of the fluorescence decay time. Curves correspond to the scatteringstrengthWgl0 /J = 10 (solid), 10

2 (dashed), and 104 (dotted). The right panel shows the fluorescence intensity as a functionof time for an aggregate of lengthN = 100 withJ = 600 cm−1 after initial excitation of thek = 7 state. The glassy-hostexciton scattering model was used with scattering strengthWgl0 /J = 10 (solid), 10

2 (dashed), and 104 (dotted).

performing the summation overk in a manner similar to the glassy-host model, one obtains

K5pl(3/2− lnKpl)5/2 =(N+1)3

60p3/2g1

Wcr0, (2.48)

whereKpl = (Tpl/J)1/2. Solving this equation forWcr0 = 102J,103J, and 104J, we find as es-

timatesTpl = 100 K, 34 K, and 11 K, respectively, in good agreement with the numerical datain Fig. 2.9(b). We note thatTpl is larger than obtained within the glassy-host model at the samescattering strengthW0. This results from the fact that within the crystalline-host model,Wkk′increases more slowly with|Ek−Ek′ | than within the glassy-host model.

B. Blue-tail excitation

Without showing figures, we mention that the temperature dependence oft for blue-side exci-tation exhibits the same general behavior as in the case of a glassy host. In particular, we againfind that (i) the zero-temperature value oft (calculated through Eq. (2.33)) deviates fromg−11and (ii) a region exists wheret decreases with increasingT. As in Sec. 2.6.1, the physical in-terpretation of these peculiarities is based on the interplay of intra-band relaxation and emissionprocesses. However, the scenario of the intra-band relaxation in the present model differs consid-erably from that in the glassy-host model. While in the latter we found that relaxation from theexcitation window directly to the bottom of the band has the highest probability, in the presentcase, the relaxation occurs preferably through several intermediate states, which may, in fact, actas a bottle-neck. We will show this below.

Let us analyze the zero-temperature relaxation rate from an initially excited exciton stateki to a low-lying statek. For the sake of simplicity, we will use several approximations: (i) thenearest-neighbor approximation for the exciton energy spectrum,Ek =−2J+JK2, (ii) f (X) = 1,

2.6 Numerical results 27

0

1

2

3

4

5

6

0 40 80 120 160

T (K)

(b)

0

1

2

3

4

5

6

(a)

γ1τ

γ1τ

Figure 2.9:As Fig. 2.2, but now using the crystalline-host model for exciton scattering. Curves correspond to scatter-ing strengthsWcr0 /J = 10

2 (solid), 103 (dashed), and 104 (dotted).

and (iii) ki ,k¿ N. Then, the scattering rate Eq. (2.28) takes the form

Wkki =16Wcr0 K

61

(N+1)2k2i k

2(k2i −k2) , (2.49)

which reaches its maximum atk2 = k2i /2: Wki/√

2,ki= [W0/4(N+1)2]K61k

6i . This scattering rate

is to be compared with the one to the superradiant state withk = 1. Neglecting 1 as compared tok2i in the last factor of Eq. (2.49) fork = 1, one obtains

W1ki =16Wcr0 K

61

(N+1)2k4i =

4

k2iWki/

√2,ki

, (2.50)

i.e.,W1ki ¿Wki/√

2,ki. From this we conclude that jumping directly to the bottom state is unlikely,

so that after excitation in a high-lying state, the exciton has to make several relaxation steps toreach the superradiantk = 1 state. In the limit of fast relaxation,W12 > g1, this will certainlyoccur and it will do so before the exciton decays through spontaneous emission. By contrast, inthe slow-relaxation limit,W12 < g1, the step-wise relaxation may be stopped at some intermediateband statek′ and the exciton may never reach the bottom state. The reason is that each step down-hill decreases the relaxation rate by a factor of eight (cf. Eq. (2.49)), so that at some point the rate

28 Intraband relaxation and fluorescence decay in homogeneous aggregates

0

0.2

0.4

0.6

0.8

1

50 150 250

Pop

ulat

ion

Time (ps)

Figure 2.10:Zero-temperature kinetics of the total populationP(t) (solid line) and the populationsP1(t) (dashed),P2(t) (dotted), andP3(t) (dash-dotted) of the three lowest exciton states for an aggregate ofN = 100 molecules withJ = 600 cm−1 after initial excitation of thek = 7 state. The crystalline-host exciton scattering model was used withscattering strengthWcr0 = 10

4J.

for making the next step,å kWkk′ , may become comparable to or smaller than the spontaneousemission rate from the corresponding state,gk′ . Then,gk′ will determine the exciton fluorescencedecay time.

Figure 2.10 nicely illustrates the above described scenario of relaxation. Here, we plottedthe numerically calculated zero-temperature kinetics of the total populationP(t) as well as thepartial populationsPk(t) for k = 1,2, and 3 using a scattering strengthWcr0 = 10

4J andki = 7 forthe initially excitated exciton state. Indeed it is seen that at the early stage of the kinetics onlya small part of the initial population is transferred to the bottom statek = 1, while the higherstatesk = 2 and 3 accumulate almost all population. After this early population transfer, thek = 3 state first relaxes to the statesk = 2 andk = 1. This becomes clear from the values ofW13 andW23, which from Eq. (2.28) with the present parameters (Wcr0 = 10

4J andN = 100) arefound to beW13 = 1.24×10−3J andW23 = 0.62×10−3J. At the same time, the radiative decayrate of thek = 3 state equalsg3 = g1/9 = 0.19×10−3J, which is an order of magnitude smallerthan the total radiationless relaxation rate of this state,W13+W23 ≈ 1.86×10−3J (0.034 ps−1in frequency units). Thus, the rateW13+W23 is expected to govern the decay of thek = 3 state,which is in perfect agreement with the numerical data in Fig. 2.10. Finally, thek = 2 state slowlyrelaxes to thek = 1 state (W−112 ≈ 10g−11 ≈ 300 ps), determining the long-tail decay of the totalpopulation. Also this time scale shows excellent agreement with the numerical data.

2.7 Summary and concluding remarks

In this chapter, we numerically studied the temperature dependence of the fluorescence decaytime of 1D Frenkel excitons in J-aggregates resulting from their vibration-assisted redistributionover the exciton band after pulsed excitation of a subset of states. The redistribution was modeledusing a Pauli master equation that accounts for the scattering between exciton levels. We con-sidered two models for the scattering rates in this equation. In one we assumed the host mediumto have a glassy (disordered) character, while in the other it was assumed to be crystalline. It ap-

2.7 Summary and concluding remarks 29

peared that the exciton scattering rates, corresponding to these two models, differ considerablyfrom each other.

We have paid particular attention to the definition of thefluorescencedecay time and its rela-tion to theradiativedecay time. We have considered two definitions. The first (t ) is the expecta-tion time of photon emission, which equals the time-integrated kinetics of the total exciton pop-ulationP(t), while the second (t e) uses the 1/e time of the fluorescence intensityI(t) = −Ṗ(t),as is often done in experiments. We have shown that for an exciton system in the presence ofintraband relaxation, it is sometimes difficult to extract information on the radiative decay timeon the basis of measurements of the fluorescence kinetics. The potential pitfalls that may maskthe real radiative decay time have been elucidated using a simple model that contains only twoexciton states as well as by analyzing numerical results for linear J-aggregates for various initialconditions and exciton-phonon scattering strengths.

We have found that, independently of the intra-band scattering model, the interplay betweenthe intra-band relaxation and the superradiant decay in combination with the type of fluorescenceexcitation (direct excitation of the superradiant bottom state or initial excitation of higher-lyingweakly radiating states), crucially affects the physics of the fluorescence kinetics. For bottomexcitation and at zero temperature, the exciton remains in the superradiant state until the radia-tive relaxation occurs. Upon increasing the temperature, population is transferred to the higher-energy weakly radiating states, giving rise to a slowing down of the radiative decay. This processonly starts when the vibration-assisted scattering rate from the bottom exciton state to the higherones approaches the superradiant emission rate. As a result, the temperature dependence ofthe fluorescence decay time shows a plateau that extends further for slower intraband scatteringstrength. In the plateau region, the superradiant damping rate determines the fluorescence decay.Beyond the plateau, the fluorescence decay time goes up with increasing temperature and even-tually approaches aT1/2-scaling. The latter reflects the fact that the excitons arrive at thermalequilibrium within the time scale of photon emission. Under these conditions, the fluorescencelifetime t reflects the excitonradiative lifetime, while the decay timet e at higher temperaturesmeasures the upward scattering rate from the bottom state, rather than the radiative decay time.

In the case of initial excitation of high-lying exciton states, the physical picture of the excitonemission process strongly depends on the ratio of the rates for intra-band scattering and excitonsuperradiance. If the former is faster than the latter, the excitons created initially in the weaklyradiating states, are rapidly (on the scale of the exciton superradiant emission) transferred tothe superradiant state. The subsequent dynamics of emission is similar to the one found fordirect excitation of the bottom state. By contrast, if the intra-band scattering is slower than thesuperradiant emission, the former represents the bottleneck