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Disorder-induced exciton scattering in molecular aggregates
Journal of Luminescence 83}84 (1999) 229}233
Disorder-induced exciton scattering in molecular aggregates J.P. Lemaistre* Laboratoire des Milieux De& sordonne& s et He& te& roge% nes, CNRS-UMR 7603, Universite& P. et M. Curie, Tour 22, 4, Place Jussieu, F-75252 Paris Ce& dex 05, France
Abstract Static and dynamic disorders are investigated to analyze the exciton scattering in molecular aggregates of "nite size like columnar aggregates or J-aggregates which exhibit strong intermolecular dipolar interactions (<). The static disorder (p), of Anderson type, stems from the inhomogeneities in the site energies while the dynamic disorder originates from the exciton-phonon stochastic coupling induced by the thermal bath. The dynamic coupling induces the exciton di!usion at rates depending on the amplitude (D) and on the correlation time (q) of the #uctuations. A theoretical model, based on the numerical calculation of the exciton eigenstates and their participation ratios is used to simulate the role of both disorders on the optical responses. It is shown that an increase of the static disorder diminishes the coherence length, i.e. the number of coherently coupled molecules and that the dynamic disorder induces a temperature-dependent scattering among the quasi-exciton band states. Simulation of the e!ects of static and dynamic disorders are presented and discussed as a function of the fundamental parameters. ( 1999 Elsevier Science B.V. All rights reserved. Keywords: Exciton; Aggregates; Coherence length; Scattering
1. Introduction Dynamical processes of excitons in mesoscopic clusters of molecules forming nanostructures and their optical line shapes are currently investigated. Experimental and theoretical studies are devoted to the exciton dynamics in columnar aggregates [1,2], linear or cyclic aggregates [3], J-aggregates [4], polymers [5] or photosynthetic units [6]. When the molecules of the aggregate exhibit strong intermolecular interactions, the cooperative e!ects are dominant and the excitons must be analyzed in terms of collective excitation states over the
* Tel.: #33-01-44-27-42-66; fax: #33-01-44-27-38-82. E-mail address: [email protected] (J.P. Lemaistre)
entire aggregate. The static disorder arising from the local inhomogeneities in the molecular energies reduces the localization length of the excitons, i.e. the number of coherently coupled molecules [7]. The dynamic disorder arises from the coupling of the chromophores to a thermal bath (phonons) acting as a dissipative medium. This stochastic coupling creates local #uctuations in the site energies which are characterized by their amplitude (D) and their correlation time (q "1/j) [3]. The dy# namics of excitation transfer is then determined by comparing the coherence time (q "+/<), where 0 < denotes the electronic interaction among the chromophores, to the exciton-phonon parameters (D, q ) [8]. We consider the case of the slow modu# lation regime for which +j;D, <. Thus, the local #uctuations of the site energies do not destroy the collective character of the exciton states.
0022-2313/99/$ - see front matter ( 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 2 3 1 3 ( 9 9 ) 0 0 1 0 3 - 9
230
J.P. Lemaistre / Journal of Luminescence 83}84 (1999) 229}233
In this short communication we analyze, through a simple model, the localization length of excitons within a linear chain of parallel molecules and the intraband scattering among their eigenmodes induced by the exciton}phonon coupling. From the calculation of the eigenstates and of their inverse participation ratios (IPR), we analyze the e!ects of the diagonal (energy) disorder on the coherence length of excitons. Thus, the dynamic disorder is studied by considering the stochastic #uctuations in the site energies due to the coupling to the thermal bath. After initial excitation of the upper state carrying the oscillator strength in the case of columnar aggregate (<'0) [1], the population relaxation is described as an incoherent energy transfer among the DkT states. An expression for the scattering rates, ;0 , is provided and used in a Maskl ter Equation to follow the time dependence of the depopulation (population) of the upper (lower) exciton state.
2. Static disorder Let us consider a linear chain of N parallel molecules coupled through long-range dipolar interactions. The electronic Hamiltonian written in the localized site representation is
sites is assumed to follow a Gaussian distribution around the mean value E : 0 1 (E!E )2 0 f (E!E )" exp! (3) 0 2p2 pJ2p with p being the standard deviation. In the delocalized representation, the electronic Hamiltonian (Eq. (1)) is conveniently described as H "+ E DkTSkD (4) 0 k k in which DkT denote the N eigenstates with energies E . On this basis set, the diagonalization of H gives k 0 the N eigenenergies, E and eigenvectors, k DkT"+ CkDiT. Then, the transition moments, k , i i k associated to the various exciton states as well as their oscillator strengths, c , are easily obtained: k k "+ CkDk T, c JDk D2c (5) k i i k k 0 i with c being the oscillator strength of the isolated 0 molecule. The localization properties of the excitons are usually described by using the inverse participation ratio (IPR) which gives, for each eigenstate, the number of coherently coupled molecules and de"ned as [9]
The E are the molecular site excitation energies i and the < describe the electronic interactions ij among the sites written in the simplest approach of the point dipole model as
(6) ¸ "+ DCkD4. i k i ¸ 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. So, the spatial extension is given by the participation ratio, 1/¸ , and k the mean exciton position is simply
< 0 . <" ij Di!jD3
Sx T"+ x DCkD2. k i i i
H "+ E DiTSiD#+@< DiTS jD. 0 i ij i i,j
(1)
(2)
The interaction energy is expressed in units of < , 0 the nearest-neighbor interaction; the distance Di!jD denotes the distance between molecules i and j with a unity lattice constant. A positive sign of < will 0 describe the case of columnar or H-aggregates while a negative sign will be used for J-aggregates. The static disorder in the site energies stems from the various local environments of the molecular
(7)
3. Dynamic disorder The dynamic disorder stems from the stochastic coupling of the exciton states to a thermal bath and is analyzed in the framework of the Kubo}Anderson's theory of the stochastic resonance [10,11] by adding to the excitonic Hamiltonian (Eq. (1)),
J.P. Lemaistre / Journal of Luminescence 83}84 (1999) 229}233
a time-dependent Hamiltonian H (t)"+ e (t)DiTSiD. (8) 1 i i The e (t) describe the site energy #uctuations of site i i due to the stochastic exciton}phonon coupling to the thermal bath assumed to have the following properties [3]: Se (t)T"0, Se (t)e (0)T"d D2e~jt. (9) i i j ij In the above equations, the energy #uctuations on di!erent molecules are assumed to be uncorrelated; D describes the amplitude of the #uctuations and j"1/q is the inverse of the correlation time q as# # sociated to the stochastic process. We will now consider the intraband scattering among the collective states as an incoherent energy transfer. The transition probability, induced by the stochastic perturbation is given, in the high-temperature approximation, by the Fourier transform of the autocorrelation function of the stochastic perturbation:
231
relaxation process is described by means of a Master Equation from which the time evolution of the eigenstates populations are obtained [2].
4. Results of the numerical simulations In order to model such a system, we simply consider in what follows a linear chain of N"100 parallel molecules coupled through long range dipolar interactions. In Fig. 1, the average number of sites covered by the excitation is represented
P
`= G (t)e*uklt dt, (10a) kl 0 G (t)"SSkDH (t)DlTSlDH (0)DkTT (10b) kl 1 1 A7. In Eqs. (10a) and (10b), u "(E !E )/+ and ;0 is kl kl k l an average transition rate which takes into account all exciton-phonon couplings. Assuming +u <+j, kl integration of Eq. (10a) leads to
;0 "(1/+2)Re kl
;0 "*2j(+ DCkD2DClD2/(+u )2). (11) kl i i kl i Furthermore, when the condition D;+u is valid, kl we get +;0 ;+j, meaning that the residence time kl of excitons (1/;0 ) is longer than q . The intraband kl # scattering between the collective states occurs by resonant transfer followed by phonon absorption or emission processes. We assume ; (u '0)";0 , kl kl kl (12) ; (u (0)";0 exp(+u /k ¹) kl kl B kl kl for downhill and uphill transfer, respectively, in order to satisfy the Boltzmann equilibrium condition. After initial excitation of the exciton band the
Fig. 1. Localization length of excitons for a linear chain of N"100 parallel molecules coupled through long-range dipolar interactions. The spatial extension of excitons at energy E is k represented by segments of length (1/¸ ) centered around the k mean exciton position. Weak disorder (p/< "0.01) and strong 0 disorder (p/< "0.1). 0
232
J.P. Lemaistre / Journal of Luminescence 83}84 (1999) 229}233
Fig. 3. Time evolution of the population decrease of the upper state (full lines) and of the corresponding population increase of the lower state (dotted lines) for D (units of < ) "0.3 (a), 0.5 (b), 0 0.7 (c), 1 (d).
Fig. 2. Participation ratio of the exciton states versus energy within the band calculated by cumulating 50 realizations of the static disorder corresponding to the p/< values of Fig. 1. 0
by segments centered around the mean exciton position for a weak disorder (p/< "0.01) and 0 a relatively strong disorder (p/< "0.1). The length 0 of the segments equals the participation ratio (1/¸ ) k for each collective state. The eigenstates energy is expressed in units of < , the nearest-neighbor inter0 action. In Fig. 2, the participation ratios of the exciton states are calculated as a function of the energy within the band by cumulating the results over 50 realizations of the disorder corresponding to the p/< values of Fig. 1. 0 After initial excitation of the upper state carrying the oscillator strength (if <'0), population relaxation occurs among the eigenstates of the exciton
band at rates proportional to *2j. Assuming the #uctuation amplitude, D, to be temperature dependent, a simulation of the temperature e!ect on the exciton relaxation can be achieved by analyzing the time dependence of the decrease (increase) of the upper (lower) exciton state within the band. Such an e!ect is illustrated in Fig. 3 for several values of D. We have presented a simple model to analyze the role of static disorder on the localization length of excitons in a simple linear chain and provide an expression for the intraband scattering induced by the dynamic disorder. Such a model could be easily used to investigate disorders e!ects and intraband scattering in molecular aggregates exhibiting strong intermolecular interactions.
References [1] C. Eco!et, D. Markovitsi, Ph. MillieH , J.P. Lemaistre, Chem. Phys. 177 (1993) 629. [2] J.P. Lemaistre, J. Lumin. 76/77 (1998) 437. [3] I. Barvik, P. Reineker, Ch. Warns, Th. Neidlinger, J. Lumin. 65 (1995) 169. [4] E.O. Potma, D.A. Wiersma, J. Chem. Phys. 108 (1998) 4894. [5] M. Shimizu, S. Suto, T. Goto, A. Watanabe, M. Matsuda, Phys. Rev. B 58 (1998) 5032.
J.P. Lemaistre / Journal of Luminescence 83}84 (1999) 229}233 [6] R. Kumble, R.M. Hochstrasser, J. Chem. Phys. 109 (1998) 855. [7] V.A. Malyshev, P. Moreno, Phys. Rev. B 51 (1995) 14 587. [8] M. Wubs, J. Knoester, Chem. Phys. Lett. 284 (1998) 63.
233
[9] M. Schreiber, Y. Toyozawa, J. Phys. Soc. Japan 51 (1982) 1537. [10] P.W. Anderson, J. Phys. Soc. Japan 9 (1954) 316. [11] R. Kubo, K. Tomita, J. Phys. Soc. Japan 9 (1954) 888.
Disorder-induced exciton scattering in molecular aggregates J.P. Lemaistre* Laboratoire des Milieux De& sordonne& s et He& te& roge% nes, CNRS-UMR 7603, Universite& P. et M. Curie, Tour 22, 4, Place Jussieu, F-75252 Paris Ce& dex 05, France
Abstract Static and dynamic disorders are investigated to analyze the exciton scattering in molecular aggregates of "nite size like columnar aggregates or J-aggregates which exhibit strong intermolecular dipolar interactions (<). The static disorder (p), of Anderson type, stems from the inhomogeneities in the site energies while the dynamic disorder originates from the exciton-phonon stochastic coupling induced by the thermal bath. The dynamic coupling induces the exciton di!usion at rates depending on the amplitude (D) and on the correlation time (q) of the #uctuations. A theoretical model, based on the numerical calculation of the exciton eigenstates and their participation ratios is used to simulate the role of both disorders on the optical responses. It is shown that an increase of the static disorder diminishes the coherence length, i.e. the number of coherently coupled molecules and that the dynamic disorder induces a temperature-dependent scattering among the quasi-exciton band states. Simulation of the e!ects of static and dynamic disorders are presented and discussed as a function of the fundamental parameters. ( 1999 Elsevier Science B.V. All rights reserved. Keywords: Exciton; Aggregates; Coherence length; Scattering
1. Introduction Dynamical processes of excitons in mesoscopic clusters of molecules forming nanostructures and their optical line shapes are currently investigated. Experimental and theoretical studies are devoted to the exciton dynamics in columnar aggregates [1,2], linear or cyclic aggregates [3], J-aggregates [4], polymers [5] or photosynthetic units [6]. When the molecules of the aggregate exhibit strong intermolecular interactions, the cooperative e!ects are dominant and the excitons must be analyzed in terms of collective excitation states over the
* Tel.: #33-01-44-27-42-66; fax: #33-01-44-27-38-82. E-mail address: [email protected] (J.P. Lemaistre)
entire aggregate. The static disorder arising from the local inhomogeneities in the molecular energies reduces the localization length of the excitons, i.e. the number of coherently coupled molecules [7]. The dynamic disorder arises from the coupling of the chromophores to a thermal bath (phonons) acting as a dissipative medium. This stochastic coupling creates local #uctuations in the site energies which are characterized by their amplitude (D) and their correlation time (q "1/j) [3]. The dy# namics of excitation transfer is then determined by comparing the coherence time (q "+/<), where 0 < denotes the electronic interaction among the chromophores, to the exciton-phonon parameters (D, q ) [8]. We consider the case of the slow modu# lation regime for which +j;D, <. Thus, the local #uctuations of the site energies do not destroy the collective character of the exciton states.
0022-2313/99/$ - see front matter ( 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 2 3 1 3 ( 9 9 ) 0 0 1 0 3 - 9
230
J.P. Lemaistre / Journal of Luminescence 83}84 (1999) 229}233
In this short communication we analyze, through a simple model, the localization length of excitons within a linear chain of parallel molecules and the intraband scattering among their eigenmodes induced by the exciton}phonon coupling. From the calculation of the eigenstates and of their inverse participation ratios (IPR), we analyze the e!ects of the diagonal (energy) disorder on the coherence length of excitons. Thus, the dynamic disorder is studied by considering the stochastic #uctuations in the site energies due to the coupling to the thermal bath. After initial excitation of the upper state carrying the oscillator strength in the case of columnar aggregate (<'0) [1], the population relaxation is described as an incoherent energy transfer among the DkT states. An expression for the scattering rates, ;0 , is provided and used in a Maskl ter Equation to follow the time dependence of the depopulation (population) of the upper (lower) exciton state.
2. Static disorder Let us consider a linear chain of N parallel molecules coupled through long-range dipolar interactions. The electronic Hamiltonian written in the localized site representation is
sites is assumed to follow a Gaussian distribution around the mean value E : 0 1 (E!E )2 0 f (E!E )" exp! (3) 0 2p2 pJ2p with p being the standard deviation. In the delocalized representation, the electronic Hamiltonian (Eq. (1)) is conveniently described as H "+ E DkTSkD (4) 0 k k in which DkT denote the N eigenstates with energies E . On this basis set, the diagonalization of H gives k 0 the N eigenenergies, E and eigenvectors, k DkT"+ CkDiT. Then, the transition moments, k , i i k associated to the various exciton states as well as their oscillator strengths, c , are easily obtained: k k "+ CkDk T, c JDk D2c (5) k i i k k 0 i with c being the oscillator strength of the isolated 0 molecule. The localization properties of the excitons are usually described by using the inverse participation ratio (IPR) which gives, for each eigenstate, the number of coherently coupled molecules and de"ned as [9]
The E are the molecular site excitation energies i and the < describe the electronic interactions ij among the sites written in the simplest approach of the point dipole model as
(6) ¸ "+ DCkD4. i k i ¸ 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. So, the spatial extension is given by the participation ratio, 1/¸ , and k the mean exciton position is simply
< 0 . <" ij Di!jD3
Sx T"+ x DCkD2. k i i i
H "+ E DiTSiD#+@< DiTS jD. 0 i ij i i,j
(1)
(2)
The interaction energy is expressed in units of < , 0 the nearest-neighbor interaction; the distance Di!jD denotes the distance between molecules i and j with a unity lattice constant. A positive sign of < will 0 describe the case of columnar or H-aggregates while a negative sign will be used for J-aggregates. The static disorder in the site energies stems from the various local environments of the molecular
(7)
3. Dynamic disorder The dynamic disorder stems from the stochastic coupling of the exciton states to a thermal bath and is analyzed in the framework of the Kubo}Anderson's theory of the stochastic resonance [10,11] by adding to the excitonic Hamiltonian (Eq. (1)),
J.P. Lemaistre / Journal of Luminescence 83}84 (1999) 229}233
a time-dependent Hamiltonian H (t)"+ e (t)DiTSiD. (8) 1 i i The e (t) describe the site energy #uctuations of site i i due to the stochastic exciton}phonon coupling to the thermal bath assumed to have the following properties [3]: Se (t)T"0, Se (t)e (0)T"d D2e~jt. (9) i i j ij In the above equations, the energy #uctuations on di!erent molecules are assumed to be uncorrelated; D describes the amplitude of the #uctuations and j"1/q is the inverse of the correlation time q as# # sociated to the stochastic process. We will now consider the intraband scattering among the collective states as an incoherent energy transfer. The transition probability, induced by the stochastic perturbation is given, in the high-temperature approximation, by the Fourier transform of the autocorrelation function of the stochastic perturbation:
231
relaxation process is described by means of a Master Equation from which the time evolution of the eigenstates populations are obtained [2].
4. Results of the numerical simulations In order to model such a system, we simply consider in what follows a linear chain of N"100 parallel molecules coupled through long range dipolar interactions. In Fig. 1, the average number of sites covered by the excitation is represented
P
`= G (t)e*uklt dt, (10a) kl 0 G (t)"SSkDH (t)DlTSlDH (0)DkTT (10b) kl 1 1 A7. In Eqs. (10a) and (10b), u "(E !E )/+ and ;0 is kl kl k l an average transition rate which takes into account all exciton-phonon couplings. Assuming +u <+j, kl integration of Eq. (10a) leads to
;0 "(1/+2)Re kl
;0 "*2j(+ DCkD2DClD2/(+u )2). (11) kl i i kl i Furthermore, when the condition D;+u is valid, kl we get +;0 ;+j, meaning that the residence time kl of excitons (1/;0 ) is longer than q . The intraband kl # scattering between the collective states occurs by resonant transfer followed by phonon absorption or emission processes. We assume ; (u '0)";0 , kl kl kl (12) ; (u (0)";0 exp(+u /k ¹) kl kl B kl kl for downhill and uphill transfer, respectively, in order to satisfy the Boltzmann equilibrium condition. After initial excitation of the exciton band the
Fig. 1. Localization length of excitons for a linear chain of N"100 parallel molecules coupled through long-range dipolar interactions. The spatial extension of excitons at energy E is k represented by segments of length (1/¸ ) centered around the k mean exciton position. Weak disorder (p/< "0.01) and strong 0 disorder (p/< "0.1). 0
232
J.P. Lemaistre / Journal of Luminescence 83}84 (1999) 229}233
Fig. 3. Time evolution of the population decrease of the upper state (full lines) and of the corresponding population increase of the lower state (dotted lines) for D (units of < ) "0.3 (a), 0.5 (b), 0 0.7 (c), 1 (d).
Fig. 2. Participation ratio of the exciton states versus energy within the band calculated by cumulating 50 realizations of the static disorder corresponding to the p/< values of Fig. 1. 0
by segments centered around the mean exciton position for a weak disorder (p/< "0.01) and 0 a relatively strong disorder (p/< "0.1). The length 0 of the segments equals the participation ratio (1/¸ ) k for each collective state. The eigenstates energy is expressed in units of < , the nearest-neighbor inter0 action. In Fig. 2, the participation ratios of the exciton states are calculated as a function of the energy within the band by cumulating the results over 50 realizations of the disorder corresponding to the p/< values of Fig. 1. 0 After initial excitation of the upper state carrying the oscillator strength (if <'0), population relaxation occurs among the eigenstates of the exciton
band at rates proportional to *2j. Assuming the #uctuation amplitude, D, to be temperature dependent, a simulation of the temperature e!ect on the exciton relaxation can be achieved by analyzing the time dependence of the decrease (increase) of the upper (lower) exciton state within the band. Such an e!ect is illustrated in Fig. 3 for several values of D. We have presented a simple model to analyze the role of static disorder on the localization length of excitons in a simple linear chain and provide an expression for the intraband scattering induced by the dynamic disorder. Such a model could be easily used to investigate disorders e!ects and intraband scattering in molecular aggregates exhibiting strong intermolecular interactions.
References [1] C. Eco!et, D. Markovitsi, Ph. MillieH , J.P. Lemaistre, Chem. Phys. 177 (1993) 629. [2] J.P. Lemaistre, J. Lumin. 76/77 (1998) 437. [3] I. Barvik, P. Reineker, Ch. Warns, Th. Neidlinger, J. Lumin. 65 (1995) 169. [4] E.O. Potma, D.A. Wiersma, J. Chem. Phys. 108 (1998) 4894. [5] M. Shimizu, S. Suto, T. Goto, A. Watanabe, M. Matsuda, Phys. Rev. B 58 (1998) 5032.
J.P. Lemaistre / Journal of Luminescence 83}84 (1999) 229}233 [6] R. Kumble, R.M. Hochstrasser, J. Chem. Phys. 109 (1998) 855. [7] V.A. Malyshev, P. Moreno, Phys. Rev. B 51 (1995) 14 587. [8] M. Wubs, J. Knoester, Chem. Phys. Lett. 284 (1998) 63.
233
[9] M. Schreiber, Y. Toyozawa, J. Phys. Soc. Japan 51 (1982) 1537. [10] P.W. Anderson, J. Phys. Soc. Japan 9 (1954) 316. [11] R. Kubo, K. Tomita, J. Phys. Soc. Japan 9 (1954) 888.