10580
J. Phys. Chem. C 2010, 114, 10580–10591
Ultrafast Dynamics of Frenkel Excitons in Tetracene and Rubrene Single Crystals Brantley A. West,† Jordan M. Womick,‡ L. E. McNeil,† Ke Jie Tan,§ and Andrew M. Moran*,‡ Department of Physics and Astronomy, UniVersity of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599, Department of Chemistry, UniVersity of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599, and School of Materials Science and Engineering, Nanyang Technological UniVersity, Nanyang AVenue, Singapore 639798 ReceiVed: February 23, 2010; ReVised Manuscript ReceiVed: May 3, 2010
Characteristics of thermally driven environmental motion (i.e., spectral densities) similarly control energy and charge-transport processes in self-assembled molecular aggregates, crystalline molecular solids, and photosynthetic antennae. A true microscopic understanding of these transport processes necessarily involves decomposing the spectral density of thermal noise into contributions from specific nuclear motions (i.e., modes). To this end, molecular solids may serve as excellent model systems due to their known crystal structures and well-defined intermolecular modes. Here, the electronic relaxation dynamics of tetracene (Tc) and rubrene (Rb) single crystals are investigated using a variety of nonlinear optical spectroscopies in conjunction with a Frenkel exciton model. Parameterization of the model is achieved by comparing simulated optical signals with those measured in experiments. In addition, electronic population transfer rates are computed with this same set of parameters using a modified Redfield theory. An important aspect of the model is its use of femtosecond stimulated Raman spectroscopies to obtain nuclear mode-specific spectral densities (i.e., polarizability spectral densities). Attainment of the spectral densities facilitates the interpretation of electronic spectroscopies, which are sensitive to both exciton delocalization and population transfer kinetics. One important prediction of the model, which is based on the comparison of low-temperature linear absorption spectra and model calculations, is that the exciton sizes for both Tc and Rb are approximately 18 molecules at 200 and 78 K, respectively. In addition, transient grating experiments detect sub-100 fs intraband population transfer processes in both crystals. Photon echo experiments and model calculations further support the assignment of these dynamics to electronic population transfer. The role of spatial correlations in the spectral densities at different molecular sites is also investigated. Calculations predict a remarkable behavior in which variation in the amount of spatial correlation for just a single mode in the spectral density, among many, causes the electronic relaxation rates to vary over an order of magnitude. This finding holds for both crystals and will likely generalize to molecular aggregates and photosynthetic antennae, thereby contributing to a microscopic understanding of phenomena that originate in spatially correlated fluctuations (e.g., coherent energy transfer). I. Introduction Transport processes and spectroscopic phenomena in molecular aggregates and crystals are intimately connected to the delocalization of electronic states.1-4 Of central importance to electronic structure in these systems is the amount of disorder imposed by thermally driven nuclear motion, which generally competes against intermolecular interactions to localize electronic states to small segments of the total system.5,6 In the limit of perfect periodicity, Bloch’s theorem yields electronic states uniformly delocalized in space, and diffusion governs energy and charge transport on macroscopic length scales.7-10 By contrast, quantum mechanical hopping mechanisms take hold in the limit of large disorder among molecular sites.11-13 This general behavior applies broadly to crystalline molecular solids,14-16 J-aggregates,17,18 and biological pigment complexes.19-25 Whereas the concepts of purely incoherent and band-like energy transport are useful for physical insight, modern models find that treatment of the intermediate regime is essential for capturing the * To whom correspondence should be addressed. † Department of Physics and Astronomy, University of North Carolina at Chapel Hill. ‡ Department of Chemistry, University of North Carolina at Chapel Hill. § Nanyang Technological University.
dynamics of real systems. For example, modified Redfield theories enable realistic simulations of electronic relaxation dynamics in photosynthesis.3,26-29 Spatially correlated environmental motion adds rich complexity to the interplay between electronic structure and lowfrequency nuclear motion.30-33 In general, it is understood that spatial correlations promote the delocalization of electronic states on a length scale comparable to the coherence sizes of intermolecular modes.34 However, conventional experiments do not readily obtain nuclear mode-specific information on the spectral densities at the root of correlation effects in electronic structure. Rather, this information is usually derived indirectly through spectroscopic line shapes (i.e., correlated line broadening).35 Signatures of correlated line broadening are subtle in linear absorption spectra but can be quite prominent in higherorder techniques such as femtosecond transient absorption and/ or photon echo spectroscopies.18,36,37 For example, long-lived intraband exciton coherences have recently been observed in conjugated polymers,38,39 molecular aggregates,40-42 and photosynthetic proteins.43,44 Achieving a microscopic understanding of correlated line broadening in these systems is a particular challenge due to heterogeneity in the liquid environments. For this reason, organic molecular crystals (OMCs) are excellent
10.1021/jp101621v 2010 American Chemical Society Published on Web 05/20/2010
Frenkel Excitons in Tetracene and Rubrene Single Crystals model systems for the development of new experimental techniques and theories. This paper investigates the effects of nuclear mode-specific fluctuations on the spectroscopy and dynamics of two OMCs.45 Tetracene (Tc) and rubrene (Rb) single crystals are used as model systems because of their relatively simple molecular optical responses, known crystal structures, and the availability of crystals with high optical quality. Similar to our earlier work involving photosynthetic proteins,46,47 here, a variety of experimental techniques are used to gather the information required to parametrize a Frenkel exciton Hamiltonian. First, polarizability spectral densities are derived from femtosecond stimulated Raman spectroscopies conducted under preresonance conditions. Linear absorption measurements then examine the influence of the spectral densities on linear and nonlinear optical line shapes. Transient grating experiments detect sub-100 fs intraband electronic relaxation processes in both Tc and Rb. Finally, photon echo experiments further support the assignment of these dynamics to electronic population transfer. Together, these measurements enable the self-consistent simulation of spectroscopic line shapes and nonradiative relaxation rates using a model with spectral densities that carry nuclear mode-specific information. With this experimentally constrained (i.e., realistic) model, we investigate the sensitivity of electronic relaxation rates to spatial correlations imposed by single nuclear modes in the spectral density. Knowledge of the crystal structures is essential to the present analysis and underscores the advantage of using OMCs as model systems for understanding the microscopic basis of phenomena that originate in spatially correlated fluctuations (e.g., coherent energy transfer).31-33,38 The present model draws on connections between stimulated Raman spectra and the spectral densities governing electronic relaxation processes established in earlier work.48,49 Essentially, the idea is that the spectrum of thermally driven solvent motion probed by stimulated Raman spectroscopies (i.e., polarizability spectral density) constitutes a natural description of the spectral density governing the spectroscopy and dynamics of a solute. The first investigations utilizing polarizability spectral densities aimed to pinpoint the microscopic origins of electronic dephasing for dye molecules in solution. Agreement between model calculations and measured photon echo signals confirmed the validity of this approach. Sophisticated laser spectroscopies have grown from these seminal experiments.50-54 Disentangling the polarizability and solute-solvent coupling strength remains a challenge to the interpretation of these techniques. Still, new insights into complex chemical dynamics in solution make their utility clear.53,55 II. Modeling Spectroscopy and Dynamics This section investigates the impact of spatially correlated environmental motion on the spectroscopy and dynamics of OMCs possessing a single molecular species (e.g., Tc and Rb). The present discussion further specializes to systems in which intermolecular interactions are predominantly electrostatic in nature (e.g., transition dipole coupling). Numerous sophisticated theoretical studies of similar systems have appeared in the literature, and the latest advances can be found in recent review articles.56-58 Below, we discuss issues specific to the present model, whose goal is to parametrize the spectral densities governing electronic relaxation dynamics using information obtained with stimulated Raman spectroscopy. An important aspect of the model is its incorporation of both nuclear modespecific and spatially correlated line broadening mechanisms.
J. Phys. Chem. C, Vol. 114, No. 23, 2010 10581
Figure 1. Crystal structures of (a) Tc and (b) Rb. In Tc, a ) 7.9 Å, b ) 6.0 Å, and c ) 13.5 Å. In Rb, a ) 7.2 Å, b ) 1.4 Å, and c ) 2.7 Å. The a-b-c convention for Rb used here is consistent with, for example, that of refs 112-114. Molecular structures of (c) Tc and (d) Rb in which the lowest-energy transition dipoles are overlaid on the molecules.
II.A. Vibronic Frenkel Exciton Hamiltonian. The Hamiltonian of the crystal is partitioned into three components representing the system, bath, and their interaction
H ) HSys + HBath + HSys-Bath
(1)
Thus, thermal fluctuations enter the model through standard line broadening effects.35 The treatment of HSys appropriate with small intramolecular vibronic coupling would view each molecular site as a two-level (purely) electronic system that interacts with its neighbors through electrostatic coupling. Such models work quite well for some photosynthetic pigment complexes20,27,59 but are inadequate for simulations of oligoacene crystals (e.g., tetracene).56-58 We therefore write HSys as a Holstein-like Hamiltonian4,60,61
HSys )
† Bmν(1 + νων) + ∑ ∑ EmBmν m
ν)0
† Bnλ〈0|ν〉〈0|λ〉 ∑ ∑ ∑ ∑ JmnBmν m
(2)
n*m ν)0 λ)0
Here, the indices m and n represent molecular sites, whereas ν and λ are vibronic levels. In this notation, the operator B†mν (Bmν) creates (annihilates) an excitation at molecule m in vibronic level ν. Equation 2 specializes the model to systems such as Tc or Rb, in which one high-frequency nuclear coordinate, which is not populated in the ground state, dominates the vibronic progression (i.e., vinyl stretching mode near 1400 cm-1).62,63 Intermolecular interactions, Jmn, are taken to occur through coupling between transition dipoles; the dipoles are superposed on the structures of Tc and Rb in Figure 1. The product of
10582
J. Phys. Chem. C, Vol. 114, No. 23, 2010
West et al. TABLE 1: Parameters of the Frenkel Exciton Model parameter
tetracene
rubrene
-1
a
19660 cm 4.5 D 1400 cm-1 1.2 19500 cm-1 Table S4 (SI) 200 K
Em |µ bm|b ωνa ∆νc Sd Ak, ωk, Γkd Td
19120 cm-1 4.5 D 1400 cm-1 1.0 19100 cm-1 Table S8 (SI) 78 K
a Parameter of eq 2. Transition dipole magnitude used to calculate Jmn in eq 2. Orientations of the vectors are shown in Figure 1. c Dimensionless displacement used to calculate vibrational overlap integrals in eq 2. Values of ∆ν are estimated using absorption spectra in solution. d Parameter of eq 6. See Supporting Information for Ak, ωk, and Γk. b
Figure 2. (a) Each molecular site, m, is a four-level system consisting of a ground state and three vibronic levels, ν, associated with the lowestenergy excited electronic state. (b) The vibronic levels delocalize in the exciton basis, where mixing between bands is a fairly minor effect because the couplings are small compared to the 1400 cm-1 vinyl stretching frequency. The dummy indices a and b correspond to the single exciton states.
overlap integrals between nuclear wave functions, 〈0|ν〉〈0|λ〉, distributes the intersite coupling, Jmn, in the vibronic basis.64-67 The nuclear wave function overlaps are computed in a harmonic basis with no Duchinsky rotation.68 The single exciton block of the Hamiltonian matrix is constructed in a basis of vibronic excitations denoted as |m,ν〉. The vibronic progressions of (molecular) Tc and Rb are dominated by the 0-0, 0-1, and 0-2 transitions.63 Therefore, the expansions in ν are truncated at ν ) 2 in the numerical calculations below to reserve computational resources for the treatment of large molecular arrays. Diagonalization of the single exciton block of the Hamiltonian for a system with finite size yields the eigenvectors
|a〉 )
∑ ∑ φa,mν|m, ν〉 m
(3)
ν)0
The eigenvectors of eq 3 enable the simulation of linear absorption spectra in addition to several nonlinear optical signal components. However, excited-state absorption nonlinearities involve an additional double exciton manifold of states, which is usually expanded in a direct product basis, |m,ν〉|n,λ〉.69 Calculation of these double exciton states challenges computational resources in simulations of large systems. For example, the number of double exciton states is N(N - 1)/2, with N states in the single exciton basis. Fortunately, the double exciton states can be neglected here because the excited-state absorption signal components do not dominate the nonlinear optical response at the field frequencies employed in our experiments. Figure 2 illustrates the present view of electronic structure and makes clear the notation used in the local and exciton bases. Because the frequency of the 1400 cm-1 vinyl stretching mode is large compared to magnitudes of the intermolecular couplings, the electronic structures of the crystals can be understood in terms of a standard Frenkel exciton model. To a good approximation, the width of a particular vibronic band is given by the size of the intermolecular coupling between nearest neighbors times 8 (e.g., eq 6.21 in ref 45). Thus, the widths of the vibronic band, ν, depend on the size of the wave function overlap, 〈0|ν〉, which governs the coupling strength through the second term in eq 2. The number of states within a particular vibronic band is simply equal to the number of molecules in the system. For example, the widths of the ν ) 0 bands for Tc and Rb are respectively 2120 and 910 cm-1 using the parameters
in Table 1. The ν ) 0 band is most relevant to the experiments presented below because laser pulses are tuned to the lowestenergy portions of the linear absorption spectra. The present model examines spatially correlated fluctuations by treating systems with finite size. Therefore, the simulations are conducted for large portions of the crystal to suppress contributions from states localized at the boundary. The system geometry should be chosen carefully for optimal use of computational resources. We model only a single a-b plane of Tc because the molecules are most closely spaced in the a-b plane; the distance between the transition dipoles of the two molecules in the same unit cell is 0.50 nm. In comparison, the distance between a-b planes is 1.35 nm.70 Moreover, our spectroscopies utilize electric field polarizations oriented in the plane of the a and b crystal axes; the projections onto the c axis of Tc are negligible. By a similar argument, we model only a single a-b plane of Rb; the present a-b-c notation is defined in Figure 1. Rb possesses orthorhombic structure in which adjacent a-b planes are separated by 1.35 nm, whereas the nearest neighbors within the same plane are separated by only 0.81 nm.71 II.B. Mode-Specific Spectral Densities. Standard treatments of spectroscopic line broadening in molecular aggregates account for thermally driven intermolecular motions (e.g., librations, center of mass displacements) through fluctuations of the site energies, Em, of eq 2.3,27,72 These fluctuations can be incorporated using time correlation functions and/or ensemble averaging in the site energy distributions. The model outlined in this section utilizes the time correlation function approach because of its ability to interpolate between the static and motionally narrowed limits. In particular, we employ an algorithm for simulating twobody correlations that can be applied to both optical response functions and a modified Redfield theory yielding electronic relaxation rates.26,28 Additionally, the spectral densities are parametrized directly with the vibrational resonances detected in stimulated Raman measurements rather than using generic distributions (e.g., Drude, Ohmic).48,49 The fundamental connection between stimulated Raman (3) (ω), and the spectral density, C˜(ω), was established spectra, χSR in earlier work involving dye molecules in liquid solution.48,49 The two are related by
[
( )]
C(ω) ) S(ω) 1 + coth
pω 2kBT
(3) Im[χSR (ω)]
(4)
(3) Here, χSR (ω) is the stimulated Raman spectrum, S(ω) is a coupling strength, and C(ω) is the polarizability spectral density (PSD). Briefly, we emphasize two principal distinctions between
Frenkel Excitons in Tetracene and Rubrene Single Crystals the PSD and the true spectral density. First, the PSD involves only Raman-active modes, which are relatively well-defined in molecular solids compared to liquids. For example, under the rigid molecule approximation, exactly 6 of the 12 intermolecular modes are Raman-active in a system with two molecules per unit cell (e.g., Tc). The second important distinction is that the mode amplitudes in the PSD reflect the magnitude of the polarizability in addition to the coupling strength, S(ω). Therefore, it is possible for highly polarizable yet weakly (3) (ω). To coupled modes to make large contributions to χSR suppress this effect, the measurements discussed in section IV enhance the contributions of coupled modes by applying laser pulses in the preresonance regime.73 The stimulated Raman spectrum, χ(3) SR(ω), generally depends on the four experimentally controlled electric field polarizations and the crystal orientation. These polarization effects are beyond the scope of this paper and are hereafter neglected. Spatially correlated fluctuations are incorporated under the assumption that each molecular site energy, Em, is modulated by identical thermally driven nuclear motion. Fluctuations at sites m and n imposed by mode k are described in the local basis by the spectral density k Cmn (ω) ) δmnLk(ω) + (1 - δmn)ηkξ(Rmn)Lk(ω)
(5)
J. Phys. Chem. C, Vol. 114, No. 23, 2010 10583 of spatially correlated nuclear motion (ηk ) 0), eq 8 essentially multiplies a sum over mode-specific line broadening functions by a quantity representing the spatial overlap in exciton states |a〉 and |b〉, that is
gab(t) ) gba(t) ) (
2 2 φb,nλ δmn)( ∑ gk(t)) ∑ ∑ φa,mν mn ν)0 λ)0
k
(10) The line broadening functions, gab(t), allow the simulation of optical line shapes and electronic population transfer dynamics. The linear absorption spectrum is given by
σA(ω) )
∑ (eˆi ·fµag)2 ∫0
∞
dt exp[i(ω - ωag)t -
a
gaa(t)]Φa(t) (11) where eˆi is the electric field polarization and Φa(t) accounts for lifetime broadening by summing the Green function, Gba(t), over all relaxation channels as46
Φa(t) )
∑ Gba(t)
(12)
b
Here, ξ(Rmn) is a spatial correlation function depending on the distance between sites m and n, Rmn, and the correlation parameter, ηk, interpolates between the fully anticorrelated (ηk ) -1) and correlated (ηk ) 1) limits. The auxiliary function Lk(ω) can assume arbitrary form. Here, it is given by
( )]
[
Lk(ω) ) S 1 + coth
pωk AkΓk 2kBT ω2 - ω2 + Γ2 k k
(6)
where ωk and Γk are mode frequencies and line widths. The parameter, S, is used to scale the overall coupling strength, whereas Ak is a mode-specific coupling obtained from fitting stimulated Raman signals (see section IV.A). Site-independent and mode-specific line broadening functions are obtained using35
gk(t) ) -
1 2π
∫-∞∞ dω
Lk(ω) ω2
[exp(-iωt) + iωt - 1]
(7) The single exciton eigenvectors in eq 3 are used to transform the mode-specific gk to the exciton basis with
gab(t) ) gba(t) )
2 2 k φb,nλ Θmn ∑ gk(t) ∑ ∑ φa,mν k
mn
(8)
ν)0 λ)0
where k Θmn ) δmn + (1 - δmn)ηkξ(Rmn)
(9)
Equation 8 writes separate summations for modes in the spectral density (k), molecular sites (m and n), and the high-frequency intramolecular modes of eq 2 (ν and λ) to best convey the k possesses all information content. The second term in Θmn information on spatial correlations. For example, in the absence
The Green function, Gba(t), describing population transfer between exciton states a and b is computed by solving a master equation with modified Redfield rate constants. The algorithm used to calculate the Green function, Gba(t), has been described elsewhere.26,27,74 Formulation of the nonlinear response function has been reviewed recently by Mukamel and co-workers.3,72 In the Appendix, it is shown how the line broadening functions, gab(t), parametrize the nonlinear response functions involving only single exciton states. It should be emphasized that all of states and relaxation dynamics involved in these nonlinearities are treated explicitly. III. Experimental Methods Tc and Rb single crystals are grown using the physical vaportransport method at Nanyang Technological University in the laboratory of Prof. Christian Kloc.75 The crystals are thin plates spanning 5-10 mm in the lateral dimensions. The faces of the Tc and Rb crystals are the a-b planes of their triclinic76 and orthorhombic structures (see notation convention for Rb in Figure 1).77 The crystal orientations are determined by measuring light transmission in the 400-800 nm range (deuterium-tungsten lamp) with “crossed” Glan Taylor calcite polarizers located before and after the crystal. One polarizer is set for transmission of p-polarized light, whereas the other is set for s-polarized transmission. The crystal is placed between the polarizers, and its orientation is varied while measuring light transmission. Minima in light transmission are measured at intervals of 90° when the incident light is polarized parallel to either the a or b crystal axes. The crystals are mounted on 1 mm thick fused silica substrates using vacuum grease as an adherent. Linear absorbance spectra are acquired with an Ocean Optics spectrometer (HR2000). A liquid-nitrogen-cooled cryostat (OptistatDN, Oxford Instruments) is used for temperature-dependent measurements. Transient grating (TG) and photon echo (PE) spectroscopies utilize a diffractive optic-based interferometer resembling those reported in several earlier publications.40,78-82 Briefly, the
10584
J. Phys. Chem. C, Vol. 114, No. 23, 2010
West et al.
Figure 3. (a) Feynman diagrams corresponding to PE signal components in the Appendix. (b) Pulse sequence used for TG and PE spectroscopies. Arrival times of the three pulses (at their peaks) are jτi. The delays, τ ) jτ2 - jτ1 and T ) jτ3 - jτ2, are experimentally controlled. Intervals between field-matter interaction times are given by t1, t2, and t3.
experiment applies four laser pulses, Ej, in a trapezoidal geometry. Figure 3 displays the three pulses that induce the nonlinear polarization; the fourth pulse is a reference field used for heterodyne detection. The signal is phase-matched in the direction kS ) -k1 + k2 + k3 and is automatically collinear with the local oscillator field, E4, after the sample. In TG, the arrival of the time-coincident E1 and E2 pulses is controlled with a motorized stage. PE additionally controls the delay between the E1 and E2 pulses by inserting prism wedges in the paths of the individual beams.80 TG measurements typically scan the experimentally controlled delay, T, over 500-800 points, where the range of the scan is set to capture the full dephasing dynamics of nuclear coherences. PE scans the delay, τ, for 125-150 fs with a step size of 0.5 fs. Scans are repeated 10-15 times and averaged. Total data acquisition times are 30-60 min. Pulse energies are 50-100 nJ, and the fwhm spot sizes at the sample are approximately 120 µm. Signals are detected by spectral interferometry using a back-illuminated CCD array (Princeton Instruments PIXIS 100B) mounted on a 0.3 m spectrograph. Integration times are 100-200 ms. Signals are processed using a Fourier transform algorithm.80,81,83-85 Two TG laser pulse configurations are used below. Stimulated Raman measurements derive the E1 & E2 and E3 & E4 pulse pairs from separate home-built noncollinear optical parametric amplifiers.86-88 Spectra of the E1 & E2 and E3 & E4 pulse pairs are respectively centered at 17500 and 15900 cm-1. The E3 & E4 pulse pair is tuned to lower frequency to suppress interference from scattered E1 and E2 light. TG measurements obtaining electronic relaxation rates in section IV.C center all four pulses at 18500 cm-1. A single color is utilized because this experiment requires optimal time resolution and cannot tolerate group velocity mismatch between the pump (E1 and E2) and probe (E3 and E4) pulses. Finally, the PE experiments in section IV.D are conducted in a one-color configuration with laser spectra centered at 18000 cm-1. For all measurements, the pulses are nearly Fourier-transform-limited with durations of 20-30 fs before interaction with the sample.
Figure 4. (a) Imaginary part of the transient grating signal field for Tc. (b) LPSVD fit (red) of the measured signal (black) from (a). (c) Imaginary part of the transient grating signal field for Rb. (d) LPSVD fit (red) of the measured signal (black) from (c). These measurements are performed at 78 K with b-polarized electric fields.
IV. Results and Discussion IV.A. Polarizability Spectral Densities. This section parametrizes the auxiliary function, Lk(ω), using the mode frequencies and line widths in the stimulated Raman spectra presented below. These quantities are obtained by fitting the imaginary (i.e., dispersive) part of the TG signal, S(T), to the phenomenological equation
S(T) )
∑ Ak cos(ωkT + φk) exp(-ΓkT)
(13)
k)1
using a linear prediction singular value decomposition (LPSVD) algorithm.89,90 Here, S(T) is a temporal slice of the TG signal field, Ak is the amplitude for LPSVD component k in eq 6, ωk is the recurrence frequency, φk is a phase shift, and Γk is a damping parameter. Figure 4 shows that LPSVD yields highquality fits with typical experimental noise levels. Generally, we find reproducible frequencies, line widths, and phases for signal components in which Ak is at least 5% of the maximum amplitude. Stimulated Raman spectra derived from the LPSVD fits below display only these robust fitting components. Signal detection by spectral interferometry disperses the TG signal field in a spectrometer.83,91,92 Thus, the temporal profile, S(T), can be defined with respect to particular emission frequencies within the bandwidth of the signal spectrum (e.g., a single pixel of the array detector). Flexibility in choosing the emission frequency can be leveraged to enhance the detection of nuclear modes that might otherwise be hidden in conventional spontaneous or coherent Raman methods.93-95 For example, Ziegler and co-workers find that the amplitudes of higher-
Frenkel Excitons in Tetracene and Rubrene Single Crystals
Figure 5. Frequency domain representation of LPSVD fitting components for Tc. Components with amplitudes at least 10% of the maximum are included. The left and right columns respectively display measurements with electric field polarizations aligned to the a and b crystal axes. Temperatures are organized as follows: (a, b) 78 K; (c, d) 190 K; (e, f) 296 K.
Figure 6. Frequency domain representation of LPSVD fitting components for Rb. Components with amplitudes at least 15% of the maximum are included. The left and right columns respectively display measurements with electric field polarizations aligned to the a and b axes. Temperatures are organized as follows: (a, b) 78 K; (c, d) 190 K; (e, f) 296 K.
frequency modes are enhanced when detection is performed at frequencies detuned from the peak of the E3 field spectrum.93 The enhancement of weak signal components is one criterion used to select the emission frequencies. Additionally, we are able to choose only from the subset of pixels for which scattered light does not dominate the signal. Therefore, the detection frequencies used to obtain the Raman spectra below represent a compromise between scattered light suppression and enhancement of vibrational resonances with small amplitudes. Shown in Figures 5 and 6 are stimulated Raman spectra of Tc and Rb measured at various temperatures. Of course, molecular bond connectivity and crystal structure govern
J. Phys. Chem. C, Vol. 114, No. 23, 2010 10585 differences in the Raman spectra of the two crystals. However, two notable similarities in the data relate to more general physical principles. First, the signal strengths are at least an order of magnitude weaker when the electric fields are aligned parallel to the a axis. Second, the number of detected resonances increases at lower temperatures. One explanation for the increasing number of resonances at lower temperatures derives from the concomitant decrease in dephasing rates; the LPSVD algorithm more readily extracts these longer-lived coherences. Strengthening of intermolecular forces promoted by contraction in the unit cell volume at lower temperatures may also contribute to these additional resonances.45 Transitions observed only at 78 or 190 K most likely represent intermolecular modes because the temperature dependence of intramolecular vibrations is usually fairly weak.96 It is also possible that the phase transition of Tc near 175 K explains the temperature dependence of its Raman spectra.71,76,97 By contrast, a recent spontaneous Raman study of Rb confirms the enhancement of certain vibrational resonances near 175 K but does not find a phase transition (in the structural sense).98 The Raman spectra of Tc display only a single peak at 296 K for both a and b polarizations. The b-polarized spectrum at 190 K reveals a new higher-frequency resonance near 175 cm-1 in addition to two transitions below 100 cm-1. Although robust assignments cannot be made at this time, large anharmonicities of the intermolecular potentials could support assignment of the transition near 175 cm-1 to a combination band with contributions from the 120 and sub-100 cm-1 modes. At 78 K, several additional transitions are observed under both polarization conditions, particularly in the b-polarized spectrum. Insight into the signal generation mechanism is derived from the fact that the b-polarized resonances near 44, 60, and 120 cm-1 are also observed in spontaneous Raman experiments employing an excitation frequency of 15300 cm-1.96 In general, the signal amplitude for a particular mode reflects both the magnitude of the polarizability and the size of its Franck-Condon factor with preresonance excitation.73 For this reason, agreement with spontaneous Raman measurements conducted further offresonance suggests that pure polarizability enhancement has a major influence on the transition amplitudes in Figure 5.96 The Raman spectra of Rb shown in Figure 6 generally exhibit higher-frequency resonances than those found in Tc. Transitions in the 100-300 cm-1 range are known to be intramolecular modes.96 As expected, the temperature dependence of this portion of the spectrum is fairly weak, whereas significant changes are observed at frequencies less than 75 cm-1. For example, the 45 cm-1 resonance in the a-polarized spectrum vanishes, and a new mode appears at 8 cm-1 when the temperature is reduced to 78 K. By contrast, the b-polarized spectrum changes little between 78 and 190 K; the main difference is a reduction in amplitude for the 15-20 cm-1 mode(s). Overall, the transition amplitudes of predominantly intermolecular modes below 75 cm-1 are smaller than those of Tc. This result is rationalized by the suppression of closepacking due to the presence of phenyl groups in Rb. In fact, the primitive unit cell of Rb is approximately 4 times larger than that of Tc. Assignments of many of the low-frequency vibrational resonances in Rb to specific nuclear modes have been given in recent literature.96,99 IV.B. Exciton Sizes and Line Narrowing in Linear Absorption Spectra. Information on electronic structure is more readily extracted with the spectroscopic line narrowing achieved at lower temperatures. In addition, lower temperatures slow down electronic relaxation, thereby promoting the resolution
10586
J. Phys. Chem. C, Vol. 114, No. 23, 2010
Figure 7. Experimental linear absorption spectra of (a) Tc and (b) Rb measured at 200 and 78 K, respectively. Calculated absorption spectra for (c) Tc and (d) Rb with two unit cells in the a and b dimensions. Calculated absorption spectra for (e) Tc and (f) Rb with six unit cells in the a and b dimensions. Calculations use eq 11 and the parameters in Table 1. In all panels, a-polarized and b-polarized spectra are displayed as black and red lines, respectively.
of dynamics in time-resolved experiments. Therefore, we have performed all electronic spectroscopies for both crystals at the lowest temperatures for which adequate signal-to-noise ratios are obtained. Experiments for Tc are performed at 200 K due to a roughening of the crystal surface induced by its phase transition near 180 K. Light scattering promoted by roughening of the crystal surface poses a significant challenge to interferometric signal detection. Stress in Tc crystallites at the phase transition is known to be quite sensitive to the details of its contact with the substrate, which could explain these observations.76 By contrast, the Rb crystallites are stable to cooling and heating within the 78-298 K temperature range. Measurements with Rb are performed at 78 K. In all simulations, the spectral densities of Tc and Rb correspond to the LPSVD components in Figures 5d and 6b, respectively. We use the b-polarized Raman spectra because the experimental errors are smaller than those associated with the weaker a-polarized signals. Linear absorption spectra of Tc measured with a-polarized and b-polarized light are shown in Figure 7. The blue shift in the b-polarized spectrum reflects the Davydov splitting, which derives from the transformation of intermolecular interactions into the exciton basis.1 The electronic structure of Tc is understood in the same framework as that of a molecular dimer because it possesses two molecules per unit cell.45 The bpolarized transition of Tc occurs at higher frequency because the molecular transition dipoles project mostly on the b axis (Figure 1). Additionally, the transition dipole coupling for a pair of molecules in the same unit cell has a positive sign; the angle between dipoles is approximately 60°. Therefore, the crystal geometry dictates that excitations with transition dipoles oriented parallel and perpendicular to the b axis respectively reside at the top and bottom of the single exciton band. Absorption spectra calculated with eq 11 capture the sign of the splitting but underestimate the magnitude. Of course, the size of the molecular dipole, |µ bm| in Table 1, can be increased to make the calculated splitting exactly reproduce the experimental splitting. However, we do not increase it beyond 4.5 D because the intermolecular coupling strengths would then be
West et al. inconsistent with earlier spectroscopic models for Tc.62,100 The underestimated splitting most likely reflects limitations of the transition dipole coupling approximation for nearest neighbors. Theoretical work confirms the importance of treating the shape of the transition density when computing electrostatic couplings at close proximity.101,102 Still, the transition dipole coupling mechanism is useful for the present model calculations because it captures the most important aspects of optical response. Absorption spectra of Tc are simulated for systems with two sizes, an array that is 2 × 2 in the a and b dimensions and an array that is 6 × 6 in the a and b dimensions. The calculations show that oscillator strength concentrates on the lowest-energy transition as the system size increases (i.e., k ) 0 state in periodic system). Dominance of the lowest-energy transition dipole is a well-known signature of exciton delocalization and is, for example, at the origin of superradiance in molecular J-aggregates.103,104 The red-shifted peak in the a-polarized absorbance spectrum near 18540 cm-1 suggests that the exciton states have achieved significant delocalization at 200 K. The model calculations find that oscillator strength begins to concentrate in the lowest-energy state for a 3 × 3 system. Therefore, the present comparison of experiment and theory suggests an exciton size of approximately 18 molecules, which is slightly larger than the estimation of 10 molecules found in ref 62. It is worth noting that an exciton size of 10 molecules was also recently found for anthracene thin films.105 The linear absorption spectra of Rb possess similar apolarized and b-polarized line shapes. Similar to Tc, the spectra reveal significant concentration of the oscillator strength in the lowest-energy single exciton transition. The transitions possess narrower line widths than those of Tc, which partly reflects the 78 K temperature at which the spectra for Rb are acquired. The smaller exciton splitting in Rb derives from the fact that its molecular transition dipoles are normal to the a-b plane, whereas those of Tc have large projections in the a-b plane. When the transition dipoles are perfectly normal to the a-b plane, the ratio of calculated a-polarized to b-polarized absorbance is much larger than that shown in Figure 7b. Therefore, to approximately reproduce the measured ratio, the molecular dipoles used to calculate the spectra in Figure 7d and f are rotated by 5° with respect to the c axis and further reoriented such that the projection in the a-b plan makes an angle of 35° with the a axis. Similar to Tc, the calculated oscillator strength of Rb concentrates on the lowest-energy transition when the system size increases. On the basis of comparison to our model, we also suggest an exciton size of 18 molecules in Rb at 78 K. We remark that the present approach to determining exciton sizes employs experimental and theoretical techniques not available during the 1960-1980s, when a significant amount of work was published in this area.1,16,45 For example, the selfconsistent treatment of electronic relaxation and spectroscopic line broadening (i.e., modified Redfield theory) is essential to our estimation of exciton sizes. IV.C. Electronic Population Relaxation Probed by Transient Grating Spectroscopy. Experimental detection of electronic relaxation in Tc and Rb is challenged by extremely fast (
