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Cooperative disordering processes in J-aggregates – difference absorption spectra caused by dynamical disorder
Journal of Luminescence 83}84 (1999) 265}270
Cooperative disordering processes in J-aggregates } di!erence absorption spectra caused by dynamical disorder K.-H. Feller!,*, E. Gaiz\ auskas" !University of Applied Sciences Jena, Tatzendpromenade 1b, 07745 Jena, Germany "Vilnius University, Laser Research Center, SauleR tekio al.10, 2054 Vilnius, Lithuania
Abstract The strong excitation-induced changes in J-aggregates, when taking into account the disorder, produced by the exciton}exciton annihilation due to the excess energy redistribution, are discussed. Theoretical analysis of the pump intensity-dependent di!erential absorption spectra on the basis of the presented o!-diagonal disordering model is made, concluding that the intensity-dependent blue shift of di!erence absorption spectra may result from the disorder induced by the degradation of the excess energy after the exciton}exciton annihilation event. ( 1999 Elsevier Science B.V. All rights reserved. Keywords: J-aggregates; Exciton}exciton annihilation; Dynamical disorder; Pump}probe spectrum
1. Introduction The structure of the aggregate chain (order/ disorder, imperfects, #uctuating energy barriers from one monomer unit to the adjacent one) is connected to a variety of e!ects, well manifested e.g. in the transient absorption or the degenerate four wave mixing spectra. Theoretical work has been done to recover the hidden e!ects in these transient spectra and to gain information on the cooperative processes (exciton delocalization length [1,2], coherent domain [3], dynamic disorder as transient e!ects [4] as well as static disorder, local structure features a!ecting the energy migration processes as static e!ects).
* Corresponding author. Tel.: #49-3641-205621; fax: #493641-205601. E-mail address: [email protected] (K.-H. Feller)
It was also suggested quite recently [5] that some rearrangement of inhomogeneity in the ensemble of J-aggregates during excitation may be responsible for the unusual spectroscopic signals obtained in pump}probe experiments. It should be stressed on this occasion, that optical line shapes of inhomogeneous (disordered) aggregates were analysed in a number of papers [6}8]. These comprehensive studies of the line shapes and pump}probe spectra of molecular aggregates with &static' disorder, however, leave many questions concerning the time- and intensity-dependent pump}probe spectra, still open. As found in Refs. [9,10], the e!ect of exciton} exciton annihilation can be of great importance in forming the optical response of the aggregate subsystem as the pump intensity rises. Evidently, annihilation a!ects the optical response a great deal under strong excitation conditions. Actually, the energy degradation processes (after the annihilation
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 5 9 - 3
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event) due to the redistribution of the excess energy over the vibrational and rotational modes, may result in the rotation of the aggregate moiety around its axis, producing change in inhomogeneity. Since the spectra of the disordered aggregates (eigenvalues of the eigenstates as well as the transition strengths to the "rst and second excited manifolds) depend on the amount of disorder, this dynamic disordering will strongly a!ect the di!erential absorption spectra of linear aggregates. The main goal of this contribution is to examine how such a dynamic disordering a!ects the pump}probe spectra after the strong excitation. For this purpose we will focus here on a simple model of the dynamic disordering, described in detail in the next section. Here, the main steps in the calculation of the spectra of the disordered aggregate relevant to the problem under investigation are also discussed. Finally, speci"c numerical calculations on the proposed model and discussions are given in the Section 3, followed by conclusions.
2. Background of the model and calculation procedure The linear excitonic spectra of the media consisting of an ensemble of aggregates, are known to be formed from collectivized eigenstates in many excitonic manifolds [11}14]. The stick spectra (eigenvalues of the collectivized states e , and correk sponding transitions) in these manifolds for N perfectly ordered (homogeneous) linear aggregates with the given number N of aggregated molecules and intermolecular interaction J are given by wellknown analytical expressions [11,12]. In fact, the aggregate chain never becomes ideal and contains a certain number of rotated molecules and molecular segments. This &static' disorder reduces to a certain extent the delocalization of excitation or, in other words, the coherence length within the aggregate. On the other hand the excess energy produced by the exciton}exciton annihilation will create &dynamic' disorder e!ects, i.e. it reduces the coherence length within strongly excited aggregates. In this short report we will restrict ourselves to a speci"c model of the correlated o!-diagonal dis-
order, assuming that the intermolecular interaction for (n ) pairs of the neighbouring molecules is $ reduced. Such a model corresponds to the cooperative rearrangement of the orientation of the molecular dipoles over the de"nite domains, unlike the uncorrelated disordering of molecular aggregate. Now, the spectra of the disordered aggregate will be calculated by considering the unperturbed Hamiltonian of the aggregate with the n o!$ diagonal defects of the form N N H " + + H DnTSmD, (1) 0 mn m/1 n/1 where H "SeTd #J and summation runs mn m,n n,m over the n (n"1, 22N) possible one-molecule excitations DnT of the aggregate. Here SeT stands for the molecular excitation energy, and J is the nm intermoleculer interaction between molecules n and m. (Further we restrict our consideration to nearest-neighbour interactions.) The n of these $ (randomly chosen) elements will be reduced to 0, according to the model of the disordering described above. Henceforth, the eigenstates and eigenvectors for the particular realization of the disorder will be found by diagonalizing the N]N matrix H nm numerically. Then the kth eigenvalue e gives k the energy of the eigenstate k, whereas the kth eigenvector / ,M/ , / ,2, / N speci"es its k k1 k1 kN wavefunction in k-space: N Dt T" + / DnT. (2) k kn n/1 It is straightforward now to evaluate the spectra resulting from transitions among ground-state, one- and two-exciton bands. Let us introduce the electric dipole moment operator k( "k+N D0TSnD, n/1 with k being transition dipole moment of the monomer molecule. Assuming that the common picture of fermionization of excitons [15,1] is valid, the explicit expressions for quantities of interest are: M "SkDk( D0T } electric dipole matrix elements 0k from the ground state to the kth state in the "rst manifold, and: M "SqDk( DkT } electric dipole kq matrix element from the kth state in the "rst manifold to the state characterized by excitonic &wave numbers' q,(k , k ) in the second manifold which 1 2
K.-H. Feller, E. Gaiz\ auskas / Journal of Luminescence 83}84 (1999) 265}270
may be calculated in a straightforward way by using the expansion coe$cients / of the Eq. (2) kn and reads as follows: N M "k + / , 0k kn n/1 N N~1 + (/ 1 #/ 2 ) M "k + k,n k,n kq n1 /1 n2 /n1 `1 ](/ 1 1 / 2 2 !/ 2 1 / 1 2 ) (3) k ,n k ,n k ,n k ,n In order to describe the light-induced changes in the system under consideration the evolution of the system after the excitation has to be considered. Taking the collectivized states Dt T, DqT together k with a ground state D0T as the basis for representation, this evolution can be described in the usual way [16}18] by the Liouville equation of the density matrix, giving the following equations for the slowly varying amplitudes of material coherences: Lo0,1 kl "iD0,1o0,1 kl kl Lt M0,1 #i+ kl E(o0,1!o0,1)!C o0,1, kk kl kl jj + k,l e0,1!e0,1 k !u, kOl"12N, q D0,1" l kl +
(4)
and populations: N Lo0 N M0 00 "2i + k0 [EHo0 !Eo0 ]#c + o0 o0 , kk kk 0k k0 Lt + k/1 k/1 (5) Lo1 N M1 00 "2i + k0 [EHo1 !Eo1 ], (6) 0k k0 Lt + k/1 Lo0 N M0 kk "2i + k0 [EHo0 !Eo0 ]#w o0 0k k`1,k k`1,k`1 k0 Lt + k/1 (7) o0 !2c + No0 o0 , !w kk kk k,k~1 kk k/1 Lo1 N M1 kk "2i + i0 [EHo1 !Eo1 ]#w o1 1k k`1,k k`1,k`1 k1 Lt + k/1 N (8) !w o1 #c + o0 o0 , kk kk k,k~1 kk k/1 when the following conservation law holds: N + + oj "1 kk k/0 j/0,1
(9)
267
and E stands for the laser (pump and probe) "elds. By deriving Eqs. (4)}(8) of the non-diagonal matrix, elements were introduced in a usual way and rotating wave approximation was made. Here again the relaxation processes (relaxation of polarization as well as population annihilation) with characteristic constants C , w and c were taken kl kl into account by adding additional relaxation terms in the form C o , w o and co0 o0 , phenomkk kk kl kl kl kk enologically. Additionally, a direct transformation of the excess energy after the annihilation in a separated aggregate, on which the annihilation event had to occur, was taken into account when writing Eqs. (4)}(8). In fact, the sudden redistribution of the excess energy in the separate aggregate during annihilation event can be considered as a creation of new (disordered) aggregate whereas the former one disappears. In order to consider such a possibility, two species of the aggregate: &hot' (vibrationally excited, the material variables of which were labelled with the index &1' in Eqs. (4)}(8) and &cold' one (labelled with the index &0'), having di!erent initial ground state populations: 0 and 1, respectively, have been included in the equations. As observed previously, we concentrate on the e!ects of strong excitation, restricting our considerations to relatively &long' (ps) pulses and fast relaxation processes. In fact, within the annihilation event a two-fold excited state occurs, i.e. a state of the second excitonic energy manifold. As the former excitation state relaxes to the "rst excitonic manifold in less then a hundred of femtoseconds, we disregard this relaxation process when treating difference absorption spectra. In other words, by considering the energy degradation from the second excited state to be on the timescale of hundred femtoseconds, the exciton}exciton annihilation event is considered here as the one during which two population quanta disappear in the "rst exited state manifold, creating at the same time the population on the ground state of the initial (&cold') aggregate and the excited state of the disordered (&hot') one. Now, by diagonalizing the matrix H for a cern,m tain number (50 000 in our calculations) of disorder realizations, the appropriate signal *A(u, q) at
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a given frequency and delay between the pump and probe pulses an determined by means of the probe pulse electric "eld and polarization of the sample as follows: *A(u, q)&ImME (u, q)SPH(u)TN, P
(10)
where E (u, q) and PH(u)) are Fourier-transforms P of the probe "eld and the polarization (giving by the summation over the o!-diagonal matrix elements), respectively. Asterisk in Eq. (10) means the complex conjugation and S2T stands for the average over the ensemble of the disordered aggregates.
3. Numerical simulations: results and discussion Let us note, "rstly, that the eigenvalues and eigenvectors of the matrix H di!er for the parnm ticular realization of the disorder, the mean values Se T of the spectral components and corresponding k mean transition strengths SDM D2T should be 0k evaluated in this case. In Fig. 1, the stick spectra averaged over the 50 000 realizations of disordered aggregate with n "1 and 10 (N"40, J" $ !700 cm~1, e"21.2]103 cm~1) are shown. It may be stated from this graph, that limited number of the eigenvalues are responsible for the spectra of the disordered aggregate under the condition considered. We have restricted ourselves to 5 of them in our calculations. Moreover, the spectra of such an aggregate are expected to be blue-shifted and broadened (preferable to the blue side also) as compared to the spectra of perfectly ordered aggregate. By the numerical analysis of the transient spectra according to the Eqs. (4)}(8) and (10), the light pulses were taken to be with intensity full-width at half-maximum (FWHM) durations q "1 ps for L the pump pulse and q "0.02 ps for the probe. We P assume carrying frequency of the pump to be in the vicinity of the dominant transition (maximum) of the J-aggregate. As the intensity measure for both pump and probe pulses the non-dimensional parameter h"fracM0 +:E dt was determined for the 01 dominant transition of the initial agregate. In our calculations these values were assumed to be rise from h"0.01 (linear excitation) up to h"1. The
Fig. 1. Stick spectra (oscillator strengths) of the initial aggregate (N"40, n "1, black columns) and those with n "10, i.e. $ $ containing o!-diagonal disorder produced by 10 rotations (grey columns, averaged over 50 000 realizations).
Fig. 2. Normalized spectra (inverted for the illustration purposes) of an initial aggregate N"40, n "1 (solid line), and $ those containing additionally induced o!-diagonal disorder: n "5, 10 (dashed and dotted lines, respectively). $
following fundamental rate parameters were also used: w "50 psv1, C "0.06 ps, C " i, i~1 0, i i, q 0.04 ps, (i"1,2N). In Fig. 2 the linear spectra (as measured with the probe pulse, without pump) for initial (n "1) $%& and disordered (n "10) aggregate (inverted and $%& normalized for illustration purposes), demonstrating the substantial blue shift due to disordering, is presented. It should be noted, that according to the
K.-H. Feller, E. Gaiz\ auskas / Journal of Luminescence 83}84 (1999) 265}270
model considered, the disordered (&hot') aggregate appears in the excited state, so that the stimulated emission from this state will contribute to the blue shift of the di!erential absorption spectra. On the other hand, the reduced population di!erence of the &cold' aggregates after the annihilation will reduce bleaching of the initial spectra, which will contribute additionally to the blue shift of the differential absorption spectra. In Fig. 3 the wavelength dependencies of the pump intensity-dependent di!erential absorption line shapes calculated from Eqs. (4)}(8) taking into account excess energy produced dynamic disorder, are presented. The calculations are made in the absence of the delay between the pump and probe pulse. The pump parameter rises here from h"0.25 (upper line) to h"2 in the order of increasing di!erence absorption depth. As predicted by considering the spectra of the disordered aggregates (see Fig. 2), the di!erence spectrum of strongly excited J-aggregates are blue shifted as a result of both reduced bleaching of the spectra of initial aggregate due to the exciton}exciton annihilation and appearance of the blue shifted stimulated emission of the disordered aggregate. Consequently, the maximum of di!erence absorption depth appears to be at the wavelength between the dominant transitions to the bottom states in the "rst excited state manifolds of the initial and disordered aggregates. It is worthwhile to note that the di!erence absorption dip in Fig. 3 starts at a lower frequency (showed by the dashed line here) and shifts to the blue side with respect to the frequency of dominant transition of the initial aggregate. This observation, as well as the considerable broadening of the di!erence absorption spectra, is quite in accordance with the above-mentioned experimental "ndings [5,19]. It is worth mentioning that the same basic tendencies of the di!erential absorption spectra at high pump intensities were mentioned in earlier papers also [20,21]. The obtained good accordance between the theoretical calculations and the experimental observations allows us to conclude that the dynamic disordering e!ects in J-Aggregates are crucial for the spectral pecularities and have to be considered at high pump intensities.
269
Fig. 3. Wavelength dependencies of the di!erential absorption line shapes calculated for the proposed excess energy degradation model. The model accounts for the n "10 rotations pro$ duced disorder after the annihilation event on the one separate aggregate with N"40 (averaged over 50 000 realizations). The pump parameter h"0.25, 0.5, 1,2 rises in the order of increasing di!erence absorption depth.
4. Conclusion We have examined the dynamic disordering effects on the di!erence absorption spectra by considering the simple model with two species of aggregates, one of which is absent initially, and appears in its excited state after the annihilation event. In fact, such a model corresponds to the direct transformation of the excess energy in a separated aggregate, on which the annihilation event had to occur. The sudden &heating' of the separate aggregate has been considered here as a creation of a new (disordered) aggregate whereas the former one disappears. On the other hand, the other pathway for the excess energy degradation after the annihilation, during which the system undergoes bath assisted relaxation (causes in the overall heating of temperature in the ensemble of aggregates), is also possible. In this case modelling of the disordering in the system is straightforward and can be achieved simply by changing the diagonal or (and) o!-diagonal dispersion parameters p , p accordE J ing to the excess energy produced in the system. In conclusion, it should be stressed on this occasion, that di!erent pathways of the energy degradation processes in an aggregate should expect to give
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K.-H. Feller, E. Gaiz\ auskas / Journal of Luminescence 83}84 (1999) 265}270
speci"c di!erence absorption spectra, which can be used to identify the energy degradation processes in the ensemble of aggregates. The consideration of these problem is under progress.
Acknowledgements This work was supported by the Federal Ministry for Education, Science, Research and Technology (BMBF) within the TRANSFORM project.
References [1] G. Juzeliunas, Z. Phys. D8 (1988) 379. [2] M. Burgel, D.A. Wiersma, K. Duppen, J. Chem. Phys. 102 (1995) 20. [3] J. Moll, S. Daehne, J.R. Durrant, D.A. Wiersma, J. Chem. Phys. 102 (1995) 6362. [4] L.D. Bakalis, M. Coca, J. Knoester, J. Chem. Phys. 110 (1999) 2208. [5] F. Sasaki, Sh. Kobayashi, J. Lumin. 72}74 (1997) 538. [6] E.W. Knapp, Chem. Phys. 67 (1984) 73.
[7] H. Fidder, J. Knoester, D.A. Wiersma, J. Chem. Phys 95 (1991) 7880. [8] Ch. Warns, I. Barvik, P. Reineker, Th. Neidlinger, Chem. Phys. 194 (1995) 117. [9] V. SundstroK m, T. Gillbro, R.A. Gadonas, A. Piskarskas, J. Chem. Phys. 89 (1988) 2754. [10] H. Stiel, K. Daehne, Teuchner, J. Lumin. 39 (1988) 351. [11] F.C. Spano, Phys. Rev. B 46 (1992) 13017. [12] F. Spano, Chem. Phys. Lett. 220 (1994) 365. [13] F. Spano, J. Chem. Phys. 103 (1995) 5939. [14] J. Knoester, F.C. Spano, in: T. Kobayashi (Ed.), J-Aggregates, World Scienti"c, Singapore, 1996. [15] D. Chesnut, A. Suna, J. Chem. Phys. 39 (1963) 146. [16] K. Blum, Density Matrix Theory and Applications, Plenum Press, New York, 1981. [17] E. Gaiz\ auskas, L. Valkunas, J. Phys. Chem. 101 (1997) 7321. [18] E. Gaiz\ auskas, K.-H. Feller, Photochem. Photobiol. 66 (1997) 611. [19] A.E. Johnson, S. Kumazaki, K. Yoshihara, Chem. Phys. Lett. 211 (1993) 511. [20] R. Gadonas, R. Danielius, A. Piskarskas, S. Rentsch, Izv. Akad. Nauk SSSR Fiz. 47 (1983) 2445 [Eng. Transl.: Bull. Acad. Sci. USSR Phys. 47 (No. 12) (1983) 151]. [21] R. Gadonas, R. Danielius, A. Piskarskas, I. Gillbro, V. SundstroK m, in: Z. Rudzikas, A. Piskarskas, R. Baltramiejunas (Eds.), Ultrafast Phenomena in Spectroscopy, World Scienti"c, Singapore, 1986, p. 411.
Cooperative disordering processes in J-aggregates } di!erence absorption spectra caused by dynamical disorder K.-H. Feller!,*, E. Gaiz\ auskas" !University of Applied Sciences Jena, Tatzendpromenade 1b, 07745 Jena, Germany "Vilnius University, Laser Research Center, SauleR tekio al.10, 2054 Vilnius, Lithuania
Abstract The strong excitation-induced changes in J-aggregates, when taking into account the disorder, produced by the exciton}exciton annihilation due to the excess energy redistribution, are discussed. Theoretical analysis of the pump intensity-dependent di!erential absorption spectra on the basis of the presented o!-diagonal disordering model is made, concluding that the intensity-dependent blue shift of di!erence absorption spectra may result from the disorder induced by the degradation of the excess energy after the exciton}exciton annihilation event. ( 1999 Elsevier Science B.V. All rights reserved. Keywords: J-aggregates; Exciton}exciton annihilation; Dynamical disorder; Pump}probe spectrum
1. Introduction The structure of the aggregate chain (order/ disorder, imperfects, #uctuating energy barriers from one monomer unit to the adjacent one) is connected to a variety of e!ects, well manifested e.g. in the transient absorption or the degenerate four wave mixing spectra. Theoretical work has been done to recover the hidden e!ects in these transient spectra and to gain information on the cooperative processes (exciton delocalization length [1,2], coherent domain [3], dynamic disorder as transient e!ects [4] as well as static disorder, local structure features a!ecting the energy migration processes as static e!ects).
* Corresponding author. Tel.: #49-3641-205621; fax: #493641-205601. E-mail address: [email protected] (K.-H. Feller)
It was also suggested quite recently [5] that some rearrangement of inhomogeneity in the ensemble of J-aggregates during excitation may be responsible for the unusual spectroscopic signals obtained in pump}probe experiments. It should be stressed on this occasion, that optical line shapes of inhomogeneous (disordered) aggregates were analysed in a number of papers [6}8]. These comprehensive studies of the line shapes and pump}probe spectra of molecular aggregates with &static' disorder, however, leave many questions concerning the time- and intensity-dependent pump}probe spectra, still open. As found in Refs. [9,10], the e!ect of exciton} exciton annihilation can be of great importance in forming the optical response of the aggregate subsystem as the pump intensity rises. Evidently, annihilation a!ects the optical response a great deal under strong excitation conditions. Actually, the energy degradation processes (after the annihilation
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 5 9 - 3
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K.-H. Feller, E. Gaiz\ auskas / Journal of Luminescence 83}84 (1999) 265}270
event) due to the redistribution of the excess energy over the vibrational and rotational modes, may result in the rotation of the aggregate moiety around its axis, producing change in inhomogeneity. Since the spectra of the disordered aggregates (eigenvalues of the eigenstates as well as the transition strengths to the "rst and second excited manifolds) depend on the amount of disorder, this dynamic disordering will strongly a!ect the di!erential absorption spectra of linear aggregates. The main goal of this contribution is to examine how such a dynamic disordering a!ects the pump}probe spectra after the strong excitation. For this purpose we will focus here on a simple model of the dynamic disordering, described in detail in the next section. Here, the main steps in the calculation of the spectra of the disordered aggregate relevant to the problem under investigation are also discussed. Finally, speci"c numerical calculations on the proposed model and discussions are given in the Section 3, followed by conclusions.
2. Background of the model and calculation procedure The linear excitonic spectra of the media consisting of an ensemble of aggregates, are known to be formed from collectivized eigenstates in many excitonic manifolds [11}14]. The stick spectra (eigenvalues of the collectivized states e , and correk sponding transitions) in these manifolds for N perfectly ordered (homogeneous) linear aggregates with the given number N of aggregated molecules and intermolecular interaction J are given by wellknown analytical expressions [11,12]. In fact, the aggregate chain never becomes ideal and contains a certain number of rotated molecules and molecular segments. This &static' disorder reduces to a certain extent the delocalization of excitation or, in other words, the coherence length within the aggregate. On the other hand the excess energy produced by the exciton}exciton annihilation will create &dynamic' disorder e!ects, i.e. it reduces the coherence length within strongly excited aggregates. In this short report we will restrict ourselves to a speci"c model of the correlated o!-diagonal dis-
order, assuming that the intermolecular interaction for (n ) pairs of the neighbouring molecules is $ reduced. Such a model corresponds to the cooperative rearrangement of the orientation of the molecular dipoles over the de"nite domains, unlike the uncorrelated disordering of molecular aggregate. Now, the spectra of the disordered aggregate will be calculated by considering the unperturbed Hamiltonian of the aggregate with the n o!$ diagonal defects of the form N N H " + + H DnTSmD, (1) 0 mn m/1 n/1 where H "SeTd #J and summation runs mn m,n n,m over the n (n"1, 22N) possible one-molecule excitations DnT of the aggregate. Here SeT stands for the molecular excitation energy, and J is the nm intermoleculer interaction between molecules n and m. (Further we restrict our consideration to nearest-neighbour interactions.) The n of these $ (randomly chosen) elements will be reduced to 0, according to the model of the disordering described above. Henceforth, the eigenstates and eigenvectors for the particular realization of the disorder will be found by diagonalizing the N]N matrix H nm numerically. Then the kth eigenvalue e gives k the energy of the eigenstate k, whereas the kth eigenvector / ,M/ , / ,2, / N speci"es its k k1 k1 kN wavefunction in k-space: N Dt T" + / DnT. (2) k kn n/1 It is straightforward now to evaluate the spectra resulting from transitions among ground-state, one- and two-exciton bands. Let us introduce the electric dipole moment operator k( "k+N D0TSnD, n/1 with k being transition dipole moment of the monomer molecule. Assuming that the common picture of fermionization of excitons [15,1] is valid, the explicit expressions for quantities of interest are: M "SkDk( D0T } electric dipole matrix elements 0k from the ground state to the kth state in the "rst manifold, and: M "SqDk( DkT } electric dipole kq matrix element from the kth state in the "rst manifold to the state characterized by excitonic &wave numbers' q,(k , k ) in the second manifold which 1 2
K.-H. Feller, E. Gaiz\ auskas / Journal of Luminescence 83}84 (1999) 265}270
may be calculated in a straightforward way by using the expansion coe$cients / of the Eq. (2) kn and reads as follows: N M "k + / , 0k kn n/1 N N~1 + (/ 1 #/ 2 ) M "k + k,n k,n kq n1 /1 n2 /n1 `1 ](/ 1 1 / 2 2 !/ 2 1 / 1 2 ) (3) k ,n k ,n k ,n k ,n In order to describe the light-induced changes in the system under consideration the evolution of the system after the excitation has to be considered. Taking the collectivized states Dt T, DqT together k with a ground state D0T as the basis for representation, this evolution can be described in the usual way [16}18] by the Liouville equation of the density matrix, giving the following equations for the slowly varying amplitudes of material coherences: Lo0,1 kl "iD0,1o0,1 kl kl Lt M0,1 #i+ kl E(o0,1!o0,1)!C o0,1, kk kl kl jj + k,l e0,1!e0,1 k !u, kOl"12N, q D0,1" l kl +
(4)
and populations: N Lo0 N M0 00 "2i + k0 [EHo0 !Eo0 ]#c + o0 o0 , kk kk 0k k0 Lt + k/1 k/1 (5) Lo1 N M1 00 "2i + k0 [EHo1 !Eo1 ], (6) 0k k0 Lt + k/1 Lo0 N M0 kk "2i + k0 [EHo0 !Eo0 ]#w o0 0k k`1,k k`1,k`1 k0 Lt + k/1 (7) o0 !2c + No0 o0 , !w kk kk k,k~1 kk k/1 Lo1 N M1 kk "2i + i0 [EHo1 !Eo1 ]#w o1 1k k`1,k k`1,k`1 k1 Lt + k/1 N (8) !w o1 #c + o0 o0 , kk kk k,k~1 kk k/1 when the following conservation law holds: N + + oj "1 kk k/0 j/0,1
(9)
267
and E stands for the laser (pump and probe) "elds. By deriving Eqs. (4)}(8) of the non-diagonal matrix, elements were introduced in a usual way and rotating wave approximation was made. Here again the relaxation processes (relaxation of polarization as well as population annihilation) with characteristic constants C , w and c were taken kl kl into account by adding additional relaxation terms in the form C o , w o and co0 o0 , phenomkk kk kl kl kl kk enologically. Additionally, a direct transformation of the excess energy after the annihilation in a separated aggregate, on which the annihilation event had to occur, was taken into account when writing Eqs. (4)}(8). In fact, the sudden redistribution of the excess energy in the separate aggregate during annihilation event can be considered as a creation of new (disordered) aggregate whereas the former one disappears. In order to consider such a possibility, two species of the aggregate: &hot' (vibrationally excited, the material variables of which were labelled with the index &1' in Eqs. (4)}(8) and &cold' one (labelled with the index &0'), having di!erent initial ground state populations: 0 and 1, respectively, have been included in the equations. As observed previously, we concentrate on the e!ects of strong excitation, restricting our considerations to relatively &long' (ps) pulses and fast relaxation processes. In fact, within the annihilation event a two-fold excited state occurs, i.e. a state of the second excitonic energy manifold. As the former excitation state relaxes to the "rst excitonic manifold in less then a hundred of femtoseconds, we disregard this relaxation process when treating difference absorption spectra. In other words, by considering the energy degradation from the second excited state to be on the timescale of hundred femtoseconds, the exciton}exciton annihilation event is considered here as the one during which two population quanta disappear in the "rst exited state manifold, creating at the same time the population on the ground state of the initial (&cold') aggregate and the excited state of the disordered (&hot') one. Now, by diagonalizing the matrix H for a cern,m tain number (50 000 in our calculations) of disorder realizations, the appropriate signal *A(u, q) at
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a given frequency and delay between the pump and probe pulses an determined by means of the probe pulse electric "eld and polarization of the sample as follows: *A(u, q)&ImME (u, q)SPH(u)TN, P
(10)
where E (u, q) and PH(u)) are Fourier-transforms P of the probe "eld and the polarization (giving by the summation over the o!-diagonal matrix elements), respectively. Asterisk in Eq. (10) means the complex conjugation and S2T stands for the average over the ensemble of the disordered aggregates.
3. Numerical simulations: results and discussion Let us note, "rstly, that the eigenvalues and eigenvectors of the matrix H di!er for the parnm ticular realization of the disorder, the mean values Se T of the spectral components and corresponding k mean transition strengths SDM D2T should be 0k evaluated in this case. In Fig. 1, the stick spectra averaged over the 50 000 realizations of disordered aggregate with n "1 and 10 (N"40, J" $ !700 cm~1, e"21.2]103 cm~1) are shown. It may be stated from this graph, that limited number of the eigenvalues are responsible for the spectra of the disordered aggregate under the condition considered. We have restricted ourselves to 5 of them in our calculations. Moreover, the spectra of such an aggregate are expected to be blue-shifted and broadened (preferable to the blue side also) as compared to the spectra of perfectly ordered aggregate. By the numerical analysis of the transient spectra according to the Eqs. (4)}(8) and (10), the light pulses were taken to be with intensity full-width at half-maximum (FWHM) durations q "1 ps for L the pump pulse and q "0.02 ps for the probe. We P assume carrying frequency of the pump to be in the vicinity of the dominant transition (maximum) of the J-aggregate. As the intensity measure for both pump and probe pulses the non-dimensional parameter h"fracM0 +:E dt was determined for the 01 dominant transition of the initial agregate. In our calculations these values were assumed to be rise from h"0.01 (linear excitation) up to h"1. The
Fig. 1. Stick spectra (oscillator strengths) of the initial aggregate (N"40, n "1, black columns) and those with n "10, i.e. $ $ containing o!-diagonal disorder produced by 10 rotations (grey columns, averaged over 50 000 realizations).
Fig. 2. Normalized spectra (inverted for the illustration purposes) of an initial aggregate N"40, n "1 (solid line), and $ those containing additionally induced o!-diagonal disorder: n "5, 10 (dashed and dotted lines, respectively). $
following fundamental rate parameters were also used: w "50 psv1, C "0.06 ps, C " i, i~1 0, i i, q 0.04 ps, (i"1,2N). In Fig. 2 the linear spectra (as measured with the probe pulse, without pump) for initial (n "1) $%& and disordered (n "10) aggregate (inverted and $%& normalized for illustration purposes), demonstrating the substantial blue shift due to disordering, is presented. It should be noted, that according to the
K.-H. Feller, E. Gaiz\ auskas / Journal of Luminescence 83}84 (1999) 265}270
model considered, the disordered (&hot') aggregate appears in the excited state, so that the stimulated emission from this state will contribute to the blue shift of the di!erential absorption spectra. On the other hand, the reduced population di!erence of the &cold' aggregates after the annihilation will reduce bleaching of the initial spectra, which will contribute additionally to the blue shift of the differential absorption spectra. In Fig. 3 the wavelength dependencies of the pump intensity-dependent di!erential absorption line shapes calculated from Eqs. (4)}(8) taking into account excess energy produced dynamic disorder, are presented. The calculations are made in the absence of the delay between the pump and probe pulse. The pump parameter rises here from h"0.25 (upper line) to h"2 in the order of increasing di!erence absorption depth. As predicted by considering the spectra of the disordered aggregates (see Fig. 2), the di!erence spectrum of strongly excited J-aggregates are blue shifted as a result of both reduced bleaching of the spectra of initial aggregate due to the exciton}exciton annihilation and appearance of the blue shifted stimulated emission of the disordered aggregate. Consequently, the maximum of di!erence absorption depth appears to be at the wavelength between the dominant transitions to the bottom states in the "rst excited state manifolds of the initial and disordered aggregates. It is worthwhile to note that the di!erence absorption dip in Fig. 3 starts at a lower frequency (showed by the dashed line here) and shifts to the blue side with respect to the frequency of dominant transition of the initial aggregate. This observation, as well as the considerable broadening of the di!erence absorption spectra, is quite in accordance with the above-mentioned experimental "ndings [5,19]. It is worth mentioning that the same basic tendencies of the di!erential absorption spectra at high pump intensities were mentioned in earlier papers also [20,21]. The obtained good accordance between the theoretical calculations and the experimental observations allows us to conclude that the dynamic disordering e!ects in J-Aggregates are crucial for the spectral pecularities and have to be considered at high pump intensities.
269
Fig. 3. Wavelength dependencies of the di!erential absorption line shapes calculated for the proposed excess energy degradation model. The model accounts for the n "10 rotations pro$ duced disorder after the annihilation event on the one separate aggregate with N"40 (averaged over 50 000 realizations). The pump parameter h"0.25, 0.5, 1,2 rises in the order of increasing di!erence absorption depth.
4. Conclusion We have examined the dynamic disordering effects on the di!erence absorption spectra by considering the simple model with two species of aggregates, one of which is absent initially, and appears in its excited state after the annihilation event. In fact, such a model corresponds to the direct transformation of the excess energy in a separated aggregate, on which the annihilation event had to occur. The sudden &heating' of the separate aggregate has been considered here as a creation of a new (disordered) aggregate whereas the former one disappears. On the other hand, the other pathway for the excess energy degradation after the annihilation, during which the system undergoes bath assisted relaxation (causes in the overall heating of temperature in the ensemble of aggregates), is also possible. In this case modelling of the disordering in the system is straightforward and can be achieved simply by changing the diagonal or (and) o!-diagonal dispersion parameters p , p accordE J ing to the excess energy produced in the system. In conclusion, it should be stressed on this occasion, that di!erent pathways of the energy degradation processes in an aggregate should expect to give
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speci"c di!erence absorption spectra, which can be used to identify the energy degradation processes in the ensemble of aggregates. The consideration of these problem is under progress.
Acknowledgements This work was supported by the Federal Ministry for Education, Science, Research and Technology (BMBF) within the TRANSFORM project.
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