- Email: [email protected]
Channels of the exciton–exciton annihilation in one-dimensional aggregates at low temperature
Chemical Physics 254 (2000) 31±38
www.elsevier.nl/locate/chemphys
Channels of the exciton±exciton annihilation in onedimensional aggregates at low temperature V.A. Malyshev a,b, G.G. Kozlov b, H. Glaeske a,*, K.-H. Feller a a b
Fachhochschule Jena, Fachbereich Medizintechnik/Physikalische Technik, Carl-Zeiss-Promenade 2, D-07745 Jena, Germany National Research Center, Vavilov State Optical Institute, Birzhevaya Liniya 12, 199034 St. Petersburg, Russian Federation Received 25 October 1999
Abstract Exciton±exciton annihilation is implied to be in fact a unique channel of exciton degradation in linear molecular aggregates and conjugated polymers as the pump intensity rises. However, at low temperatures, excitons represent rather immobile than movable quasi-particles due to the localization eect. We analyze possible channels of exciton± exciton annihilation, which seem to be natural ones under the conditions of localization and low temperature, and calculate the corresponding rates. Ó 2000 Elsevier Science B.V. All rights reserved.
1. Introduction In the last decade, quasi-one-dimensional structures, such as linear molecular aggregates and conjugated polymers, attracted much attention due to the extraordinary features of their photoresponse: short spontaneous emission time and giant values of high-order susceptibilities both scaled with the size of an aggregate [1±5] (see also the reviews [6,7] and references therein), making them promising candidates in developing devices of optical logic. Recently, spectral narrowing [8,9] and cooperative emission [10,11] in p-conjugated polymer thin ®lms as well as super-radiant lasing from J-aggregated cyanine dye molecules adsorbed onto colloidal silica and silver [12,13] were re-
* Corresponding author. Tel.: +49-3641-205630; fax: +493641-205601. E-mail address: [email protected] (H. Glaeske).
ported, the eects being of great importance from the viewpoint of laser applications. Another group of interesting species is represented by aggregates of biological molecules (such as chlorophyll a, for instance) which are involved in the excitation energy transfer and charge separation in the natural photosynthetic systems [14± 17]. A common peculiarity of all these systems is that their lowest electronic excitations are Frenkel excitons mainly determining optical and transport properties of these objects. It was discovered [18,19] that with increasing pump intensities, exciton±exciton annihilation becomes more and more important, so that it strongly in¯uences the optical response of the excitonic system [18±27]. The usual framework for the analysis of this process is the bimolecular theory using rate equations, in which the rate of annihilation is proportional to the concentration of excitations [18,19,22,23]. In addition, it is assumed that two excitations approach each other before the annihilation starts.
0301-0104/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 0 1 0 4 ( 0 0 ) 0 0 0 2 0 - 3
32
V.A. Malyshev et al. / Chemical Physics 254 (2000) 31±38
This concept seems to be a good framework for the description of exciton±exciton annihilation at higher (room) temperatures when the exciton coherence length is rather short (several lattice constants [4,18,19,22,23]). The approach of the two excitons can then be treated as a series of incoherent jumps thus allowing the rate equation framework. At low temperatures, the coherence length can cover several tens of sites [1±3]. Clearly, the rate equation can no longer be used for the description of motion of excitons created within the same coherent domain. An adequate picture of the annihilation process under the condition of a suciently large exciton coherence length is then in order. In this article, we consider the case of low temperatures when the exciton coherence length is determined by static diagonal disorder. The latter results in localization of the excitonic states so that the exciton coherence length is nothing but the exciton localization length. By the term localization, we mean that an excitonic state is spread over a part of the aggregate (segment of localization) rather than over the whole one. In addition, we will assume that the strength of disorder is relatively small (the quantitative formulation will be done later), so that the localization length contains a large number of molecules (weak localization). Recall that from the viewpoint of the exciton optical dynamics, it is of importance that close to the bottom of the exciton band (the range of optically active exciton states), one may classify the localized exciton states, into several groups of few states (two or sometimes three) such that the states of each group are localized within a particular segment of the typical size N (frequently called the number of coherently bound molecules) and do not overlap with the others [28±30]. Since in the process of excitonic annihilation two excitons take part, one should take into consideration twoexciton states. As is well known (see, for instance, Ref. [6]), one-dimensional Frenkel excitons are weakly interacting fermions so that the eigenfunctions of states with two excitons can be composed of Slater determinants of single-exciton eigenfunctions. Under the condition of localization, two dierent types of two-exciton states then
appear: (i) with two excitons belonging to the same localization segment and (ii) with those localized on dierent ones. Notice that the exciton±exciton annihilation, in the former case (hereafter, intrasegment annihilation), does not require any approach of two excitons since they occupy the same segment of the aggregate. This is not the case when excitons are created on dierent segments of localization (intersegment annihilation). Here, excitons may either approach one another, jumping over dierent segments of localization, and then annihilate or directly annihilate, without any approaching step involved. There are rather clear indications that at low temperatures, the above noticed movement is rather improbable (see the discussion in Ref. [31]). Thus, the unique possibility to annihilate for the two excitons created on separated segments is the direct annihilation (without approaching). In our previous short paper [31], we estimated the rate of an elementary event for the intrasegment exciton±exciton annihilation. The goal of the present article is to make a comprehensive analysis of both channels and to discuss their relevance to the experiments. The remainder of the paper is organized as follows: In Section 2, we derive a general formula for the rate of an elementary event of annihilation in terms of the two-exciton eigenfunctions of the one-dimensional Frenkel Hamiltonian with disorder, not restricting ourselves to the nearest-neighbor terms, as previously [31], but taking into account the whole dipole±dipole interaction in the annihilation channel. Section 3 deals with the calculations of the intra and intersegment annihilation rates. Experimental data are discussed in Section 4, from the viewpoint of the results obtained in Section 3. Finally, Section 5 summarizes the article. 2. Model In this section, we address the problem of the elementary event of exciton±exciton annihilation, considering a linear chain of a ®nite number (N) of three-level molecules with diagonal disorder. Two lower states of molecules take part in forming excitonic states due to the intermolecular dipole±
V.A. Malyshev et al. / Chemical Physics 254 (2000) 31±38
dipole interaction. The high-lying molecular term will serve as an intermediate one through which annihilation occurs. We will assume that this (electronic±vibrational) term is in resonance with two-exciton states as well as that it is strongly coupled to vibrations, undergoing as a result, an ecient multiphonon assisted relaxation. With respect to the other possible schemes for two excitons to annihilate, we refer the reader to Ref. [31]. Initially, let the chain be excited to a two-exciton state, which means that any two of N molecules are in their ®rst excited states. Our subject of interest will be the disappearance of both excitations by any process except the spontaneous emission of each exciton. Under the conditions adopted, the process of annihilation consists of three steps. First, the energy of the two-exciton state is transferred to the high-lying electronic vibrational term of one of the two excited molecules. This excited state then rapidly relaxes to the ground vibrational state due to the intramolecular vibrational relaxation, so that the back conversion to a two-exciton state is impossible. The ®nal step consists in the multiphonon relaxation from the high-lying molecular term either to the ground state (both excitons disappear) or to the one exciton state (one exciton disappears). Fig. 1 demonstrates the annihilation scheme for a system of only two molecules. Our task is to calculate the rate of the ®rst step consisting of the energy transfer from the twoexciton state to the high-lying excited state of a molecule. We assume that the process of interest originates from the dipole±dipole intermolecular interaction. The corresponding Hamiltonian within the site state representation reads Va
X
V
m
jn ÿ mj
3
b1n b1m b 2n b2m h:c:;
1
where V hn2jVa jn1; n 1; 1i is the matrix element of the annihilation operator for nearest neighbors; jn1; m1i represent the ket vector of state with molecules n and m in their ®rst excited states; jn2i is the ket vector of state with molecule n in the higher state; b an ban ; a 1; 2 denote the Pauli creation (annihilation) operator of the ath molecular state of molecule n (the state with all the
33
Fig. 1. Scheme of the exciton±exciton annihilation process through a high-lying electronic-vibrational molecular term (resonance case, x21 x32 ). The ®rst step consists of simultaneous transitions of molecule a to the ground state and molecule b to the high-lying term initiated by the intermolecular interaction Va . The second step presents the fast vibrational relaxation within high-lying electronic-vibrational sub-levels characterized by a rate C Va . Finally, the radiationless multiphonon relaxation from the high-lying term occurs either to the second level (one excitation disappears) or to the ground one (both excitations disappear) with rates C2 and C1 , respectively.
molecules in their ground states serves as the vacuum state j0i with zero energy). Note that, in contrast to our previous article [31], the interactions of all molecules with one another are taken into account. The magnitude of V will be considered further to be much less than the rate of vibrational relaxation of the higher state C. Under such a condition, the rate of the ®rst annihilation step can be calculated by making use of the perturbation theory. The corresponding expression is given by the ``golden rule'' wlm a
N 2p X jhn2jVa jlmij2 q Ef h n1 N 2p X 2 jhn2jVa jlmij : hC n1
2
34
V.A. Malyshev et al. / Chemical Physics 254 (2000) 31±38
Here, we substituted the density of ®nal states q Ef by C. The two-exciton states jlmi, where indexes l and m range from 1 to N, are composed as a solution of the eigenvalue problem for the exciton Hamiltonian Uex ÿU
N ÿ1 N X X b E1n b 1n b1;n1 b1;n1 b1n 1n b1n ; n1
n1
3 treated hereafter in the nearest-neighbor approximation. 1 The sign of this interaction (negative, ÿU < 0) is chosen as it takes place for J-aggregates [1,4±7] as well as for chlorophyll±water aggregates [17]. The second term in Eq. (3) represents the Hamiltonian of noninteracting molecules, where E1n E1 Dn is the energy of the ®rst excited state of molecule n with E1 and Dn being the average value of energy and a static random oset with zero mean distributed within an interval of width D, respectively. The Dn describes diagonal disorder resulting in localization of the exciton states. In the limit of D U (weak localization), which will be further assumed, the exciton eigenfunctions are localized on rather large chain segments of size given by [28,29,32] 2=3 U : 4 N 1 3p2 D Concluding this section, we stress that the dipole±dipole interaction in the annihilation channel (Va ) cannot be taken in the nearest-neighbor approximation since annihilation of two excitons localized on separate localization segments is determined namely by the coupling to far neighbors. 3. Rate of annihilation from a two-exciton state As is well known, in the frame of the nearestneighbor approach, one-dimensional Frenkel excitons are noninteracting fermions (see, for instance,
1 As was pointed out in Refs. [2,3,30], the eigenstates, in fact, do not change when adding the long-range dipole±dipole terms to the nearest-neighbor Hamiltonian (3).
Refs. [33±37]). This allows one to compose the eigenfunctions of states with n-excitons as Slater determinants of those obtained from the solution of the one-exciton problem. They are given by jmi
N X n1
umn jn1i;
5
where jn1i is the state vector of molecule n being in its ®rst excited state and the column-vector fumn g is the eigenvector of the one-exciton state m obeying the equation N X n1
mn Uex umn Em umm :
6
mn hm1jUa jn1i and Em is the eigen energy Here, Uex of the one-exciton state m. The two-exciton states jlmi, we are mainly interested in, can be composed now as
jlmi
N X m
wlm;mn jm1; n1i;
wlm;mn ulm umn ÿ uln umm :
7a 7b
Substituting Eq. (7b) into Eq. (2), one obtains " #2 N X wlm;mn X wlm;mn 2pV 2 X lm : wa ÿ hC n1 mn
m ÿ n3
8 This formula is the basis of our analysis of the exciton±exciton annihilation of weakly localized Frenkel excitons. Note that we are basically interested in the exciton±exciton annihilation of exciton states coupled to the light. This implies that the indexes l and m in Eq. (8) number in fact the lowest exciton states (close to the bottom of the exciton band). 3.1. Intrasegment annihilation Recall that in the case of intrasegment exciton± exciton annihilation, both eigenfunctions ulm and umn in Eq. (8) are localized on the same aggregate segment, which can be approximately considered as a regular chain of size N [28±30]. For an estimate, one can then replace N in Eq. (8) by N as
V.A. Malyshev et al. / Chemical Physics 254 (2000) 31±38
well as ulm and umn by the corresponding eigenfunctions 1=2 2 pmn sin ; 9 umn N 1 N 1 where m takes, in fact, values 1 and 2 since the two exciton states, composed of these eigenfunctions, are dominantly coupled to the light (see, for instance, Ref. [6]). Now, consider the expression in square parentheses in Eq. (8) for an arbitrary 1 < n < N . Introducing the ®rst and second sum new summation indexes l n ÿ m and l m ÿ n, respectively, one can rewrite this expression in the form: Sn
ÿn n N X X 1 1 w ÿ w : 1;nÿl;2n 3 l l3 1;nl;2n l1 l1
10
A rather fast decrease of the factor lÿ3 upon increasing l allows one, to ®rst replace the upper limits in sums in Eq. (10) by N (except at a few points n close to the segment ends), and second, to represent the appearing dierence w1;nÿl;2n ÿ w1e;nl;2n as ÿ ow1;l;2n =olln 2l. After these transformations, Sn turns out to be X N 8p pn 1 3 sin : 11 Sn 2 N 1 l1 l2 N 1 Now, substituting the expression in square parentheses in Eq. (11) with Sn given by Eq. (11) and accounting for that PN 6 n1 sin pn= N 1 5 N 1=16, one arrives at !2 N X 40p3 V 2 1 intra : 12 wa 3 l2 hC N 1 l1 The part of Eq. (12) before the parentheses represents the intrasegment annihilation rate obtained in the nearest-neighbor approximation [31]. We see that the coupling to far neighbors gives rise to an increase of the intrasegment annihilation rate by a PN 2 numerical factor l1 1=l2 , preserving however its N -dependence. At a large N , one can replace the upper limit by in®nity and ®nd for this factor a value p4 =36 ' 2:71. Numerical calculations based on the exact formula (8) not presented here con-
35
®rm this result. As was already noted, the simpli®cations made, when deriving Eq. (12), are not valid for few points close to the segment ends, but, as can be checked, those terms have an order of magnitude of N 1ÿ8 and thus really can be neglected. As follows from Eq. (12), the annihilation rate for the excitonic (extended) states steeply scales with the localization segment length N thus greatly distinguishing from the corresponding rate for a system of only two molecules 2 hC. It is worth mentioning the origin w12 a 4pV = of such a scaling. One factor of type N 1ÿ1 re¯ects simply the probability of ®nding any two nearest-neighbor molecules to be excited. Indeed, the probability to ®nd two excitations on any two molecules within a localization segment is proportional, in accordance with Eq. (9), to ÿ2 N 1 . Accounting further for that, two excitations annihilate, in fact, when they are nearest neighbors and making the summation over the possible number of nearest-neighbor pairs, we ®nally get for the above probability a value of the ÿ1 order of N 1 . The additional factor ÿ2 N 1 originates from the fermionic nature of one-dimensional Frenkel excitons, lowering the probability to ®nd two excitations on adjacent molecules.
3.2. Intersegment annihilation When analyzing the N -scaling of the intersegment annihilation rate, we will assume that the exact exciton eigenfunctions have the typical extension N and do not overlap if they are centered at dierent localization segments. A simple analysis of the general formula (8) shows that under the assumption of no overlapping, it is converted into winter a
8 " #2 2pV 2 < X 2 X umn u hC : m2S1 lm n2S2 jn ÿ mj3 " #2 9 = X ulm X u2mn ; 3 ; n2S m2S jn ÿ mj 2
1
13
36
V.A. Malyshev et al. / Chemical Physics 254 (2000) 31±38
where the summation over m and n run over the ®rst (S1 ) and second (S2 ) segments of localizations, respectively. Assume ®rst that two excitons are created on two segments separated by a distance R (between the centers of localization segments and counted in units of the lattice constant), which is much larger that the segment size N . In this limit, we can replaceP jn ÿ mj by R P in Eq. (13) and account for both that m2S1 u2lm n2S2 u2mn 1 (according to the normalization condition of eigenfunctions) and P P 2 2 Fl m2S1 ulm and Fm n2S2 umn are the oscillator strengths of states jli and jmi, respectively. It seems to be natural to assume that both ulm and umn correspond to the ground states of exciton created on separated segments (l m 1). Then, typically, F1 F2 N . With these simpli®cations, Eq. (13) takes the form winter a
4pV 2 N : hCR6
14
is proportional to the localization Note that winter a segment size N . It originates of that the transition dipole moment of the excited segment, which p is depopulated to the ground state, scales as N . Strictly speaking, Eq. (14) is not applicable for adjacent localization segments, i.e., at R N . However, it can certainly serve as an estimate of . Replacing R in Eq. (14) by the upper limit of winter a N , one gets winter a
4pV 2 : hCN 5
15
Comparing this result with that obtained for the intrasegment annihilation, Eq. (12), we conclude that the latter process has dominating probability since N 1. Moreover, this conclusion seems to be valid for N not too large due to the large difference in the numerical factors 40p3 and 4p in the and winter , respectively. expressions for wintra a a 4. Discussion First of all, we would like to mention the fact winter does not lead to exciton±exciton that wintra a a annihilation occuring uniquely through the intrasegment channel. The latter may appear to be in-
eective in the sense of aecting the total population of excited states. Indeed, the intrasegment channel of exciton±exciton annihilation requires the creation of two excitons on the same localization segment, the probability of which depends on the pump power and decreases as the intensity decreases. Hence, only a part of the population will disappear through this channel. In such a case, the intersegment channel of annihilation, being free of the above limitation, will come to its own, despite its rate being much lower. In addition, with rising temperature and shortening the exciton coherence length, the intersegment channel will survive while the intrasegment one will disappear due to blurring of the local (hidden) structure of the exciton energy spectrum [28±30] necessary for this mechanism. It is worthwhile to get an idea as to how much the magnitudes of both annihilation rates for some real systems are. In this sense, J-aggregates of pseudo-isocyanine represent the most studied species of interest. In order to make the necessary estimates, we need the experimental data concerning the parameters U, V, N and C. Since we have no information with respect to the magnitude of interaction in the annihilation channel, we assume it as being of dipole±dipole nature and consider V of the same order of magnitude as the dipole±dipole interaction U responsible for the formation of the J-band. The value U 600 cmÿ1 is widely admitted for J-aggregates of pseudo-isocyanine [1±3,6]. Substituting V by 600 cmÿ1 , N by 20 (low temperature, data taken from Ref. [21]) and the relaxation constant C U [20,21], we , according to Eq. (12), corrected obtain for wintra a by the above-mentioned factor 2.71, wintra a 8 1012 sÿ1 ' 130 fsÿ1 . The corresponding estimate for winter reads winter 7 107 sÿ1 ' a a 14 nsÿ1 . Kinetics of the exciton±exciton annihilation at low temperatures was studied in Ref. [21]. Two well separated components of the annihilation curves were reported: 200 fs related to the annihilation within a localization segment and 1:5 ps prescribed to the annihilation between separated segments. Unfortunately, the corresponding weights were not provided. As is seen, our estimate is in good qualitative agreement with the of wintra a
V.A. Malyshev et al. / Chemical Physics 254 (2000) 31±38
fastest experimentally observed annihilation condiers drasstant. However, the estimate of winter a tically from the slower one. A possible explanation for the slower annihilation component can be, in principle, done in the frame of the intrasegment mechanism itself. Recall that the sizes of localization segments undergo ¯uctuations while the annihilation rate depends on size very steeply. In fact, the twofold increase of the segment size relative to the typical one N 20 (a reasonable ¯uctuation) matches the experimentally observed domain of the annihilation time constants. In opposite, the intersegment mechanism needs at least a sixfold decrease of the segment size (which is rather improbable at low temperatures) in order to arrive at the slower experimental magnitude of 1:5 ps. A number of experiments were devoted to the exciton±exciton annihilation in linear molecular aggregates at room temperature [18±20,22,23]. As we already mentioned earlier, the intersegment channel at room temperature is the unique one through which excitons annihilate. It is reasonable to assume (and what is usually done [18±20,22,23]) that at room temperature, excitons are highly movable quasi-particles and being created on two separate segments of the aggregate then rapidly approach each other. Under such conditions, the rate of their annihilation will be determined by for adjacent segments, i.e., by Eq. (15). winter a Taking N 5 (the typically reported value at room temperature) and holding the rest of parameters to be the same as above, we arrive at an estimate for the annihilation rate ÿ1 7 1010 sÿ1 14 ps which has the orwinter a der of that observed in the experiments [18±20,23]. 5. Summary The problem of exciton±exciton annihilation in one-dimensional molecular aggregates is considered under the conditions of weak localization of the exciton states. The perturbation theory is used in order to estimate rates of the intrasegment and intersegment exciton±exciton annihilation under the assumption of annihilation channel involving multiphonon relaxation from a high-lying molec-
37
ular electronic-vibrational term. It is shown that the rates of intrasegment annihilation wintra scale a with the typical size of the localization range as ÿ3 N while the scaling law of the intersegment annihilation rate has the form winter N =R6 , where a R is the distance between two excited segments (in units of the lattice constant). For two adjacent localization segments R N , the latter rate scales ÿ5 as N .
Acknowledgements This work was supported by the Federal Ministry of Education, Science, Research and Technology under the project ``Bistable optical switching elements on the basis of organic macromolecular materials.'' G.G.K. and V.A.M. greatly acknowledge supports both from the German Federal Ministry of Education, Science, Research and Technology within the TRANSFORM ± programme for support of science (project no. 01 BP 820/7) and from the INTAS (project no. 97-10434). V.A.M. also acknowledges the University of Juvaskyla where this work was started.
References [1] S. de Boer, D.A. Wiersma, Chem. Phys. Lett. 165 (1990) 45. [2] H. Fidder, J. Terpstra, D.A. Wiersma, J. Chem. Phys. 94 (1991) 6895. [3] H. Fidder, J. Terpstra, D.A. Wiersma, J. Chem. Phys. 95 (1991) 7880. [4] V.I. Bogdanov, E.N. Viktorova, S.V. Kulya, A.S. Spiro, JETP Lett. 53 (1991) 105 (original work, Pis'ma Zh. Eksp. Teor. Fiz. 53 (1991) 100). [5] Y. Wang, J. Opt. Soc. Am. B 8 (1991) 981. [6] F.C. Spano, J. Knoester, in: W.S. Warren (Ed.), Advances in Magnetic and Optical Resonance, vol. 18, Academic Press, New York, 1994, p. 117. [7] J. Knoester, F.C. Spano, in: T. Kobayashi (Ed.), J-aggregates, World Scienti®c, Singapore, 1996, p. 111. [8] N. Tessler, G.J. Denton, R.H. Friend, Nature (London) 382 (1996) 695. [9] F. Hide, B.J. Schwartz, M.A. Diaz-Garsia, A.J. Heeger, Chem. Phys. Lett. 256 (1996) 424. [10] S.V. Frolov, W. Gellermann, M. Ozaki, K. Yoshino, Z.V. Vardeny, Phys. Rev. Lett. 78 (1997) 729.
38
V.A. Malyshev et al. / Chemical Physics 254 (2000) 31±38
[11] S.V. Frolov, Z.V. Vardeny, K. Yoshino, Phys. Rev. B 57 (1998) 9141. ßelik, D.L. Akins, Appl. Phys. Lett. 71 (1997) 3057. [12] S. Ozc ßelik, I. Ozc ßelik, D.L. Akins, Appl. Phys. Lett. 73 [13] S. Ozc (1998) 1949. [14] V.M. Helenius, J.O. Siikki, P.H. Hynninen, J.E.I. KorppiTommola, Chem. Phys. Lett. 226 (1994) 137. [15] J.A.I. Oksanen, E.I. Zenkevich, V.N. Knyukshto, S. Pakalnis, P.H. Hynninen, J.E.I. Korppi-Tommola, Biochem. Biophys. Acta 1321 (1997) 165. [16] R. Monshouwer, M. Abrahamsson, F. van Mourik, R. van Grondelle, J. Phys. Chem. 101 (1997) 7241. [17] J. Linnanto, V.M. Helenius, J.A.I. Oksanen, T. Peltola, J.-L. Garaud, J.E.I. Korppi-Tommola, J. Phys. Chem. A 102 (1998) 4337. [18] V. Sundstrom, T. Gillbro, R.A. Gadonas, A. Piskarskas, J. Chem. Phys. 89 (1988) 2754. [19] H. Stiel, S. Daehne, K. Teuchner, J. Lumin. 39 (1988) 351. [20] R. Gagel, R. Gadonas, A. Laubereau, Chem. Phys. Lett. 217 (1994) 228. [21] K. Minoshima, M. Taiji, K. Misawa, T. Kobayashi, Chem. Phys. Lett. 218 (1994) 67. [22] M. van Burgel, D.A. Wiersma, K. Duppen, J. Chem. Phys. 102 (1995) 20.
[23] R. Gadonas, K.-H. Feller, A. Pugzlys, G. Jonusauskas, J. Oberle, C. Rulliere, J. Chem. Phys. 106 (1997) 8374. [24] E. Gaizauskas, K.-H. Feller, R. Gadonas, Opt. Commun. 118 (1995) 360. [25] V.A. Malyshev, H. Glaeske, K.-H. Feller, Opt. Commun. 140 (1997) 83. [26] V.A. Malyshev, H. Glaeske, K.-H. Feller, J. Lumin. 76, 77 (1998) 455. [27] V.A. Malyshev, H. Glaeske, K.-H. Feller, Phys. Rev. A 58 (1998) 670. [28] V.A. Malyshev, Opt. Spectr. 71 (1991) 505 (original work, Opt. Spektr. 71 (1991) 873. [29] V.A. Malyshev, J. Lumin. 55 (1993) 225. [30] V. Malyshev, P. Moreno, Phys. Rev. B 51 (1995) 14587. [31] V.A. Malyshev, H. Glaeske, K.-H. Feller, Chem. Phys. Lett. 305 (1999) 117. [32] L.D. Bakalis, J. Knoester, J. Phys. Chem. 103 (1999) 6620. [33] D.B. Chesnut, A. Suna, J. Chem. Phys. 39 (1963) 146. [34] Yu.A. Avetisyan, A.I. Zaitsev, V.A. Malyshev, Opt. Spectr. 59 (1985) 582 (original work, Opt. Spektr. 59 (1985) 967). [35] G. Juzeliunas, Z. Phys. D 8 (1988) 379. [36] F.C. Spano, Phys. Rev. Lett. 67 (1991) 3424. [37] F.C. Spano, Phys. Rev. Lett. E 68 (1992) 2976.
www.elsevier.nl/locate/chemphys
Channels of the exciton±exciton annihilation in onedimensional aggregates at low temperature V.A. Malyshev a,b, G.G. Kozlov b, H. Glaeske a,*, K.-H. Feller a a b
Fachhochschule Jena, Fachbereich Medizintechnik/Physikalische Technik, Carl-Zeiss-Promenade 2, D-07745 Jena, Germany National Research Center, Vavilov State Optical Institute, Birzhevaya Liniya 12, 199034 St. Petersburg, Russian Federation Received 25 October 1999
Abstract Exciton±exciton annihilation is implied to be in fact a unique channel of exciton degradation in linear molecular aggregates and conjugated polymers as the pump intensity rises. However, at low temperatures, excitons represent rather immobile than movable quasi-particles due to the localization eect. We analyze possible channels of exciton± exciton annihilation, which seem to be natural ones under the conditions of localization and low temperature, and calculate the corresponding rates. Ó 2000 Elsevier Science B.V. All rights reserved.
1. Introduction In the last decade, quasi-one-dimensional structures, such as linear molecular aggregates and conjugated polymers, attracted much attention due to the extraordinary features of their photoresponse: short spontaneous emission time and giant values of high-order susceptibilities both scaled with the size of an aggregate [1±5] (see also the reviews [6,7] and references therein), making them promising candidates in developing devices of optical logic. Recently, spectral narrowing [8,9] and cooperative emission [10,11] in p-conjugated polymer thin ®lms as well as super-radiant lasing from J-aggregated cyanine dye molecules adsorbed onto colloidal silica and silver [12,13] were re-
* Corresponding author. Tel.: +49-3641-205630; fax: +493641-205601. E-mail address: [email protected] (H. Glaeske).
ported, the eects being of great importance from the viewpoint of laser applications. Another group of interesting species is represented by aggregates of biological molecules (such as chlorophyll a, for instance) which are involved in the excitation energy transfer and charge separation in the natural photosynthetic systems [14± 17]. A common peculiarity of all these systems is that their lowest electronic excitations are Frenkel excitons mainly determining optical and transport properties of these objects. It was discovered [18,19] that with increasing pump intensities, exciton±exciton annihilation becomes more and more important, so that it strongly in¯uences the optical response of the excitonic system [18±27]. The usual framework for the analysis of this process is the bimolecular theory using rate equations, in which the rate of annihilation is proportional to the concentration of excitations [18,19,22,23]. In addition, it is assumed that two excitations approach each other before the annihilation starts.
0301-0104/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 0 1 0 4 ( 0 0 ) 0 0 0 2 0 - 3
32
V.A. Malyshev et al. / Chemical Physics 254 (2000) 31±38
This concept seems to be a good framework for the description of exciton±exciton annihilation at higher (room) temperatures when the exciton coherence length is rather short (several lattice constants [4,18,19,22,23]). The approach of the two excitons can then be treated as a series of incoherent jumps thus allowing the rate equation framework. At low temperatures, the coherence length can cover several tens of sites [1±3]. Clearly, the rate equation can no longer be used for the description of motion of excitons created within the same coherent domain. An adequate picture of the annihilation process under the condition of a suciently large exciton coherence length is then in order. In this article, we consider the case of low temperatures when the exciton coherence length is determined by static diagonal disorder. The latter results in localization of the excitonic states so that the exciton coherence length is nothing but the exciton localization length. By the term localization, we mean that an excitonic state is spread over a part of the aggregate (segment of localization) rather than over the whole one. In addition, we will assume that the strength of disorder is relatively small (the quantitative formulation will be done later), so that the localization length contains a large number of molecules (weak localization). Recall that from the viewpoint of the exciton optical dynamics, it is of importance that close to the bottom of the exciton band (the range of optically active exciton states), one may classify the localized exciton states, into several groups of few states (two or sometimes three) such that the states of each group are localized within a particular segment of the typical size N (frequently called the number of coherently bound molecules) and do not overlap with the others [28±30]. Since in the process of excitonic annihilation two excitons take part, one should take into consideration twoexciton states. As is well known (see, for instance, Ref. [6]), one-dimensional Frenkel excitons are weakly interacting fermions so that the eigenfunctions of states with two excitons can be composed of Slater determinants of single-exciton eigenfunctions. Under the condition of localization, two dierent types of two-exciton states then
appear: (i) with two excitons belonging to the same localization segment and (ii) with those localized on dierent ones. Notice that the exciton±exciton annihilation, in the former case (hereafter, intrasegment annihilation), does not require any approach of two excitons since they occupy the same segment of the aggregate. This is not the case when excitons are created on dierent segments of localization (intersegment annihilation). Here, excitons may either approach one another, jumping over dierent segments of localization, and then annihilate or directly annihilate, without any approaching step involved. There are rather clear indications that at low temperatures, the above noticed movement is rather improbable (see the discussion in Ref. [31]). Thus, the unique possibility to annihilate for the two excitons created on separated segments is the direct annihilation (without approaching). In our previous short paper [31], we estimated the rate of an elementary event for the intrasegment exciton±exciton annihilation. The goal of the present article is to make a comprehensive analysis of both channels and to discuss their relevance to the experiments. The remainder of the paper is organized as follows: In Section 2, we derive a general formula for the rate of an elementary event of annihilation in terms of the two-exciton eigenfunctions of the one-dimensional Frenkel Hamiltonian with disorder, not restricting ourselves to the nearest-neighbor terms, as previously [31], but taking into account the whole dipole±dipole interaction in the annihilation channel. Section 3 deals with the calculations of the intra and intersegment annihilation rates. Experimental data are discussed in Section 4, from the viewpoint of the results obtained in Section 3. Finally, Section 5 summarizes the article. 2. Model In this section, we address the problem of the elementary event of exciton±exciton annihilation, considering a linear chain of a ®nite number (N) of three-level molecules with diagonal disorder. Two lower states of molecules take part in forming excitonic states due to the intermolecular dipole±
V.A. Malyshev et al. / Chemical Physics 254 (2000) 31±38
dipole interaction. The high-lying molecular term will serve as an intermediate one through which annihilation occurs. We will assume that this (electronic±vibrational) term is in resonance with two-exciton states as well as that it is strongly coupled to vibrations, undergoing as a result, an ecient multiphonon assisted relaxation. With respect to the other possible schemes for two excitons to annihilate, we refer the reader to Ref. [31]. Initially, let the chain be excited to a two-exciton state, which means that any two of N molecules are in their ®rst excited states. Our subject of interest will be the disappearance of both excitations by any process except the spontaneous emission of each exciton. Under the conditions adopted, the process of annihilation consists of three steps. First, the energy of the two-exciton state is transferred to the high-lying electronic vibrational term of one of the two excited molecules. This excited state then rapidly relaxes to the ground vibrational state due to the intramolecular vibrational relaxation, so that the back conversion to a two-exciton state is impossible. The ®nal step consists in the multiphonon relaxation from the high-lying molecular term either to the ground state (both excitons disappear) or to the one exciton state (one exciton disappears). Fig. 1 demonstrates the annihilation scheme for a system of only two molecules. Our task is to calculate the rate of the ®rst step consisting of the energy transfer from the twoexciton state to the high-lying excited state of a molecule. We assume that the process of interest originates from the dipole±dipole intermolecular interaction. The corresponding Hamiltonian within the site state representation reads Va
X
V
m
jn ÿ mj
3
b1n b1m b 2n b2m h:c:;
1
where V hn2jVa jn1; n 1; 1i is the matrix element of the annihilation operator for nearest neighbors; jn1; m1i represent the ket vector of state with molecules n and m in their ®rst excited states; jn2i is the ket vector of state with molecule n in the higher state; b an ban ; a 1; 2 denote the Pauli creation (annihilation) operator of the ath molecular state of molecule n (the state with all the
33
Fig. 1. Scheme of the exciton±exciton annihilation process through a high-lying electronic-vibrational molecular term (resonance case, x21 x32 ). The ®rst step consists of simultaneous transitions of molecule a to the ground state and molecule b to the high-lying term initiated by the intermolecular interaction Va . The second step presents the fast vibrational relaxation within high-lying electronic-vibrational sub-levels characterized by a rate C Va . Finally, the radiationless multiphonon relaxation from the high-lying term occurs either to the second level (one excitation disappears) or to the ground one (both excitations disappear) with rates C2 and C1 , respectively.
molecules in their ground states serves as the vacuum state j0i with zero energy). Note that, in contrast to our previous article [31], the interactions of all molecules with one another are taken into account. The magnitude of V will be considered further to be much less than the rate of vibrational relaxation of the higher state C. Under such a condition, the rate of the ®rst annihilation step can be calculated by making use of the perturbation theory. The corresponding expression is given by the ``golden rule'' wlm a
N 2p X jhn2jVa jlmij2 q Ef h n1 N 2p X 2 jhn2jVa jlmij : hC n1
2
34
V.A. Malyshev et al. / Chemical Physics 254 (2000) 31±38
Here, we substituted the density of ®nal states q Ef by C. The two-exciton states jlmi, where indexes l and m range from 1 to N, are composed as a solution of the eigenvalue problem for the exciton Hamiltonian Uex ÿU
N ÿ1 N X X b E1n b 1n b1;n1 b1;n1 b1n 1n b1n ; n1
n1
3 treated hereafter in the nearest-neighbor approximation. 1 The sign of this interaction (negative, ÿU < 0) is chosen as it takes place for J-aggregates [1,4±7] as well as for chlorophyll±water aggregates [17]. The second term in Eq. (3) represents the Hamiltonian of noninteracting molecules, where E1n E1 Dn is the energy of the ®rst excited state of molecule n with E1 and Dn being the average value of energy and a static random oset with zero mean distributed within an interval of width D, respectively. The Dn describes diagonal disorder resulting in localization of the exciton states. In the limit of D U (weak localization), which will be further assumed, the exciton eigenfunctions are localized on rather large chain segments of size given by [28,29,32] 2=3 U : 4 N 1 3p2 D Concluding this section, we stress that the dipole±dipole interaction in the annihilation channel (Va ) cannot be taken in the nearest-neighbor approximation since annihilation of two excitons localized on separate localization segments is determined namely by the coupling to far neighbors. 3. Rate of annihilation from a two-exciton state As is well known, in the frame of the nearestneighbor approach, one-dimensional Frenkel excitons are noninteracting fermions (see, for instance,
1 As was pointed out in Refs. [2,3,30], the eigenstates, in fact, do not change when adding the long-range dipole±dipole terms to the nearest-neighbor Hamiltonian (3).
Refs. [33±37]). This allows one to compose the eigenfunctions of states with n-excitons as Slater determinants of those obtained from the solution of the one-exciton problem. They are given by jmi
N X n1
umn jn1i;
5
where jn1i is the state vector of molecule n being in its ®rst excited state and the column-vector fumn g is the eigenvector of the one-exciton state m obeying the equation N X n1
mn Uex umn Em umm :
6
mn hm1jUa jn1i and Em is the eigen energy Here, Uex of the one-exciton state m. The two-exciton states jlmi, we are mainly interested in, can be composed now as
jlmi
N X m
wlm;mn jm1; n1i;
wlm;mn ulm umn ÿ uln umm :
7a 7b
Substituting Eq. (7b) into Eq. (2), one obtains " #2 N X wlm;mn X wlm;mn 2pV 2 X lm : wa ÿ hC n1 m
V.A. Malyshev et al. / Chemical Physics 254 (2000) 31±38
well as ulm and umn by the corresponding eigenfunctions 1=2 2 pmn sin ; 9 umn N 1 N 1 where m takes, in fact, values 1 and 2 since the two exciton states, composed of these eigenfunctions, are dominantly coupled to the light (see, for instance, Ref. [6]). Now, consider the expression in square parentheses in Eq. (8) for an arbitrary 1 < n < N . Introducing the ®rst and second sum new summation indexes l n ÿ m and l m ÿ n, respectively, one can rewrite this expression in the form: Sn
ÿn n N X X 1 1 w ÿ w : 1;nÿl;2n 3 l l3 1;nl;2n l1 l1
10
A rather fast decrease of the factor lÿ3 upon increasing l allows one, to ®rst replace the upper limits in sums in Eq. (10) by N (except at a few points n close to the segment ends), and second, to represent the appearing dierence w1;nÿl;2n ÿ w1e;nl;2n as ÿ ow1;l;2n =olln 2l. After these transformations, Sn turns out to be X N 8p pn 1 3 sin : 11 Sn 2 N 1 l1 l2 N 1 Now, substituting the expression in square parentheses in Eq. (11) with Sn given by Eq. (11) and accounting for that PN 6 n1 sin pn= N 1 5 N 1=16, one arrives at !2 N X 40p3 V 2 1 intra : 12 wa 3 l2 hC N 1 l1 The part of Eq. (12) before the parentheses represents the intrasegment annihilation rate obtained in the nearest-neighbor approximation [31]. We see that the coupling to far neighbors gives rise to an increase of the intrasegment annihilation rate by a PN 2 numerical factor l1 1=l2 , preserving however its N -dependence. At a large N , one can replace the upper limit by in®nity and ®nd for this factor a value p4 =36 ' 2:71. Numerical calculations based on the exact formula (8) not presented here con-
35
®rm this result. As was already noted, the simpli®cations made, when deriving Eq. (12), are not valid for few points close to the segment ends, but, as can be checked, those terms have an order of magnitude of N 1ÿ8 and thus really can be neglected. As follows from Eq. (12), the annihilation rate for the excitonic (extended) states steeply scales with the localization segment length N thus greatly distinguishing from the corresponding rate for a system of only two molecules 2 hC. It is worth mentioning the origin w12 a 4pV = of such a scaling. One factor of type N 1ÿ1 re¯ects simply the probability of ®nding any two nearest-neighbor molecules to be excited. Indeed, the probability to ®nd two excitations on any two molecules within a localization segment is proportional, in accordance with Eq. (9), to ÿ2 N 1 . Accounting further for that, two excitations annihilate, in fact, when they are nearest neighbors and making the summation over the possible number of nearest-neighbor pairs, we ®nally get for the above probability a value of the ÿ1 order of N 1 . The additional factor ÿ2 N 1 originates from the fermionic nature of one-dimensional Frenkel excitons, lowering the probability to ®nd two excitations on adjacent molecules.
3.2. Intersegment annihilation When analyzing the N -scaling of the intersegment annihilation rate, we will assume that the exact exciton eigenfunctions have the typical extension N and do not overlap if they are centered at dierent localization segments. A simple analysis of the general formula (8) shows that under the assumption of no overlapping, it is converted into winter a
8 " #2 2pV 2 < X 2 X umn u hC : m2S1 lm n2S2 jn ÿ mj3 " #2 9 = X ulm X u2mn ; 3 ; n2S m2S jn ÿ mj 2
1
13
36
V.A. Malyshev et al. / Chemical Physics 254 (2000) 31±38
where the summation over m and n run over the ®rst (S1 ) and second (S2 ) segments of localizations, respectively. Assume ®rst that two excitons are created on two segments separated by a distance R (between the centers of localization segments and counted in units of the lattice constant), which is much larger that the segment size N . In this limit, we can replaceP jn ÿ mj by R P in Eq. (13) and account for both that m2S1 u2lm n2S2 u2mn 1 (according to the normalization condition of eigenfunctions) and P P 2 2 Fl m2S1 ulm and Fm n2S2 umn are the oscillator strengths of states jli and jmi, respectively. It seems to be natural to assume that both ulm and umn correspond to the ground states of exciton created on separated segments (l m 1). Then, typically, F1 F2 N . With these simpli®cations, Eq. (13) takes the form winter a
4pV 2 N : hCR6
14
is proportional to the localization Note that winter a segment size N . It originates of that the transition dipole moment of the excited segment, which p is depopulated to the ground state, scales as N . Strictly speaking, Eq. (14) is not applicable for adjacent localization segments, i.e., at R N . However, it can certainly serve as an estimate of . Replacing R in Eq. (14) by the upper limit of winter a N , one gets winter a
4pV 2 : hCN 5
15
Comparing this result with that obtained for the intrasegment annihilation, Eq. (12), we conclude that the latter process has dominating probability since N 1. Moreover, this conclusion seems to be valid for N not too large due to the large difference in the numerical factors 40p3 and 4p in the and winter , respectively. expressions for wintra a a 4. Discussion First of all, we would like to mention the fact winter does not lead to exciton±exciton that wintra a a annihilation occuring uniquely through the intrasegment channel. The latter may appear to be in-
eective in the sense of aecting the total population of excited states. Indeed, the intrasegment channel of exciton±exciton annihilation requires the creation of two excitons on the same localization segment, the probability of which depends on the pump power and decreases as the intensity decreases. Hence, only a part of the population will disappear through this channel. In such a case, the intersegment channel of annihilation, being free of the above limitation, will come to its own, despite its rate being much lower. In addition, with rising temperature and shortening the exciton coherence length, the intersegment channel will survive while the intrasegment one will disappear due to blurring of the local (hidden) structure of the exciton energy spectrum [28±30] necessary for this mechanism. It is worthwhile to get an idea as to how much the magnitudes of both annihilation rates for some real systems are. In this sense, J-aggregates of pseudo-isocyanine represent the most studied species of interest. In order to make the necessary estimates, we need the experimental data concerning the parameters U, V, N and C. Since we have no information with respect to the magnitude of interaction in the annihilation channel, we assume it as being of dipole±dipole nature and consider V of the same order of magnitude as the dipole±dipole interaction U responsible for the formation of the J-band. The value U 600 cmÿ1 is widely admitted for J-aggregates of pseudo-isocyanine [1±3,6]. Substituting V by 600 cmÿ1 , N by 20 (low temperature, data taken from Ref. [21]) and the relaxation constant C U [20,21], we , according to Eq. (12), corrected obtain for wintra a by the above-mentioned factor 2.71, wintra a 8 1012 sÿ1 ' 130 fsÿ1 . The corresponding estimate for winter reads winter 7 107 sÿ1 ' a a 14 nsÿ1 . Kinetics of the exciton±exciton annihilation at low temperatures was studied in Ref. [21]. Two well separated components of the annihilation curves were reported: 200 fs related to the annihilation within a localization segment and 1:5 ps prescribed to the annihilation between separated segments. Unfortunately, the corresponding weights were not provided. As is seen, our estimate is in good qualitative agreement with the of wintra a
V.A. Malyshev et al. / Chemical Physics 254 (2000) 31±38
fastest experimentally observed annihilation condiers drasstant. However, the estimate of winter a tically from the slower one. A possible explanation for the slower annihilation component can be, in principle, done in the frame of the intrasegment mechanism itself. Recall that the sizes of localization segments undergo ¯uctuations while the annihilation rate depends on size very steeply. In fact, the twofold increase of the segment size relative to the typical one N 20 (a reasonable ¯uctuation) matches the experimentally observed domain of the annihilation time constants. In opposite, the intersegment mechanism needs at least a sixfold decrease of the segment size (which is rather improbable at low temperatures) in order to arrive at the slower experimental magnitude of 1:5 ps. A number of experiments were devoted to the exciton±exciton annihilation in linear molecular aggregates at room temperature [18±20,22,23]. As we already mentioned earlier, the intersegment channel at room temperature is the unique one through which excitons annihilate. It is reasonable to assume (and what is usually done [18±20,22,23]) that at room temperature, excitons are highly movable quasi-particles and being created on two separate segments of the aggregate then rapidly approach each other. Under such conditions, the rate of their annihilation will be determined by for adjacent segments, i.e., by Eq. (15). winter a Taking N 5 (the typically reported value at room temperature) and holding the rest of parameters to be the same as above, we arrive at an estimate for the annihilation rate ÿ1 7 1010 sÿ1 14 ps which has the orwinter a der of that observed in the experiments [18±20,23]. 5. Summary The problem of exciton±exciton annihilation in one-dimensional molecular aggregates is considered under the conditions of weak localization of the exciton states. The perturbation theory is used in order to estimate rates of the intrasegment and intersegment exciton±exciton annihilation under the assumption of annihilation channel involving multiphonon relaxation from a high-lying molec-
37
ular electronic-vibrational term. It is shown that the rates of intrasegment annihilation wintra scale a with the typical size of the localization range as ÿ3 N while the scaling law of the intersegment annihilation rate has the form winter N =R6 , where a R is the distance between two excited segments (in units of the lattice constant). For two adjacent localization segments R N , the latter rate scales ÿ5 as N .
Acknowledgements This work was supported by the Federal Ministry of Education, Science, Research and Technology under the project ``Bistable optical switching elements on the basis of organic macromolecular materials.'' G.G.K. and V.A.M. greatly acknowledge supports both from the German Federal Ministry of Education, Science, Research and Technology within the TRANSFORM ± programme for support of science (project no. 01 BP 820/7) and from the INTAS (project no. 97-10434). V.A.M. also acknowledges the University of Juvaskyla where this work was started.
References [1] S. de Boer, D.A. Wiersma, Chem. Phys. Lett. 165 (1990) 45. [2] H. Fidder, J. Terpstra, D.A. Wiersma, J. Chem. Phys. 94 (1991) 6895. [3] H. Fidder, J. Terpstra, D.A. Wiersma, J. Chem. Phys. 95 (1991) 7880. [4] V.I. Bogdanov, E.N. Viktorova, S.V. Kulya, A.S. Spiro, JETP Lett. 53 (1991) 105 (original work, Pis'ma Zh. Eksp. Teor. Fiz. 53 (1991) 100). [5] Y. Wang, J. Opt. Soc. Am. B 8 (1991) 981. [6] F.C. Spano, J. Knoester, in: W.S. Warren (Ed.), Advances in Magnetic and Optical Resonance, vol. 18, Academic Press, New York, 1994, p. 117. [7] J. Knoester, F.C. Spano, in: T. Kobayashi (Ed.), J-aggregates, World Scienti®c, Singapore, 1996, p. 111. [8] N. Tessler, G.J. Denton, R.H. Friend, Nature (London) 382 (1996) 695. [9] F. Hide, B.J. Schwartz, M.A. Diaz-Garsia, A.J. Heeger, Chem. Phys. Lett. 256 (1996) 424. [10] S.V. Frolov, W. Gellermann, M. Ozaki, K. Yoshino, Z.V. Vardeny, Phys. Rev. Lett. 78 (1997) 729.
38
V.A. Malyshev et al. / Chemical Physics 254 (2000) 31±38
[11] S.V. Frolov, Z.V. Vardeny, K. Yoshino, Phys. Rev. B 57 (1998) 9141. ßelik, D.L. Akins, Appl. Phys. Lett. 71 (1997) 3057. [12] S. Ozc ßelik, I. Ozc ßelik, D.L. Akins, Appl. Phys. Lett. 73 [13] S. Ozc (1998) 1949. [14] V.M. Helenius, J.O. Siikki, P.H. Hynninen, J.E.I. KorppiTommola, Chem. Phys. Lett. 226 (1994) 137. [15] J.A.I. Oksanen, E.I. Zenkevich, V.N. Knyukshto, S. Pakalnis, P.H. Hynninen, J.E.I. Korppi-Tommola, Biochem. Biophys. Acta 1321 (1997) 165. [16] R. Monshouwer, M. Abrahamsson, F. van Mourik, R. van Grondelle, J. Phys. Chem. 101 (1997) 7241. [17] J. Linnanto, V.M. Helenius, J.A.I. Oksanen, T. Peltola, J.-L. Garaud, J.E.I. Korppi-Tommola, J. Phys. Chem. A 102 (1998) 4337. [18] V. Sundstrom, T. Gillbro, R.A. Gadonas, A. Piskarskas, J. Chem. Phys. 89 (1988) 2754. [19] H. Stiel, S. Daehne, K. Teuchner, J. Lumin. 39 (1988) 351. [20] R. Gagel, R. Gadonas, A. Laubereau, Chem. Phys. Lett. 217 (1994) 228. [21] K. Minoshima, M. Taiji, K. Misawa, T. Kobayashi, Chem. Phys. Lett. 218 (1994) 67. [22] M. van Burgel, D.A. Wiersma, K. Duppen, J. Chem. Phys. 102 (1995) 20.
[23] R. Gadonas, K.-H. Feller, A. Pugzlys, G. Jonusauskas, J. Oberle, C. Rulliere, J. Chem. Phys. 106 (1997) 8374. [24] E. Gaizauskas, K.-H. Feller, R. Gadonas, Opt. Commun. 118 (1995) 360. [25] V.A. Malyshev, H. Glaeske, K.-H. Feller, Opt. Commun. 140 (1997) 83. [26] V.A. Malyshev, H. Glaeske, K.-H. Feller, J. Lumin. 76, 77 (1998) 455. [27] V.A. Malyshev, H. Glaeske, K.-H. Feller, Phys. Rev. A 58 (1998) 670. [28] V.A. Malyshev, Opt. Spectr. 71 (1991) 505 (original work, Opt. Spektr. 71 (1991) 873. [29] V.A. Malyshev, J. Lumin. 55 (1993) 225. [30] V. Malyshev, P. Moreno, Phys. Rev. B 51 (1995) 14587. [31] V.A. Malyshev, H. Glaeske, K.-H. Feller, Chem. Phys. Lett. 305 (1999) 117. [32] L.D. Bakalis, J. Knoester, J. Phys. Chem. 103 (1999) 6620. [33] D.B. Chesnut, A. Suna, J. Chem. Phys. 39 (1963) 146. [34] Yu.A. Avetisyan, A.I. Zaitsev, V.A. Malyshev, Opt. Spectr. 59 (1985) 582 (original work, Opt. Spektr. 59 (1985) 967). [35] G. Juzeliunas, Z. Phys. D 8 (1988) 379. [36] F.C. Spano, Phys. Rev. Lett. 67 (1991) 3424. [37] F.C. Spano, Phys. Rev. Lett. E 68 (1992) 2976.