Vibronic Coupling in J-Aggregates and Beyond: A ... - ACS Publications

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Vibronic Coupling in J-Aggregates and Beyond: A Direct Means of Determining the Exciton Coherence Length from the Photoluminescence Spectrum Frank C. Spano* and Hajime Yamagata Department of Chemistry, Temple University, Philadelphia, Pennsylvania 19122 ABSTRACT: Exciton coherence in a J-aggregate with exciton-phonon coupling involving a single intramolecular vibration is studied. For linear aggregates with no disorder and periodic boundary conditions, the 0-0 to 0-1 line strength ratio, SR, corresponding to the low-temperature photoluminescence spectrum is rigorously equal to N/λ2, where N is the number of chromophores comprising the aggregate and λ2 is the Huang-Rhys factor of the coupled vibrational mode. The result is independent of exciton bandwidth and therefore remains exact from the weak to strong exciton-phonon coupling regimes. The simple relation between SR and N also holds for more complex morphologies, as long as the transition from the lowest exciton state to the vibrationless ground state is symmetry-allowed. For example, in herringbone aggregates with monoclinic unit cells, the line 0-1 (where I0-0 and I0-1 correspond to the b-polarized 0-0 and 0-1 line strengths, strength ratio, defined as SR I0-0 b /Ib b b 2 respectively) is rigorously equal to N/λ . In the presence of disorder and for T > 0 K, λ2SR is closely approximated by the exciton coherence number Ncoh, thereby providing a simple and direct way of extracting Ncoh from the photoluminescence spectrum. Increasing temperature in linear J-aggregates (and herringbone aggregates) generally leads to a demise in SR and therefore also the exciton coherence size. When no disorder is present, and under the fast scattering and thermodynamic limits, SR is equal to NT/λ2, where the thermal coherence size is given by NT = 1 þ [4πωc/kbT]d/2 for an aggregate of dimension d, where ωc is the exciton band curvature at k = 0.

I. INTRODUCTION The concept of coherence is central to exciton transport in molecular crystals, films, and aggregates as it derives from the fundamental wave-like nature of delocalized excited states in marked opposition to incoherent transport, which relies on diffusive hopping. Exciton coherence is normally measured in terms of the number of coherently connected molecules, Ncoh, which can easily be transformed into a coherence length or volume. From an applications standpoint, a large Ncoh is highly prized since transport within the coherence domain is far more rapid than incoherent hopping and would lead, for example, to more efficient organic-based photovoltaic devices. The difficulty in achieving large coherence domains is the sensitivity of the latter to static disorder and thermal fluctuations. Large Ncoh requires low temperature and ultrapure samples. An increase in the exciton coherence length impacts the optical response as well, through enhanced radiative decay rates in cationic J-aggregates,1-5 LH2 ring aggregates,6,7 and polyacene thin films and crystals,8-11 as well as reduced splittings between the bleaching and induced absorption bands in aggregate pump-probe spectra.12-15 Such experiments therefore provide a means of determining the exciton coherence length. Mukamel and co-workers have developed a useful framework based on the exciton density matrix with which to understand exciton coherence and its relation to the aforementioned observables.16-19 r 2010 American Chemical Society

It is the purpose of the present work to introduce a novel and direct means of determining exciton coherence in molecular aggregates from the shape of the photoluminescence spectrum. The method is applicable when the constituent chromophores exhibit a vibronic progression in their solution (isolated molecule) PL spectrum. The distortion of the vibronic progression upon aggregation provides a direct means of determining the coherence length. Fortunately, the great majority of conjugated molecules exhibit well-resolved progressions involving the symmetric vinyl stretching mode with energy pω0 ≈ 0.18 eV and Huang-Rhys (HR) factor λ2 in the range of 0.6-1.2. In effect, this ubiquitous vibration serves as a probe for the coherence length. We will show that when the optical transition from the lowest exciton to the vibrationless electronic ground state (the “0-0” transition) is symmetry-allowed, as in a J-aggregate,20 the ratio of the 0-0 to 0-1 transition line strengths (SR) in the low-temperature PL spectrum is simply equal to Ncoh/λ2, where Ncoh is the exciton coherence number determined largely by static and dynamic disorder. The dependence on Ncoh is carried entirely by the 0-0 line strength; the Special Issue: Shaul Mukamel Festschrift Received: May 24, 2010 Revised: August 20, 2010 Published: October 19, 2010 5133

dx.doi.org/10.1021/jp104752k | J. Phys. Chem. B 2011, 115, 5133–5143

The Journal of Physical Chemistry B 0-1 line strength is not coherently enhanced and is independent of Ncoh.21,22 We treat in detail two limiting cases, aggregates with diagonal disorder at T = 0 K and defect-free aggregates at a nonzero temperature. The general case for disordered aggregates with T > 0 K is also considered. In many ways, the present work serves as a companion to our prior work on exciton coherence in linear H-aggregates,23 where the radiative decay of the exciton at the band bottom to the vibrationaless ground state (0-0) is symmetry-forbidden. In this case, SR is zero when T = 0 K, that is, there is no 0-0 peak. Extracting the coherence size is therefore only possible when the 0-0 transition is given oscillator strength through the introduction of disorder and the resulting symmetry breaking. In the present case of J-aggregates, the analysis is much more straightforward since in the ideal limit of no disorder, SR = N/λ2, where N is the actual number of chromophores in the aggregate. In marked contrast to the case of ideal H-aggregates, adding disorder reduces SR as the coherence number shrinks to reflect exciton localization. The paper is structured as follows. In the following section, we present the Holstein Hamiltonian24 in the multiphonon basis set and introduce the form of one-exciton wave functions. In section III, the multiphonon eigenstates are used to rigorously derive the relation SR = N/λ2 for a linear J-aggregate with periodic boundary conditions. In such an ideal J-aggregate, the lowest exciton has wave vector k = 0, resulting in a 0-0 transition which consumes all of the oscillator strength to the vibrationless ground state. The 0-0 oscillator strength scales with N, leading to superradiance at low temperatures.1,2,5,6 We prove, quite remarkably, that SR = N/λ2 is independent of the exciton bandwidth, the HR factor λ2, and the vibrational frequency ω0 and is thus valid throughout the entire regime of exciton-phonon coupling. In section IV, we consider the localizing influence of diagonal disorder,25 demonstrating that SR = Ncoh/λ2, making SR a direct probe of the exciton coherence number. Ncoh is determined from the exciton coherence function, C(s), defined in terms of exciton-vacuum phonon operators, in contrast to the more conventional pure exciton forms of Mukamel and co-workers16-19 and Sundstrom and coworkers.14,15,18 In section V, we treat emission from a thermalized Boltzmann distribution of ordered J-aggregates, followed in section VI with an analysis of the more general case of disordered J-aggregates with T > 0 K. More complex morphologies with two or more molecules per unit cell are discussed in section VII, where it is shown that the polarization component of the PL spectrum associated with the lower Davydov components retains the same basic temperature and disorder dependences found for ideal J-aggregates, while the component of polarization corresponding to the upper Davydov component mimics H-aggregate behavior. In the final section, we conclude our findings and discuss applications to polyacene and oligothiophene assemblies.

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Hamiltonian reads26 X X H ¼ ω0 b†n bn þ ω0 λ jnæÆnjfbn þ b†n g n

þ

X

n

Jnm jnæÆmj þ ω0 λ2 þ ω0-0 þ D

ð1Þ

n,m

where we have set p = 1. The first term represents the vibrational energy, with the operators b†n and bn, respectively, creating and annihilating vibrational excitations on the nth molecule. The vibrational mode is taken to be the symmetric vinyl stretching mode with energy ω0 ≈ 0.17 eV (1400 cm-1). The corresponding Huang-Rhys factor is λ2, which lies in the range of 0.6-1.2 for most conjugated molecules. The second term in eq 1 represents on-site vibronic coupling, while the third represents excitonic coupling, with the state |næ indicating that molecule n (=1, 2, ..., N) is electronically excited while all others remain in the electronic ground state. Jmn is the excitonic coupling between molecules at m and n. Finally, ω0-0 þ D represents the molecular 0-0 transition frequency plus the gas-to-crystal shift sourced by nonresonant interactions. Rewriting the Hamiltonian in eq 1 in k,q space, where k (q) is the wavenumber of the exciton (phonon), allows the separation of the q = 0 symmetric phonon, yielding the form27,28 H ¼ H1 þ H2

ð2aÞ

where ω0 λ ω0 λ2 H1 ¼ ω0 b†q¼0 bq¼0 þ pffiffiffiffi ðbq¼0 þ b†q¼0 Þ þ N N

ð2bÞ

contains the full dependence on the q = 0 phonon and X ω0 λ X H2 ¼ ω0 b†q bq þ pffiffiffiffi fjk þ qæÆkjbq þ h:c:g N k, q6¼ 0 q6¼ 0 X 1 ~ þ J k jkæÆkj þ 1 ð2cÞ ω0 λ2 þ lω0-0 þ D N k accounts for the interaction (scattering) between all non-totally symmetric (q 6¼ 0) phonons and excitons. Here, the delocalized excitons with dimensionless wave vector k are given by X jkæ ¼ N -1=2 eikn jnæ n

k ¼ 2πl=Nðl ¼ 0, 1, ..., N - 1Þ

ð3Þ

and the phonons with dimensionless wave vector q are defined through the operators X eiqn b†n ð4aÞ b†q ¼ N -1=2 n

and bq ¼ N -1=2

X

e-iqn bn

ð4bÞ

n

II. LINEAR AGGREGATES WITHOUT DISORDER: MULTIPHONON BASIS SET The coupled exciton-phonon system is represented by the usual site-based Holstein Hamiltonian,24 where the nuclear potentials for molecular vibrations in the ground (S0) and excited (S1) electronic states are shifted harmonic wells of identical curvature. In the vector space containing a single electronic excitation in an aggregate consisting of N molecules, the

with q = 2πl/N(l = 0, 1, ..., N - 1). The electronic coupling between molecules is diagonal in the delocalized exciton basis with X ~ Jmn cos½kðn - mÞ Jk n

Equations 2a and 2b show that only the non-totally symmetric phonons participate in the nonadiabatic coupling between excitons with different k. 5134

dx.doi.org/10.1021/jp104752k |J. Phys. Chem. B 2011, 115, 5133–5143

The Journal of Physical Chemistry B

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The eigenstates and energies of H1 are readily determined by rewriting H1 in diagonal form, H1 = ω0~b†q=0~bq=0, where the shifted q = 0 phonon creation and annihilation operators are λ b~†q¼0 b†q¼0 þ pffiffiffiffi N

and

λ b~q¼0 bq¼0 þ pffiffiffiffi N

ð5Þ

The eigenstates of H1, denoted as |~nq=0æ, correspond to the usual harmonic oscillator number states, which satisfy ~b†q=0~bq=0|~nq=0æ = ~nq=0|~nq=0æ with energy ~nq=0ω0. Here, ~nq=0 (=0, 1, 2, ...) is the number of totally symmetric phonons. Because of the separability of the totally symmetric phonon, ~nq=0 is a good quantum number. Moreover, since we have assumed periodic boundary conditions, the total momentum k ¼ k þ nq1 q1 þ nq2 q2 þ ... þ nqN-1 qN-1

ð6Þ

(modulo 2π) is also a good quantum number, where nql is the number of non-totally symmetric phonons with wave vector ql = 2πl/N (l = 1,2, ..., N - 1). Hence, the Rth eigenfunction (R = 1, 2, ... in order of increasing energy) of H in eq 2a with total momentum κ and ~nq=0 totally symmetric phonons takes on the general form jΨðk, R, ~nq¼0 Þæ ¼ X X ðk,RÞ 0 ck;nq ,nq ,... nq jk; nq1 , nq2 , ..., nqN-1 æ j~nq¼ 0 æ X 1 2 N-1 k nq1,nq2 ,... nqN-1 ð7Þ where |k;nq1,nq2, ..., nqN-1æ is a product state containing an exciton of wave vector k and nql non-totally symmetric phonons with wave vector ql. The prime on the summations in eq 7 indicates the constraint of eq 6. The energy of states in eq 7 with ~nq=0 = 0 are denoted as ωκ,R. States with ~nq=0 > 0 are higher in energy by ~nq=0ω0.

III. EMISSION FROM LINEAR ORDERED J-AGGREGATES AT T = 0 K In this section, we consider a linear J-aggregate (J0 < 0) with one molecule per unit cell, as depicted in the inset of Figure 1. The electronic transition S0 f S1 is optically allowed with transition dipole μ. In all calculations to follow, we take nearest-neighbor-only coupling and maintain periodic boundary conditions. The exciton couplings are therefore given by Jmn ¼ J0 fδn,mþ1 þ δn,m-1 þ δn,N δm,1 þ δn,1 δm,N g

Figure 1. (a) The reduced PL spectrum for ordered linear J-aggregates of several sizes N and λ2 = 1 at T = 0 K. (b) PL spectra for N = 10 and several HR factors. In (a) and (b), the exciton bandwidth is W = ω0 = 1400 cm-1. In (c), the ratio SR is shown as a function of exciton bandwidth. Solid circles are the exact results; open circles are obtained using the two-particle approximation.

nuclear potential. The dimensionless line strength that governs emission from |Ψ(R=1) κ=0 æ to the vibrationless ground state is then ðR¼1Þ

I 0-0

jg; vacæ jg; nq¼0 ¼ 0, nq1 ¼ 0, ..., nqN-1 ¼ 0æ where the states |nq=0æ (without the overstrike on n) are eigenfunctions of b†q=0bq=0, corresponding to the ground (S0) unshifted

ð9Þ

where μ̂ is the transition dipole moment operator with matix elements Ægj^ μ jkæ ¼ Ækj^ μ jgæ pffiffiffiffi ¼ N μδk,0

ð8Þ

with J0 < 0 required for a J-aggregate. In this simple coupling scheme, the free exciton bandwidth is W = 4|J0|. In a J-aggregate, the lowest-energy exciton-phonon state from eq 7 has κ = 0, R = 1, and ~nq=0 = 0. This state serves as the origin of emission at T = 0 K. For notational simplicity, we drop the quantum number for the totally symmetric phonons and denote the emitting state as, |Ψ(R=1) κ=0 æ, which correlates to the free exciton, |k = 0æ|vacæ in the limit of strong excitonic coupling, W . λ2ω0. The optical selection rule dictates that only states with κ = 0 have oscillator strengths to the vibrationless ground electronic state. The latter is represented as

jÆg; vacj^ μ jΨk¼0 æj2 μ2

After inserting the wave function from eq 7 into eq 9, we obtain the simple result for the 0-0 line strength I 0-0 ¼ NFðR¼1Þ

ð10Þ

F(R) is a generalized Franck-Condon (FC) factor 2 ðk¼0,RÞ F ðRÞ e-λ =N jck¼0;0,0:::0 j2

ð11Þ

which includes the FC factor from symmetric phonons, jÆnq¼0 ¼ 0j~nq¼0 ¼ 0æj2 ¼ e-λ =N 2

The factor F(R=1) varies from exp(-λ2) in the weak exciton coupling limit (W , λ2ω0) to exp(-λ2/N), which is essentially unity for large N, in the strong exciton coupling limit (W . λ2ω0). The first emission sideband is comprised of a sum over all transitions, terminating on states containing a total of one 5135



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phonon. There are a total of N such states, abbreviated as |g;1q0,0,...0æ, |g;0,1q1,...0æ, and so forth, indicating the presence of only one phonon with wave vector ql. We take the energy of such states to be simply ω0, that is, we neglect dispersion as for Einstein phonons. The 0 - 1 line strength I0-1 is therefore

P

I 0-1

ðR¼1Þ

jÆg; 0, 1q , ...0j^ μ jΨk¼0 æj2 μ2

ð12Þ

where the sum is over all N one-phonon ground states. Upon inserting the wave function from eq 7 into eq 12, the sum is easily performed since the only nonzero contribution arises from the terminal state |g;1q0,0,...0æ, the state with one totally symmetric phonon and no nonsymmetric phonons. We therefore obtain ðk¼0,R¼1Þ 2 I 0-1 ¼ NjÆnq¼0 ¼ 1j~nq¼0 ¼ 0æj2 jck¼0;0 ð13Þ ,...0 j which reduces to I 0-1 ¼ λ2 F ðR¼1Þ 2

2

ð14Þ -λ2/N

and utilizing after inserting |Ænq=0 = 1|~nq=0 = 0æ| = (λ /N)e eq 11. The form of the 0-1 line strength in eq 14 is quite different from I0-1 in H-aggregates, which strongly attenuates with increasing exciton bandwidth, W.21,23 For the linear J-aggregates under consideration, the ratio of the line strengths in eqs 10 and 14 is simply SR

I 0-0 N ¼ 2 I 0-1 λ

ð15Þ

which is the main result of this paper. Note that the ratio precludes any knowledge of the generalized FC factors. Hence, with the HR factor λ2 known from solution spectra, the ratio SR directly yields the number of coherently coupled chromophores, which, in the present case, is the total number of chromophores, N. The robustness of eq 15 is quite remarkable as it remains exact over the entire range of exciton-phonon coupling. It is entirely independent of exciton bandwidth, W, as well as the vibrational frequency, ω0. Moreover, eq 15 is not limited to nearestneighbor coupling, as long as the k = 0 exciton remains the lowest in energy. Equation 15 can also be extended to more complex morphologies, as we show in section VII. To demonstrate the generality of eq 15, we show in Figure 1 the reduced PL spectrum for linear J-aggregates with nearest-neighbor coupling (|J0| = ω0/4) as a function of N and λ2 (Figure 1a and b, respectively). The reduced PL spectrum is defined as X I 0-v Γðω - ωem þ vω0 Þ ð16Þ SðωÞ ¼ v ¼ 0,1,::: where ωem = ωκ=0,R=1 is the emission frequency corresponding to the state |Ψκ=0(R=1)æ. In eq 16, we omit the cubic frequency dependence of the vibronic peaks as well as a frequencydependent index of refraction present in the actual PL spectrum in order to focus on the line strengths alone. The line shape function in eq 16 is taken to be a peak-normalized Gaussian, Γ(ω) = exp(-ω2/σH2) with the homogeneous line width σH set to 0.2ω0 in all calculations to follow. In addition, the calculations employ the two-particle approximation (TPA), as discussed in detail in several previous works.22,29,30 Figure 1a shows how the 0-0 peak is uniquely sensitive to the number of chromophores, N, as only I0-0 is coherently enhanced. The 0-0 peak triples in intensity as N increases from 5 to 15, while the 0-1 peak undergoes no discernible change.

Figure 1b shows that the 0-0 intensity is favored with decreasing HR factor, as one would expect, since an HR factor of zero would result in only the 0-0 peak. As observed in Figure 1a and b, the ratio of the 0-0 to 0-1 peak heights, or SR, remains solidly equal to N/λ2, despite a considerable variation in the individual 0-0 and 0-1 intensities. Figure 1c further demonstrates the independence of SR on the exciton bandwidth, W = 4|J0|. Note the slight artificial deviation from N/λ2 when W > ω0 as a consequence of the TPA. The behaviors shown in Figure 1 are quite remarkable, but to be useful, the behavior of SR needs to be understood in more practical situations involving the localizing effect of disorder as well as higher temperatures.

IV. J-AGGREGATES WITH SITE DISORDER; T = 0 K In this section, we consider emission from disordered linear J-aggregates at T = 0 K. Disorder will localize excitations near the band bottom to a coherence range Ncoh(σ,Τ = 0) smaller than the physical number of chromophores N.4,25,31 Here, σ is a parameter representing the magnitude of the disorder. For the remainder of this section, we will drop the arguments in Ncoh(σ,Τ=0) and simply refer to the coherence size as Ncoh. We aim to show that Ncoh replaces N in eq 15, so that the spectral line strength ratio SR is simply given by Ncoh SR ¼ 2 ð17Þ λ when disorder is present. In the simplest disorder model, each chromophore within a particular aggregate is randomly detuned from the central absorption frequency with the detuning value chosen from a Gaussian distribution of width σ. Denoting Δn as the detuning of the nth chromophore, the occurrence probability of a particular configuration, [Δ1, Δ2, ..., ΔN], is PðΔ1 , ..., ΔN Þ ¼

N Y expð-Δ2n =σ2 Þ pffiffiffi πσ n¼1

ð18Þ

In what follows, we consider the simplest case of no spatial correlation, ÆΔmΔnæ = (σ2/2)δmn, but our results remain valid for spatially correlated distributions as well. In order to treat diagonal disorder, we consider the modified Holstein Hamiltonian X Δn jnæÆnj ð19Þ Hdis ¼ H1 þ H2 þ n

with H1 and H2 defined in eqs 2b and 2c. As translational symmetry no longer exists, the lowest-energy (emitting) exciton takes the form jψðemÞ æ ¼ j~nq¼0 ¼ 0æ X X ðemÞ X ck;nq ,nq ,:::nq jk; nq1 , nq2 , ..., nqN-1 æ 1 2 N-1 nq1,nq2,... nqN-1 k ð20Þ where the lack of a prime on the summation as compared with eq 7 indicates that all exciton-multiphonon states with variable total momentum κ can, in principle, contribute. Hence, eq 6 is no longer a constraint. Ncoh is determined by the extent of the generalized coherence function introduced in ref 32. For a cylic aggregate, the coherence 5136



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disorder, the coherence function narrows; in the limit of strong disorder, the coherence function is completely localized 2 and takes on P the isolated molecule value, ÆC(em)(s)æC = e-λ δs,0. Because n Bþ n Bn is not the usual exciton number operator, C(em)(0) is not generally unity. As discussed in more detail in ref 23, C(em)(0) provides additional information about the extent of nuclear relaxation at the site of the vibronically excited molecule. On the basis of the coherence function, the coherence number Ncoh for a generally disordered aggregate is defined as Ncoh

Figure 2. The averaged exciton coherence function for several values of disorder width σ in linear J-aggregates containing N = 60 chromophores. In all cases, the exciton bandwidth is W = ω0 = 1400 cm-1, and the HR factor is λ2 = 1.

function is given by CðemÞ ðsÞ

X

ðemÞ ÆψðemÞ jBþ æ n Bnþs jψ

n

s ¼ 0, (1, ..., (ðN=2 - 1Þ, N=2

ð21Þ

where the allowed values of s assume N to be even. It is also understood that n þ s is replaced by n þ s - N when n þ s > N and n þ s is replaced by n þ s þ N when n þ s < 1. In eq 21, |ψ(em)æ is the lowest-energy exciton from eq 20, and Bþ n is the |n;vacæÆg;vac|. local exciton creation operator, given by Bþ n Here, |n;vacæ represents the vibronic state with molecule n excited electronically and with all molecular vibrations (on all sites) remaining in the vacuum state pertaining to the unshifted (ground-state) nuclear potential. Note that in our definition, and Bn are not pure electronic operators; hence Bþ n (em) æ in eq 21 is not identical to the one-exciton Æψ(em)|Bþ n Bnþs|ψ density matrix element utilized by Mukamel and coworkers16-18 to define exciton coherence. Aside from this difference, the form of the coherence function in eq 21 closely resembles that utilized by Kuhn and Sundstrom15 in their analysis of light-harvesting complexes. Inserting the wave function in eq 20 into eq 21 and averaging over disorder gives the average coherence function in the multiphonon representation X ðemÞ 2 Æjck;0,0,... 0 j2 æC cos ks ÆCðemÞ ðsÞæC ¼ e-λ =N ð22Þ k

where Æ...æC indicates the configurational average using the distribution in eq 18. Figure 2 shows the average coherence function for several values of disorder width, σ, for J-aggregates with N = 60 chromophores; 104 configurations were utilized to obtain the average, and the TPA was once again employed in numerically manipulating the Hamiltonian. For a disorder-free aggregate, only the k = 0 term contributes to the sum in eq 22. ÆC(em)(s)æC is therefore independent of s and equal to the generalized FC factor F(R=1) in eq 11. This is represented by the horizontal curve in Figure 2. With increasing

X 1 jÆCðemÞ ðsÞæC j ÆCðemÞ ð0ÞæC s

ð23Þ

From the behavior of ÆC(em)(s)æC in Figure 2, it follows that Ncoh = N in the limit of no disorder and reduces to unity in the limit of strong disorder. The absolute value appearing on the rhs ensures that the coherence size is not affected by phase oscillations.15 This is not an issue for J-aggregates near the exciton band bottom but is necessary in H-aggregates where the band bottom exciton has k = π, leading to a (-1)s phase oscillation in C(em)(s).23 Our definition of coherence number is intimately connected to physical observables. It is straightforward to show that Ncoh represents the enhancement in the radiative decay rate leading to superradiance in J-aggregates (where the absolute value in eq 23 can be omitted).16 Ncoh also represents the coherent enhancement of the 0-0 line strength in the photoluminescence spectrum, as we will now show. On the basis of the definitions of I0-0 and C(em)(s) in eqs 9 and 21, respectively, it readily follows that23 X ÆI 0-0 æC ¼ ÆCðemÞ ðsÞæC ð24Þ s

The 0-0 line strength is essentially the “integral” over the coherence function. Note that the sum in eq 24 can approach N (multiplied by an FC factor) only in disorder-free J-aggregates. In H-aggregates, the sum in eq 24 is exactly zero when disorder is absent due to the aforementioned (-1)s phase oscillation in the coherence function. Adding disorder in H-aggregates increases the 0-0 peak through symmetry breaking but only to a 2 maximum of e-λ (the single molecule value) in the strong disorder limit. Hence, eq 24 shows that there can be no coherent enhancement of the PL emission in an H-aggregate. In J-aggregates, the absolute value appearing in eq 23 can be removed since the coherence function as seen in Figure 2 has uniform phase. Additional use of eq 24 gives ÆI 0-0 æC ¼ ÆCðemÞ ðs ¼ 0ÞæC Ncoh

J-aggregates

ð25Þ

where ÆC(em)(s = 0)æC can be understood as a generalized FC factor, varying from exp(-λ2) for weak exciton coupling to unity in the limit of strong exciton coupling. Equation 25 shows a direct scaling of the 0-0 line strength with the coherence size. It is important to note that eq 25 is not valid for H-aggregates. The 0-0 line strength and coherence size can also be written entirely in terms of the multiphonon wave function coefficients. Inserting the wave function (eq 20) into the definitions of I0-0 and Ncoh gives ðemÞ

ÆI 0-0 æC ¼ Ne-λ =N Æjck¼0;0,0,:::j2 æC 2

5137

J-aggregates

ð26Þ



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and ðemÞ

Ncoh ¼

NÆjck¼0;0,0,...j2 æC Æ

P k

ðemÞ

jck;0,0,...j2 æC

J-aggregates

ð27Þ

Equations 26 and 27 show that the coherence is limited by the increasing admixture of k 6¼ 0 excitons (with no phonons) to the emitting wave function with increasing disorder. In the limit of no disorder, only the k = 0 exciton is present and the coherence number is Ncoh = N. In the opposite limit of very strong disorder, 2 all excitons are equally represented; Æ|c(em) k;0,0,...| æC becomes independent of k, and Ncoh reduces to unity. The expression of Ncoh in terms of excitons with vacuum (q 6¼ 0) phonons is entirely consistent with an equivalent expression in terms of only one-particle states appearing in previous works.23,32 In contrast to the 0-0 line strength, the 0-1 line strength is not coherently enhanced in any aggregate type. Inserting the wave function (eq 20) into eq 12 gives ðemÞ

ÆI 0-1 æC ¼ λ2 e-λ =N Æjck¼0;0,0,... j2 æC X ðemÞ 2 þ Ne-λ =N Æjck¼0;0,1q ,... j2 æC 2

ð28Þ

q6¼ 0

The second term is a source of two-particle states, making the 0 - 1 intensity depend on both one- and two-particle states.21,22 When σ , ω0, that is, when the disorder cannot significantly couple excitons with different numbers of total vibrational quanta, eq 28 simplifies to X ðemÞ 2 2 Æjck;0,0,... j æC ð29Þ ÆI 0-1 æC λ2 e-λ =N

Figure 3. (a) Reduced PL spectrum for an ensemble of N = 60 linear J-aggregates for several values of disorder width σ. The spectra were obtained by averaging eq 16 over 104 configurations using a homogeneous line width of 0.2ω0. The exciton bandwidth is W = 4|J0| = ω0 (=1400 cm-1) and λ2 = 1 in all cases. (b) λ2SR (solid dots) and Ncoh () as functions of N for several values of σ. Ncoh was evaluated using eq 23. The PL spectra and the line strengths in SR were determined using the TPA.

Alternatively, eq 29 holds whenever the vibronic peaks in the emission spectrum are well-resolved. (If this were not the case, SR would be irrelevant because it could not be extracted from the PL spectrum). Using expressions 26 and 29 for the 0-0 and 0-1 line strengths and eq 27 for Ncoh, we obtain our final result for J-aggregates

In Figure 3b, λ2SR and Ncoh are plotted as functions of the physical size N for the three values of σ from Figure 3a. The figure shows λ2SR and Ncoh to be practically identical; both initially increase with N and then level off and become constant once the physical size of the aggregate surpasses the (converged) coherence length. Finally, we have also investigated the HR factors λ2 = 0.5 and 2.0 for reasonable values of σ ( 0 K In this section, we consider emission from an ordered linear J-aggregate as a function of increasing temperature. Much of the theoretical development parallels that presented for two-dimensional herringbone aggregates studied in refs 33 and 34 and will not be reproduced here. As temperature increases, excitons become thermally excited, and as long as the lattice phonon scattering is sufficiently rapid compared with the radiative decay rate, a steady-state Boltzmann distribution of excitons near the band bottom is established. Emission then proceeds from all excitons in the Boltzmann distribution. However, only states with κ = 0 contribute to the 0-0 emission because the vibrationless ground state carries no momentum and the optical selection rule requires Δκ = 0. By contrast, sideband (0-1, 0-2, ...) emission is sourced by all thermally excited excitons independent of κ.33,34 In cases where κ is not zero, the optical selection rule is satisfied by terminating on the ground electronic state with one or more intramolecular phonons (i.e., those appearing in eq 2a) with total wave vector κ. Such states are denoted as |g;nq0,nq1,...nqN-1æ with nq1q1 þ nq2q2 þ .... = κ. From here on, we assume that the temperature is low enough such that

k

SR

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excitation to states with ~nq0 > 0 in eq 7 is negligible. This requires kbT , ω0, where kb is the Boltzmann constant. For ω0 = 1400 cm-1, this allows us to confidently explore temperatures from 0 K up to room temperature. As the temperature increases, the 0-0 line strength is diminished by the fraction of population remaining in the lowest-energy (κ=0,R=1) exciton. Using eq 10, we obtain ÆI 0-0 æT ¼

NF ðR¼1Þ ZðN, TÞ

ð31Þ

where Æ...æT indicates an average over a Boltzmann distribution of (κ,R) emitting excitons (from here on, we assume ~nq0 = 0) with energies ωκ,R. The partition function is given by X ZðN, TÞ ¼ exp½-ðωk,R - ωk¼0,R¼1 Þ=kb T ð32Þ k, R The Boltzmann constant is kb = 0.695 cm-1/K when ωκ,R is expressed in units of cm-1. Equation 31 can be rewritten in terms of a thermal coherence number, Ncoh(σ=0,T), which we abbreviate as simply NT for the remainder of this section. Rewriting eq 31 as ÆI 0-0 æT ¼ NT F ðR¼1Þ

ð33Þ

the thermal coherence number becomes NT

N ZðN, TÞ

ð34Þ

Equation 34 shows that when T = 0 K, NT is simply N since the partition function is unity. In the opposite thermodynamic limit (large N, high T), NT becomes independent of N. In this limit, the κ sum in eq 32 can be converted to an integral, giving ( )-1 Z 1 X π -ðωk,R-ωk¼0,R¼1 Þ=kb T NT ¼ e dk ð35Þ 2π R -π which is independent of the number N of chromophores comprising the aggregate. By analogy with the configurational averaging of the last section, the 0-0 line strength and coherence number may also be recast in terms of a thermally averaged coherence function. The analogous forms of eqs 24 and 23 are, respectively X ÆCðemÞ ðsÞæT ð36Þ ÆI 0-0 æT ¼ s and X 1 ÆCðemÞ ðsÞæT ð37Þ NT ¼ ðemÞ ÆC ð0ÞæT s Equation 36 contains an average over coherence functions corresponding to all emitting excitons within the Boltzmann distribution. The only coherence functions which contribute to the sum over s correspond to the nodeless κ = 0 excitons (with R = 1, 2, ...). Consistent with our taking kbT , ω0, the only state which significantly contributes is the lowest-energy state with R = 1. Hence, eq 36 agrees with eq 31. The definition of NT in eq 37 agrees with eq 34 when ÆC(em)(0)æT = F(R=1), a condition satisfied when kbT , ω0, as was assumed from the outset. Figure 4 shows the thermally averaged coherence function for several temperatures. The coherence function becomes increasingly localized with increasing T. The coherence size as defined in eq 37 falls from NT = N at T = 0 K to approximately 3 at room

Figure 4. Thermally averaged exciton coherence function as a function of s for several temperatures. Aggregates contain N = 60 chromophores. In all cases, the exciton bandwidth is W = ω0 = 1400 cm-1, and the HR factor is λ2 = 1.

temperature. At much higher temperatures, NT will approach unity. We now consider sideband emission. In marked contrast to 0-0 emission, sideband emission does not diminish with increasing T. This arises because the optical selection rule for emission from an exciton with wave vector κ can be satisfied by termination on a ground electronic state containing phonons with total wave vector κ. Hence, the 0-1 line strength is practically independent of T,33 and from eq 14, we obtain ÆI 0-1 æT λ2 F ðR¼1Þ

kb T , ω0

ð38Þ

Figure 5a shows how the reduced PL spectrum for an ordered linear J-aggregate with N = 10 chromophores behaves with increasing temperature. The spectra were obtained by averaging emission from all thermally excited excitons of the form given by eq 7 according to a Boltzman distribution. The figure shows the 0-0 peak to be uniquely sensitive to temperature, while the sidebands are practically unchanged except for a small blue shift with increasing T. Hence, as with the case of disorder, the coherence size can be extracted from the peak ratios. Using eqs 33 and 38 in the definition of SR gives SR ¼

NT λ2

kb T , ω0

ð39Þ

The behavior of λ2SR and NT with increasing N is depicted in Figure 5b. In a manner reminiscent of Figure 3b, SR initially increases with N but quickly levels off to an N-independent thermal coherence number in the thermodynamic limit. Also plotted in Figure 5b is the thermal coherence number NT evaluated from eq 37. The agreement with λ2SR is excellent, even at T = 300 K. The temperature dependence of NT in the thermodynamic limit can be approximately understood under the parabolic band approximation (PBA), where the integral defining the thermal coherence size in eq 35 can be evaluated analytically.33,34 The PBA assumes that near the exciton band bottom, the exciton 5139



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Figure 6. λ2SR (solid dots) and Ncoh (asterisks) as a function of temperature for disordered aggregates containing N = 20 chromophores. Ncoh was evaluated using eq 43. The line strengths in SR were obtained from spectra averaged over configurational and Boltzmann distributions. The exciton bandwidth is W = 4|J0| = ω0 (=1400 cm-1) and λ2 = 1 in all cases. Figure 5. (a) Reduced PL spectrum for an ensemble of N = 10 linear J-aggregates at several temperatures. The exciton bandwidth is W = 4|J0| = ω0 (=1400 cm-1) and λ2 = 1 in all cases. (b) λ2SR (solid dots) and NT () as functions of N for several temperatures. NT was evaluated using eq 37. The PL spectra and the line strengths in SR were determined using the TPA.

determined from the configurationally and thermally averaged exciton coherence function X 1 ÆÆCðemÞ ðsÞæC æT ð43Þ Ncoh ðσ, TÞ ðemÞ ÆÆC ð0ÞæC æT s

dispersion is approximately given by,

The coherence size also depends on N but levels off for sufficiently large N to a value independent of N, as observed in Figures 3 and 5. In this section, we demonstrate that to an excellent approximation, the coherence size in eq 43 can be extracted from the emission 0-0/0-1 ratio, that is, the relation

ωk,R¼1 ωk¼0,R¼1 þ ωc k2

ð40Þ

where ωc is the band curvature. In the strong exciton coupling limit where the band dispersion is ωk,R=1 = 2J0 cos k, the band curvature is simply ωc = |J0|. For weaker exciton coupling, the curvature is modulated by the generalized FC factor33,34 ωc F

ðR¼1Þ

jJ0 j

SR ¼

ð41Þ

Inserting eq 40 into eq 35 gives the thermal coherence size for a linear J-aggregate in the thermodynamic limit as rffiffiffiffiffiffiffiffiffiffiffi 4πωc NT ¼ 1 þ ð42Þ kb T where we have neglected the high-energy excitons (R = 2, 3, ...) in eq 35. Equation 42 shows that NT approaches unity for very high temperatures, kbT . 4πωc. At low temperatures, the expression is no longer valid since the continuum approximation inherent in eq 35 breaks down. In a two-dimensional herringbone lattice, the thermal coherence size has the same form as that in eq 42 but without the square root.33,34 Hence, the general dependence of the thermal coherence size on dimensionality d takes the form NT = 1 þ (4πωc/kbT)d/2.

VI. GENERAL CASE: DISORDERED J-AGGREGATES WITH T > 0 K In the previous two sections, we showed that the coherence size for the emitting exciton (or thermally averaged group of excitons) is a strong function of the disorder width σ (when T = 0 K) and the temperature (when σ = 0). In the most general case, the coherence size Ncoh(σ,T) is a function of both σ and T and is

Ncoh ðσ, TÞ λ2

ð44Þ

continues to hold for a linear J-aggregate as long as kbT,σ , ω0. To this end, we plot in Figure 6 λ2SR and Ncoh(σ,T) as a function of temperature for weakly (σ = 0.01ω0) and moderately (σ = 0.1ω0) disordered J-aggregates. The aggregates contain N = 20 molecules, and the exciton bandwidth is set to W = ω0. The spectral ratio SR was evaluated from the configurationally and thermally averaged emission spectrum. The figure shows that eq 44 remains remarkably accurate, yielding the coherence size to within 5% at the highest value of disorder. The two configuration ensembles (σ = 0.01ω0 and 0.1ω0) yield essentially identical coherence sizes at sufficiently high temperature, although at low temperatures, the effect of disorder is quite pronounced. In any case, the spectral ratio SR remains an accurate measure of the (emitting) exciton coherence size even in the most general case of disordered J-aggregates with T(K) > 0.

VII. MORE COMPLEX MORPHOLOGIES It is straightforward to extend the results of the previous sections to higher dimensions and to cases with several molecules per unit cell. In ordered two-dimensional J-aggregates with periodic boundary conditions and one molecule per unit cell, the derivation proceeds along identical lines as in the 5140



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Figure 7. Simplified 3 3 (square) herringbone lattice. The herringbone angle is φ = 45.

one-dimensional case outlined in section II. The wave vectors k and q are good “quantum numbers”, and the totally symmetric phonon with q = 0 is separable. Hence, λ2SR remains rigorously equal to N. Most organic molecules crystallize in structures with unit cells containing two or more molecules. Consider the very common herringbone (HB) packing exhibited by a variety of organic molecules including anthracene and quaterthiophene. A simplified two-dimensional herringbone lattice is shown in Figure 7. The lines indicate the directions of the transition dipole moments projected onto the ab plane. Such lattices cannot be classified as ideal J- or H-aggregates since oscillator strength is generally divided among the two Davydov components. In HB aggregates based on a monoclinic unit cell containing two molecules, 0-0 emission originates from a lower Davydov component and is polarized either along the unique axis (b) or perpendicular to it. Conversely, sideband emission can be polarized in either direction, depending on the symmetry of the terminal phonon(s).35 A straightforward derivation exploiting the separability of the q = 0 totally symmetric phonon shows that the dimensionless b-polarized line strengths are Ib0-0 ¼ N cos2 jF ðR¼1Þ

ð45aÞ

Ib0-1 ¼ λ2 cos2 jF ðR¼1Þ

ð45bÞ

where we have taken the lower Davydov component to be symmetry-allowed along the b direction. Hence, the ratio of the b-polarized line strengths is SR

Ib0-0 N 0-1 ¼ 2 Ib λ

ð46Þ

The simple form of eq 15 is once again obtained, as long as SR is constructed from the components of the 0-0 and 0-1 line strengths corresponding to the polarization of the lower Davydov component. Equation 46 was originally derived for herringbone aggregates in the limit of weak exciton-phonon coupling.27 Here, we have shown that eq 46 applies over the entire range of exciton-phonon coupling. Figure 8a shows the corresponding PL spectra for several size square aggregates identified by the number of unit cells along each direction and by the total number of chromophores N. Figure 8b shows the dependence of the PL spectrum on the HR factor, while Figure 8c shows the dependence of λ2SR on the exciton bandwidth, W = 8|J0|. We have again assumed

Figure 8. (a) The reduced PL spectrum for square herringbone aggregates for several N with λ2 = 1. The inset shows the magnified sideband region where the spectrum becomes independent of N for both polarizations. (b) The reduced PL spectrum for several HR factors for N = 18. In (a) and (b), the exciton bandwidth is W = 8|J0| = ω0 = 1400 cm-1. In (c), the ratio SR is shown as a function of exciton bandwidth. Solid circles are the exact results; open circles are obtained using the TPA.

nearest-neighbor coupling only with J0 taken to be nonzero between the nearest “edge-to-face” neighbors. Note that the sign of J0 is immaterial and serves only to determine the polarization directions of the Davydov components. The behavior displayed in Figure 8 closely mimics that of the linear J-aggregates from Figure 1. In fact, our calculations show that the exciton coherence number Ncoh and thermal coherence number NT (generalized to two dimensions) are also simply related to SR according to eqs 30 and 39, and we therefore expect the most general relations for disordered herringbone aggregates with T > 0 K (eqs 43 and 44) to hold as well. Hence, one can immediately deduce the coherence number from λ2SR using eqs 30 and 39 as long as SR is defined as the ratio of b-polarized 0-0 and 0-1 line strengths. We hasten to add that one can also deduce the coherence number from the “mixed” polarization ratio SM R

Ib0-0 0-1 I^b

since the 0-1 line strength polarized along the direction perpendicular to b is not coherently enhanced by Ncoh.21,22,33,36 The inset of Figure 8a shows the spectrum polarized along the direction perpendicular to b, that is, within the ac plane. It has no 0-0 component, and although it is independent of N, it substantially differs from the b-polarized sidebands. In fact, the spectrum polarized in the ac plane mimics the emission spectrum of an ideal H-aggregate which lacks a 0-0 component and contains sidebands which, unlike 5141



NF R¼1 μ2^ cos2 j 0-1 ðW, λ2 Þμ2 Iac )

SM R ¼

where μ^(μ ) is the short (long)-axis component of the OT4 transition dipole moment with μ^ , μ and φ is the herringbone 2 angle. The ac-polarized sideband line strength, I0-1 ac (W,λ ), being characteristic of H-aggregate emission, decreases strongly with increasing exciton bandwidth W.33,34 In packing arrangements in which the transition between the band bottom exciton and the vibrationless ground state is symmetry-forbidden, the simple relation between SR and Ncoh in eqs 30 and 44 no longer holds. Such is the case for ideal H-aggregates, where there is no 0-0 emission when disorder is absent. In such aggregates, the coherence function for the emitting (k = π) exciton contains a factor (-1)s which causes the sum in eq 24, and therefore the 0-0 emission, to vanish exactly. Note that eq 23 still predicts a large coherence size (equal to N) due to presence of the absolute value. However, eq 25, whose derivation assumes a uniformly positive coherence function (as for J-aggregates) no longer holds. Therefore SR = Ncoh/λ2 is not valid for H-aggregates. When disorder is present in H-aggregates, the symmetry is broken, and the 0-0 intensity gains oscillator strength as the coherence number shrinks. Hence, the spectral ratio SR is inversely related to the coherence size, in marked contrast to the behavior in J-aggregates presented here. However, it is still possible to extract Ncoh from SR in H-aggregates, as shown in detail in ref 23 for P3HT π-stacks and in refs 32 and 47 for chiral MOPV4 aggregates. Finally, we point out that in cases where the radiative decay rate is much faster than the time scale of population relaxation within the exciton band, there may be insufficient time to establish a Boltzmann distribution of exciton emitters. Such an interesting nonequilibrium situation was treated in a series of papers by Knoester and co-workers,48-50 where it was shown that under certain conditions, the time dependence of the fluorescence decay from J-aggregates reflects population decay dynamics (the bottleneck) and not superradiance. In such cases, the spectral ratio SR generally serves as a time-dependent probe )

VIII. DISCUSSION/CONCLUSION The ratio SR between the 0-0 and 0-1 line strengths in the PL spectrum of a disorder-free linear J-aggregate is rigorously equal to N/λ2 when T = 0 K (and periodic boundary conditions are employed). The result is quite robust and is completely independent of the exciton bandwidth and frequency of the progression-forming vibrational mode. Thus, this simple relationship is valid throughout the entire range of exciton-phonon coupling and can be readily extended to more complex morphologies such as the common herringbone packing found in many organic molecular crystals. What is perhaps equally gratifying is that in the general case of increasing (site) disorder and rising temperature, SR remains a direct measure of the (emitting) exciton coherence number, Ncoh(σ,T), as defined in eq 43, thereby providing a simple means for extracting the exciton coherence number from the PL spectrum. Our main result in this regard is embodied in eq 44. We note that in extracting the line strength ratio SR from the experimental PL spectra, one must utilize the 0-0 and 0-1 spectral areas and not peak intensities. In addition, one must remove the dependence on the photon density of states by eliminating the cubic frequency dependence, which is straighforward.37 Furthermore, in order to ensure that the emission corresponds to an intrinsic exciton and not a defect, one must perform the analysis with an early time PL spectrum, before defects can be populated (but presumably after a Boltzmann distribution of emitters has been established). In the polyacene experiments, this required emission spectra measured within 50 ps after excitation.8,9 Finally, one must be careful to eliminate self-absorption, which acts to diminish the 0-0 line peak and therefore give artificially lower values for the coherence number. This requires working with optically thin samples and utilizing front-face emission.37,38 The validity of eq 44 extends to more complex morphologies as long as the (0-0) optical transition from the vibrationless ground state to the band bottom exciton (from which emission originates) is allowed by symmetry. In aggregates and crystals which pack with two or more molecules in a unit cell, this is equivalent to an optically allowed lower Davydov component. In such cases, eq 44 applies as long as SR is constructed from the components of the 0-0 and 0-1 line strengths polarized like the lower Davydov component. Analysis of the measured PL spectra from anthracene thin films9 shows that at T = 10 K the spectral ratio SR is roughly 4, therefore indicating a coherence number of approximately the same number, since λ2 ≈ 1. (In ref 9, we calculated NT ≈ 4 from the ratio, SR, in small, disorder-free clusters containing N = 8 molecules with open boundary conditions. Such boundary conditions mimic inhomogeneous broadening for small cluster sizes leading to a coherence size NT less than N even as T approaches 0 K.) Tetracene aggregates and thin films show similar behavior with a low-temperature SR of approximately 5.8,11 We are currently conducting more sophisticated calculations including disorder and charge transfer to better understand exciton coherence in polyacene crystals. Like the polyacenes, aggregates and thin films of the oligophenylenevinylenes and oligothiophenes also pack in a herringbone fashion. However, unlike the polyacenes, the vast

majority of oscillator strength resides in the upper Davydov component, resulting in a very large aggregation-induced blue shift of the main absorption peak and the designation of such aggregates as H-aggregates.39-42 Nevertheless, the lower Davydov component remains optically allowed, although very weakly so, along the b axis.43 Hence, b-polarized emission mimics J-aggregate emission, most notably, the 0-0 peak is superradiantly enhanced at low temperatures.44 Conversely the ac-polarized component of emission, which dominates the sidebands (0-1, 0-2, ...), mimics that of an ideal linear H-aggregate, and therefore, when disorder is lacking the 0-0 emission has no ac-polarized component. These properties have been confirmed experimentally.44-46 In our previous analysis of quaterthiophene (OT4) assemblies,33,34,36 we extracted the T-dependent exciton 0-0 0-1 coherence number from the mixed intensity ratio SM R Ib /Iac due to the extreme dominance of the ac-polarized component of the 0-1 emission (in polyacenes, the b-polarized component remains dominant for all vibronic peaks). Although this is a valid way to determine coherence size, it is a good deal more complex because for mixed polarizations, the simple form of eq 46 no longer applies, and much more information is required. For example, in an ordered OT4 lattice at T = 0 K, eq 46 is replaced by

)

the b-polarized sidebands, strongly diminish with increasing exciton bandwidth. An account of the differences between J- and H-aggregate emission has recently been published.21

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The Journal of Physical Chemistry B of the emitting exciton’s 0-0 oscillator strength. In J-aggregates, the oscillator strength generally increases in going from the top to the bottom of the exciton band, and therefore, SR should increase with time following impulsive excitation above the band edge. Exactly the opposite behavior is expected in H-aggregates, and this has been observed in polythiophene π-stacks.51,52 In future studies, we need to incorporate the coupling of two or more vibrational modes to the J-aggregate exciton. The latter has been studied in H-aggregates,53 where it was shown that the quantum yield can be synergistically enhanced by the coupling to fast and slow modes. A more sophisticated model would also include coupling to lattice phonons or, more generally, to a bath in which spatial and temporal fluctuations induce dephasing. Then, in addition to the line strengths studied here, one could also analyze the impact of vibronic coupling and disorder on the line widths.54

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Vibronic Coupling in J-Aggregates and Beyond: A ... - ACS Publications

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