© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

phys. stat. sol. (b) 245, No. 12, 2667–2670 (2008) / DOI 10.1002/pssb.200879861 p s sbasic solid state physics

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Exciton energy transfer in mixed J-aggregates

J. P. Lemaistre*

INSP, Université Pierre et Marie Curie and CNRS, UMR 7588, Campus Boucicaut, 140, rue de Lourmel, 75015 Paris, France

Received 16 June 2008, revised 24 July 2008, accepted 25 July 2008

Published online 22 October 2008

PACS 36.20.Kd, 71.35.Aa, 78.20.Bh, 78.30.Ly

* e-mail lemaistre@insp.jussieu.fr, Phone: +33 (1) 44 27 42 66, Fax: +33 (1) 43 54 28 78

© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction Many experimental and theoretical studies are currently investigated in molecular J-aggregates because of their remarkable properties. Because of the short stacking distance between the molecules, as com-pared to the molecule size, the intermolecular interactions, due to the transition dipole-transition dipole couplings, are relatively large. Thus, the photophysical properties of such organized molecular assemblies are strongly de-pendent on their geometrical configuration (linear, two dimensional, circular, cylindrical or tubular). The most ex-tensively studied system in the literature is certainly the pseudoisocyanine (PIC) aggregate in the one dimensional configuration. Recently, absorption and emission studies on a linear chain of PIC molecules, taking into account the dihedral angle between the planes of the PIC molecule, was reported and interpreted by using a unit cell with four molecules [1]. Furthermore, the PIC aggregates ex-hibit two close transition bands corresponding to two different structures called blue and red. The close resem-blance between the absorption and emission spectra of the blue and red structures could be due to the interac-

tion between two linear J-aggregates, the red band act- ing as an energy trap in this donor–acceptor configuration [2]. Another class of donor–acceptor system is provided by a two dimensional monolayer of oxacyanine (S9) mole-cules doped with a concentration of thiacyanine (S11) molecules of a similar structure [3, 4]. A two dimensional model for excitation energy transfer in pure and mixed J-aggregates was recently proposed to describe the absorp-tion and the fluorescence spectra, the localization/delocali-zation behaviour of the exciton states and the intraband and interband scatterings after initial excitation [5]. In this short communication, we present a computa-tional method to describe the excitation energy transfer and trapping mechanisms in mixed J-aggregates featuring a donor–acceptor system. We use this simulation method to analyze the energy transfer between the exciton bands of two closely spaced linear parallel chains featuring the blue and red structures of PIC aggregates. We also analyze the concentration dependence of the fluorescence emissions of the (S9/S11) mixed J-aggregate.

A computational method to analyze the exciton energy trans-

fer and trapping mechanisms in mixed J-aggregates featuring

a donor–acceptor system is provided. For a given geometri-

cal configuration, the intermolecular interactions are calcu-

lated using the extended dipole method. Numerical diagonali-

zation of the aggregate Hamiltonian is performed taking into

account the static disorder. The density of states, the absorp-

tion lineshapes and the relative number of coherently coupled

molecules within the donor and acceptor bands are obtained.

The excitation energy transfer processes among the excited

states of the donor–acceptor system (intraband and interband

scatterings) are described through an exciton-phonon cou-

pling model. The scattering rates are calculated and used in a

Master Equation to obtain the time evolution of the exciton

populations after initial excitation. Our simulation method is

used to analyze the excitation energy transfer between two

kinds of closely spaced linear parallel chains featuring the so-

called blue (donor) and red (acceptor) structures of pseudo-

isocyanine molecules (PIC). It is also applied to describe the

energy transfer in the mixed J-aggregate of oxacyanine (do-

nor) and thiacyanine (acceptor) molecules.

2668 J. P. Lemaistre: Exciton energy transfer in mixed J-aggregates

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2 Simulation method Let us consider a mixed ag-gregate of N molecules (donors and acceptors) in a given geometrical configuration. The Hamiltonian of the aggre-gated system is written in the localized site representation as

0

,

| | | | .i ij

i i j

H E i i V i j= Ò · + Ò ·Â Â (1)

Ei are the molecular site excitation energies and Vij de-scribes the electronic interactions among the sites |i⟩ and |j⟩. In the extended dipole method [6, 7],

2

0

1 1 1 15.04 ( / ) ,ij

ij ij ij ij

V ld d d d

µ++ -- +- -+

È ˘= + - -Í ˙

Î ˚ (2)

where µ0 is the amplitude of the transition moment of the molecule. The electronic transition is represented by two opposite charges +q and –q, at a distance l, so that µ0 = ql; the dij denote the distances between the charges on mole-cules i and j. In Eq. (2) all the distances are expressed in nm, the transition moments in Debye and the interactions in wavenumbers. The dipole length is a parameter which can be adjusted to fit the experimental observations. In the

delocalized representation | | |k

i

i

k C iÒ = ÒÂ of the exciton

states with energy Ek the electronic Hamiltonian reads

0| | .

k

k

H E k k= Ò ·Â (3)

The numerical diagonalization of H0 gives the eigen-states from which the oscillator strengths, fk, are obtained. Furthermore, the localization behaviour of the |k⟩ states can be analyzed through the inverse participation ratio

4| |kk i

i

L C=Â which gives the relative number rk = 1/NLk

of molecules participating to the excitation. The line shape of the absorption spectra is given by

( )abs 2 2

av

1( ) ,

π

k k

k k k

fI E

N E E

Γ

Γ

Ê ˆ= Ë ¯ - +Â (4)

where Γk is the homogeneous linewidth of the exciton state |k⟩ and the average takes into account all the realizations of the static disorder. The excitation energy transfer among the eigenstates of the donor–acceptor system (intraband and interband inco-herent diffusion) is induced by the stochastic exciton-phonon couplings. A schematic representation of the energy transfer mechanisms in the donor–acceptor system is shown in Fig. (1). In the weak and linear exciton-phonon coupling approximation, the one phonon transition rates are calculated according to the Fermi Golden Rule as

( ) ( ) if 0 ,

( ) [1 ( )] if 0 .

k l kl kl kl kl

k l kl kl kl kl

U S n

U S n

ρ ω ω ω

ρ ω ω ω

�

�

= >

= + <

(5)

Figure 1 A schematic representation of the energy transfer

mechanisms in the donor–acceptor system. Under experimental

conditions (excitation on the blue tail of the donor exciton band

and observation of the fluorescence at the bottom of the acceptor

band), the energy transfer within and between the donor and ac-

ceptor band states occurs at rates U(k, l).

Skl is a factor related to the overlap of the exciton prob-abilities and ρ(ω) = χλω/(λ2 + ω2) is the spectral density at the frequency ω in which λ describes the frequency width of the phonon modes and χ represents a mean value of the exciton-phonon coupling. n(ωkl) is the mean occupation number of the phonon mode ωkl at the temperature T. The time evolution of the exciton populations, pk (t), af-ter an initial excitation, is obtained from the solutions of the following Master equation.

d( ) ( ) ( ) ,

dk k k l k l k l

l k l k

p t U p t U p tt

γ� �

π π

Ê ˆ= - + +Á ˜Ë ¯Â  (6)

where γk is the spontaneous emission rates of the exciton state. The solutions of Eq. (6), calculated for each realiza-tion of the disorder, gives the time dependent fluorescence spectra

2

fl

av

( ) ( ) | | .k k

k

I t p t µ= Â (7)

3 Simulation results Our simulation model is now applied to describe the excitation energy transfer between two closely spaced linear chains of PIC molecules and to a two-dimensional monolayer composed of a mixing of oxa-cyanine and thiacyanine molecules. 3.1 Interacting red and blue chains We describe the PIC linear chains by using a unit cell with four mole-cules as described in [1] which takes into account the dihe-dral angle between the planar parts of the PIC molecule. We then represent the aggregate as a chain of dimers [8]. They are separated by 1.4 nm with an intermolecular dis-tance of 1.2 nm between the two molecules of the dimer.

phys. stat. sol. (b) 245, No. 12 (2008) 2669

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Original

Paper

0

0.2

0 .4

0 .6

0 .8

1 .0

425 465 505 545 585

Energy (nm)

Intens

ity(a.u.)

Figure 2 Simulated density of states (dashed line) and absorption

line shape of a linear PIC chain in the configuration with 4 mole-

cules per unit cell taking into account the dihedral angle between

the planar parts of the molecule.

We use as parameters a dipole moment of 10D, a tilt angle of 22°4, a dihedral angle of 54° and a dipole length of 0.89 nm in order to fit a nearest neighbour interaction of –600 cm–1 for the red configuration. The calculated density of states and the absorption lineshape is shown in Fig. (2). Such a structure allows reproducing the three observed peaks of the exciton band with their relative transition in-tensities [1]. Let us now consider two closely spaced blue and red linear chains. Using the same parameters with a slightly different tilt angle of 26°4 for the blue chain al-lows reproducing the measured energy separation between the two k = 0 states of the blue and red chains. We show in Fig. (3) the calculated absorption lineshapes of the two structures of the red and blue structures of PIC aggregates for several values of the distance between the chains. When they are close enough, the inter chain coupling cre-ates a mixing of the two exciton bands at an extent which depends on their relative distance, d [9]. After an initial excitation close to the k = 0 state of the blue chain acting as a donor, the excitation energy is trans-ferred to the lowest exciton state (k = 0) of the red chain acting as an acceptor. Taking a radiative lifetime of 15 ps at low temperature [2], the relative populations of the do-

0

50

100

150

200

540 560 580 600

d=1.5 nmd=2.5 nmd=5 nm

Energy (nm)

Abso

rptionintensity

(a.u.)

Figure 3 Absorption line shapes of the two interacting chains as

a function of their relative distance.

0

0.2

0.4

0.6

0.8

1.0

0 10 20 30 40 50

d=1.5 nmd=2.5 nmd=5 nm

time (ps)

populations

Figure 4 Decreasing donor and increasing acceptor populations

of two interacting linear parallel chains for various values of the

inter chain distance.

nor and acceptor exciton states are calculated from Eq. (6) and is illustrated in Fig. (4). This allows estimating the rates of energy transfer from the blue band to the red one to be 20 ps, 12 ps and 7 ps at distances 5 nm, 2.5 nm and 1.5 nm respectively.

3.2 Mixed J-aggregates of (S9/S11) molecules

We describe the mixed J-aggregate of oxacyanine (S9) and thiacyanine (S11) as a square arrangement of the mole-cules. In the (S9) donor matrix, a concentration, c, of (S11) acceptor is randomly substituted. Both (S9) and (S11) molecules have a similar structure with the thiacyanine monomer transition energy lower than that of oxacyanine of 2900 cm–1 [4]. We use as parameters for both cyanine molecules a dipole moment of 8.1 D, a tilt angle of 40°, an intermolecular distance of 1 nm and a dipole length of 0.3 nm [10]. The measured absorption spectra at room tem-perature are simulated and shown in Fig. (5), over the full range of concentrations, by using a value of 500 cm–1 for the static disorder as well as the concentration dependence of the lineshifts. Furthermore, the number of coherently coupled molecules was calculated within the exciton bands for both quantum sub systems [5]. Excitation energy transfer among all eigenstates can be calculated using Eq. (5) for each configuration of the static disorder and for each concentration. Then, the time evolu-tion of the exciton populations is obtained from the Master equation given by Eq. (6).

0

5

1 0

1 5

2 0

2 5

3 0 0 3 5 0 4 0 0 4 5 0 5 0 0

c = 0 .0 1c = 0 .6c = 0 .2

E n e r g y (n m )

Abso

rptionintensity

(a.u.)

Figure 5 Concentration dependence of the absorption line shapes

of the (S9/S11) mixed J-aggregate.

2670 J. P. Lemaistre: Exciton energy transfer in mixed J-aggregates

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0

5

10

375 400 425 450 475 500

c=0.01

c=0.08

c=0.4

c=0.2

Energy (nm)

Fluoresc

ence

intensity

(a.u.)

Figure 6 Simulated fluorescence line shapes of the (S9/S11)

mixed J-aggregate at various trap concentrations (low concentra-

tion).

After excitation of the upper lying states of the donor exciton band, a fast depopulation occurs followed by a re-distribution of the populations within the donor and accep-tor bands. Under experimental conditions (excitation on the blue tail of the donor exciton band and observation of the fluorescence at the bottom of the acceptor band) the fluorescence intensity of the mixed J-aggregate, calculated from Eq. (7) is simulated as a function of the concentration, c, and depicted in Figs. (6) and (7). We have also simu-lated, at low concentration of acceptors, the fluorescence intensity decay of the k = 0 donor state. The concentration dependence of the fluorescence decay is illustrated in Fig. (8). It can be seen that the decay of the fluorescence of the donor becomes faster when the concentration of accep-tors increases due to the donor to acceptor energy transfer as it is experimentally observed [11].

0

5

10

375 400 425 450 475 500

c=0.8

c=0.6

c=0.4

c=0.3

c=0.2

Energy (nm)

Fluoresc

ence

intensity

(a.u.)

Figure 7 Simulated fluorescence line shapes of the (S9/S11)

mixed J-aggregate at various trap concentrations (high concentra-

tion).

0

0 .2

0 .4

0 .6

0 .8

1 .0

0 200 4 00 600 800 1000

c=0 .08c=0 .03c=0 .01c=0

tim e (ps)

Fluoresc

ence

intensity

Figure 8 Simulated fluorescence decay curves of (S9/S11)

mixed J-aggregate at the k = 0 donor energy at various concentra-

tions.

It is shown that the fluorescence intensity increases from low concentration until a maximum is reached at c = 0.2. Then, it decreases with increasing concentration. At low concentration, below c = 0.2, the fluorescence intensity originates from isolated molecules or small ag-gregates. Above c = 0.2 the decrease of the fluorescence intensity is probably due to the increase of the number of coherently coupled acceptor molecules. At c = 0.2, a transition occurs in the nature of the acceptor eigenstates from localized (molecular-type) to delocalized states (band-type).

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