J. Phys. Chem. 1993,97, 11603-11610
11603
Exciton Dynamics in Disordered Molecular Aggregates: Dispersive Dephasing Probed by Photon Echo and Rayleigh Scattering He& Fidder and Douwe A. Wienma’ Ultrafast Laser and Spectroscopy Laboratory, Department of Chemistry, Materials Science Centre, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands Received: July 8, 1993’
Results are presented of absorption, fluorescence, accumulated photon echo, and resonance Rayleigh scattering experiments on aggregates of a thiacarbocyanine (TC) dye. In the echo experiments the dephasing time constant is found tovary strongly with excitation wavelength; a similar effect is observed in resonance Rayleigh scattering. All data can be interpreted using a one-dimensional Frenkel exciton Hamiltonian with diagonal disorder, In this model the dispersion of the dephasing time constant is due to an energy-dependent exciton-phonon scattering process. The effective frequency-dependent electron-phonon coupling constant is found to be a fractal power of frequency.
I. Introduction Research on collective optical response in mesoscopicstructures is presently a very active field. Interest in these systems was fueled by the prediction that such structures could have large optical nonlinearities,’making them potential candidatesfor many technological applications. Furthermore, new developments in preparation techniques of these materials, such as epitaxial deposition2and the Langmuir-Blodgett method,’ enable one now to quite readily fabricate both inorganic and organic structures of any desired composition. Examples of such systems are quantum dots: multiple quantum well^,^.^ Langmuir-Blodgett films of molecular or polymeric materia1,a and molecular aggregates.” 1 In the past few years we have been engaged in an extensive study of the optical dynamics of aggregates. The model system we have chosen is the well-known aggregate of pseudoisocyanine (PIC) dissolved in a waterlethylene glycol (1: 1 volume mixture) glass.ell As probes for the exciton dynamics, we used the accumulated photon echo, resonance light scattering, and timecorrelated single-photon counting. Ever since its discovery in 1936 by Jelley12and Scheibe,” the aggregateof PIC has been the subject of numerous investigations. Until recently, most of the research interest was directed at the understandingof the dramatic changesin the absorption spectrum that occurred upon aggregation of the PIC molecules. An important step forward to the grasp of the spectroscopy of PIC was made by Scheibe, whoconcluded fromseveraldifferent studies that the structure of the PIC aggregate is one-dimensional.14 Using a one-dimensionalmodel with periodic boundary conditions, Knappls demonstrated that exchange narrowing is the cause for the observed narrowing of the spectrum on aggregation, while Scherer and Fischer16 tried, using the same model, to explain the notable changes in the vibrational spectrum on aggregation. In the past decade the radiative and dephasing dynamics of PIC aggregates have been at the center of research.”JJ7J* Sundstr6m et aL1*showed that very low excitation densities are required to prevent the room-temperature fluorescence lifetime of about 400 ps being determined by exciton-excitonannihilation effects. De Boer and Wiersma,”Jhowever, reported that below 50 K the fluorescence lifetime was about a factor of 6 shorter than at room temperature, even though that very low excitation densities were used and that the quantum yield of emission was found to be substantiallyhigher at low temperature. They further showed that the fluorescence lifetimein PIC aggregatesmarkedly ~~~
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Abstract published in Advance ACS Absrracrs. October
IS, 1993.
0022-365419312097-11603$04.00/0
increases with temperature above 50 K despite a diminishing quantum yield. On the theoretical front the field has mainly advanced through the work of Spano and Mukamel.1e2z They showed that the observed superradiant behavior of aggregatesZ2is destroyed by exciton-phonon scattering and also by diagonal disorder.19 They also emphasized the importance of the one-exciton to two-exciton transition with respect to the expected size enhancementof optical nonlinearitieszo in aggregates of mesoscopic size. Recently, we reported a thorough numerical investigation of the effects of disorder on the linear optical properties of onedimensional molecular aggregate^.^' Among other things, we showed that scaling can be used to predict the dependence of the absorption bandwidth and bandshift, and the superradiant enhancement, on disorder and dipolar intermolecular interactions. Using the Frenkel exciton Hamiltonian with Gaussian diagonal disorder, we also demonstrated that the PIC aggregate’s absorption line shape can be simulated very well. The same calculations also predicted a 50-fold enhancement in the rate of radiative decay to occur. This prediction was in perfect agreement with the observed ratio of the monomer and aggregate’s fluorescence lifetimes at low temperature.” While basic research on aggregates is flourishing, as of today many aggregates have been examined only by classical spectroscopic methods, both in solution and on AgBr ~urfaces.2~ Therefore, littleinformationis available on the dynamical behavior of aggregates other than PIC. To test the applicability of the disordered Frenkel exciton concept to the description of the spectroscopy and dynamics in aggregates other than PIC, we decided to investigate some other systems. In this paper we report on a study of 6,6’-dimethoxy-3,3’disulfopropyl-9-ethyIthiacarbocyanine~~ (TC), a dye that can also form mixed aggregates.26Unlike the case of PIC, the absorption band of the TC aggregates is very broad, which makes it more than PIC a possible model system for biological ~ystems.2~ We present and discuss results of steady-stateabsorption, fluorescence, fluorescence lifetime measurements, wavelength- and temperature-dependent accumulated photon echo experiments, and resonance Rayleigh scattering. Compared to PIC, many differences in dynamic behavior are observed; a prime example is the temperature dependence of the fluorescence lifetime. While in PIC the lifetime lengthens with increasing temperature in TC the lifetime decreases. Also, in contrast to PIC it is found that the optical dephasing time constant (T2) varies strongly with wavelength. We show that both the wavelength and the temperature dependence of Tz in TC can be described by linear 0 1993 American Chemical Society
11604 The Journal of Physical Chemistry, Vol. 97, No. 45, 1993
(CHZIS , O;
CH213SOSH
Figure 1. Molecular structure of 6,6’-dimethoxy-3,3’-disulfopropyl-9cthylthiacarbocyaninc(in the paper referred to as TC).
exciton-phonon scattering using the information of the densityof-states function from numerical calculations. We also demonstrate that resonance Rayleigh scattering2*is not only capable of providing information on the dephasing dynamics but also can beused as a sensitive test for the assumed model of the aggregate. We further show that the rise of the accumulated signal observed for PIC and TC aggregates6s9is related to the echo detection scheme rather than to an interference effect between coherences on different excitonic transitions.29 11. Experimental Section
The molecular structure of the thiacarbocyanine (TC) dye (CAS No. 21521-28-8) used in our study is depicted in Figure 1. Samples were made from dye solutions at concentrations of 10-3to 5 X 10-3 M, in a 1:1 volume mixture of water and ethylene glycol, withO.l M KCIadded. Theadditionofpotassiumchloride is essential, as otherwise no aggregation is observed. The dye solutionswere stored in the dark, because the dye slowly degrades when exposed to light. This is evidenced by the appearance of a new absorption band near 420 nm and a yellowish color for solutions kept in the light for a few weeks. As upon standing the solution already shows aggregation and precipitation, the solution is heated before use to remove existingaggregates or solid particles. Samples were made by putting a drop of the dye solution between 0.1-mm glass slides. The samples are then cooled to about -50 OC by putting them in a precooled cryostat. This way the cooling is too rapid for aggregates to be formed. At this stage the samples usually show a new broad absorption band at 625 nm, which we attribute to a dimeric species. Next the temperature is slowly raised to between-20 and-10 “C, where aggregation takes place. This process is followed by monitoring the transmission at 670 nm. Then the cryostat is cooled to 77 K as quickly as possible, whereby the liquid turns into a glass. The described procedure waschosen to reduce theamount of light scatter, due toaggregates of a size comparable to or larger than the wavelength of the exciting light. Also, absorption and fluorescence spectra (see next section) indicate that at least two aggregate structures are present. By following the procedure described we were able to create, most of the time, samples with one aggregate form dominating the spectrum. Time-correlated single photon counting (TCSPC) measurements were done using a cavity-dumped synchronously pumped dye laser, operating at 94 kHz. The fluorescence was passed through a Zeiss M4 Q I11 prism monochromator and detected with a Hamamatsu 1534401 microchannel plate detector. The excitation wavelength was about 550 nm (fluorescein as lasing dye), while the excitation densities used were typically 1011 photons/(pulse cm2). The full width at half-maximum (fwhm) of the system response is about 30 ps. Accumulated photon echoes (APE) were performed using a 94-MHz train of ‘stochastic” excitation pulses, derived from a synchronously-pumped dye laser with sulforhodamine 640 as lasing medium. The output was made stochastic and tunable by inserting a pellicle (of 2-pm thickness) in the dye laser cavity while removing all other tuning elements. In this manner stochastic optical pulses were obtained of =10-ps duration, which had an excitation field correlation time of =0.5 ps. The total excitation power used in these measurements varied from 10 to 100 p W , depending on the wavelength and on the temperature at which the experiment was done.
Fidder and Wiersma Resonance Rayleigh scattering (RRS) experiments were performed with the same dye laser, but now a two-plate Lyot filter was used as tuning element. The Rayleigh scattering was collected perpendicular to the exciting laser beam, using an amplified OMA system (Princeton Instruments) equipped with a Spex 1877 triple-mate monochromator, ofwhich the filter stage was bypassed.
III. Model and General Remarks The most direct indication of aggregation is provided by the appearance of new and much sharper absorption bands in the optical spectrum. These bands appear because in the aggregate the optical excitations are delocalized as a result of strong intermolecular interactions. When the dye molecules are treated as two-level systems and when only one excitation per aggregate is allowed, the Hamiltonian for the aggregate is23 n
n.m. n#m
Here In) denotes the state of the system where molecule n (n = 1, ..., N) is excited, and all other molecules are in the ground state. ( e ) stands for the average molecular excitation energy, and D,, is the inhomogeneous offset energy of molecule n, which reflects the effect of the disordered (glassy) environment on the molecule. For the offset values D,, (diagonal disorder) we have taken a Gaussian distribution, with standard deviation D. The second term in eq 1 describes the interaction between molecules n and m, which we assume to be of dipolar nature. In absence of these interactions the single molecule states In) are the eigenstates with energy ( e ) + D,,, and the absorption spectrum would be governedby the inhomogeneous(Gaussian) distribution, as is the case for glasses formed from very dilute dye solutions. The intermolecular interactions cause the excitation to be delocalized over the molecules making up the aggregate, thereby averaging the local inhomogeneities. This delocalization effect, however, is counteracted by disorder, which can be either diagonal or off-diagonal.23 The delocalized eigenstates of the exciton Hamiltonian form a band,30 whose width is related to the magnitude of the interactions and the dimensionalityof the aggregate. In the case of one-dimensional aggregates, and for a parallel alignment of the transition dipoles, the total exciton bandwidth is about 4.W, where J is the nearest-neighbor dipolar interaction.23 In this situation only one aggregate absorption band is observed, which is located at the bottom of the exciton band for negative values of J and at the top for positive J. When the transition dipoles in the aggregate are nonparallel, additional absorption bands appear in the spectrum.30 In a previous paper we reported on the effects of disorder on the absorption spectrum of the aggregate and on several other quantities related to the delocalization length of the exciton eigenstate~.~~ We have also shown that scaling applies for the relation between absorption bandwidth and bandshift and the ratio of the disorder parameter D and the dipolar interaction parameter J. We have also argued that the dependence of the bandwidth and of the radiative enhancement, expressed in powers of the ratio D/J, can qualitatively be related using the concept of exchangenarrowing, first introduced by Knappls in this context. He showed that a Gaussian distribution of oscillator frequencies is narrowed by a factor NI2,when the excitation is delocalized over N molecules and assuming that the local inhomogeneities areuncorrelated. While this concept qualitativelyis very powerful in explaining many effects of disorder on excitonic properties, it cannot provide quantitative results, one of the reasons being that it only takes into account the oscillator strength on the lowest eigenstate of the chain of N molecules. The concept also fails
The Journal of Physical Chemistry, Vol. 97, No. 45, 1993 11605
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to describe the Lorentzian tail of the line shape on the intraband side of the absorption spectrum. For a more detailed discussion and numerical calculation of the effects of disorder on (Frenkel) excitons, we refer the reader to ref 23. Recently, it was suggested that the structure of self-assembled aggregates is more likely to be fractal than linear.31 This conjecture was based on results of computer simulations, which used a diffusion-limited model to describe aggregati~n.~'The observation of a fractal structure of gold colloids32 in transmission electron microscopy seemed to provide strong support for this model. A linear chain model as used by us to mimic molecular aggregates seemed therefore farfetched. Although intuitively a perfectly linear structure for molecular aggregates may indeed seem unlikely, there are several points in favor of this model. First, the molecular geometry of dye molecules is far from spherical, which imposes a strong directional preference for the way aggregates will grow. Actually, the dye investigated here is essentially planar; the crystal structure of a related dye, 3,3'diethylthiacarbocyanine bromide,33 suggests that the molecules will pack in stacks, with an angle of about 60° between the molecular and linear chain axis. For the aggregate of pseudoisocyanine (PIC), a one-dimensionalstructure has been inferred from the fact that the aggregate absorption band (J-band) is polarized parallel to the flow direction of streaming solutions of these aggregates.14 Although this is not an infallible proof of a one-dimensional structure, it does show that the gross features of the structure are one-dimensional. Compelling evidence34for a linear chain structure of the PIC aggregate was provided by a recent spectroscopic study of the lineshape of the J-band in "isotopically" mixed systems of PIC-I and its aza analogue. It was found that the line shape in these mixed aggregates changes dramatically with the addition of "aza-PIC", in a fashion which is in completeagreement with calculations based on a linear chain model. The assumption of a linear structure for the TC aggregate therefore seems reasonable as well; in fact, it turns out that a linear structure of the aggregate over only about 20 monomer units suffices to support the analysis.
IV. Results and Discussion A. Steady-State Spectroscopy. Figure 2 shows the monomer (at room temperature) and aggregate (at 1.5 K) absorption and fluorescence spectra (at 1.5 K) of TC for excitation at 514.5 nm. Upon aggregation, the absorption maximum is red-shifted by approximately2500cm-1 (from 565 to659 nm), and theabsorption bandwidth (fwhm) is found to reduce from 1150 to 240 cm-1. The presence of only one band in the spectrum clearly suggests that the transition dipoles in the aggregate have a parallel alignment. This arrangement is also observed in the crystal structure of the
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Figure 2. Absorption spectra of the TC monomer (at rmm temperature, dashed line) and aggregate (at 1.5 K, solid line). The dotted line is the fluorescence spectrum of the aggregate at 1.5 K, for excitation at 514.5
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Figure 3. (a, top) Line shape simulation of the TC aggregate absorption spectrum. The solid line is the experimental curve; the dotted line is obtained from numerical calculations based on the Hamiltonian in q 1, by averaging over lo00 spectra of randomly-generated one-dimensional chains of 250 molecules, with D / J = 0.36 and J = -lo00 cm-I. (b, bottom) Density-of-states function near the bottom of the band and corresponding to the line shape simulation in (a).
closely related 3,3'-diethylthiacarbocyanine bromide." The fact that the low-energy side of the absorption spectrum is well described by a Gaussian indicates that the disorder, which determines the width of the aggregate absorption band, is likely to be of energetic origin only (diagonal).*3 It also implies that the J-band is located at the bottom of the exciton band, which implies that J i s negative. The absence of other absorption bands in the spectrum of the TC aggregate presents a striking difference with the PIC case.14 A fluorescence excitation spectrum of the TC aggregate (not shown), from 500 to 650 nm with detection at the aggregate fluorescencemaximum (662 nm), also shows no other aggregate absorption bands with significant intensity.Apart from the slowly decreasing high-energy absorption tail, only a very tiny, 20-nm-broad shoulder is observed, which peaks at 590 nm. This feature is presumably related to a vibronic transition in the aggregate. In most samples the absorption spectra are broader (300400 cm-I) than shown in Figure 2 and have a "Lorentzian-like" tail on the low-energyside. These broader spectra are probably related to the formationofanotheraggregate. The fact that thespectrum of this other aggregate exhibits a Lorentzian red edge suggests that it cannot be described by a chain model with Gaussian diagonal disorder. As was shown in a previous paper,23 offdiagonal disorder caused, for instance, by random fluctuations in the regular chain positions can lead to Lorentzian wings. It is unlikely, however, that such disorder would only emerge in a late stage of aggregate growth. We therefore conjecture that the observed spectral change is related to the formation of a more extended structure of the aggregate. Before continuing our discussion of the experimental data, we present the results of numerical calculations based on a linear chain exciton model for the TC aggregate. Figure 3a presents both the low-temperature absorption and a computer-simulated spectrum of the aggregate based on the Hamiltonian of eq 1. Only Gaussian diagonal disorder is assumed with D / J = 0.36 and J = -1000 cm-I. On basis of the simulation, an average reduction of the radiative lifetime by a factor 20 is predicted. (The average enhancement peaks about 100 cm-l below the
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11606 The Journal of Physical Chemistry, Vol. 97, No. 45, 1993
absorption maximum.) Using the earlier derived “scaling lawnz3 for the relation between the bandwidth (Ao)and dipolar coupling strength (a,Ao = 0.94J(D/J)1.34, it is clear that a fit to the experimental line shape can be obtained for many combinations of D and J. The values of D and J presented, however, not only provide a good fit to the absorption spectrum but also yield the observed red shift of 2500 cm-l of the J-band. Quite noticeable is also the fact that the width of the disorder distribution used in the calculations (850 cm-l) differs by only a factor 1.35 from the width of the monomer spectrum in solution; in the case of PIC11 this difference was a factor 8. Apparently, aggregation in the case of PIC leads to a different type of disorder than for TC. A possible cause might be the difference in temperature at which aggregation occurs. Although both aggregates were studied in the same glass, the T C solutions show no aggregation below -20 OC, whereas aggregation of PIC still takes place at -80 OC. It is also remarkable that the absorption spectra of the PIC aggregate in LB films6 and highly concentrated solution^'^ (aggregation occurs at room temperature) are 6-7 times broader than those obtained in a glass.9-11 A similar effect is not observed for TC aggregates. Very recent nonlinear optical studies of several aggregates34 confirm the fact that the degree of correlation in local inhomogeneity can be very different. Figure 3b shows the low-energy part of the density-of-states (DOS)function, calculated with the parameters derived from the line shape simulation. This DOS function will be of great importance in the analysis of the frequency-dependent dephasing data to be discussed in section C. For further details of the effect of disorder (diagonal and off-diagonal) on the exciton’s line shape and the density-of-states function, we refer the reader to ref 23. We now continue with a discussion of the fluorescence spectrum displayed in Figure 2. The fluorescence maximum is found to be red-shifted by about 3 nm with respect to the absorption. Furthermore, the 60-cm-I line width of the fluorescence is 4 times narrower than the absorption bandwidth. In the disorderedFrenkel-exciton model these observations indicate that the intraband relaxation of the excitonic population is very fast compared to the fluorescent decay. The fluoresence therefore comes from states near the bottom of the band, which can be thermally populated within the fluorescence lifetime of the excitons. In contrast, for PIC aggregates the absorption and emission spectra of the J-band in PIC aggregates almost completely overlap.’ The extreme narrowing of the T C fluorescence cannot be understood if one assumes that no communication exists over a larger part of the chain than is set by the delocalization length. Even more important is the fact that the width of the T C fluorescence is much narrower than the distribution of bottom states of the “delocalization segments”, based on exchange narrowing arguments15J3 (approximately 70% of the fwhm). The large reduction of the line width in fluorescencetherefore indicates that the aggregate chain is much longer than the delocalization length and that, by exciton-phonon scattering, the excitation can reach states on a much larger part of the aggregate chain than given by the delocalization length. The temperature-dependent fluorescence spectra, after excitation at 630 nm, are given in Figure 4. The observed red shift of the fluorescence maximum at higher temperatures is caused by increased exciton-phonon scatteringand is matched by a shift of the absorption s p e c t r ~ m . 3 ~Comparison .~~ of the fluorescence spectra at 1.6 and 83.5 K shows no differences in line shape at the low-energy side, but at the high-energy side a broadening is observed at 83.5 K, which is attributed to thermalization of the exciton population. The small bump in the fluorescence spectrum a t 669 nm is probably related to a different aggregate structure, the presence of which was also noted in the absorption spectrum of this sample.
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Figure 4. Fluorescence spectra of the TC aggregate at 1.6 (solid), 83.5 (dotted), and 172.5 K (dashed), when excited at 630 nm.
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Figure 5. Temperature dependence of the fluorescence lifetime of the aggregate of TC up to 80 K. The fit assumes a Boltzmann-activated nonradiative decay channel with an activation energy of 90 cm-l.
B. Time-Resolved Fluorescence Decay. Figure 5 displays the results of fluorescence lifetime measurements in the temperature range 1.5-80 K. The excitation of the aggregate was performed at 550 nm. The detection bandwidth was =lo nm, covering all of the aggregate fluorescence. The first thing to note is that the fluorescence lifetime of 18 ps a t 1.5 K differs by a factor 9 from the radiative lifetime expected on the basis of the 20-fold reduction of the monomer’s lifetime (3.2 ns) as discussed in the previous section. The temperature dependence of the fluorescence is also completely different from PIC; instead of an increase of the fluorescence lifetime with rising temperature, a decrease is found. The fit shown in Figure 5 assumes exponential activation of a radiationless decay channel, with an activation energy of 90 f 40 cm-1. We note that Kemnitz et al.36 found for aggregates of a similar dye a similar type of fluorescence dynamics. The physics behind this radiationless process is presently unknown. We therefore speculate that at the activation energy above the bottom of the exciton band a strongly-coupled triplet-
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The Journal of Physical Chemistry, Vol. 97, No. 45, 1993 11607
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Figure 6. Accumulated photon echo decays measured at 1.5 K. The excitation wavelength, excitation bandwidth ("(8 In 2)1/2 in eq 7) and T2/2values obtained are, from top to bottom, (661.5 nm, 3.8 X 10'2 rad/s, 6.4 ps), (658.5 nm, 2.5 X 10'2 rad/s, 3.2 ps), and (652.5 nm, 4.3 X 10'2 rad/s, 1.0 ps). The fits are calculated from eqs 12 and 13 with the above values of W(8In 2)1/2 and T2/2.
state level" is located. A more mundane interpretation of the activation energy is that at this energy the exciton band of another aggregate structure begins, which then could function as a "black hole" for excitons promoted to this level. More work is clearly necessary to elucidate the cause for the unexpected temperature dependence of the fluorescence lifetime in TC. C. Exciton D e p W i g . Figure 6 shows the effect of the excitation wavelength on the decay of the accumulated photon echo'* (APE) at 1.5 K. Stochastic excitation pulses39were used to increase the effective time resolution of the setup from 3 to 0.5 ps. We have shown previously that these short decays are not affected by spectral diffusion.'0 The procedure used to fit these echo signals is described in the Appendix. The results in Figure 6 clearly show a strong decrease of the dephasing time constant T2 at shorter excitation wavelengths. The echo decay times, however, are found to be temperature independent up to 10 K. In contrast to PIC the echo decay can be fitted well by a single exponential.9 In view of the observed strong wavelength dependence of T2 the decays represent an average decay over an energy rangeof about 20 wavenumbers, which is the excitation bandwidth. The observed exponentiality of the echo decays therefore merely demonstrates that the range over which T2 varies within a given wavelength interval is much smaller for aggregates of TC than of PIC. Using hole burning, Tilgner et a1.* showed that in the excitonic transition of polysilane the dephasing time constant changes also across the band. Dispersive dephasing in excitonic systems therefore seems to be the rule. The exceptional behavior of PIC in this regard seems related to the narrow spectral width over which the exciton states can be optically accessed. The wavelength dependence of the dephasing time constant T2 in relation to the absorption spectrum is presented in Figure 7. We first note that a decrease of T2 at shorter wavelengths cannot be associated with a decrease of the radiative lifetime at these energies in the band, because the most radiative states are found near the bottom of the band. In view also of the fluorescence spectrum of the aggregate, there seems little doubt that the dispersive nature of T2 is related to the population dynamics in the band, which is such that exciton decay increases rapidly with increasing energy above the bottom of the band. This conclusion accords also with previous resonance light scattering experiments on PIC aggregates, which showed that intraband exciton-phonon
Figure 7. Wavelength dependenceof the total dephasing time Tz (0)at 1.5 K. The solid line is the aggregate absorption spectrum from Figure 2.
scattering, by all (thermally) accessible phonons, is the main source of the temperature-activated dephasing.11 For aggregates of TC we have been able to measure both a temperature (performed at a single wavelength, i.e., 661.5 nm) and a wavelength dependence (at 1.5 K) of the dephasing dynamics. Figure 8a shows the wavelength dependence of T2 at 1.5 K, after subtraction of the 18-ps fluorescence lifetime (TI) contribution:
1 1 -- 1 Tnth(@) T2(w,T=1.5K) 2Tl
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Figure 8b depicts the temperature-activated change of TZat 661.5 nm, Le.
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1 (3) Tth(w,T) Tz(w,T) T*(@,T=1*5K) We have fitted both sets of data simultaneously,assuming that the exciton dynamics is dominated by linear electron-phonon coupling.30 I -
wm = ~ P ( Qc )( m m w + ~ )n ( ~ +) D(w-Q)[n(Q) + 111 dQ (4) HereD(wff2) represents theexciton density of states at frequency w f Q, n(f2) the BoseEinstein occupation number at phonon frequency 0,p ( Q ) the phonon density of states, and c(0.Q) the exciton-phonon interaction parameter. Using this expression, the temperature dependence of the dephasing time constant (Figure 8b) is calculated to be
D(o-Q))n(Q)dQ (5a)
whereas the temperature-independent exciton scattering time constant (Figure 8a) is
In eqs 4 and 5 we have left out the momentum conservation condition.30 Justification for neglect of this conditioncomes from the lack of long-range order in amorphous solids, which implies that the phonons cannot be labeled with a specific wave vector or momentum. In fact, the disorder will cause a wavelengthdependent localization of the phonons.4143 Also, the aforementioned fact that exciton dephasing in PIC aggregates is caused by all thermally populated phonons11indicatesthat the momentum conservation condition does not apply for the disordered aggregates.
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Fidder and Wiersma both the exciton-phonon interaction parameter and the phonon (vibrational) density of states; so before definite conclusions can be drawn as to the relevance of the function @3s, independent information on both of these quantities is needed. D. Resonance Rayleigb Scattering. A different option for obtaining information on exciton dephasing is by performing resonance Rayleigh scattering (RRS) experimentsa2*The main advantage of this technique is that only very low excitation powers are needed, which makes it ideal for samples that show efficient hole burning. A distinct disadvantage of this technique is that it yields only information on the relative dephasing time constant, unless the ratio of Rayleigh versus fluorescence can be measured and the fluorescence lifetime is known. Hegarty et a1.28 were the first to use RRS to study exciton dephasing in GaAs/AlGaAs multiple quantum wells; we recently showed that RRS can be applied successfully also to the study of molecular excitonic systemse6Although RRSexperimentsqualitativelyyield the same information as APE, in excitonic systems the interpretation of the Rayleigh data relies upon knowledge of the distribution of oscillator strength across the exciton band. When the homogeneous line width is much narrower than the inhomogeneous line width, the RRS intensity can be shown to be determined by the following expression6,2*
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Figure 8. (a, top) Nonthermally activated contribution to the total
dephasing time as a function of wavelength (0),obtained from the T2 data of Figure 7, by using eq 2. (b, bottom) The temperature-activated contributionto the dephasing time ( O ) , obtained by applying eq 3 to the data obtained at 661.5 nm. The fits through the data in both (a) and (b) are obtained from eq 5 , with p ( Q ) c(w,Q) a QO.35 and D(wAQ) the density-of-statesfunction in Figure 3b. In fitting the data displayed in Figure 8, we have taken p ( Q ) c(w,Q) a Qa. For crystalline systems Davydov30derived equations for the exciton-phonon coupling parameter for both acoustic and optical phonons. For small values of the exciton wave vector k, which are found at the bottom of the exciton band, both coupling parameters were found to be independent of k. It seems therefore warranted to neglect thew dependence in c(w,Q). Furthermore, in crystalline systems the exciton-phonon coupling strength is constant for high-frequency optical phonons and changes linear with phonon frequency for low-frequency acoustic phonons. As theexciton density of states D(w*Q) is obtained from a numerical shape simulation (see Figure 3b), a is the only fit parameter. The fits presented in Figure 8 use a = 0.35 f 0.1. The shape of both fits does not change dramatically when a is varied. However, accordingtoeq4 both the wavelengthand temperaturedependence of T2 carry the same proportionality constant, and this prerequisite makes the simultaneous fitting of the two different sets of data sensitive to the precise value of a. The shape of the wavelength dependence fit (eq 5b) is strongly influenced also by the exact shape of the low-energy tail of the density-of-states (DOS) function. Note that a DOS obtained with off-diagonal disorder does not yield the strong decrease of T2 across the band. The temperature dependence (eq 5a), however, is fairly insensitive to this detail, as it experiences the DOS through a thermal window (the Bose-Einstein factor). The quality of the obtained fits demonstrates most vividly that the Frenkel exciton model with inclusion of diagonal disorder provides an excellent description of the dynamics in aggregates of TC. The function Qo.35 used in the fitting of our frequency-dependent dephasingdata matches the function 52113, which isoftenconnected to the vibrational density of states in fractal s0lids.~3For instance, a vibrational density of states has been found for fractal silica aerogels4*with covering the range of 20 GHz to 10 THz, a range comparable to the one used in the fit of the wavelength dependence of T2. On the other hand, our fit function includes
In this expression a ( w ) is the absorption coefficient, d the sample thickness, T2(0) the frequency-dependent dephasing time constant, and (p4(w)) ( (p2(w))) the frequency-dependent average of the fourth (second) power of the transition dipole moment. The term 1 - exp(-2a(o)d) stands for the loss of Rayleigh scatteringdue to absorption of the signal and of the exciting light field. K is a constant that is proportional to the volume of the scattering entity. It is clear from eq 6 that, in case the transition dipole moment averages do not vary with frequency, extraction of the qualitative frequency dependence of T2 from the RRS intensity data is straightforward. If the averages do change, as is clearly the case for this will be reflected in the frequency dependence of Zw(w). Conversely, by using T2 values obtained from photon echo experiments, RRS measurements can be used as a critical test of band structure calculations. The RRS technique was previously applied by Terpstra et a1.6 to study exciton dynamics in Langmuir-Blodgett bilayers of PICI. For these systems however no reliable photon echo data could be obtained; therefore, no decisive comparison between theory and experiment could be made. As shown above, a detailed photon echo study could be performed on TC aggregates. We therefore attempted a RRS study of these aggregates, to test our exciton model and also to illustrate the power of this probe for the study of exciton dynamics. The results of the RRS experiments are depicted in Figure 9. The upper figure shows the sample’s absorption spectrum, and the measured Rayleigh intensity as a function of wavelength, both at 1.5 K. The absorption spectrum differs somewhat from the one in Figure 2, which we ascribe to the presence of another aggregate structure. The lower figure presents the comparison between the Rayleigh data and the results from the photon echo experiments. We note that the data presented here are corrected for stray light. (We hereby assume that the signal a t 630 nm is mainly determined by stray light). The solid line in Figure 9b is calculated according to eq 6, with T2(w) taken from photon echo experiments and (p4(u))/(p2(u)) from numerical calculations of the distribution of oscillator strength in the band. Although the agreement between theory and experiment is not perfect, it is good enough to be assured of the fact that RRS is an extremely useful probe to test the validity of the exciton dimensionality and to ascertain whether or not the dephasing
Exciton Dynamics in Disordered Molecular Aggregates r
I
c;
.-c
625
635
645
655
665
.675
605
662
664
Wavelength (nm)
-7
J
cd
Y
c;
K .-
652
654
656
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Figure 9. (a, top) Relative Rayleigh intensity as a function of laser wavelength (0). The solid line is the 1.5 K absorption spectrum of the sample. (b, bottom) Comparison of the Rayleigh data from (a) ( 0 )with the dephasing data obtained by the accumulated photon echo. The solid line is based on eq 6, using the T2 data of Figure 7, while taking the values of (p4(u)) and ( p 2 ( u ) ) from the line shape simulation data that were used to calculate the spectrum displayed in Figure 3.
dynamics across an absorption band is dispersive. The results presented clearly support the linear exciton model for aggregates of TC. V. Summary and Conclusions We have shown that the spectroscopy and dynamics of aggregates of TC can be modeled well by using a one-dimensional Frenkel exciton model with diagonal disorder. Especially supporting our analysis is the fact that band structure calculations together with accumulated photon echo measurements predict a wavelength dependence of resonance Rayleigh scattering which is found to be in good agreement with experiment. The cause of the observed temperature dependenceof the fluorescencelifetime in aggregates of TC remains to be elucidated. The fact that several other aggregates show similar fluorescence dynamics suggests that this effect is, most likely, not a peculiarity of TC but related to the presence of a hereto unidentified state in these aggregates.
Acknowledgment. We gratefully acknowledge Polaroid for the generous gift of the thiacarbocyanine dye. We also thank Foppe de Haan for providing the software for instrument control and data analysis. The investigations were supported by the Netherlands Foundation for Chemical Research (SON), with financial aid from the Netherlands Organization for the Advancement of Science (NWO). Appendix. Rise of the Accumulated Photon Echo Figure 6shows that the accumulated photon echo signals exhibit their maximum intensity for delays beyond zero delay time. This peculiar effect needs to be addressed, especially in view of some recent theories that relate such behavior to interference between one-exciton and two-exciton coherences.29 First, we note that the signal grows in with the field coherence correlation time and not with the intensity correlation, which shows that we are dealing
The Journal of Physical Chemistry, Vol. 97, No. 45, 1993 11609 with an (coherent) echo instead of an (incoherent) absorption recovery signal. Spano showedz9that the two-pulse photon echo of molecular aggregates may be expected to exhibit a delayed rise due to the aforementioned interference effect. It therefore seems appealing to ascribe the observed ingrowth of the accumulated echo to this effect. However, the APE technique only samples contributions to the polarization that survive after 10 ns, the repetition time of the laser system. In practice, only agroundstate grating (frequency and/or spatial) remains as a result of the relaxation of the excited-state population into the bottleneck state. Because APE measurements are performed with very low pulse energies, leading to small rotations of the optical Bloch vector, this ground-state grating cannot contain a contribution related to theone-exciton to two-excitons+oherence. Therefore, in the accumulated echo the interferencebetween the twoexcitonic transitionsis not observable. Another effect must thereforecause the observed ingrowth of the APE signal. The accumulated photon echo is basically a stimulated photon echo. For two-pulse and three-pulse stimulated photon echoes, it is well-known that the maximum echo signal occurs at zero delay. Thesignal at T = 0, however, isentirely caused by scattering off a spatial grating. As the delay time T between the first two excitation pulses is increased,the macroscopicpolarization created by the first pulse decreases due to the free induction decay of the excited inhomogeneous ensemble, and therefore the modulation depth of the spatial grating decreases. Simultaneously, however, a frequency grating grows in with a complementary amplitude (when neglecting depha~ing).~s Weobserve, however, zerosignals near zero delay time in the accumulated photon echo. In these experiments the echo signal is detected after interference (in the sample) with a probe beam, which leads to measurement of the echo's amplitude instead of its intensity.38 This probe beam gains intensity scattered off the spatial grating in its direction by the (preceding) echo generating beam but loses intensity also as it self-diffracts from this grating. In case of equal powers in both beams these two effects cancel, and only the frequency grating contribution remains. Note that the scattering off the spatial grating is instantaneous, whereas the scattering off the frequency grating yields a maximum signal after rephasing (i.e., a time T later). This implies also that interference with the probe is imperfect for spatial grating scattering when T # 0, so that observation of a (purely) spatial grating signal is restricted to the time range associated with the field correlation time. In deriving the equation to describe the ingrowth, we treat the aggregate as an effective two-level system. Justification for this approach is found in the effectiveseparationof -200 cm-I between the first two optically allowed exciton levels in J-aggregates of TC,&compared to theaverage excitation bandwidthof -2Ocm-1. We disregard the small change of the absorption profile over the excitation range. We also assume impulsive excitation; only in the interference step of the echo with the fourth pulse is the time profile of the pulse taken into account. Neglect of the excitation pulse time profiles replaces the signal by that generated at the average delay value T. Because of the small pulse intensities, the polarization P(o) created by the first pulse as a function of detuning A from the pulse center frequency (which we set to zero) is linear in the Gaussian laser field amplitude (7) with W(8 In 2)'i2 the full width at half-maximum of the laser (intensity) spectrum (in rad/s). After the application of two pulses separated by a delay T, the ground-state absorption spectrum contains a tiny modulation proportional to
M(A,T) 0: (P(A)l2COS(AT)exp(-7/T2)
(8) It is obvious from this equation that for T 99. (10) de Boer, S.;Wiersma, D. A. Chem. Phys. Lett. 1990, 165, 45. 1/ W.The relevant portion of the polarization as a function of (11) Fidder, H.; Wiersma, D. A. Phys. Reu. Lett. 1991,66,1501. Fidder, time t after the third pulse (given at t = 0) is given by H.; Terptra, J.; Wiersma, D. A. J. Chem. Phys. 1991, 94, 6895. (12) Jelley, E. E. Nuture (London) 1936,138, 1009; 1937, 139,631. P(A,t ,T) a (P(A)l3 exp(-[ 1 / T,)(exp(iA[ 2-T] ) (13) Scheibe, G. Angew. Chem. 1936, 49, 563; 1937.50, 212. (14) Scheibe, G. In Optische Anregungen Orgunischer Systeme; Fdrst, exp(iAt)] (10) W., Ed.; Verlag Chemie: Weinheim, 1966; p 109 ff. (15) Knapp, E. W. Chem. Phys. 1984,85,73. Interference with the temporal profile of the fourth pulse, that (16) Scherer, P. 0. J., Fisher, S . F. Chem. Phys. 1984, 86, 269. is P(A) exp(-Wr[t-r]2), and restraining to the real part of the (17) Brumbaugh, D. V.; Muenter, A. A.; Knox, W.; Mourou, G.; Wittmenhaus, B. J. Lumin. 1984, 31-32, 783. signal, the following expression is obtained (18) SundstrBm, V.; Gillbro, T.; Gadonas, R. A,; Piskankas, A. J. Chem. Phys. 1988,89, 2754. P(A,t,T) 0: (P(A)l4 exp(-[t+r]/T2){cos(A[t-~])(19) Spano, F. C.; Mukamel, S.J . Chem. Phys. 1989, 91, 683. (20) Spano, F. C.; Mukamel, S.Phys. Rev. A 1989,40, 5783. cos(At)) exp(-W2[t-~I2) (11) (21) Spano, F. C.; Mukamel, S.Phys. Rev. Left. 1991, 66, 1197. (22) Spano, F. C.; Kuklinski, J. R.; Mukamel, S.Phys. Rev. Lett. 1990, After performing the spectral integration, the contribution to the 65, 211. signal generated at time t is obtained (23) Fidder, H.; Knoester, J.; Wiersma, D. A. J. Chem. Phys. 1991, 95, 7880. (24) Herz, A. H. Photogr. Sci. Eng. 1974, 18,323. & ( t , T ) a ( e x p ( - ~ ’ [ t - ~ ] ~ / 4) (25) Rosenoff, A. E.; Norland, K. S.;Amcs, A. E.; Walworth, V. K.; Bird, exp(-~’t’/4)) e x p ( - ~ ~ [ t - ~ ] ’e)x p ( - [ t + ~ ] / ~ , ) (12) G. R. Photogr. Sci. Eng. 1968,12, 185. (26) Gilman, P. B. Photogr. Sci. Eng. 1974, 18, 418. Performing an integration over time yields the total signal as a (27) Secforinstance: Miller,M.;Cox,R. P.;Gillbro,T. Biochem.Biophys. Acta 1991. 1057. 187. function of the pulse delay T (28) Hegarty; J.;Sturge, M. D.; Weisbuch, C.; Gossard, A. C.; Wiegmann, W. Phys. Rev. Lett. 1982, 49, 930. (29) Spano, F. C. J. Phvs. Chem. 1992, 96,2843. S(T)= K & ( t , T ) dt (13) (30) Davydov, A. S.Theory of Molecular Excitons; Plenum Press: New York, 1971. The fits in Figure 6 are calculated with this function, taking the (31) Wales, D. J.; Ewing, G. E. J. Chem. Soc., Faruduy Trans. 1992.88, laser line widths used in the experiments and the best fit values 1359. of the total dephasing time T2. The quality of the fits strongly (32) Weitz, D. A.; Oliveria, M. Phys. Rev. Lett. 1984, 52, 1433, supports our analysis. The ingrowth behavior, which was also (33) Smith, D. L. Phorogr. Sci. Eng. 1974, 18, 309. (34) Unoublished results of this laboratorv. observed in accumulated photon echoes of PIC aggregates in a (35j Toiozawa, Y. Prog. Theor. Phys. lk8, 20, 53. glass9and Langmuir-Blodgett films,6 is thus a peculiarity of the (36) Kemnitz, K.; Yoshihara, K.;Tani, T. J . Phys. Chem. 1990,94,3090. accumulated photon echo technique. See also: Tani, T.; Suzumoto, T.; Kemnitz, K.; Yoshihara, K. J . Phys. Chem. 1992. 96. 2778. We would like to end by noting that our interpretation in no (37) Trommsdorff, H. P.; et al. Paper presented at the symposium on way invalidates Spano’s We conclude, however, that ‘Dynamical Processes in Condensed Molecular Systems”, Garchy, France, only the accumulated photon echo truly measura the one-exciton May 1993. (38) Hesselink, W. H.; Wiersma, D. A. Phys. Rev. Lett. 1979,13, 1991; dephasing, whereas the two-pulse and three-pulse photon echo J. Chem. Phvs. 1991. 75. 4192. contain contributions from both the one-exciton and one-exciton (39) A s a h , S.;Nakatsuka, H.; Fujiwara, M.;Matsuola, M. Phys. Rev. to two-exciton coherences. A 1984. 29. 2286. (40)’de’Boe.r, S.;Wierrma, D. A. Chem. Phys. 1989, 131, 135. (41) Weaire, D.; Taylor, P. C. Dynumical Properties of Solids; Horton, References and Notes G. K., Maradudin, A. A., Eds.; North-Holland: Dordrecht, 1980; Vol. 4, pp 1-62. (1) Hanamura, E. Phys. Rev. B 1988, 37, 1273. (2) Rickert, S.E.; Lando, J. B.; Ching, S.In Nonlineur Optical Properties (42) Vacher, R.; Courtens, E.; Coddens, G.; Heidemann, A.; Tsujima, Y.; Pelour, J.; Foret, M. Phys. Reu. Lett. 1990,65, 1008. Courtens, E.; Vacher, of &&!a& and Polymeric Materials; ACS Symposium Series 233; American Chemical Society: Washington, DC, 1983; Chapter 11. R.; Stoll, E. Physicu D 1989, 38, 41. (3) Kuhn, H.; Mdbius, D.; BUcher, H. In Techniques in Chemistry; (43) Alexander,S.;Orbach, R. J . Phys. (Puris)1982,43, L62S. Aharony, Wiley-Interscience: New York, 1972; Vol. 1, Part IIIB, p 577 ff. A.; Entin-Wohlman, 0.;Alexander, S.; Orbach, R. Philos. Mug.B 1987,56, (4) Kempa, K.; Broido, D. A.; Bahhi, P. Phys. Rev. B 1991,43, 9343. 949. (5) Feldmann, J.;Peter,G.;Gdbel,E.O.;Dawson,P.;Moore,K.;Foxon, (44) Fidder, H.; Knoester, J.; Wiersma, D. A. J . Chem. Phys. 1993.98, C.; Elliott, R. J. Phys. Rev. Lett. 1987, 59, 2337. 6564. (6) Terptra, J.; Fidder, H.; Wiersma, D. A. Chem. Phys. Lett. 1991, (45) Duppen, K.; Wiersma, D. A. J. Opt. Soc. Am. B 1986, 3,614. 179, 349. (46) Knoester, J. Phys. Rev. A 1993, 47, 2083.
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