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Intraband relaxation and dephasing of Frenkel exciton states in one-dimensional J-aggregates
ARTICLE IN PRESS
Journal of Luminescence 107 (2004) 332–338
Intraband relaxation and dephasing of Frenkel exciton states in one-dimensional J-aggregates J.P. Lemaistre* Laboratoire des Milieux D!esordonn!es et H!et!erog"enes, CNRS-UMR 7603, Universit!e P. et M. Curie, Tour 22, 4, Place Jussieu, Paris F-75252, C!edex 05, France
Abstract A theoretical model is proposed to analyse the relaxation and the dephasing mechanisms of Frenkel excitons in onedimensional J-aggregates. At low temperature, the exciton dynamics originates from the coupling of excitons to phonons which creates temperature dependent local fluctuations in the site energies of the molecules. A microscopic model of exciton–phonon coupling is used assuming a Bose–Einstein distribution for the phonon occupation density. Intraband scattering rates among the exciton states are calculated and used in a master equation to obtain the time evolution of the exciton populations after initial excitation of the optically allowed state. At higher temperatures, the dephasing of the exciton states originates from the coupling of excitons to the vibrational modes. Both processes (intraband scattering and dephasing) contribute to the lengthening of the radiative exciton lifetime with increasing temperature. Using a generalized participation ratio taking into account the homogeneous width allows to define, at each temperature, the number of coherently coupled molecules and to interpret the anomalous observed temperature dependence of the radiative lifetime of excitons in PIC aggregates at low, intermediate and high temperature. r 2003 Elsevier B.V. All rights reserved. Keywords: Exciton; Aggregate; Scattering; Dephasing
1. Introduction Dynamical processes of Frenkel excitons in mesoscopic aggregates of molecules forming nanostructures are currently investigated. Many experimental and theoretical works are devoted to the exciton dynamics in molecular systems of low dimensionality like J [1–7] or H [8–10] aggregates, dye aggregates exhibiting a Davydov splitting [11,12], or cyclic aggregates [13]. In general, these aggregates of finite size, are characterized by relatively strong intermolecular inter*Tel.: +33-01-44-27-42-66; fax: +33-01-44-27-38-82. E-mail address: [email protected] (J.P. Lemaistre).
actions due to the long range dipole–dipole couplings. Thus, the cooperative effects are dominant (even at room temperature) and are generally discussed in terms of collective excitation states. The dynamic disorder stems from the coupling of the chromophores to a thermal bath (phonons) acting as a dissipative medium. This stochastic coupling creates local fluctuations in the site energies characterized by their amplitude ðDÞ and their correlation time ðtc Þ: The exciton dynamics is determined by comparing the coherence time ðt0 ¼ _=V0 Þ; where V0 denotes the electronic interaction between the molecular sites, to the exciton–phonon parameters ðD; tc Þ: When the
0022-2313/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jlumin.2003.12.016
ARTICLE IN PRESS J.P. Lemaistre / Journal of Luminescence 107 (2004) 332–338
exciton–phonon correlation time is shorter than the coherence time ðtc pt0 Þ; the excitation is temporarily localized on sites and moves incoherently from site to site. The exciton motion is described according to the so-called hopping model, with transfer rates given by Fermi’s golden rule formulae. In the opposite case, for which the exciton–phonon correlation time is longer than the coherence time ðtc Xt0 Þ; the local fluctuations in the site energies do not destroy the collective character of the exciton states. Thus, the energy fluctuations of the sites are distributed over all exciton states inducing an intraband scattering [5–8]. In this paper we present a theoretical model based on numerical calculations of the exciton eigenstates and their participation ratios, in order to analyze the photophysical properties of collective states in linear J-aggregates. The intraband scattering is studied by considering the stochastic exciton–phonon couplings. After initial excitation of the optically allowed bottom state of the quasiexciton band, population relaxation occurs according to a jkS2jlS scattering mechanism among the eigenstates. This energy relaxation, induced by the coupling of the exciton states to a thermal bath is described as an incoherent energy transfer. An expression for the scattering rates is provided assuming that the exciton-bath coupling is described by means of two parameters: D; the amplitude of the fluctuations and tc the correlation time associated to the stochastic process. The transfer rates are then used in a master equation to describe the time evolution of the eigenstates populations and that of the optical lineshapes. Such an approach was recently used to analyze the temperature dependence of the exciton radiative lifetime at low temperatures [7]. At higher temperatures, the dephasing of the collective excitation states by molecular vibrations become the dominant mechanism. Increasing the temperature destroys the phase coherence among the sites and decreases the number of coherently coupled molecules. The role and the importance of the homogeneous broadening was pointed out by Leegwater [14]. Using a generalized average participation ratio which takes into account the homogeneous width allows to obtain the tempera-
333
ture dependence of the number of coherently coupled molecules.
2. Model Hamiltonian We model an aggregate by a linear chain of N identical two-level molecules. In the localized site representation the electronic Hamiltonian is written as H0 ¼
X
Ei jiS/ij þ
0 X
Vij jiS/jj:
ð1Þ
i;j
i
The Ei are the molecular site excitation energies and the Vij describe the electronic interactions among the sites. We will consider a linear chain of parallel molecules featuring J-aggregates ðVij o0Þ like PIC [1–4,6,7] or BIC [15] systems. They are assumed to be coupled through dipolar interactions and are represented in the simplest approach of the point dipoles as Vij ¼ V0 =ji jj3 ;
ð2Þ
where V0 ðV0 o0Þ denotes the nearest neighbor interaction and ji jj is the distance between molecules i and j with a unity lattice spacing. The molecular aggregates that we consider being characterized by strong intermolecular interactions, we use the delocalized representation of the Hamiltonian X H0 ¼ Ek jkS/kj; ð3Þ k
in which jkS denotes the N eigenstates with energies Ek and eigenvectors X jkS ¼ Cik jiS: ð4Þ i
The transition moments of the delocalized eigenstates are X mk ¼ Cik mi ; ð5Þ i
giving for the oscillator strengths Fk pEk jmk j2 :
ð6Þ
ARTICLE IN PRESS J.P. Lemaistre / Journal of Luminescence 107 (2004) 332–338
334
The localized/delocalized behaviour is analyzed by using the inverse participation ratio defined as X jCik j4 : ð7Þ Lk ¼ i
Lk equals unity for an exciton localized on a single molecule and goes to zero as 1=N for an exciton evenly delocalized over N sites.
energy fluctuations and l ¼ 1=tc is the inverse of the correlation time associated to the stochastic process. The rates of the intraband scattering are given by the Fourier transform of the autocorrelation function of the stochastic perturbation. Thus, the transition probability among the exciton states is given by Z þN Ukl0 ¼ ð1=_2 ÞRe Gkl ðtÞeiokl t dt; ð10Þ 0
3. Intraband relaxation and homogeneous broadening
where
Excitation transfer in molecular aggregates is induced by the dynamical coupling of the molecular sites to a thermal bath. This stochastic coupling induces local fluctuations in the site energies characterized by their amplitude and their correlation time. In J-aggregates, the relatively strong electronic coupling creates delocalized excitons forming a quasi-exciton band. Thus, the stochastic coupling of excitons to phonons induces the diffusion among the delocalized jkS states. This mechanism is an incoherent energy transfer process among the chain eigenmodes and may be described in the framework of the Kubo–Anderson’s theory of the stochastic resonance [16,17]. The transfer rates of the intraband scattering will depend on the exciton–phonon coupling, the correlation time associated to the stochastic fluctuation, the temperature and also on the energy separation of the eigenmodes. We add to the excitonic Hamiltonian H0 of Eq. (1) a time dependent Hamiltonian, H1 ðtÞ; which describes the stochastic coupling X ei ðtÞjiS/ij; ð8Þ H1 ðtÞ ¼
Ukl0 is an energy transition rate taking into account all exciton–phonon couplings at frequencies okl ¼ ðEk El Þ=_: On the basis set of the delocalized states jkS; eigenstates of H0 ; the matrix elements of the stochastic perturbation are X ekl ðtÞ ¼ /kjH1 ðtÞjlS ¼ Cik Cil ei ðtÞ: ð12Þ
Gkl ðtÞ ¼ //kjH1 ðtÞjlS/ljH1 ð0ÞjkSSAv :
i
where the ei ðtÞ describe the site energy fluctuations of site i due to the exciton–phonon coupling. These fluctuations are assumed to have the following properties [18–20]: /ei ðtÞS ¼ 0; /ei ðtÞej ð0ÞS ¼ dij D2 elt :
ð9Þ
In the above equations, it is assumed that the energy fluctuations on different molecules are uncorrelated. D is the amplitude of the local
ð11Þ
i
It can be pointed out that for k ¼ l; the fluctuations in the exciton energies are simply X ekk ðtÞ ¼ jCik j2 ei ðtÞ: ð13Þ i
For for an evenly delocalized state, Cik ¼ pexample, ffiffiffiffiffi 1= N ; the energy fluctuations are reduced by a factor of N relatively to those of the molecules. Averaging over the thermal fluctuations and using the properties of Eq. (9) leads to X Gkl ðtÞ ¼ D2kl ðtÞ ¼ jCik j2 jCil j2 D2 elt i
¼ Skl D2 elt ; ð14Þ P k2 l2 where Skl ¼ i jCi j jCi j is a factor directly related to the exciton wavefunctions overlap. It can also be noticed that, for k ¼ l; D2kk ðtÞ ¼ D2 Lk elt ;
ð15Þ
where Lk is the localization index defined by Eq. (7). In the case pffiffiffiffiffiffiffiffiffiffi ffi pofffiffiffiffiffi slow fluctuations ðl-0Þ Dkk ¼ D Lk ¼D= N describes the inhomogeneous broadening of the exciton states. In the opposite case of fast fluctuations ðl-NÞ; the transition frequency of each molecule vary rapidly and the molecule absorption line exhibits the well-known motional narrowing effect with a
ARTICLE IN PRESS J.P. Lemaistre / Journal of Luminescence 107 (2004) 332–338
homogeneous width given by D2 =_l [19,20]. For an exciton evenly delocalized over the N molecules, as it is the case for the k ¼ 1 bottom state of the J-aggregate, the p amplitude of the fluctuation is ffiffiffiffiffi reduced by p a factor N from that of the molecule ffiffiffiffiffi ðDk¼1 ¼ D= N Þ and the homogeneous width becomes D2k¼1 =_l ¼ D2 =N_l because of the renormalization of the wavefunction and not because of the exchange narrowing among the exciton states. The integration of Eq. (10) gives the scattering rates _Ukl0 ¼
D2 ð_lÞ2 Skl : _l ð_lÞ2 þ ð_okl Þ2
ð16Þ
The scattering among the exciton states occurs by phonon absorption or emission processes. We set, for the uphill and downhill transfer rates, respectively, Ukl ¼ Ukl0 nðokl Þ; Ulk ¼ Ukl0 ð1 þ nðokl ÞÞ;
ð17Þ
in which nðokl Þ ¼ 1=½expð_okl =kB TÞ 1
ð18Þ
is the thermal occupation, at temperature T; of the phonon mode with energy _okl : The transfer rates of Eq. (17) satisfy the Boltzmann equilibrium condition. After an initial excitation of the exciton band, the population relaxation among the eigenstates can be described by means of a master equation as follows: X d Pk ðtÞ ¼ gk Pk ðtÞ þ ðUkl Pl ðtÞ dt l Ulk Pk ðtÞÞ;
ð19Þ
in which the gk are the rate constants associated to the radiative lifetimes of the states. The solutions of the master equation, which depend on the excitation initial conditions, allow to calculate the time dependent emission spectra: X IðE; tÞp Fk Pk ðtÞdðE Ek Þ: ð20Þ k
At low temperature, the contribution of the pure dephasing is very small. At intermediate and high temperatures (say 40 KoTo200 K), the pure dephasing of the exciton states originating from
335
the coupling to the vibrational modes becomes the dominant mechanism of the homogeneous line broadening. Assuming the coupling of excitons to one vibrational Raman mode ðovib Þ; the homogeneous width can be described by G ¼ jwvib j2 nðovib Þðnðovib Þ þ 1Þ
ð21Þ
with nðovib Þ ¼ 1=½expð_ovib =kB TÞ 1 ;
ð22Þ
which for _ovib bkT reduces to G ¼ jwvib j2 expð_ovib =kTÞ;
ð23Þ
where wvib describes the exciton–vibration coupling. The exciton dephasing process destroys the phase coherence among the molecular sites and strongly affects the number of coherently coupled molecules. The role of the homogeneous broadening on the exciton localization properties was pointed out in Ref. [14] by using a generalized average participation ratio. It gives, in the limit of zero homogeneous width, ðG-0Þ; the usual participation ratio (Eq. (7)) while it goes to one for large values of G:
4. Results and discussion The calculations are performed for a linear chain of N ¼ 80 molecules featuring the PIC Jaggregate for which experimental data is available. The intermolecular electronic interaction between the molecules is V0 ¼ 600 cm1 and the monomer decay rate is g0 ¼ 3:7 ns: The bottom state ðk ¼ 1Þ of the quasi-exciton band carries most of the total oscillator strength (90%). For a linear chain of N ¼ 80 molecules, the energy splitting between the two lowest states is around 10 cm1 and the number of coherently coupled molecules at zero temperature is 52 molecules. The scattering rates among the exciton states are calculated according the Eq. (16) and (18) using _l ¼ 200 cm1 and a fluctuation amplitude D ¼ 120 cm1 : We show in Fig. 1 the calculated time evolution of the total exciton population and that of the bottom radiating state after initial excitation of the k ¼ 1 state for temperatures ranging from T ¼ 10 to 200 K: The fluorescence intensity decay
ARTICLE IN PRESS J.P. Lemaistre / Journal of Luminescence 107 (2004) 332–338
336 1.0
Exciton population
0.8
0.6
0.4
(d)
0.2 (c) (b) (a)
(d ← a)
0 0
0.1
0.2
0.3
0.4
0.5
Time (ns)
Fig. 1. Time evolution of the eigenstates populations at different temperatures: T ¼ 10 K (a), 50 K (b), 100 K (c) and 200 K (d). Total population decay (full lines) and population decay of the bottom state (dotted lines).
Fluorescence Intensity
10
1
(d) (c) (b)
0.1
(a)
0.01 0
0.1
0.2
0.3
0.4
temperatures which is not consistent with the experimental data. Let us consider now the role of the homogeneous dephasing on the temperature dependence of the radiative lifetime and on the number of coherently coupled molecules. The temperature dependence of the homogeneous linewidth, GðTÞ; was described by thermal activation of the exciton–phonon scattering by using three phonons of 9, 305 and 973 cm1 with appropriate prefactors associated to the exciton–phonon coupling parameters in Ref. [2] and by using an optical phonon of 240 cm1 in Ref. [4]. The knowledge of GðTÞ does not allow to determine unambiguously the phonon frequencies involved in the scattering mechanisms. Nevertheless, at low temperature, the temperature dependence of the dephasing time can be well reproduced by using a low frequency phonon of 9 cm1 [2]. This frequency is close to the energy separation of the two lowest exciton states. At higher temperatures, we suggest that the dephasing originates from the coupling of excitons to the Raman mode of 607 cm1 [1]. We show in Fig. 3 the temperature dependence of the homogeneous width as arising from acoustic phonon (a), vibrational dephasing setting a coupling constant parameter of 3600 cm1 (b) and the sum of the two contributions (c) in order to fit the experimental data of Ref. [2]. The homogeneous broadening creates fast fluctuations in the transition frequencies and is
0.5
50
Time (ns) (c)
Fig. 2. Decay of the fluorescence intensity at the temperatures defined in Fig. 1.
30
Γ
is shown, for the same temperatures, on Fig. 2. It can be seen that upon increasing the temperature, the population of the k ¼ 1 state is transferred to higher states within the exciton band, thus increasing the radiative lifetime. The interplay between the intraband scattering and the superradiant emission decay was previously analyzed in Ref. [7]. The intraband relaxation model allows to reproduce the temperature dependence of the exciton radiative lifetimepat ffiffiffiffi low temperatures (up to 40 K) but predicts a T dependence at higher
40
(b)
20
10 (a)
0 0
50
100
150
200
Temperature (˚K)
Fig. 3. Temperature dependence of the homogeneous width of the k ¼ 1 bottom state.
ARTICLE IN PRESS J.P. Lemaistre / Journal of Luminescence 107 (2004) 332–338
conveniently described in the motional narrowing regime. In this fast fluctuations limit, the homogeneous linewidth of the k ¼ 1 exciton radiating state is given by GðTÞ ¼ G0 =Ncoh ðTÞ; where G0 is the homogeneous width associated to the monomer and Ncoh ðTÞ is the number of coherently coupled molecules in the aggregate. It can be also
1000
Radiative lifetime (ps)
800
600
400
200
0 40
60
80
100 120 140 160
Temperature (˚K)
(a)
60
337
defined as the relative ratio of the radiative lifetimes, Ncoh ðTÞ ¼ t0 =tðTÞ; based on the relative oscillator strengths of the monomer and exciton states. The temperature dependence of the radiative lifetime of the k ¼ 1 exciton state can be written as tðTÞ ¼ ðt0 =G0 ÞGðTÞ and is depicted in Fig. 4. At T ¼ 40 K; GðTÞ ¼ 1:35 cm1 and using the number of coherently coupled molecules Ncoh ¼ 52 at low temperature, we set G0 ¼ 70 cm1 : This strong non linear temperature dependence of the radiative lifetime, experimentally observed at relatively high temperatures, can be interpreted as arising mainly from the strong dephasing processes. We show in Fig. 5 the number, Ncoh ðTÞ ¼ G0 =GðTÞ; of coherently coupled molecules decreasing from 52 to 4 molecules when the temperature increases from 40 to 160 K: In this simple approach, we set G0 as a temperature-independent parameter. Obviously, the knowledge of this temperature dependence will be of interest to get a better fitting of the experimental data. It is concluded that both processes, the intraband scattering induced by the phonons, at low temperatures, and the vibrational dephasing, at higher temperatures, must be taken into account to explain the anomalous temperature dependence of the exciton radiative lifetime observed in PIC J-aggregates.
References Ncoh
40
20
0 40 (b)
60
80
100 120 140 160
Temperature (˚K)
Fig. 4. (a) Temperature dependence (intermediate and hightemperature range) of the relative radiative lifetime. The experimental points are from Ref. [2]; (b) temperature dependence (intermediate and high-temperature range) of the number of coherently coupled molecules.
[1] H. Fidder, J. Knoester, D.A. Wiersma, Chem. Phys. Lett. 171 (1990) 529. [2] H. Fidder, D.A. Wiersma, Phys. Stat. Sol. B 188 (1995) 285. [3] E.O. Potma, D.A. Wiersma, J. Chem. Phys. 108 (1998) 4894. [4] F.C. Spano, J.R. Kuklinski, S. Mukamel, Phys. Rev. Lett. 65 (1990) 211. [5] M. Bednarz, J.P. Lemaistre, Int. J. Mod. Phys. B 15 (2001) 3761. [6] M. Bednarz, V.A. Malyshev, J.P. Lemaistre, J. Knoester, J. Lumin. 94/95 (2001) 271. [7] M. Bednarz, V.A. Malyshev, J. Knoester, J. Chem. Phys. 117 (2002) 6200. [8] J.P. Lemaistre, Chem. Phys. 246 (1999) 283. [9] C. Ecoffet, D. Markovitsi, Ph. Milli!e, J.P. Lemaistre, Chem. Phys. 177 (1993) 629. [10] D. Markovitsi, L.K. Gallos, J.P. Lemaistre, P. Argyrakis, Chem. Phys. 269 (2001) 147.
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[11] I.G. Scheblykin, O.Y. Sliusarenko, L.S. Lepnev, A.G. Vitukhnovsky, M. Van der Auweraer, J. Phys. Chem. B 105 (2001) 4636. [12] D.M. Basko, A.N. Lobanov, A.V. Pimenov, A.G. Vitukhnovsky, Chem. Phys. Lett. 369 (2003) 192. [13] L.D. Bakalis, M. Coca, J. Knoester, J. Chem. Phys. 110 (1999) 2208. [14] J.A. Leegwater, J. Phys. Chem. 100 (1996) 14403.
[15] V.K. Kamalov, I.A. Struganova, K. Yoshihara, J. Phys. Chem. 100 (1996) 8640. [16] P.W. Anderson, J. Phys. Soc. Japan 9 (1954) 316. [17] R. Kubo, K. Tomita, J. Phys. Soc. Japan 9 (1954) 888. [18] I. Barvik, P. Reineker, Ch. Warns, Th. Neidlinger, J. Lumin. 65 (1995) 169. [19] M. Wubs, J. Knoester, Chem. Phys. Lett. 284 (1998) 63. [20] M. Wubs, J. Knoester, J. Lumin. 76&77 (1998) 359.
Journal of Luminescence 107 (2004) 332–338
Intraband relaxation and dephasing of Frenkel exciton states in one-dimensional J-aggregates J.P. Lemaistre* Laboratoire des Milieux D!esordonn!es et H!et!erog"enes, CNRS-UMR 7603, Universit!e P. et M. Curie, Tour 22, 4, Place Jussieu, Paris F-75252, C!edex 05, France
Abstract A theoretical model is proposed to analyse the relaxation and the dephasing mechanisms of Frenkel excitons in onedimensional J-aggregates. At low temperature, the exciton dynamics originates from the coupling of excitons to phonons which creates temperature dependent local fluctuations in the site energies of the molecules. A microscopic model of exciton–phonon coupling is used assuming a Bose–Einstein distribution for the phonon occupation density. Intraband scattering rates among the exciton states are calculated and used in a master equation to obtain the time evolution of the exciton populations after initial excitation of the optically allowed state. At higher temperatures, the dephasing of the exciton states originates from the coupling of excitons to the vibrational modes. Both processes (intraband scattering and dephasing) contribute to the lengthening of the radiative exciton lifetime with increasing temperature. Using a generalized participation ratio taking into account the homogeneous width allows to define, at each temperature, the number of coherently coupled molecules and to interpret the anomalous observed temperature dependence of the radiative lifetime of excitons in PIC aggregates at low, intermediate and high temperature. r 2003 Elsevier B.V. All rights reserved. Keywords: Exciton; Aggregate; Scattering; Dephasing
1. Introduction Dynamical processes of Frenkel excitons in mesoscopic aggregates of molecules forming nanostructures are currently investigated. Many experimental and theoretical works are devoted to the exciton dynamics in molecular systems of low dimensionality like J [1–7] or H [8–10] aggregates, dye aggregates exhibiting a Davydov splitting [11,12], or cyclic aggregates [13]. In general, these aggregates of finite size, are characterized by relatively strong intermolecular inter*Tel.: +33-01-44-27-42-66; fax: +33-01-44-27-38-82. E-mail address: [email protected] (J.P. Lemaistre).
actions due to the long range dipole–dipole couplings. Thus, the cooperative effects are dominant (even at room temperature) and are generally discussed in terms of collective excitation states. The dynamic disorder stems from the coupling of the chromophores to a thermal bath (phonons) acting as a dissipative medium. This stochastic coupling creates local fluctuations in the site energies characterized by their amplitude ðDÞ and their correlation time ðtc Þ: The exciton dynamics is determined by comparing the coherence time ðt0 ¼ _=V0 Þ; where V0 denotes the electronic interaction between the molecular sites, to the exciton–phonon parameters ðD; tc Þ: When the
0022-2313/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jlumin.2003.12.016
ARTICLE IN PRESS J.P. Lemaistre / Journal of Luminescence 107 (2004) 332–338
exciton–phonon correlation time is shorter than the coherence time ðtc pt0 Þ; the excitation is temporarily localized on sites and moves incoherently from site to site. The exciton motion is described according to the so-called hopping model, with transfer rates given by Fermi’s golden rule formulae. In the opposite case, for which the exciton–phonon correlation time is longer than the coherence time ðtc Xt0 Þ; the local fluctuations in the site energies do not destroy the collective character of the exciton states. Thus, the energy fluctuations of the sites are distributed over all exciton states inducing an intraband scattering [5–8]. In this paper we present a theoretical model based on numerical calculations of the exciton eigenstates and their participation ratios, in order to analyze the photophysical properties of collective states in linear J-aggregates. The intraband scattering is studied by considering the stochastic exciton–phonon couplings. After initial excitation of the optically allowed bottom state of the quasiexciton band, population relaxation occurs according to a jkS2jlS scattering mechanism among the eigenstates. This energy relaxation, induced by the coupling of the exciton states to a thermal bath is described as an incoherent energy transfer. An expression for the scattering rates is provided assuming that the exciton-bath coupling is described by means of two parameters: D; the amplitude of the fluctuations and tc the correlation time associated to the stochastic process. The transfer rates are then used in a master equation to describe the time evolution of the eigenstates populations and that of the optical lineshapes. Such an approach was recently used to analyze the temperature dependence of the exciton radiative lifetime at low temperatures [7]. At higher temperatures, the dephasing of the collective excitation states by molecular vibrations become the dominant mechanism. Increasing the temperature destroys the phase coherence among the sites and decreases the number of coherently coupled molecules. The role and the importance of the homogeneous broadening was pointed out by Leegwater [14]. Using a generalized average participation ratio which takes into account the homogeneous width allows to obtain the tempera-
333
ture dependence of the number of coherently coupled molecules.
2. Model Hamiltonian We model an aggregate by a linear chain of N identical two-level molecules. In the localized site representation the electronic Hamiltonian is written as H0 ¼
X
Ei jiS/ij þ
0 X
Vij jiS/jj:
ð1Þ
i;j
i
The Ei are the molecular site excitation energies and the Vij describe the electronic interactions among the sites. We will consider a linear chain of parallel molecules featuring J-aggregates ðVij o0Þ like PIC [1–4,6,7] or BIC [15] systems. They are assumed to be coupled through dipolar interactions and are represented in the simplest approach of the point dipoles as Vij ¼ V0 =ji jj3 ;
ð2Þ
where V0 ðV0 o0Þ denotes the nearest neighbor interaction and ji jj is the distance between molecules i and j with a unity lattice spacing. The molecular aggregates that we consider being characterized by strong intermolecular interactions, we use the delocalized representation of the Hamiltonian X H0 ¼ Ek jkS/kj; ð3Þ k
in which jkS denotes the N eigenstates with energies Ek and eigenvectors X jkS ¼ Cik jiS: ð4Þ i
The transition moments of the delocalized eigenstates are X mk ¼ Cik mi ; ð5Þ i
giving for the oscillator strengths Fk pEk jmk j2 :
ð6Þ
ARTICLE IN PRESS J.P. Lemaistre / Journal of Luminescence 107 (2004) 332–338
334
The localized/delocalized behaviour is analyzed by using the inverse participation ratio defined as X jCik j4 : ð7Þ Lk ¼ i
Lk equals unity for an exciton localized on a single molecule and goes to zero as 1=N for an exciton evenly delocalized over N sites.
energy fluctuations and l ¼ 1=tc is the inverse of the correlation time associated to the stochastic process. The rates of the intraband scattering are given by the Fourier transform of the autocorrelation function of the stochastic perturbation. Thus, the transition probability among the exciton states is given by Z þN Ukl0 ¼ ð1=_2 ÞRe Gkl ðtÞeiokl t dt; ð10Þ 0
3. Intraband relaxation and homogeneous broadening
where
Excitation transfer in molecular aggregates is induced by the dynamical coupling of the molecular sites to a thermal bath. This stochastic coupling induces local fluctuations in the site energies characterized by their amplitude and their correlation time. In J-aggregates, the relatively strong electronic coupling creates delocalized excitons forming a quasi-exciton band. Thus, the stochastic coupling of excitons to phonons induces the diffusion among the delocalized jkS states. This mechanism is an incoherent energy transfer process among the chain eigenmodes and may be described in the framework of the Kubo–Anderson’s theory of the stochastic resonance [16,17]. The transfer rates of the intraband scattering will depend on the exciton–phonon coupling, the correlation time associated to the stochastic fluctuation, the temperature and also on the energy separation of the eigenmodes. We add to the excitonic Hamiltonian H0 of Eq. (1) a time dependent Hamiltonian, H1 ðtÞ; which describes the stochastic coupling X ei ðtÞjiS/ij; ð8Þ H1 ðtÞ ¼
Ukl0 is an energy transition rate taking into account all exciton–phonon couplings at frequencies okl ¼ ðEk El Þ=_: On the basis set of the delocalized states jkS; eigenstates of H0 ; the matrix elements of the stochastic perturbation are X ekl ðtÞ ¼ /kjH1 ðtÞjlS ¼ Cik Cil ei ðtÞ: ð12Þ
Gkl ðtÞ ¼ //kjH1 ðtÞjlS/ljH1 ð0ÞjkSSAv :
i
where the ei ðtÞ describe the site energy fluctuations of site i due to the exciton–phonon coupling. These fluctuations are assumed to have the following properties [18–20]: /ei ðtÞS ¼ 0; /ei ðtÞej ð0ÞS ¼ dij D2 elt :
ð9Þ
In the above equations, it is assumed that the energy fluctuations on different molecules are uncorrelated. D is the amplitude of the local
ð11Þ
i
It can be pointed out that for k ¼ l; the fluctuations in the exciton energies are simply X ekk ðtÞ ¼ jCik j2 ei ðtÞ: ð13Þ i
For for an evenly delocalized state, Cik ¼ pexample, ffiffiffiffiffi 1= N ; the energy fluctuations are reduced by a factor of N relatively to those of the molecules. Averaging over the thermal fluctuations and using the properties of Eq. (9) leads to X Gkl ðtÞ ¼ D2kl ðtÞ ¼ jCik j2 jCil j2 D2 elt i
¼ Skl D2 elt ; ð14Þ P k2 l2 where Skl ¼ i jCi j jCi j is a factor directly related to the exciton wavefunctions overlap. It can also be noticed that, for k ¼ l; D2kk ðtÞ ¼ D2 Lk elt ;
ð15Þ
where Lk is the localization index defined by Eq. (7). In the case pffiffiffiffiffiffiffiffiffiffi ffi pofffiffiffiffiffi slow fluctuations ðl-0Þ Dkk ¼ D Lk ¼D= N describes the inhomogeneous broadening of the exciton states. In the opposite case of fast fluctuations ðl-NÞ; the transition frequency of each molecule vary rapidly and the molecule absorption line exhibits the well-known motional narrowing effect with a
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homogeneous width given by D2 =_l [19,20]. For an exciton evenly delocalized over the N molecules, as it is the case for the k ¼ 1 bottom state of the J-aggregate, the p amplitude of the fluctuation is ffiffiffiffiffi reduced by p a factor N from that of the molecule ffiffiffiffiffi ðDk¼1 ¼ D= N Þ and the homogeneous width becomes D2k¼1 =_l ¼ D2 =N_l because of the renormalization of the wavefunction and not because of the exchange narrowing among the exciton states. The integration of Eq. (10) gives the scattering rates _Ukl0 ¼
D2 ð_lÞ2 Skl : _l ð_lÞ2 þ ð_okl Þ2
ð16Þ
The scattering among the exciton states occurs by phonon absorption or emission processes. We set, for the uphill and downhill transfer rates, respectively, Ukl ¼ Ukl0 nðokl Þ; Ulk ¼ Ukl0 ð1 þ nðokl ÞÞ;
ð17Þ
in which nðokl Þ ¼ 1=½expð_okl =kB TÞ 1
ð18Þ
is the thermal occupation, at temperature T; of the phonon mode with energy _okl : The transfer rates of Eq. (17) satisfy the Boltzmann equilibrium condition. After an initial excitation of the exciton band, the population relaxation among the eigenstates can be described by means of a master equation as follows: X d Pk ðtÞ ¼ gk Pk ðtÞ þ ðUkl Pl ðtÞ dt l Ulk Pk ðtÞÞ;
ð19Þ
in which the gk are the rate constants associated to the radiative lifetimes of the states. The solutions of the master equation, which depend on the excitation initial conditions, allow to calculate the time dependent emission spectra: X IðE; tÞp Fk Pk ðtÞdðE Ek Þ: ð20Þ k
At low temperature, the contribution of the pure dephasing is very small. At intermediate and high temperatures (say 40 KoTo200 K), the pure dephasing of the exciton states originating from
335
the coupling to the vibrational modes becomes the dominant mechanism of the homogeneous line broadening. Assuming the coupling of excitons to one vibrational Raman mode ðovib Þ; the homogeneous width can be described by G ¼ jwvib j2 nðovib Þðnðovib Þ þ 1Þ
ð21Þ
with nðovib Þ ¼ 1=½expð_ovib =kB TÞ 1 ;
ð22Þ
which for _ovib bkT reduces to G ¼ jwvib j2 expð_ovib =kTÞ;
ð23Þ
where wvib describes the exciton–vibration coupling. The exciton dephasing process destroys the phase coherence among the molecular sites and strongly affects the number of coherently coupled molecules. The role of the homogeneous broadening on the exciton localization properties was pointed out in Ref. [14] by using a generalized average participation ratio. It gives, in the limit of zero homogeneous width, ðG-0Þ; the usual participation ratio (Eq. (7)) while it goes to one for large values of G:
4. Results and discussion The calculations are performed for a linear chain of N ¼ 80 molecules featuring the PIC Jaggregate for which experimental data is available. The intermolecular electronic interaction between the molecules is V0 ¼ 600 cm1 and the monomer decay rate is g0 ¼ 3:7 ns: The bottom state ðk ¼ 1Þ of the quasi-exciton band carries most of the total oscillator strength (90%). For a linear chain of N ¼ 80 molecules, the energy splitting between the two lowest states is around 10 cm1 and the number of coherently coupled molecules at zero temperature is 52 molecules. The scattering rates among the exciton states are calculated according the Eq. (16) and (18) using _l ¼ 200 cm1 and a fluctuation amplitude D ¼ 120 cm1 : We show in Fig. 1 the calculated time evolution of the total exciton population and that of the bottom radiating state after initial excitation of the k ¼ 1 state for temperatures ranging from T ¼ 10 to 200 K: The fluorescence intensity decay
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336 1.0
Exciton population
0.8
0.6
0.4
(d)
0.2 (c) (b) (a)
(d ← a)
0 0
0.1
0.2
0.3
0.4
0.5
Time (ns)
Fig. 1. Time evolution of the eigenstates populations at different temperatures: T ¼ 10 K (a), 50 K (b), 100 K (c) and 200 K (d). Total population decay (full lines) and population decay of the bottom state (dotted lines).
Fluorescence Intensity
10
1
(d) (c) (b)
0.1
(a)
0.01 0
0.1
0.2
0.3
0.4
temperatures which is not consistent with the experimental data. Let us consider now the role of the homogeneous dephasing on the temperature dependence of the radiative lifetime and on the number of coherently coupled molecules. The temperature dependence of the homogeneous linewidth, GðTÞ; was described by thermal activation of the exciton–phonon scattering by using three phonons of 9, 305 and 973 cm1 with appropriate prefactors associated to the exciton–phonon coupling parameters in Ref. [2] and by using an optical phonon of 240 cm1 in Ref. [4]. The knowledge of GðTÞ does not allow to determine unambiguously the phonon frequencies involved in the scattering mechanisms. Nevertheless, at low temperature, the temperature dependence of the dephasing time can be well reproduced by using a low frequency phonon of 9 cm1 [2]. This frequency is close to the energy separation of the two lowest exciton states. At higher temperatures, we suggest that the dephasing originates from the coupling of excitons to the Raman mode of 607 cm1 [1]. We show in Fig. 3 the temperature dependence of the homogeneous width as arising from acoustic phonon (a), vibrational dephasing setting a coupling constant parameter of 3600 cm1 (b) and the sum of the two contributions (c) in order to fit the experimental data of Ref. [2]. The homogeneous broadening creates fast fluctuations in the transition frequencies and is
0.5
50
Time (ns) (c)
Fig. 2. Decay of the fluorescence intensity at the temperatures defined in Fig. 1.
30
Γ
is shown, for the same temperatures, on Fig. 2. It can be seen that upon increasing the temperature, the population of the k ¼ 1 state is transferred to higher states within the exciton band, thus increasing the radiative lifetime. The interplay between the intraband scattering and the superradiant emission decay was previously analyzed in Ref. [7]. The intraband relaxation model allows to reproduce the temperature dependence of the exciton radiative lifetimepat ffiffiffiffi low temperatures (up to 40 K) but predicts a T dependence at higher
40
(b)
20
10 (a)
0 0
50
100
150
200
Temperature (˚K)
Fig. 3. Temperature dependence of the homogeneous width of the k ¼ 1 bottom state.
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conveniently described in the motional narrowing regime. In this fast fluctuations limit, the homogeneous linewidth of the k ¼ 1 exciton radiating state is given by GðTÞ ¼ G0 =Ncoh ðTÞ; where G0 is the homogeneous width associated to the monomer and Ncoh ðTÞ is the number of coherently coupled molecules in the aggregate. It can be also
1000
Radiative lifetime (ps)
800
600
400
200
0 40
60
80
100 120 140 160
Temperature (˚K)
(a)
60
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defined as the relative ratio of the radiative lifetimes, Ncoh ðTÞ ¼ t0 =tðTÞ; based on the relative oscillator strengths of the monomer and exciton states. The temperature dependence of the radiative lifetime of the k ¼ 1 exciton state can be written as tðTÞ ¼ ðt0 =G0 ÞGðTÞ and is depicted in Fig. 4. At T ¼ 40 K; GðTÞ ¼ 1:35 cm1 and using the number of coherently coupled molecules Ncoh ¼ 52 at low temperature, we set G0 ¼ 70 cm1 : This strong non linear temperature dependence of the radiative lifetime, experimentally observed at relatively high temperatures, can be interpreted as arising mainly from the strong dephasing processes. We show in Fig. 5 the number, Ncoh ðTÞ ¼ G0 =GðTÞ; of coherently coupled molecules decreasing from 52 to 4 molecules when the temperature increases from 40 to 160 K: In this simple approach, we set G0 as a temperature-independent parameter. Obviously, the knowledge of this temperature dependence will be of interest to get a better fitting of the experimental data. It is concluded that both processes, the intraband scattering induced by the phonons, at low temperatures, and the vibrational dephasing, at higher temperatures, must be taken into account to explain the anomalous temperature dependence of the exciton radiative lifetime observed in PIC J-aggregates.
References Ncoh
40
20
0 40 (b)
60
80
100 120 140 160
Temperature (˚K)
Fig. 4. (a) Temperature dependence (intermediate and hightemperature range) of the relative radiative lifetime. The experimental points are from Ref. [2]; (b) temperature dependence (intermediate and high-temperature range) of the number of coherently coupled molecules.
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